Systematic return-strategies-cs

Management of Systematic
Return Strategies
A Primer
For Qualified Investors / Institutional Clients Only
For Investors in Spain: This Document Is Provided to the Investor at His / Her Request
Asset Management

2 / 54
Dr. Dietmar Peetz, Dr. Daniel Schmitt and Ozan Akdogan, the authors of this report, are portfolio managers and form the
Systematic Return team at Credit Suisse. Dietmar Peetz, the Head of the Systematic Return team, has a background as a fixed
income derivatives trader, financial engineer and absolute return portfolio manager. He works primarily in the design and
management of robust systematic return portfolios. Daniel Schmitt is a senior portfolio manager responsible for managing derivative
portfolios in the equities, fixed income, alternatives and multi-asset risk premium space. In his doctoral thesis in Theoretical Physics,
Dr. Schmitt focused on applying interdisciplinary concepts from complex systems to understand dynamics in financial markets.
Ozan Akdogan is a portfolio manager and specialist in financial derivatives and volatility management. Prior to this position he held
several roles in quantitative analytics and risk management.
The image concept symbolizes the robust design of our framework for systematic return strategies and the meticulous analysis that
it is based upon. The interlocking gearwheels emphasize the interaction of the various systematic return strategies in the portfolio.
From left to right: Ozan Akdogan, Dr. Dietmar Peetz, Dr. Daniel Schmitt.

Management of Systematic Return Strategies 3 / 54
Foreword
Dear Reader
Just as with natural phenomena, rapid changes in financial markets, as well as in the financial industry, are very often hard to
explain or comprehend from an equilibrium-centered worldview. Danish theoretical physicist Per Bak identified a simple but
successful theory to help improve our understanding of such occurrences. His concept of “self-organized criticality” showed how
wild fluctuations arise even in his oversimplified sandpile models.
Armed with this intuition, one cannot expect the real world to follow well-behaved equilibrium dynamics, be it in nature or within
financial markets. One consequence, for example, is that a single event can dominate all previous fluctuations. In such an uncertain
world, it is therefore better to try to identify robust solutions rather than solutions optimized to address historically observed
fluctuations. For several decades, the classic buy-and-hold strategy was one such solution, but a growing number of investors
think that this may no longer be an optimal allocation in an age when markets are being driven more by central-bank policy and
less by pure fundamentals.
Investors are currently stuck between a rock and a hard place. On the one hand, they do not want to miss out on profit opportunities
in today’s low-yield environment, but they equally fear capital losses if markets experience another shock not consistent with an
equilibrium-centered worldview. While we accept that the complexities of financial markets are here to stay, investors need to be
addressing these concerns with more simplified solutions. Systematic return strategies are a particularly viable option because they
extract value from structural return sources that are largely independent of central-bank action. At Credit Suisse, we see very
interesting opportunities for investors to build more robust portfolios with the help of systematic return funds.
The numerous research papers that investors have to read nowadays are the direct result of the increasing complexities and
changes in our investment industry. What those changes require is a broader perspective away from linear relationships and
incorporation of real-world uncertainties into the realm of the investment practitioner.
This report distills some of the most relevant research findings in the field of systematic return strategies. It highlights the practical
areas that should be of central concern when it comes to using them within the asset allocation framework of investment
professionals. The paper also draws heavily from the day-to-day experience of our Systematic Return team at Credit Suisse Asset
Management Switzerland and thus delivers interesting insights for academics and practitioners alike.
We wish you an interesting and entertaining read, and we hope that this report will be useful for your sphere of activity in the
financial markets.
Michael Strobaek
Global Chief Investment Officer
& Head Asset Management Switzerland

4 / 54
“We cannot solve our problems with
the same thinking we used when we
created them.” – Albert Einstein

Contents
Executive Summary 7
1. The Fundamentals of Systematic Return Strategies 9

1.1 Introduction 10
1.2 Definition 11
1.3 A Simple Classification Scheme for Systematic Return Strategies 13
2. Portfolio Construction Using Systematic Return Strategies 25

2.1 The Role of Systematic Return Strategies in Institutional Portfolios 26
2.2 Guiding Principles for Selecting Systematic Trading Strategies 31
2.3 Portfolio Construction Using Systematic Return Strategies 33
2.4 Case Study 1: Portfolio Diversification with Entropy Measures 35
2.5 Case Study 2: The Effect of Adding Systematic Return Strategies to a Balanced Portfolio 39
3. Implications for Investors 43

3.1 Overview 44
3.2 Conclusion 45
Appendix 46
Literature 51

Executive Summary
This report contains three sections, all of which can be read independently of one
another. At the beginning of each section, a summary page highlights the main
points.
1. The Fundamentals of Systematic Return Strategies
In the past, the classic equity/bond mixture within the traditional
balanced concept was sufficient to enable many investors to
achieve their return targets. The world is changing, however.
With interest rates at record lows, investors may not be enjoying
the diversification and capital preservation properties of global
bonds like they did in the past. So, global bonds are no longer
the answer. In search of potential solutions, a number of
investors have turned to systematic return strategies. Systematic
return strategies are fully transparent, objective and directly
investable strategies that aim to monetize risk premia. They
consist of a set of trading rules created to capture specific risk
premia embedded in traditional and nontraditional asset classes.
However, most investments (traditional or systematic return
strategies) behave similarly during risk-off periods. Therefore,
diversification, which is normally a powerful risk control, leads to
unsatisfactory results in market downturns. We address this
issue by introducing a simple but robust classification scheme
for almost all systematic return strategies, aiming to identify
truly diversifying investments.
2. Portfolio Construction Using Systematic Return
Strategies
The main benefit of portfolio construction stems from the notion
of diversification. The idiosyncratic risk of individual assets can
be substantially reduced if the portfolio contains a sufficient
number of assets that are not perfectly correlated. So, adding
systematic return strategies to a balanced portfolio can increase
diversification of the resulting portfolio because these strategies
tend to exhibit low correlations to bonds and equities. Portfolio
construction often involves some sort of portfolio optimization.
Therefore, a risk model and an objective are chosen and inputs
need to be estimated. Both the choice of model and the input
parameter estimation are subject to errors, which introduces
additional risks. These risks can be particularly prominent for
systematic return strategies since they often possess only a
limited set of live data. We discuss these issues in depth and
provide a straightforward, effective solution. Our guiding
principle here is that there is a trade-off between the ex-ante
optimality and the robustness of the optimization results.
3. Implications for Investors
Systematic return strategies can provide investors with more
direct access to the return drivers, and at the same time they
can share the liquid tradability and thus the flexibility of
traditional asset classes. Furthermore, they can give investors
access to a larger opportunity set than traditional investment
strategies and can therefore increase diversification when
added to an existing portfolio. Another advantage for investors
is that these strategies do not require explicit forecasts of
returns and risks for asset classes or securities. The
methodologies underlying these systematic return strategies
rest on publicly available market information. Our proposed
robust allocation minimizes the necessity of forecasting individual
returns, so investors do not have to rely on forecasting skills.
Investors will find systematic return strategies a viable alternative
to balanced portfolios during market-correction periods.

1. The Fundamentals of Systematic Return Strategies
Traditional asset classes such as bonds, equities and foreign exchange (FX)
go hand in hand with a number of risk premia that are persistent and attached
to certain economically well-understood and empirically documented sources
of risk. Extracting these risk premia often involves specific methodological,
nondiscretionary investment rules known as systematic return strategies.
Systematic return strategies are fully transparent, objective and directly
investable. Their aim is to capture specific risk premia embedded in traditional and
nontraditional assets. The resulting distinct statistical properties of a strategy’s
return can differ substantially from those of the underlying asset classes. Many
systematic return strategies show similar behavior in risk-off market situations.
Diversification is therefore an issue.
In this section, we study the benefits of systematic return strategies for risk-
averse investors. First, we provide an overview of systematic return strategies.
Next, we introduce a simple but effective classification scheme that can help
investors build “all-weather portfolios” that retain some diversification benefits
even during times of crisis. We show that systematic strategies offer unique risk/
return characteristics that can help to improve portfolio efficiency. Therefore, we
encourage a paradigm shift toward investing in portfolios of systematic strategies
as opposed to portfolios of traditional assets.

10 / 54
1.1 Introduction
For many years, investors in balanced portfolios relied on fixed-
income markets to provide yield income, diversification and
elements of capital protection.
The chart below shows the performance of US equities relative
to US government bonds during various crisis episodes. We see
that whenever there was a significant equity crisis, the bond
market, on average, delivered exactly what investors were
expecting of it, i.e. protection.
However, after three decades of prolonged yield compression,
many institutional and private investors are concerned about
how they will achieve their investment objectives going forward.
Unprecedented central-bank action forced investors to move up
the risk curve, which in turn depressed yields further and
stretched valuations of risky assets. There is now growing
evidence that investors are feeling increasingly uncomfortable
with the elevated risk levels of the fixed-income holdings in their
portfolios. They are particularly concerned about the potential
drawdown risks in a rising yield environment.
Looking ahead, the historical trend of fixed-income investments
providing downside protection and diversification benefits to
balanced portfolios is unlikely to continue to the same extent as
before. In short, a simple buy-and-hold strategy when investing
in fixed income within balanced portfolios may not work when
yields start rising from their multi-decade lows.
There is a growing trend that more and more investors are
considering replacing part of their traditional portfolio allocations
with income solutions that make greater use of investing in
uncorrelated risk premia. According to Kaya et al. (2012), the
idea of risk premium investing has received a lot of attention,
especially after the last financial crisis, when an increasing
number of investors focused on risk classes rather than asset
classes. Extracting these risk premia often involves
nondiscretionary investment rules, which we call systematic
return strategies.
Figure 1: Equities and Bonds during Crisis Time
Source: Datastream, own calculations. Both indices are not directly investable. Historical performance indications and financial market scenarios are not reliable indicators of current or future
performance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/or redemption.
150
140
130
120
110
100
90
80
70
60
50
04/56 – 03/57
-8.4% / -0.5%
07/57 – 12/57
-16.9% / 8.9%
12/61 – 06/62
-22.5% / 3.9%
12/68 – 06/70
-29% / -4.5%
01/73 – 12/74
-43.4% / 6.6%
11/80 – 07/82
-19.4% / 15.5%
08/87 – 12/87
-26.8% / 4.6%
06/90 – 10/90
-14.8% / 0.1%
07/98 – 09/98
-11.8% / 8.6%
08/00 – 02/03
-43.7% / 32.6%
10/07 – 03/09
-50.8% / 22.6%
Equities (Dow Jones Industrial Average Index) Bonds (10y US Government Bond Index)

1.2 Definition
A systematic return strategy is an investment strategy
that invests according to transparent, predefined
nondiscretionary rules based on public information
available at the time of investment. For a long-only
investment in a certain stock, such a rule could be very simple:
invest all capital in this one stock and never change. The rules
for long-only investments in an equity index (for example
through exchange-traded funds) would be more complicated.
Usually, stocks are added or excluded from an index according
to market capitalization and many other criteria defined in the
index rules.
The index rules are predefined, but input variables like the
market capitalization of each stock in the universe cannot be
known in advance.
An investment in an active fund would not necessarily be
considered a systematic return strategy since one does not
know what a fund manager will do given a certain set of
information. In other words, the rules are not predefined.
Investing is about taking risks. It is a well-established paradigm
in finance that every investment that is expected to deliver an
excess return above the risk-free rate has to be exposed to
some additional risks. The expected excess return over the risk-
free rate is known as the risk premium. Risk premia are usually
not directly (individually) tradable but can be monetized via
systematic return strategies. Returns from systematic strategies
are expected to reflect the respective risk premia but are
affected by market fluctuations. This means that return
realizations can turn out to be negative, depending on the time
period investigated.
Risk-averse investors typically construct portfolios with a
positive aggregated risk premium, i.e. they expect an excess
return above the risk-free rate for bearing additional risks. There
are different ways for investors to look at risk premia:
ƁƁ One can define a risk premium based on the type of
investment. For example, the expected excess return
of equities is called the equity risk premium.1
ƁƁ One can define risk premia based on the source of risk.
In this case, the equity risk premium could be viewed
as a combination of a business risk premium, a recession
risk premium, a liquidity risk premium, a country-specific
risk premium and potentially many more.
ƁƁ Statistical methods like principal component analysis (PCA)
can be used to separate and isolate different “abstract” risk
premia for an investment.
No matter which way one dissects risk premia, the requirements
from a practical portfolio management point of view are:
ƁƁ Risk premia should be investable.
ƁƁ A larger number of sustainable positive risk premia is
preferable (all else equal).
ƁƁ More economically and statistically independent risk
premia are better (all else equal).
Traditional asset classes such as equities, bonds, foreign
exchange, commodities and their derivatives can be considered
baskets of risk premia. The allocation of these baskets of risk
premia is, in general, not optimal.
With the help of systematic return strategies, it is possible to
gain more direct access to less correlated risk premia. In
Figure 2, we show this for some sample systematic strategies2
and a typical portfolio of traditional assets.3
We can see that the average correlations are significantly lower
for the strategies compared to those for the traditional assets.
Moreover, the average correlations are more stable as well. This
was particularly significant during the collapse of Lehman
Brothers in 2008, when the average correlations of traditional
assets spiked and remained at high levels for several years,
while the correlations of systematic strategies remained virtually
unchanged.
Figure 2: Average Two-Year Correlations of Some
Systematic Return Strategies Compared to Investments
in Sample Traditional Assets
Source: Bloomberg L.P., own calculations. As from 30.11.2001 to 29.08.2014, based
on monthly data. Historical performance indications and financial market scenarios are
not reliable indicators of current or future performance. Performance indications do
not consider commissions, fees and other charges, including commissions levied at
subscription and/or redemption.
Traditional assets Systematic return strategies
20142001 2003 2005 2007 2008 2010 2012
40%
35%
30%
25%
20%
15%
10%
5%
0%
1
The notion of risk premia is closely related to the famous capital asset
pricing model (CAPM), where only one risk factor, namely the market beta,
is considered. This model was subsequently extended by Fama, French and
Sharpe to multiple premia.
2
Systematic return strategies included were: CBOE SP 500 PutWrite Index,
SP 500 Pure Value Total Return Index minus SP 500 Total Return Index,
MSCI World Small Cap Index minus MSCI World Large Cap Index, UBS
American Volatility Arbitrage Index, BofA Merrill Lynch US High Yield Index
minus BofA Merrill Lynch US Corporate Index, J.P. Morgan G10 FX Carry Index,
Barclay Systematic Traders Index.
3
For traditional assets we have chosen the SP 500 Index, SP GSCI Excess
Return Index, Barclays GlobalAgg Total Return Index, Dollar Index, DAX Index,
SMI Index, MSCI Total Return Emerging Markets Index, Nikkei Index, EURO
STOXX 50 Index.

12 / 54
On average, systematic return strategies are less correlated
among each other than traditional asset classes, which can
improve portfolio diversification. Since systematic strategies can
target individual risk premia more directly, they enable portfolio
managers to come closer to an optimal portfolio.4
Therefore, we
argue that systematic strategies using simple quantitative
investment rules based on straightforward economic reasoning
are better portfolio building blocks than traditional asset classes.
Extracting risk premia with systematic investment strategies is
a very well-established and documented concept among
investment professionals, and it can be found in all asset
classes. To provide a better and more practical understanding
of systematic return strategies, we list a few common ones
along with their most important risk premia in the table below.
Table 1: Illustrative Examples of Risk Premia in Various Asset Classes
Asset Classes Examples of Systematic Return Strategies Risk Premia (Examples)
Equity ƁƁ Value stocks versus benchmark
ƁƁ Small-cap stocks versus benchmark
ƁƁ High-dividend versus low-dividend stocks
ƁƁ Covered put writing/covered call writing
ƁƁ Calendar effects in equity indices
ƁƁ Merger arbitrage
ƁƁ Volatility arbitrage
ƁƁ Value risk premium
ƁƁ Small-cap risk premium
ƁƁ Dividend risk premium
ƁƁ Equity-protection risk premium
ƁƁ Equity-liquidity risk premium
ƁƁ Liquidity and deal risk premium
ƁƁ Equity-volatility risk premium
Fixed Income ƁƁ High-yield versus investment-grade bonds
ƁƁ New on-the-run issues versus off-the-run bonds
ƁƁ Convertible bond optionality versus listed options
ƁƁ Default risk premium
ƁƁ Liquidity risk premium
ƁƁ Volatility and liquidity risk premium
Currencies ƁƁ High-yielding currencies versus low-yielding FX
ƁƁ FX-implied versus realized volatility spread
ƁƁ Liquidity and inflation risk premium
ƁƁ Currency-volatility risk premium
Commodities ƁƁ Preroll commodity indices versus benchmark
ƁƁ Deferred indices versus benchmark
ƁƁ Implied versus realized commodity volatility
ƁƁ Backwardated versus contangoed commodities
ƁƁ Index-liquidity risk premium
ƁƁ Supply/demand risk premium
ƁƁ Commodity-volatility risk premium
ƁƁ Inventory risk premium
Source: Credit Suisse AG.
4
In a traditional balanced portfolio of equities and bonds, it can be a challenge
to reduce exposure to rising interest rates or rising equity volatility, for example,
without causing many other (sometimes unintended) changes to other risk
factors in the portfolio.

The difference between systematic return strategies and other
quantitative strategies or strategies based on technical analysis
is mainly an ideological one: when investing in systematic return
strategies based on risk premia, investors expect to be
compensated for certain risks that they are willing to bear. Our
philosophy is that a well-diversified portfolio of systematic return
strategies makes it possible to better diversify those risks
compared to a portfolio of traditional asset classes. We do not
believe that superior forecasting or information-processing
abilities are the driver of performance without additional risks, as
may be suggested by some strategies based on quantitative or
technical analysis.
Since systematic return strategies rest on historical market
information and do not require any kind of return or risk
forecasts, they could be interpreted as being mainly passive
strategies. However, the investor still has to make an active
decision when he or she selects a strategy from the overall
universe of available strategies. What is the exposure, how do
the strategies behave during risk-off markets, and do they
overlap in terms of tail-risk behavior? Those are just some of the
questions that need to be answered within the context of an
investment process for a portfolio of systematic return strategies.
1.3 A Simple Classification Scheme for Systematic
Return Strategies
To help answer those questions, we introduce a simple
classification scheme for systematic return strategies. This
particularly presents a challenge because in the field of
systematic return strategies, there is no commonly agreed upon
scientific terminology yet. The classification scheme has three
main purposes: to organize the strategies we deal with, to see
how strategies are related to each other, and to evaluate the
appropriateness of new strategies. Besides the simplicity of our
classification scheme that uses only two categories, we assert
that it can be a powerful tool for understanding diversification,
especially in extreme market environments.
Explaining the nature of investment strategies has been a major
topic of academic literature. Sharpe (1992), for example,
showed that returns from strategies employed by mutual funds
in the US were highly correlated with standard asset classes,
and that the performance differences of these strategies could
be explained by different styles or asset class exposures. Other
authors expanded Sharpe’s model by adding additional factors
in order to analyze the investment strategies of hedge funds.
However, many questions remain open. Given the theoretically
unlimited universe of possible strategies, not all of them can be
captured by the style factors of the authors. In addition, very
often there is not enough empirical data available to draw the
correct conclusion about the factor exposure of systematic
strategies. The nonlinear relationship between “style factors”
and the corresponding asset classes is not always so easily
captured. Our classification scheme therefore goes beyond the
classical (linear) factor model approach. It is motivated by the
idea that systematic strategies should be interpreted as a
derivative of traditional asset classes due to their option-type
payoffs (Perold and Sharpe 1988).
We use a taxonomy that applies specific criteria to distinguish
between two categories: “carry” and “trend-following.” Carry
strategies provide income in stable market environments,
whereas trend-following strategies aim to act as a return
diversifier, especially amid unstable market environments.
The idea is quite simple. Although we acknowledge that our
classification scheme is a simplification of the real world, we
believe that our approach can help absolute-return investors to
better allocate strategies in a market where exposure to tail risk
cannot be diversified away.
Carry Strategies
The first of our categories is carry strategies. Here, strategies
are classified based on their nonlinear behavior toward the
broader market, with a particular focus on negative market
returns.
Typically, the carry of an asset is the return obtained from
holding it. A classic example is high-yield bonds, where
investors can collect income from coupon payments as
compensation for issuer default risk. Carry strategies –
sometimes also called relative value strategies – are also used
to extract a risk premium by holding two offsetting positions in
similar instruments or asset classes where one of the positions
creates a price return or cash flow that is greater than the
obligations of the other. An example of such a strategy would
be going long high-yield bonds and, at the same time, short
government bonds to extract a default risk premium.
Many risk premium strategies that are commonly classified as
carry, income or relative value strategies can be seen as
compensation for investors for assuming some form of
systematic risk.5
In such cases, the investment provides
insurance against systematic risks. Such risk premium strategies
have risk profiles that are similar to put-selling strategies (selling
direct insurance against price risks).
The common characteristic of carry strategies is that they have
a positive expected return. However, during sharp market
corrections, these strategies can suffer as well. Similarly, selling
out-of-the-money puts gives investors the right to sell stocks at
a price below the current level. Selling puts is usually profitable
in rising or range-bound markets, but can become very loss-
making if equity prices move sharply lower and volatility rises
significantly.6
5
In some cases, the size of the risk premia is also attributed to what market
participants consider to be some kind of “market inefficiency.” In recent years,
a number of articles have suggested that these “inefficiencies” can be traced
back to behavioral bias or structural imbalances. Please refer to Appendix 3 for
further elaboration.
6
The insurance premium is particularly high for short-dated volatility, which is due
to the well-known phenomenon in option markets that short-dated options trade
very rich in terms of implied volatility, since the nonhedgeable jump risk plays a
decisive role here.

14 / 54
How can we evaluate the generic risk behavior of carry
strategies? From option pricing theory we know that the delta is
the first derivative of the option price in response to market
changes. As the underlying market moves, the option price is
not likely to change in the same fashion, but instead changes
over some curved function. The delta of the option position –
and in our case the delta of the strategy – can therefore only be
a first linear approximation of the price change in the systematic
return strategy when there is a small change in the underlying
market factor. In order to capture more of the dynamics of the
systematic trading strategies, we have to go farther and look for
the convexity of the strategy. Convexity is a measure of the
sensitivity of the delta of an option – in our case the delta of the
strategy – to changes in the underlying.7
The risk behavior of systematic return strategies can be
approximated as a function of equity returns. We consider carry
strategies to be negative convexity strategies: covered call or
put writing, going long equities with certain stop-loss rules,
corporate bonds, high-yield bonds, emerging-market bonds,
relative value, equity long/short strategies and certain portfolio
construction and rebalancing techniques all fall into this
category.8
If the dependence on the equity market is concave
(negative convexity), then we put the strategy in the carry
category.
Popular strategies capitalizing on monetizing the insurance
premium via option writing include the covered-call strategy and
the aforementioned put-writing strategy. The tracked and
independently calculated CBOE SP 500 PutWrite Index is an
example of the latter strategy. The CBOE SP 500 PutWrite
Index measures the performance of a hypothetical portfolio that
sells SP 500 put options against a cash reserve. The index
rules determine the number of options to sell each month, their
strike price and their maturity, and accordingly are independent
of the views of an investment manager.
The issue many investors are facing is that selling insurance
suggests low risk when applying standard metrics such as the
Sharpe ratio or alpha, for example. However, many practitioners
and researchers argue that this standard approach gives too
narrow a perspective in that it does not fully reflect the true risk
content of all the different risk premia that are related with this
strategy.
Figure 3: Performance of CBOE SP 500 PutWrite Index
versus SP 500 Total Return Index
Source: Bloomberg L.P, own calculations. As from 29.01.1999 to 29.08.2014, based
on monthly data. Both indices are not directly investable. Historical performance
indications and financial market scenarios are not reliable indicators of current or
future performance. Performance indications do not consider commissions, fees and
other charges, including commissions levied at subscription and/or redemption.
CBOE SP 500 PutWrite Index SP 500 Total Return Index
20141999 2000 2002 2003 2005 2006 2008 2010 2011
350
300
250
200
150
100
50
0
When we look at Figure 4, we see from the return distribution
that the CBOE SP 500 PutWrite Index returns show a much
lower skewness and considerably higher kurtosis compared to
the SP 500 Total Return Index.9
Skewness is a measure that
indicates that the tail on one side of the distribution is longer
than the other (i.e. the distribution is asymmetrical). The fourth
standardized moment is kurtosis, which is an indicator for
distributions with more extreme deviations from the mean (e.g.
infrequent but very large losses) than would be expected by a
normal distribution with the same variance.
Even though the beta of the CBOE SP 500 PutWrite Index is
lower than one, it still outperforms the market, contradicting the
Efficient Market Hypothesis (EMH), which is usually attributed
to alpha.10
7
The price process of the option is said to be convex in the underlying if the
second derivative with respect to the price of the underlying is positive.
8
By applying this logic, it should not come as a surprise that we also classify
momentum portfolios that go long past winners and short past losers in the
equity markets as negative convexity trades. Momentum and trend-following
often seem similar strategies, but in reality they are not because they exhibit
very different empirical behaviors. This assertion is corroborated by studies by
Daniel et al. (2012) and Avramov et al. (2014), which found that the empirical
return distribution of a momentum portfolio between 1927 and 2010 has both
strong excess kurtosis and strong negative skewness.
9
The Jarque-Bera test for normality shows a value of 17 × 103
for the CBOE
SP 500 PutWrite Index strategy, whereas the critical value is 5.99 at the 95%
significance level.
10
As introduced in the CAPM framework.

Statistics SP 500 Total Return Index CBOE SP 500 PutWrite Index
Total Return 109.4% 187.5%
Return p.a. 4.8% 7.0%
Volatility 18.5% 12.4%
Sharpe Ratio 0.35 0.61
Skewness -0.51 -2.49
Excess Kurtosis 5.37 22.91
Maximum Drawdown 54.7% 36.4%
Table 2: Return Statistics of the SP 500 Total Return Index and the CBOE SP 500 PutWrite Index
Source: Bloomberg L.P., own calculations. Weekly data as from 29.01.1999 to 29.08.2014. Both indices are not directly investable. Historical performance indications and financial market
scenarios are not reliable indicators of current or future performance. Performance indications do not consider commissions, fees and other charges, including commissions levied at
Figure 4: Return Distribution of SP 500 Total Return Index and CBOE SP 500 PutWrite Index
Source: Bloomberg L.P. As from 29.01.1999 to 29.08.2014. Both indices are not directly investable. Historical performance indications and financial market scenarios are not reliable
indicators of current or future performance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/or redemption.
SP 500 Total Return Index return frequencies
Fitted normal probability density function
15%-20% -15% -10% -5% 0% 10%5%
30
25
20
15
10
5
0
CBOE SP 500 PutWrite Index return frequencies
Fitted normal probability density function
60
50
40
30
20
10
0
15%-20% -15% -10% -5% 0% 10%5%

16 / 54
In many popular market models (see, for example, Fang and Lai
1997), skewness and the kurtosis are dedicated risk factors,
and the superior performance of the CBOE SP 500 PutWrite
Index could be partly explained in this model with the exposure
to unfavorable higher moments of the return distribution. The
CBOE SP 500 PutWrite Index has a different skewness than
that of the SP 500 Total Return Index. This can be explained
by the fact that short put options have a first-order sensitivity to
moves in the underlying (delta) between 0% and 100%,
whereas at-the-money forward (ATMF) options have a sensitivity
of around 50%. As the market drops, the first-order sensitivity
(delta) increases, and as the market rises the delta decreases.
This explains the asymmetry of the return distribution. The
maximum drawdown of the CBOE SP 500 PutWrite Index
stands at 36.4%, compared to 56.2% for the equity index. This
is due to the fact that options have a delta of less than 100%,
and at each monthly option rebalancing the delta is set back to
approximately 50% while at the same time locking in gains from
monetizing option premia, which can cushion potential earlier
losses.
Therefore, it can be seen from the CBOE SP 500 PutWrite
Index example that by gaining exposure to new sources of risk
in a systematic way, the traditional risk-adjusted performance
on the surface looks superior when compared to the underlying
market. In our example, the Sharpe ratio for the CBOE SP
500 PutWrite Index is more than twice that of the SP 500
Total Return Index (Table 2). This figure looks so impressive that
it is tempting to invest all of one’s assets in such a systematic
return strategy.
However, attributing the outperformance of put writing to
alpha might be misleading because it might derive from
exposure to higher-order risks that an underspecified
model will not be able to address.
The example of systematic put writing shows that systematic
return strategies can have exposure to higher-order risks.11,12
Let us illustrate this for some carry strategies. In Figure 5, we
have plotted the empirical functional dependencies of six
exemplary carry strategies with respect to the broad equity
market index (we are using the SP 500 Total Return Index as
a proxy for equity market risk).
11
Another example is the well-known merger arbitrage strategy. Here, one goes
long in shares of the target company after a deal is announced and holds them
until completion or termination of the deal and, at the same time, hedges the
portfolio of target-company shares with shares of the acquirer or with the
equity market (to make the strategy beta-neutral). The objective is to capture
the difference between the acquisition price and the target’s stock price
before completion of the merger. The performance of this strategy is positively
correlated with market returns in severely falling markets, but uncorrelated in
flat or rising markets. When risk aversion increases across the board, credit
conditions deteriorate and merger deals thus tend to fail. Mitchell and Pulvino
(2001) interpret the returns of the merger arbitrage strategy as similar to
those obtained from selling uncovered index put options because they show a
nonlinear relationship with market returns. In essence, the authors found that
excess returns can be interpreted as a compensation for providing liquidity,
especially in negative market regimes. Och and Pulvino (2004) show that this
strategy can be seen as selling insurance to shareholders against the risk that
the deal may fail.
12
Many popular carry strategies in the fixed-income space can be characterized
as synthetic option positions, as Fung and Hsieh (2002) show.

The first strategy is the CBOE SP 500 PutWrite Index that we
have already discussed. This strategy collects premium income
by monetizing the volatility risk premium options’ implied
volatility. The second strategy is a combination of a long position
in the SP 500 Pure Value Total Return Index and a short
position in the SP 500 Total Return Index.
This strategy benefits from the value risk premia in the equity
markets. The third strategy shows a combination of a long
position in the MSCI World Small Cap Index and a short position
in the MSCI World Large Cap Index. Here, the goal is to benefit
from the small-cap risk premia in equity markets.
13
We have used this method because it fits smooth curves to local subsets of the
empirical data and thus does not require specification of a global function to fit
the data set. The dark blue line fits 85% of the data and disregards data points
where absolute market returns are extreme. The light blue line fits the entire
data set. We chose to highlight the center portion because the tails only contain
few data points and the explanatory power decreases markedly.
Source: Bloomberg L.P. Monthly data as from 31.12.1999 to 29.08.2014. Indices are not directly investable. Historical performance indications and financial market scenarios are not reliable
Figure 5: A Robust Local Regression (LOESS)13
of Monthly Returns for Six Systematic Return Strategies versus the
SP 500 Total Return Index
LOESS fit all data
LOESS fit center data
CBOE SP 500 PutWrite Index returns
LOESS fit all data
MSCI World Small Cap Index returns –
MSCI World Large Cap Index returns
LOESS fit all data
SP 500 Pure Value Total Return Index –
SP 500 Total Return Index returns
LOESS fit all data
UBS American Volatility Arbitrage
Index returns
LOESS fit all data
J.P. Morgan G10 FX Carry Index returns
LOESS fit all data
BofAML 1- to 10-year US High-Yield
Index – BofAML 1- to 10-year US
Corporate Government Bond Index returns
-20% -15% -10% -5% 0% 5% 10% 15%
-20% -15% -10% -5% 0% 5% 10% 15%
-20% -15% -10% -5% 0% 5% 10% 15%
-20% -15% -10% -5% 0% 5% 10% 15%
-20% -15% -10% -5% 0% 5% 10% 15%
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The fourth strategy is the UBS American Volatility Arbitrage
Index. This strategy consists of short exposure to one-month
variance swaps on the SP 500 Total Return Index, with the
aim of monetizing the spread between the implied and realized
volatility of the SP 500 Total Return Index constituents
(volatility risk premium). The fifth strategy is a long position in
the Bank of America Merrill Lynch 1- to 10-year US High-Yield
Index and a short position in the Bank of America Merrill Lynch
1- to 10-year US Corporate Government Bond Index, which
aims to monetize the liquidity and default risk premia of high-
yield bonds.14
Finally, the J.P. Morgan G10 FX Carry strategy aims to exploit
the empirically observed fact that currencies with comparatively
higher interest rates do not tend to depreciate (as implied by
currency forwards) by selecting four G10 currency pairs based
on interest-rate differentials on a monthly basis.
When looking at the relationship between our sample strategies
and the SP 500 Total Return Index as a proxy for market risk
in Table 3, we can make three observations. First, the common
feature of these strategies is the fact that the investor should
expect to earn a positive return (positive carry) from holding the
positions in the longer run, as evidenced by the positive
annualized return for all of the strategies. Second, the
relationship shows negative convexity, indicated by higher
negative skewness in the return distribution for most of the
carry strategies. In the case of a falling SP 500 Total Return
Index, the sensitivity, or the delta, to the market increases
drastically. In this case, the short put is “in the money” and the
position is accumulating losses.15
Theincomefrommonetizingtheriskpremiumisovercompensated
by mark-to-market losses from the underlying price movement.
This has serious consequences for absolute-return investors.
Having a portfolio of 30 carry strategies diversified across asset
classes means that the portfolio is still not properly diversified
since it shows concentrated tail-risk exposure. Third, the
dispersion in the scatterplot around the LOESS fit in Figure 5 is
an indication that there are other explanatory factors besides
the underlying price. These additional factors provide even more
diversification potential for an investor than a simple replication
would suggest.
Table 3: Statistics for Six Systematic Return Strategies
Source: Bloomberg L.P., own calculations. Monthly data as from 31.12.1999 to 29.08.2014. Indices are not directly investable. Historical performance indications and financial market scenarios
are not reliable indicators of current or future performance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/or
redemption.
Statistics CBOE
SP 500
PutWrite Index
SP 500 Pure
Value Total Return –
SP 500
Total Return
Index
MSCI World
Small Cap Index –
MSCI World
Large Cap Index
UBS American
Volatility
Arbitrage
Index
BofAML
US High-Yield –
BofAML US
Corporate
Government Bond
Index
J.P. Morgan
G10
FX Carry USD
Index
Total Return 144.5% 196.7% 164.6% 106.7% 35.9% 65.3%
Return p.a. 6.3% 7.7% 6.9% 5.1% 2.1% 3.5%
Volatility 11.4% 13.2% 8.2% 8.3% 9.6% 9.3%
Sharpe Ratio 0.59 0.63 0.86 0.64 0.27 0.42
Skewness -1.80 0.86 0.07 -4.27 -1.19 -1.19
Excess Kurtosis 7.36 6.60 3.39 28.28 6.87 5.71
Maximum Drawdown 32.7% 38.7% 17.4% 31.5% 37.2% 34.6%
14
The spread duration was roughly 3.5 years for the High-Yield Index versus
around 3.9 years for the Corporate Government Bond Index as of the end
of March 2014.
15
Taleb (1997), Brunnermeier et al. (2008), Melvin and Taylor (2009) and
Menkhoff et al. (2012) discuss the behavior of carry trades’ dynamic trading
strategies and analyze their exposure to crash risk.

Using the delta and convexity together gives a better
approximation of the change in the strategy value given a
change in the market than using delta alone.16
However, for
professional risk management purposes, this is not sufficient
because they ignore the sensitivity of the portfolio to other
dynamic features (especially to volatility).17
Trend-Following Strategies
Trend-following strategies form the second category in our
classification scheme. Almost 200 years ago, the British
economist David Ricardo (1772–1823) phrased the golden
rules of investing as: “Cut short your losses” and “Let your
profits run on” or, in other words, “The trend is your friend.” In
rising markets, Ricardo suggested investing more, while
recommending exiting or changing sides in falling markets.
Investors who follow this advice do so by going long in rising
markets or short in falling markets in the anticipation that those
trends will continue into the future.18
A large body of empirical
literature has been published over the last decades to support
the notion that segments of financial markets do indeed trend
over identifiable periods.19
The empirical justification for these
types of strategies is based on the existence of significant
autocorrelation in the asset return’s time series (see, for
example, Lo and MacKinlay 1990). The existence of these
trends can often be traced to some behavioral patterns.
This includes an initial underreaction and then a delayed
overreaction compared to the classical rational investor with
unlimited resources and unconstrained borrowing capabilities as
the main explanations for its existence.
An example of “overreaction” is the decision by investors to cut
losses after an asset portfolio has dropped to a critical value.
With unchanged liabilities, the leverage then increases, which
could threaten the survival of the business. When more
investors are forced to cut back on losses, this can initiate a
positive feedback mechanism that can drive prices lower, thus
causing more investors to sell assets, which in turn depresses
prices further. This is a classic situation where, for example,
value stocks can become even more valuable.
Trend-followers believe that prices tend to move persistently
upward or downward over time. When a trend-follower expects
autocorrelation in returns, he follows the strategy “Buy high, buy
higher or sell short and sell shorter.” Typically, this kind of
autocorrelation in returns can best be monetized during larger
market moves in either direction. During quiet periods, returns
tend to be rather small and may even be negative.
16
In general, the larger the move in the underlying, the larger the error term of a
linear approximation.
17
A thorough discussion is beyond the scope of this report. Interested readers
are recommended to consult the relevant literature (see, for example, Taleb
1997).
18
Trend-following strategies operate by using rules such as moving averages or
moving average crossovers or other more complex approaches to signal when
to buy or sell based on underlying trends.
19
See Moskowitz et al. (2012) for a comprehensive analysis – the authors find
a significant time series momentum effect that is consistent across 58 equity,
currency, commodity and bond futures over a time span of 25 years. Miffre
and Rallis (2007), Menkhoff et al. (2012), Moskowitz et al. (2012) and, more
recently, Hutchinson and O’Brien (2014) provide further evidence of the
consistently high long-term performance of trend-following strategies.
Figure 6: Performance of Barclay Systematic Traders Index versus SP 500 Total Return Index
Source: Bloomberg L.P. Monthly data as from 31.01.1999 to 29.08.2014. Both indices are not directly investable. Historical performance indications and financial market scenarios are
not reliable indicators of current or future performance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/
or redemption.
Barclay Systematic Traders Index SP 500 Total Return Index
01.1999 05.2005 12.2006 07.200803.2002 10.200308.2000 02.2010 09.2011 04.2013
250
200
150
100
50
0

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A widely accepted benchmark index for measuring the
performance of trend-following strategies is the Barclay
Systematic Traders Index. The index represents an equally
weighted composite of managed programs whose approach is
at least 95% systematic. At the start of 2014, there were 482
systematic programs included in the index. Figure 6 compares
the SP 500 Total Return Index to the Barclay Systematic
Traders Index for the period from 31.01.1999 to 29.08.2014.
Obviously this is not a fair comparison since the SP 500 Total
Return Index is a basket of 500 liquid stocks, whereas
systematic programs can typically diversify across a larger
number of liquid assets in different markets.
This, in part, explains why the returns of the Barclay Systematic
Traders Index are less erratic than the returns of the equity
index.20
Therefore, the risk/return relationship as measured by
the Sharpe ratio is higher for the systematic strategy. Both the
total return and the volatility are lower for the Barclay Systematic
Traders Index compared to the SP 500 Total Return Index.
However, the reduction in volatility by almost 50% leads to
significantly higher risk-adjusted returns.
Table 4: Return Statistics of SP 500 Total Return Index and Barclay Systematic Traders Index
Source: Bloomberg L.P., own calculations. Monthly data as from 31.01.1999 to 29.08.2014. Both indices are not directly investable. Without costs or fees. Historical performance indications and
financial market scenarios are not reliable indicators of current or future performance. Performance indications do not consider commissions, fees and other charges, including commissions
levied at subscription and/or redemption.
Statistics SP 500 Total Return Index Barclay Systematic Traders Index
Total Return 109.4% 78.9%
Return p.a. 4.9% 3.8%
Volatility 15.3% 8.1%
Sharpe Ratio 0.39 0.50
Skewness -0.56 0.29
Excess Kurtosis 0.90 0.51
Maximum Drawdown 50.9% 11.8%
20
The index reports monthly returns only, which can hide the true volatility of
returns. This smoothing effect of monthly returns can lead to an upward bias
in performance measures. For more details, see, for example, Huang et al.
(2009).

The advantage of investing in trend-following strategies is
demonstrated when combined with traditional assets and many
other popular systematic return strategies. When this is done,
trend-following strategies show interesting risk-mitigation
properties in times of market stress. In fact, according to
Ilmanen (2011), trend-following strategies perform well during
periods of sharp equity-market corrections and rising volatility.
They have therefore been very good diversifiers for risky assets.
This can be confirmed when we compare the price behavior of
the Barclay Systematic Traders Index with that of the SP 500
Total Return Index during past crisis periods (see Figure 7).
Fung and Hsieh (2001) and Fung and Hsieh (2002) use a
portfolio of options to model the nonlinear payoff from trend-
following strategies. Other authors interpret trend-following
strategies as an approximation of a long-straddle position (a
combination of a long call and a long put position) because the
strategy gains from large underlying market movements in
either direction.21
From option theory, we know that long option
positions exhibit positive convexity. This is because a long option
position can only lose the premium (with, theoretically, unlimited
gain potential), whereas a short position, in theory, can incur
unlimited losses.
Accordingly, we classify any systematic trading strategy
that resembles a payout with positive convexity (long
optionality) as a trend-following strategy.
In Table 5, we have summarized the return statistics of three
exemplary positive convexity strategies: the SP 500 VIX
Short-Term Futures Index,22
the SP 500 VIX Futures Tail Risk
Index23
and the Barclay Systematic Traders Index.
The negative return for the SP 500 VIX Short-Term Futures
Index and the SP 500 VIX Futures Tail Risk Index appears to
be the price that an investor has to pay for the asymmetry
(positive convexity or skewness of the return distribution) of the
two strategies. Although the skewness of the return distribution
is positive and high, the investor has not gained anything from
this tail-risk insurance when we look at the total return numbers.
The investor had to pay a high price for gaining exposure to
positive convexity. On the other hand, the Barclay Systematic
Traders Index has a higher annualized return and a lower
skewness. The index itself is a diversified basket of strategies
(across instruments and markets). Underlying trend-following
programs apply various risk management and money
management techniques, which themselves can be a source of
positive convexity.
In Figure 8, we show the return distributions of these strategies
and the functional dependence with respect to a broad equity
market. The local regression shows positive convexity for the
center of the market returns.
The local regression shows that there is an imperfect fit to the
empirical data because other factors still play a role that may not
be neglected. In general, this can be viewed as positive for the
investor because those additional parameters can have
additional diversification benefits.
21
See Ilmanen (2011). He emphasizes that trend-following benefits from changes
in realized volatility and not from market-implied volatility.
22
For more information, see http://us.spindices.com. The index aims to have a
constant one-month rolling long position in the first two VIX futures contracts.
This means that the strategy benefits from an increase in the VIX. However,
depending on the shape of the VIX futures curve, rebalancing can create
negative roll yield.
23
The SP 500 VIX Futures Tail Risk Index provides long volatility exposure. The
index tries to mitigate the negative impact of roll yield via a rebalanced short
exposure. See http://us.spindices.com for more information.
Figure 7: Performance of Barclay Systematic Traders
Index versus SP 500 Total Return Index during
Selected Periods of High Market Volatility
Source: Bloomberg L.P., own calculations. Both indices are not directly investable.
Historical performance indications and financial market scenarios are not reliable
indicators of current or future performance. Performance indications do not consider
commissions, fees and other charges, including commissions levied at subscription
and/or redemption.
Barclay Systematic Traders Index SP 500 Total Return Index
Black Friday
October 1987
Bursting of technology
bubble 2000 –2002
Financial crisis 2008
40%
30%
20%
10%
0%
-10%
-20%
-30%
-40%
-50%

22 / 54
24
For the VIX futures strategies, the excess return index is used because it is
often used as an overlay in an unfunded format. Total return is excess return
plus the yield on short-term liquidity (often the three-month federal funds
futures rate is used).
When comparing Table 5 and Figure 8, we see that trend-
following and carry strategies can have complementary risk/
return profiles that argue in favor of our classification scheme.
We believe that its main advantage is that it helps investors with
anabsolute-returnobjectivenottooverestimatethediversification
effects in their portfolios during crisis periods. By explicitly
focusing on the risk-off behavior, we purposely disregard many
other properties that may be important in normal markets but
might diminish in times of crisis.
Table 5: Return Statistics for Alternative Trend-Following Strategies24
Source: Bloomberg L.P., own calculations. As from 31.12.2005 to 29.08.2014. Indices are not directly investable. Historical performance indications and financial market scenarios are not reliable
Statistics SP 500 VIX Short-Term
Futures Index
SP 500 VIX Futures
Tail Risk Index
Barclay Systematic
Traders Index
Total Return -99.3% -14.6% 26.9%
Return p.a. -43.8% -1.8% 2.8%
Volatility 70.4% 61.9% 6.5%
Sharpe Ratio -0.51 0.15 0.45
Skewness 2.62 7.46 0.44
Excess Kurtosis 11.89 66.28 0.32
Maximum Drawdown 99.6% 72.7% 11.8%

25
LOESS is also known as locally weighted scatterplot smoothing (LOWESS).
This robust version of LOESS assigns zero weight to data outside six mean
absolute deviations. Here we used the robust LOESS with a span of 60%.
Figure 8: Return Histograms and Robust Local Regression (LOESS)25
of Monthly Returns of Three Not Directly
Investable Systematic Return Strategies versus the SP 500 Total Return Index
Source: Bloomberg L.P. Monthly data as from 31.12.1999 to 29.08.2014. Historical performance indications and financial market scenarios are not reliable indicators of current or future per-
formance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/or redemption.
LOESS fit all data LOESS fit center data
SP 500 VIX Futures Tail Risk Index returns
SP 500 VIX Short-Term Futures Index returns
Barclay Systematic Traders Index returns
-20%
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Frequencies
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SP 500 VIX Short-Term Futures Index returns
Frequencies
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25
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0
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25
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10
5
0
20%

2. Portfolio Construction Using Systematic Return Strategies
Many investors are not fully free in their investment approach. This becomes
apparent especially during crisis periods. Although it seems to be a rational
intention to limit losses during crisis periods, if too many market participants are
forced to do this at the same time (e.g. due to regulatory constraints), this can
create a feedback loop in the system with very negative consequences. Portfolios
once considered optimal will turn out to be less resilient to systemic shocks. So,
what investors ultimately are looking for are robust portfolios that can cope better
with shocks in the market.
In this section, we analyze how a smart combination of trend-following and carry
strategies can make portfolios more robust. In addition, we introduce some guiding
principles for selecting the most appropriate systematic return strategies. We then
analyze innovative portfolio construction methods for systematic strategies with
a focus on associated estimation and assumption risks. We additionally illustrate
how these risks can have adverse impacts on the out-of-sample performance of
optimized portfolios. We discuss portfolio construction methods that take these
risks into account and can lead to a more robust allocation. As a particularly simple
and straightforward method, we present the constrained entropy approach, which
leads to less concentrated and more robust portfolios. The aim is to show how
this approach can deliver superior out-of-sample risk-adjusted performance,
which is relevant for both relative-return and absolute-return investors.

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2.1 The Role of Systematic Return Strategies in
Institutional Portfolios
In theory, institutional investors such as insurance companies or
pension funds are thought to have a long-term investment
horizon and, in turn, a higher tolerance for short-term market
fluctuations. In reality, though, institutional investors have
shorter-term performance reporting requirements and regulatory
capital constraints for their shareholders, clients and regulators.
All of these factors can lead to procyclical investment behavior.
This is especially true during general market declines because
preservation of capital is a priority in order for retirees to cover
their living expenses. Also, the time that they can wait for the
market to recover the losses is limited. The larger the
drawdowns, the more difficult it becomes for the investor to
break even to previous levels. So, for example, an investment
that loses 20% requires a gain of 25% to break even, and a
loss of 50% requires 100%. With a loss of 100%, the entire
portfolio is wiped out and business has thus ended.
Withdrawals driven by the need for capital preservation can
become a vicious circle for institutional investors. Market losses
can lead to client redemptions, which in turn can lead to further
asset price declines due to forced unwinding of portfolio
positions in markets with reduced liquidity. This can then lead to
even more redemptions. These client redemptions ultimately
have the same effect as a margin call, which typically comes at
the worst time and prevents the investment manager from
participating in any subsequent recovery to the full extent after
he was stopped. Hence, we can interpret the position of the
institutional investor as a writer of a down-and-out American
barrier option with a rebate and negative interest rate.26
The strike price of the barrier option puts a floor under
performance and can restrict the investment manager from
participating in investments that may look attractive. However,
since such an investment may potentially entail short-term
losses, the investment manager may become more risk-averse
and thus focus on capital preservation to increase his survival
probability. One risk measure closely related to the idea of
capital preservation is drawdown. Drawdown answers the
question: “What would my losses have been if I had entered the
market at the worst possible time?”27
In Figure 9, we see on the left-hand side the historical
drawdown of the SP 500 Total Return Index (percentage price
decline from the recent high to the current value). On the right-
hand side we see the subsequent drawup (percentage price
increase from the current value to the subsequent high of the
remaining period). We can see that steep drawdowns are
typically followed by sharp recoveries.
Therefore, an investor who was stopped either by risk
management policies or his or her own drawdown aversion will
underperform a benchmark investor who is continuously
invested.
26
See Ekström and Wanntorp (2000). A down-and-out put option (also known as
a knock-out put) works like an ordinary put option unless the barrier is breached
during the term of the contract, otherwise the option expires worthless.
27
Drawdowns can reveal how successive price drops culminate in a persistent
process that cannot be captured by standard risk measures such as variance
of returns.

Figure 9: SP 500 Total Return Drawdowns and
Subsequent Drawups
2013
2011
2009
2007
2005
2003
2001
1999
-100% 0 100%50%-50% 150%
Source: Bloomberg L.P., own calculations. Based on weekly data as from 31.12.1999
to 29.08.2014. Historical performance indications and financial market scenarios are
not reliable indicators of current or future performance. Performance indications do
not consider commissions, fees and other charges, including commissions levied at
Drawdown Subsequent drawup
Typically, investors try to avoid drawdowns by simply cutting
back on risks with the aim of increasing their short-term survival
probability. Over the longer term, however, this is not a real
option for the majority of institutional and private investors
because, for instance, the liability side might be growing at a
constant rate. This may be a guaranteed interest rate with
insurance companies or the inflation rate that a private investor
wants to keep up with. Therefore, an investment strategy that
is primarily based on avoiding risk cannot be an ideal long-term
solution.
Positive portfolio convexity can help investors to stick to their
investments even in times of increased market stress, and can
prevent them from missing market opportunities that are
typically most attractive during those periods. However, in the
absence of market-forecasting abilities, the investor has to be
careful not to overpay for positive convexity.
In the following example, we will illustrate how a combination of
carry and trend-following strategies can significantly reduce the
risk of a higher portfolio drawdown without running the risk of
being stopped out. This is possible by means of the adaptive
“cushioning effect” provided by trend-following strategies.
Let us consider two systematic return strategies (the CBOE
SP 500 PutWrite Index and the Barclay Systematic Traders
Index) combined in an equally weighted portfolio. The resulting
portfolio in Figure 10 can be decomposed into (ignoring
transaction costs and fees)
ƁƁ a cash position and a short put on the SP 500 Total
Return Index; and
ƁƁ a long put option on the broad market (tail-risk insurance)
plus a long call option on the broad market (uncapped
upside potential).28
The combined exposure with respect to the SP 500 Total
Return Index is therefore roughly equivalent to a call option.
However, the portfolio still keeps its long volatility exposure to
the broad market thanks to the second component.
28
The long straddle position from the trend-following index could also be
replicated by some dynamic option strategy. The main advantage would be
that the floor would be known and the strategy could be easily implemented.
However, this would be very costly. Many investors are not willing to pay such
high costs. Kulp et al. (2005) show that by investing in a managed futures
index, an investor can replicate a long volatility exposure 9.5% cheaper than it
can be bought in the option market.

28 / 54
Figure 10: Return Histograms and Robust Local Regression (LOESS) of Monthly Returns of Three Not Directly
Investable Systematic Return Strategies versus the SP 500 Total Return Index
Source: Bloomberg L.P. Monthly data as from 30.12.1999 to 29.08.2014. Historical performance indications and financial market scenarios are not reliable indicators of current or future per-
formance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/or redemption.
Frequencies
Equally weighted (CBOE SP 500 PutWrite and
Barclay Systematic Traders Index) returns
Equally weighted (CBOE SP 500 PutWrite and
Barclay Systematic Traders Index) returns
-20%
-20%
-20%
-10%
-10%
-10%
5%
5%
5%
-15%
-15%
-15%
-5%
-5%
-5%
0%
0%
0%
10%
10%
10%
15%
15%
15%
25
20
15
10
5
0
10%
5%
0%
-5%
-10%
15%
10%
5%
0%
-5%
-10%
-15%
-20%
10%
5%
0%
-5%
-10%
Frequencies
-10% -5% 0% 5% 10%
25
20
15
10
5
0
Frequencies
60
50
40
30
20
10
0
15% 20%-20% -15% -10% -5% 0% 10%5%
-10% -5% 0% 5% 10%

From Figure 10 and Table 6, we see that the total return of the
combined portfolio falls between that of the carry and the trend-
following strategies. Interestingly, although both individual
strategies have significant drawdowns (32.7% and 11.8%,
respectively), the maximum drawdown for the combined
portfolio is reduced to 13.8% as a result of the beneficial
covariance properties of the two strategies.
In Table 6, we can see that excess kurtosis and negative
skewness are reduced when compared with the values
presented for the carry strategy. Hence, we should expect more
stable returns in terms of lower drawdowns for the overall
portfolio, which plays in favor of absolute-return-oriented
investors.
Table 6: Return Statistics of CBOE SP 500 PutWrite Index, the Barclay Systematic Traders Index and a 50 : 50
Combination
Source: Bloomberg L.P., own calculations. Monthly data as from 31.12.1999 to 29.08.2014. Historical performance indications and financial market scenarios are not reliable indicators of current
or future performance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/or redemption.
Statistics CBOE SP 500
PutWrite Index
Barclay Systematic
Traders Index
50 : 50
Combination
Total Return 144.5% 77.6% 117.7%
Return p.a. 6.3% 4.0% 5.4%
Volatility 11.4% 8.1% 6.3%
Sharpe Ratio 0.59 0.53 0.87
Skewness -1.80 0.33 -0.47
Figure 11: Performance of CBOE SP 500 PutWrite Index, Barclay Systematic Traders Index, SP 500 Total Return
Index and a 50 : 50 Combination of the PutWrite Index and Systematic Traders Index
Source: Bloomberg L.P., own calculations. Monthly data as from 31.12.1999 to 29.08.2014. Indices are not directly investable. Historical performance indications and
financial market scenarios are not reliable indicators of current or future performance. Performance indications do not consider commissions, fees and other charges,
including commissions levied at subscription and/or redemption.
CBOE SP 500 PutWrite Index Barclay Systematic Traders Index SP 500 Total Return Index 50 : 50 combination
20131999 2001 2003 2005 2007 2009 2011
250
230
210
190
170
150
130
110
90
70
50

30 / 54
It is well known that in risk-off markets, the correlations
between asset classes rise, and this usually implies that large
drawdowns may be highly correlated across asset classes. This
confirms the dilemma that diversification benefits typically
diminish exactly when investors need them the most.
In Figure 12, we look at the correlation coefficients of six
systematic return strategies – three carry strategies29
(blue
diamonds) and three trend-following strategies30
(gray
diamonds) over three time periods: the full period, the crisis
period of the Lehman Brothers collapse and the subsequent
recovery.
Over the full period, we see that our carry strategies show a
high correlation to the equity market. The correlation moves
closer to 100% during the crisis period, while in the subsequent
recovery the carry strategies, as expected, are also highly
correlated to the equity market. On the other hand, trend-
following strategies show a somewhat low correlation to the
equity market over the full period, but a high negative correlation
during the crisis period. During the recovery period, trend-
following strategies are again positively correlated to the equity
market due to the adaptive nature of their market exposure.
These findings lead to two conclusions. First, we see the
empirical behavior again as a confirmation of the effectiveness
of our simple classification scheme. Second, it does not pay for
absolute-return-oriented investors to place too much confidence
in the diversification effect of a basket of carry strategies. As we
have shown, during a crisis period, all strategies that are “short
a put” exhibit an almost perfect downside correlation to the
equity market. On the other hand, a basket of trend-following
strategies could add diversification benefits to the portfolio
during such crisis times.
Now let us take a look at how the strategies contributed to the
bottom line. During the crisis period, the equity index lost half
its value, whereas the three carry strategies lost 30%, 23% and
31%, respectively. Trend-followers, on the other hand, gained
an impressive 28%, 48% and 64%, respectively, when
measured by total returns. When looking at the recovery period,
the SP 500 shows a return of +102%, which means that all
of the early losses were fully recovered. The carry strategy also
posted strong returns (+74% for the CBOE SP 500 PutWrite
Index, +34% for the UBS American Volatility Arbitrage Index
and +33% for the J.P. Morgan G10 FX Carry Index). The
trend-following strategies appreciated as well.
29
CBOE SP 500 PutWrite Index, UBS American Volatility Arbitrage Index,
J.P. Morgan G10 FX Carry Index.
30
Winton Futures Fund, Credit Suisse Tail Risk Strategy Index, J.P. Morgan Mean
Reversion Index.
Figure 12: Correlation Coefficients of Six Exemplary Systematic Return Strategies with the SP 500 Total Return
Index for Full Period, Crisis Period and Recovery Period (Compare with Table 7)
Source: Bloomberg L.P. Not directly investable. Without fees or costs. Full period (28.09.2007–31.07.2012), Crisis period (28.09.2007–27.02.2009) and Recovery period
(27.02.2009–31.07.2012). Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/or redemption.
Carry strategies Trend-following strategies
Full period
100%80%60%40%20%-20%-40%-60%-80%-100% 0%
Crisis period
100%80%60%40%20%-20%-40%-60%-80%-100% 0%
Recovery period
100%80%60%40%20%-20%-40%-60%-80%-100% 0%

Hence, we see that the strategies with the highest drawdowns
during crisis periods are typically the ones that deliver the
highest drawups during recovery periods. The implications of
our statistical analysis for absolute-return investors now become
apparent. Investors should strive for a balance between
concave and convex strategies to lower the overall
drawdown risk of their portfolios.
2.2 Guiding Principles for Selecting Systematic Trading
Strategies
In practice, it is impossible to consider the entire universe of
systematic return strategies for portfolio construction. Since
systematic strategies are dynamic combinations of assets
traded in the market, there is an unlimited number of strategies.
They often differ only in minor details such as signal filters,
volatility targets, etc. Therefore, preselection becomes an
important step in building a systematic return portfolio. In this
section of the primer, we summarize guiding principles.
Table 7: Return Statistics of SP 500 Total Return Index and Three Concave and Three Convex Systematic Return
Strategies
Source: Bloomberg L.P., own calculations. Monthly data as from 28.09.2007 to 27.02.2009 (top) and as from 27.02.2009 to 31.07.2012 (bottom). Historical performance indications and
financial market scenarios are not reliable indicators of current or future performance. Performance indications do not consider commissions, fees and other charges, including commissions
levied at subscription and/or redemption.
LINEAR CONCAVE (Carry) CONVEX (Trend-Following)
Statistics
SP 500
Total Return
Index
CBOE
SP 500
PutWrite Index
UBS American
Volatility
Arbitrage Index
J.P. Morgan
G10 FX
Carry Index
Winton
Futures
Fund
Credit Suisse
Tail Risk
Strategy Index
J.P. Morgan
Mean Reversion
Index
CRISIS PERIOD
Total Return -50.2% -29.7% -23.1% -30.7% 28.3% 48.4% 64.0%
Return p.a. -38.8% -22.0% -16.9% 22.8% 19.3% 32.2% 41.8%
Volatility 19.6% 18.8% 20.5% 15.5% 10.8% 19.5% 18.2%
Sharpe Ratio -2.36 -1.21 -0.79 -1.58 1.69 1.54 2.03
Maximum Drawdown 50.9% 32.7% 31.5% 33.6% 7.9% 9.2% 1.6%
RECOVERY PERIOD
Total Return 101.6% 74.5% 33.8% 32.8% 15.3% 42.1% 4.4%
Return p.a. 22.8% 17.7% 8.9% 8.7% 4.2% 10.8% 1.3%
Volatility 16.3% 12.7% 7.2% 11.8% 8.5% 7.4% 4.0%
Sharpe Ratio 1.35 1.35 1.22 0.76 0.53 1.43 0.33
Maximum Drawup 88.0% 61.1% 35.7% 26.2% 27.5% 44.4% 6.9%

32 / 54
A Sound Economic Rationale Can Help to Select More
Sustainable Return Sources
The more that potential strategies are tested against historical
data, the greater the likelihood of finding one that looks very
attractive – most probably by chance alone. One effective way
to protect oneself against these false positives is to require a
sound economic rationale that can explain why a specific
strategy should be sustainable. The probability of a systematic
strategy having both an attractive backtest31
and a suitable
economic rationale is significantly lower than for it having just an
attractive backtest. Of course, this leaves the possibility of an
attractive strategy getting discarded because one was not able
to see the economic rationale at the time. However, not
rejecting them in these instances would most likely result in
more severe consequences for investors. It should also be
noted that the negative impact of selecting an unattractive
systematic strategy with both an attractive backtest and
economic rationale is much smaller compared to selecting an
unattractive strategy based on a positive backtest alone.32
Preference for Simple Strategies
When selecting strategies, Occam’s razor33
is another useful
guiding principle. Strategy developers like investment banks, for
example, usually develop complex systematic return strategies.
It is easier to find attractive strategies since the search space of
more complex strategies is bigger than that of simple strategies.
Second, the complexity obscures transparency and causes
higher implementation and trading costs. The more free
parameters a strategy has, the higher the risk of overfitting to
historical data. This means that a longer live track record is
needed to decide whether a strategy really delivers what it is
promising.
Costs Are Often Not Negligible
Paying attention to costs is also crucial when selecting
systematic strategies. There are many direct and indirect costs
associated with a systematic strategy, for example trading
costs, structuring costs, market impact costs (slippage), etc.34
Trading costs increase with turnover and bid-ask spreads.
Often, choosing strategies with lower turnover or more liquid
underlyings can add value. For example, the same strategy with
an equivalent fixed exposure is often more attractive when
compared to its volatility-controlled35
version.
Transparency and competition between strategy providers can
also help to keep costs down. So, choosing generic strategies
over proprietary ones and using a best-in-class selection
approach should be preferred. Transparency (required by
regulators for example) can also have negative side effects. The
US Natural Gas Fund, for example, used to publish the exact
roll dates and specific future contracts of its systematic natural
gas roll schedule. With more than USD 4 billion in assets under
management at the beginning of 2010, it made for an easy
target for front running.
Awareness of Capacity, Liquidity and Market Impact
Risk premia are the rewards for taking certain risks that other
market participants are not willing to take or not capable of
bearing. Risk premia returns are risky and cannot be seen as
pure arbitrage opportunities.36
This also means that risk premia
are not constant. The price of risks changes over time due to
the arrival of new information. However, sometimes one can
observe cycles where money is rushing into a certain risk
premium chasing returns. It is important to recognize such
phases and to adapt to the new risk/return regime. This is
because during periods when a risk premium is stretched thin,
even small external shocks can lead to sudden reversals and
losses.
31
The danger is that backtested performance figures are the result of unintended
data snooping, especially when there are many parameters. Any backtest model
that is only in-sample (with perfect hindsight) and lacks out-of-sample test
qualities should be rejected out of hand.
32
See Bailey et al. (2014) for a comprehensive discussion of backtesting
problems.
33
Occam’s (sometimes Ockham’s) razor is a principle attributed to the English
Franciscan friar William of Ockham (1287–1347). The original principle states
that “Pluralitas non est ponenda sine necessitate,” which translates to “Entities
should not be multiplied unnecessarily.” It is usually interpreted as preferring
simpler over more complex theories/explanations if they make the same
predictions.
34
As Frazzini et al. (2012) show, real trading costs make short-term reversal
strategies unprofitable. However, size, value and momentum are profitable
after adjusting for trading costs. They conclude that those return sources can
be implemented and scaled up. Lesmond et al. (2004) argue that returns from
momentum strategies (buying past winners and selling past losers) do not
exceed trading costs. They claim that abnormal returns create an illusionary
profit opportunity.
35
Many systematic trading strategies involve unfunded instruments. The decision
of how much notional value to employ therefore becomes a bit arbitrary. One
way to determine the leverage is to target a specific volatility and dynamically
set the notional value accordingly.
36
See also Appendix 3.

Inflows into risk factors can have very different effects,
depending on whether returns are driven mainly by mark-to-
market effects or realized cash flows. This is mainly related to
the time horizon of the instruments employed. We consider
some examples below.
Thirty-year US Treasury bonds mainly have interest-rate risk.
Their yield is the reward for bearing this risk. If the US Treasury
market experiences large money inflows, one can expect yields
to decrease. This means mark-to-market gains, which can be
viewed as advance coupon payments. Since the time horizon of
the instrument is rather long, the mark-to-market effects
outweigh the effect that realized cash flows in the future will be
lower. Large inflows in such a market will lead to above-average
performance gains.
However, if we consider a portfolio of short-term high-yield
bonds, the effect is less pronounced or even the opposite. With
short-term high-yield bonds, credit risks dominate. Large
money inflows will compress credit spreads. The mark-to-
market effect will be small due to the short duration, but the
coupon payments of new bonds will be much lower and over the
period of a year or so will be below average. A similar effect for
merger arbitrage strategies is explained by Mitchell and Pulvino
(2001).37
The same is true for more nontraditional risk premia such as
volatility risk premia, for example, where the underlying
systematic return strategy involves the selling of variance
swaps. A short variance swap position accumulates the
difference between implied and realized variance typically for
major equity indices. The profit and loss (PnL) consists of a
realized part since inception and an expected value of the
nonrealized part of the remaining life of the swap. Large inflows,
i.e. from sellers of variance/volatility, can depress the implied
volatility. This is beneficial for the mark-to-market of the
nonrealized part, which is more pronounced the longer the
remaining life span of the swap is. Given that the realized
volatility remains constant, a lower implied volatility lowers the
future PnL. Depending on the maturity of the swaps, inflows
into this strategy can lead to increasing or diminishing returns.
Understanding and monitoring such movements are crucial
because real money flows are often not directly observable.
Active management in this regard can prevent investors from
riding an overextended horse.
2.3 Portfolio Construction Using Systematic Return
Strategies
Earlier in this report, we argued that systematic return strategies
are more suitable for portfolio construction due to their very
attractive correlation properties (see Figure 2 and Figure 12).
Therefore, the full potential of systematic return strategies can
only emerge when they are combined within a portfolio in a
suitable way.
The portfolio construction issue is usually embedded into a
probabilistic framework by regarding the future returns as
random variables with unknown probability distributions. If the
investor is aware of his or her utility function
Management of Systematic Return Strategies – A Primer
1/21
1 Portfolio construction with systematic return strategies
In the introduction, we argued that Systematic Return strategies are
portfolio construction due to their very attractive correlation prope
Reference source not found. and Error! Reference source not foun
full potential of systematic return strategies can only emerge when com
in a suitable way.
The portfolio construction problem is usually embedded into a probabil
regarding the future returns as random variables with unknown probab
the investor is aware of his/her utility function , the portfolio weights
given by the solution of the following optimization problem:
[ ]
In words, the weights are chosen such that expected utility of the inve
The utility function corresponding to the prominent mean-variance va
defined as with denoting the random vector of
random covariance matrix.
This approach to portfolio construction is well-established and market pr
is not without certain pitfalls which can be particularly severe for portfo
strategies. Here we want to draw the attention to the so-called model r
estimation risks.
In order to solve the portfolio optimization problem, the probability distrib
returns is required. The usual approach is to choose a parameteriz
returns such that it captures relevant statistical properties of the return
of the model are estimated such that the model provides a good fit to th
data. Once the model is calibrated the probability distribution of th
specified.
While the process seems to be straight-forward it introduces the ris
inappropriate model with potentially severe consequences (misalloca
pitfall is to choose a model that is too simple and hence ignoring
properties of the historical return time series. On the other hand, si
usually more tractable and the estimation risk tends to be s
choosing a model that is very complex usually increase the est
introduces the risk of overfitting.
The situation is particularly simple, if the returns are assumed to be
identically normally distributed. In this case, the required parameters to c
are the expected values of the returns and the covariance matrix.
sample returns and sample covariance matrix correspond to the so
likelihood estimates for the parameters and accordingly it is enoug
optimization with these sample estimates from historical values. Insp
, the portfolio
weights
1/21
In the introduction, we argued that Systematic Return strategies are more suitable for
portfolio construction due to their very attractive correlation properties (see Error!
Reference source not found. and Error! Reference source not found.). Therefore, the
full potential of systematic return strategies can only emerge when combined to a portfolio
in a suitable way.
The portfolio construction problem is usually embedded into a probabilistic framework by
regarding the future returns as random variables with unknown probability distributions. If
the investor is aware of his/her utility function , the portfolio weights are
[ ]
In words, the weights are chosen such that expected utility of the investor is maximized.
The utility function corresponding to the prominent mean-variance variance portfolio is
defined as with denoting the random vector of returns and the
This approach to portfolio construction is well-established and market practice. However, it
is not without certain pitfalls which can be particularly severe for portfolios of systematic
strategies. Here we want to draw the attention to the so-called model risk and associated
estimation risks.
In order to solve the portfolio optimization problem, the probability distribution of the future
returns is required. The usual approach is to choose a parameterized model for the
returns such that it captures relevant statistical properties of the returns. The parameters
of the model are estimated such that the model provides a good fit to the observed returns
data. Once the model is calibrated the probability distribution of the returns is fully
specified.
While the process seems to be straight-forward it introduces the risk of choosing an
inappropriate model with potentially severe consequences (misallocations). A common
pitfall is to choose a model that is too simple and hence ignoring several statistical
properties of the historical return time series. On the other hand, simple models are
usually more tractable and the estimation risk tends to be smaller. Similarly,
choosing a model that is very complex usually increase the estimation risk and
The situation is particularly simple, if the returns are assumed to be independent and
identically normally distributed. In this case, the required parameters to calibrate the model
are the expected values of the returns and the covariance matrix. Here the average
sample returns and sample covariance matrix correspond to the so-called maximum
likelihood estimates for the parameters and accordingly it is enough to perform the
optimization with these sample estimates from historical values. Inspired by this, some
are given by the solution of the
following optimization problem:
1/21
In the introduction, we argued that Systematic Return strategies are more suitabl
portfolio construction due to their very attractive correlation properties (see E
Reference source not found. and Error! Reference source not found.). Therefore
full potential of systematic return strategies can only emerge when combined to a por
in a suitable way.
The portfolio construction problem is usually embedded into a probabilistic framewo
regarding the future returns as random variables with unknown probability distributio
the investor is aware of his/her utility function , the portfolio weights
[ ]
In words, the weights are chosen such that expected utility of the investor is maxim
The utility function corresponding to the prominent mean-variance variance portfo
defined as with denoting the random vector of returns and
This approach to portfolio construction is well-established and market practice. Howev
is not without certain pitfalls which can be particularly severe for portfolios of system
strategies. Here we want to draw the attention to the so-called model risk and assoc
estimation risks.
In order to solve the portfolio optimization problem, the probability distribution of the f
returns is required. The usual approach is to choose a parameterized model fo
returns such that it captures relevant statistical properties of the returns. The param
of the model are estimated such that the model provides a good fit to the observed re
data. Once the model is calibrated the probability distribution of the returns is
specified.
While the process seems to be straight-forward it introduces the risk of choosin
inappropriate model with potentially severe consequences (misallocations). A com
pitfall is to choose a model that is too simple and hence ignoring several statis
properties of the historical return time series. On the other hand, simple models
usually more tractable and the estimation risk tends to be smaller. Simi
choosing a model that is very complex usually increase the estimation risk
The situation is particularly simple, if the returns are assumed to be independent
identically normally distributed. In this case, the required parameters to calibrate the m
are the expected values of the returns and the covariance matrix. Here the ave
sample returns and sample covariance matrix correspond to the so-called maxi
likelihood estimates for the parameters and accordingly it is enough to perform
optimization with these sample estimates from historical values. Inspired by this, s
In words, the weights are chosen in such a way that the
expected utility for the investor is maximized. The utility function
corresponding to the prominent mean-variance portfolio
is defined as
1/21
1 Portfolio construction with systematic return strategie
In the introduction, we argued that Systematic Return strategies
portfolio construction due to their very attractive correlation
Reference source not found. and Error! Reference source not
full potential of systematic return strategies can only emerge when
in a suitable way.
The portfolio construction problem is usually embedded into a pro
regarding the future returns as random variables with unknown pr
the investor is aware of his/her utility function , the portfolio weigh
[ ]
In words, the weights are chosen such that expected utility of the
The utility function corresponding to the prominent mean-varian
defined as with denoting the random vec
This approach to portfolio construction is well-established and mar
is not without certain pitfalls which can be particularly severe for
strategies. Here we want to draw the attention to the so-called mo
estimation risks.
In order to solve the portfolio optimization problem, the probability
returns is required. The usual approach is to choose a param
returns such that it captures relevant statistical properties of the r
of the model are estimated such that the model provides a good fit
data. Once the model is calibrated the probability distribution
specified.
While the process seems to be straight-forward it introduces t
inappropriate model with potentially severe consequences (misa
pitfall is to choose a model that is too simple and hence ign
properties of the historical return time series. On the other han
usually more tractable and the estimation risk tends to
choosing a model that is very complex usually increase the
The situation is particularly simple, if the returns are assumed
identically normally distributed. In this case, the required paramete
are the expected values of the returns and the covariance ma
sample returns and sample covariance matrix correspond to t
likelihood estimates for the parameters and accordingly it is e
optimization with these sample estimates from historical values.
, with
1/21
1 Portfolio construction with systematic return strateg
In the introduction, we argued that Systematic Return strateg
portfolio construction due to their very attractive correlation
Reference source not found. and Error! Reference source n
full potential of systematic return strategies can only emerge whe
in a suitable way.
The portfolio construction problem is usually embedded into a p
regarding the future returns as random variables with unknown
the investor is aware of his/her utility function , the portfolio wei
[ ]
In words, the weights are chosen such that expected utility of t
The utility function corresponding to the prominent mean-varia
defined as with denoting the random ve
This approach to portfolio construction is well-established and ma
is not without certain pitfalls which can be particularly severe fo
strategies. Here we want to draw the attention to the so-called m
estimation risks.
In order to solve the portfolio optimization problem, the probabilit
returns is required. The usual approach is to choose a para
returns such that it captures relevant statistical properties of the
of the model are estimated such that the model provides a good
data. Once the model is calibrated the probability distributio
specified.
While the process seems to be straight-forward it introduces
inappropriate model with potentially severe consequences (m
pitfall is to choose a model that is too simple and hence ig
properties of the historical return time series. On the other ha
usually more tractable and the estimation risk tends to
choosing a model that is very complex usually increase th
The situation is particularly simple, if the returns are assumed
identically normally distributed. In this case, the required parame
are the expected values of the returns and the covariance m
sample returns and sample covariance matrix correspond to
likelihood estimates for the parameters and accordingly it is
optimization with these sample estimates from historical value
denoting the
random vector of returns and
1/21
In the introduction, we argued that Systematic Return strategies are more suitable for
portfolio construction due to their very attractive correlation properties (see Error!
Reference source not found. and Error! Reference source not found.). Therefore, the
full potential of systematic return strategies can only emerge when combined to a portfolio
in a suitable way.
The portfolio construction problem is usually embedded into a probabilistic framework by
regarding the future returns as random variables with unknown probability distributions. If
the investor is aware of his/her utility function , the portfolio weights are
[ ]
In words, the weights are chosen such that expected utility of the investor is maximized.
The utility function corresponding to the prominent mean-variance variance portfolio is
defined as with denoting the random vector of returns and the
This approach to portfolio construction is well-established and market practice. However, it
is not without certain pitfalls which can be particularly severe for portfolios of systematic
strategies. Here we want to draw the attention to the so-called model risk and associated
estimation risks.
In order to solve the portfolio optimization problem, the probability distribution of the future
returns is required. The usual approach is to choose a parameterized model for the
returns such that it captures relevant statistical properties of the returns. The parameters
of the model are estimated such that the model provides a good fit to the observed returns
data. Once the model is calibrated the probability distribution of the returns is fully
specified.
While the process seems to be straight-forward it introduces the risk of choosing an
inappropriate model with potentially severe consequences (misallocations). A common
pitfall is to choose a model that is too simple and hence ignoring several statistical
properties of the historical return time series. On the other hand, simple models are
usually more tractable and the estimation risk tends to be smaller. Similarly,
choosing a model that is very complex usually increase the estimation risk and
The situation is particularly simple, if the returns are assumed to be independent and
identically normally distributed. In this case, the required parameters to calibrate the model
are the expected values of the returns and the covariance matrix. Here the average
sample returns and sample covariance matrix correspond to the so-called maximum
likelihood estimates for the parameters and accordingly it is enough to perform the
optimization with these sample estimates from historical values. Inspired by this, some
the random covariance matrix.
This approach to portfolio construction is well-established and
standard market practice. However, it is not without certain
pitfalls – and these can be particularly severe for portfolios of
systematic strategies. Here we would like to draw attention to
the model risk and associated estimation risks.
In order to solve the portfolio optimization problem, the
probability distribution of future returns is required. The usual
approach is to choose a parameterized model for the returns in
such a way that it captures relevant statistical properties of the
returns. The parameters of the model are estimated so that the
model provides a good fit to the observed return data. Once the
model is calibrated, the probability distribution of the returns is
fully specified.
Although the process seems to be straightforward, it introduces
the risk of choosing an inappropriate model with potentially
severe consequences (misallocations). A common pitfall is to
choose a model that is too simple and thus ignores several
statistical properties of the historical return time series. However,
simple models are usually more tractable and the estimation risk
tends to be smaller. Similarly, choosing a model that is very
complex usually increases the estimation risk and introduces
the risk of overfitting.
37
Mitchell and Pulvino (2001) show that merger arbitrage generates excess
returns of +4% per year after transaction costs. Jetley and Ji (2010), however,
find that the merger arbitrage spread has declined by more than 400 basis
points since 2002. The authors attribute this to increased inflows into merger
arbitrage hedge funds and reduced transaction costs.

34 / 54
The situation is particularly easy if the returns are assumed to
be independent and identically normally distributed. In this case,
the required parameters to calibrate the model are the expected
values of the returns and the covariance matrix. Here, the
average sample returns and sample covariance matrix
correspond to the maximum likelihood estimates for the
parameters, and, accordingly, it is enough to perform the
optimization with these sample estimates from historical values.
Inspired by this, some investment managers perform the
portfolio optimization with sample estimates from historical
values and implicitly assume that the resulting weights will be
optimal for the out-of-sample period. However, if the model
assumptions are not satisfied, this estimator will generally be
biased (model error).
In the sample-based approach, the maximum likelihood
estimators as given by the sample values agree with the “true”
values only asymptotically, that is only if the sample size tends
to infinity and, accordingly, the size of the estimation error is
particularly high for small sample sizes. This can be a serious
problem for systematic return strategies because the available
historical time series, in general, are much shorter than those
for traditional assets. Some instruments used to construct
certain systematic return strategies have only been introduced
over the past four decades. For example, exchange-traded
options and VIX futures started trading on the Chicago Board
Options Exchange (CBOE) in 1973 and 2004, respectively.
The realized estimation error is given by the difference
ptimization with sample estimates from
esulting weights will be optimal for the out-
ptions are not satisfied, this estimator will
difference ̂ ̂ , where ̂ ̂
he estimated parameters, that is, ̂ ̂
s and covariance matrix given by ̂ and ̂.
uel, 2009) compared the out-of-sample
izers and found that none of them is
ed performance measures like the Sharpe
n strategy given by the equally weighted
ve strategy, the benefit from the portfolio
on errors.
elihood estimators as given by the sample
ptotically, that is, only if the sample size
imation error is particularly high for small
for systematic return strategies, as the
uch shorter when compared to traditional
tain systematic return strategies have only
exchange-traded options and VIX futures
Exchange (CBOE) in 1973 and in 2004
ndix 4 and investigated the relationship
e risk-adjusted portfolio performance. We
form naïve diversification if the estimation
asymmetry of the effect: estimation errors
, since, trivially, every deviation from the
al., 1989) described this effect as an “error
ion procedures especially mean-variance,
ardize the whole optimization endeavor.
o calibrate, and a large number of assets
ment professional employing optimization
alyze potential negative consequences.
away from the in-sample optimal portfolio
of-sample performance.
n with entropy measures
, where
timization with sample estimates from
sulting weights will be optimal for the out-
tions are not satisfied, this estimator will
difference ̂ ̂ , where ̂ ̂
e estimated parameters, that is, ̂ ̂
and covariance matrix given by ̂ and ̂.
el, 2009) compared the out-of-sample
zers and found that none of them is
d performance measures like the Sharpe
strategy given by the equally weighted
e strategy, the benefit from the portfolio
n errors.
lihood estimators as given by the sample
totically, that is, only if the sample size
mation error is particularly high for small
for systematic return strategies, as the
ch shorter when compared to traditional
in systematic return strategies have only
exchange-traded options and VIX futures
Exchange (CBOE) in 1973 and in 2004
dix 4 and investigated the relationship
risk-adjusted portfolio performance. We
orm naïve diversification if the estimation
symmetry of the effect: estimation errors
since, trivially, every deviation from the
., 1989) described this effect as an “error
on procedures especially mean-variance,
rdize the whole optimization endeavor.
calibrate, and a large number of assets
ment professional employing optimization
yze potential negative consequences.
way from the in-sample optimal portfolio
-sample performance.
with entropy measures
corresponds to the maximized
utility based on the estimated parameters, that is,
urn Strategies – A Primer
s perform the portfolio optimization with sample estimates from
mplicitly assume that the resulting weights will be optimal for the out-
wever, if the model assumptions are not satisfied, this estimator will
model error).
ion error is given by the difference ̂ ̂ , where ̂ ̂
aximized utility based on the estimated parameters, that is, ̂ ̂
estimated expected returns and covariance matrix given by ̂ and ̂.
be quite severe. (DeMiguel, 2009) compared the out-of-sample
ple-based portfolio optimizers and found that none of them is
erms of popular risk adjusted performance measures like the Sharpe
to the naïve diversification strategy given by the equally weighted
es that, relative to the naïve strategy, the benefit from the portfolio
more than offset by estimation errors.
approach, the maximum likelihood estimators as given by the sample
e “true” values only asymptotically, that is, only if the sample size
accordingly the size of estimation error is particularly high for small
an be a serious problem for systematic return strategies, as the
e series are in general much shorter when compared to traditional
ents used to construct certain systematic return strategies have only
e past four decades; e. g., exchange-traded options and VIX futures
Chicago Board of Option Exchange (CBOE) in 1973 and in 2004
milar experiment in Appendix 4 and investigated the relationship
he estimation error and the risk-adjusted portfolio performance. We
optimization can underperform naïve diversification if the estimation
ge. Most important is the asymmetry of the effect: estimation errors
affect the optimal portfolio, since, trivially, every deviation from the
rior results1
. (Michaud et. al., 1989) described this effect as an “error
inherent in many optimization procedures especially mean-variance,
stimation errors can jeopardize the whole optimization endeavor.
series, many parameters to calibrate, and a large number of assets
mation errors. Every investment professional employing optimization
aware of this effect and analyze potential negative consequences.
on we show how moving away from the in-sample optimal portfolio
folio entropy can help out-of-sample performance.
Portfolio diversification with entropy measures
investment managers perform the portfolio optimization with sample estimates from
historical values and implicitly assume that the resulting weights will be optimal for the out-
of-sample period. However, if the model assumptions are not satisfied, this estimator will
generally be biased (model error).
The realized estimation error is given by the difference ̂ ̂ , where ̂ ̂
corresponds to the maximized utility based on the estimated parameters, that is, ̂ ̂
̂ ̂ ̂ ̂ ̂ for the estimated expected returns and covariance matrix given by ̂ and ̂.
This difference can be quite severe. (DeMiguel, 2009) compared the out-of-sample
performance of sample-based portfolio optimizers and found that none of them is
consistently better in terms of popular risk adjusted performance measures like the Sharpe
ratio when compared to the naïve diversification strategy given by the equally weighted
portfolio. He concludes that, relative to the naïve strategy, the benefit from the portfolio
optimization is often more than offset by estimation errors.
In the sample based approach, the maximum likelihood estimators as given by the sample
values agree with the “true” values only asymptotically, that is, only if the sample size
tends to infinity and accordingly the size of estimation error is particularly high for small
sample sizes. This can be a serious problem for systematic return strategies, as the
available historic time series are in general much shorter when compared to traditional
assets. Some instruments used to construct certain systematic return strategies have only
been introduced in the past four decades; e. g., exchange-traded options and VIX futures
started trading at the Chicago Board of Option Exchange (CBOE) in 1973 and in 2004
respectively.
We conducted a similar experiment in Appendix 4 and investigated the relationship
between the size of the estimation error and the risk-adjusted portfolio performance. We
find that that portfolio optimization can underperform naïve diversification if the estimation
errors become too large. Most important is the asymmetry of the effect: estimation errors
can only negatively affect the optimal portfolio, since, trivially, every deviation from the
optimum leads to inferior results1
. (Michaud et. al., 1989) described this effect as an “error
maximizing” property inherent in many optimization procedures especially mean-variance,
where even small estimation errors can jeopardize the whole optimization endeavor.
Especially short time series, many parameters to calibrate, and a large number of assets
tend to increase estimation errors. Every investment professional employing optimization
techniques needs be aware of this effect and analyze potential negative consequences.
In the following section we show how moving away from the in-sample optimal portfolio
while maximizing portfolio entropy can help out-of-sample performance.
1.1 Case Study 1: Portfolio diversification with entropy measures
for the estimated expected returns and
covariance matrix given by
o optimization with sample estimates from
he resulting weights will be optimal for the out-
sumptions are not satisfied, this estimator will
the difference ̂ ̂ , where ̂ ̂
on the estimated parameters, that is, ̂ ̂
turns and covariance matrix given by ̂ and ̂.
eMiguel, 2009) compared the out-of-sample
ptimizers and found that none of them is
djusted performance measures like the Sharpe
cation strategy given by the equally weighted
naïve strategy, the benefit from the portfolio
mation errors.
m likelihood estimators as given by the sample
symptotically, that is, only if the sample size
estimation error is particularly high for small
lem for systematic return strategies, as the
l much shorter when compared to traditional
certain systematic return strategies have only
. g., exchange-traded options and VIX futures
ption Exchange (CBOE) in 1973 and in 2004
ppendix 4 and investigated the relationship
d the risk-adjusted portfolio performance. We
rperform naïve diversification if the estimation
the asymmetry of the effect: estimation errors
olio, since, trivially, every deviation from the
et. al., 1989) described this effect as an “error
mization procedures especially mean-variance,
eopardize the whole optimization endeavor.
ers to calibrate, and a large number of assets
vestment professional employing optimization
d analyze potential negative consequences.
ng away from the in-sample optimal portfolio
out-of-sample performance.
ation with entropy measures
and
io optimization with sample estimates from
he resulting weights will be optimal for the out-
ssumptions are not satisfied, this estimator will
the difference ̂ ̂ , where ̂ ̂
on the estimated parameters, that is, ̂ ̂
eturns and covariance matrix given by ̂ and ̂.
eMiguel, 2009) compared the out-of-sample
optimizers and found that none of them is
djusted performance measures like the Sharpe
cation strategy given by the equally weighted
e naïve strategy, the benefit from the portfolio
imation errors.
m likelihood estimators as given by the sample
symptotically, that is, only if the sample size
f estimation error is particularly high for small
blem for systematic return strategies, as the
al much shorter when compared to traditional
t certain systematic return strategies have only
e. g., exchange-traded options and VIX futures
ption Exchange (CBOE) in 1973 and in 2004
Appendix 4 and investigated the relationship
nd the risk-adjusted portfolio performance. We
erperform naïve diversification if the estimation
the asymmetry of the effect: estimation errors
folio, since, trivially, every deviation from the
et. al., 1989) described this effect as an “error
mization procedures especially mean-variance,
jeopardize the whole optimization endeavor.
ers to calibrate, and a large number of assets
nvestment professional employing optimization
d analyze potential negative consequences.
ing away from the in-sample optimal portfolio
out-of-sample performance.
ation with entropy measures
. This difference can be
quite severe. DeMiguel et al. (2009) compared the out-of-
sample performance of sample-based portfolio optimizers and
found that none of them were consistently better in terms of
popular risk-adjusted performance measures such as the
Sharpe ratio when compared to the naïve diversification strategy
given by the equally weighted portfolio. He concluded that
relative to the naïve strategy, the benefit from the portfolio
optimization was often more than offset by estimation errors.
We conducted a similar experiment in Appendix 4 and
investigated the relationship between the size of the estimation
error and the risk-adjusted portfolio performance. We found
that portfolio optimization can underperform naïve diversification
if the estimation errors become too large. Most important is the
asymmetry of the effect: estimation errors can only negatively
affect the optimal portfolio since, trivially, every deviation from
the optimum leads to inferior results.38
Michaud (1989) described this effect as an “error-maximizing”
property inherent in many optimization procedures, especially
mean-variance, where even small estimation errors can
jeopardize the entire optimization endeavor. In particular, short
time series, many parameters to calibrate and a large number
of assets tend to increase estimation errors. Investment
professionals employing optimization techniques should be
aware of this effect and should analyze potential negative
consequences.
In the next section, we show how moving away from the
in-sample optimal portfolio while maximizing portfolio entropy
can help out-of-sample performance.
38
Of course, any deviation from the optimal portfolio is ex-ante (i.e. in expectation)
inferior, but ex-post (i.e. a realization) might be superior.

2.4 Case Study 1: Portfolio Diversification with Entropy
Measures
So far, we have discussed potential pitfalls that arise in the
construction of “optimal” portfolios of systematic return
strategies by introducing the notions of model risk and
estimation risk. The purpose of this case study is twofold. First,
we quantify the effect of those risks on optimized portfolios in
terms of various performance and risk measures, and second,
we introduce a simple but highly effective method to mitigate
some of those risks.
It is well known that portfolio optimizers are very sensitive to the
input parameters39
and, accordingly, minor model
misspecifications and estimation errors can lead to drastically
altered “optimal” allocations. In layman’s terms, the portfolio
optimizer has 100% confidence in the input data and,
accordingly, has no issues with assigning extreme weights.
There are several methods that address this problem by
including an additional input parameter, which reflects the
uncertainty incorporated in the estimates. Popular examples
include the Black-Litterman model, Meucci’s entropy-pooling
approach and several shrinkage methods.40
The common
underlying idea of these methods is the introduction (explicitly
or implicitly) of a neutral probability distribution of the returns (a
neutral benchmark) that is blended with the input data.
The basic intention is to obtain weights that are in between the
neutral benchmark and the weights that correspond to the
estimated return distribution such that the deviation from the
neutral benchmark should be in an inverse relationship to the
uncertainty with respect to the estimated quantities.
In other words, the optimization result is being pulled toward
the neutral benchmark. In the event that the estimates are very
uncertain, it essentially results in weights of the neutral
benchmark. At the other extreme, if the estimates are certain,
the neutral benchmark has no effect and the optimization is
entirely based on the estimated parameters. In general, the
estimated parameters will be blended with the neutral
benchmark.
It should be noted that the highlighted words require the notion
of a distance between potential sets of weights. If we rule out
the possibility of short positions and leverage, the weights will
lie between 0% and 100% and will add up to 100%, just like a
probability distribution. And for these, a well-understood notion
of distance is given by the entropy (or cross-entropy).41
Below,
we demonstrate how entropy can be used to efficiently deal
with estimation errors.
For traditional assets, the market portfolio is a natural choice as
a neutral benchmark. Since no such market portfolio exists
for systematic return strategies, the maximum entropy, i.e.
the equally weighted portfolio, is a common choice. Here,
the entropy can be interpreted as a measure of diversification
since – in the absence of any reliable estimates – the equally
weighted portfolio promises the maximum out-of-sample
diversification.42
Any deviation from this portfolio leads to a
potentially smaller diversification and should be justified by the
existence of reliable estimates.
39
See Best and Grauer (1991).
40
See also Stefanovits et al. (2014) for a very recent account of this and the
references therein.
41
Note that short and leveraged positions can both be incorporated by using the
generalized entropy measure.
42
Trivially, if reliable information on the distribution exists, the maximum diversified
portfolio will, in general, differ from the maximum entropy portfolio. For
example, if two of the assets in a portfolio are linearly dependent (with high
confidence), the equally weighted portfolio would assign a combined weight
to those assets that is too high when compared to the weights of the maximal
diversified portfolio.

36 / 54
Next, we will discuss a particularly simple and straightforward
method to employ this notion, which is inspired by the Optimal
Portfolio Diversification Using Maximum Entropy Principle
authored by Bera and Park (2008). The key finding is that
sacrificing in-sample optimality in favor of increased entropy (in
the weight’s space) potentially increases the out-of-sample
performance.
We consider the following portfolio optimization problem: given
the in-sample optimal weights
4/21
equally weighted portfolio promises the maximum out-of-sample diversification 4
. Any
deviation from this portfolio leads to a potentially smaller diversification and should be
justified by the existence of reliable estimates.
In the following, we will discuss a particularly simple and straightforward method to employ
this notion, which is inspired by “Optimal Portfolio Diversification Using Maximum Entropy
Principle” by Bera/Park (2008). The key finding is that, sacrificing in-sample-optimality in
favor of increased entropy (in the weight’s space) potentially increases the out-of-sample
performance.
We consider the following portfolio optimization problem:
Given the in-sample optimal weights and the corresponding in-
sample maximal utility the constraint maximum entropy weights are given by
subject to
where the confidence parameter is a number between 0 and 15
and is the entropy6
corresponding to the weights The parameter indicates the confidence
in the estimates of the risk-model: the closer to one, the less is the portfolio optimizer
allowed to deviate from the in-sample optimum. An of zero would be appropriate if there
is a complete disbelief in the in-sample estimates, while a value of one would be
appropriate if there is full confidence in the in-sample estimates.
The impact of model misspecifications and estimation errors can be demonstrated by
comparing the utility of in-sample and out-of-sample optimized portfolios using the same
sample-estimators7
. In fact, if there were no model or estimation errors, the optimal in-
sample weights should trivially be also optimal for the out-of-sample period. This implies
that the out-of-sample utility should increase with increasing values of the parameter .
However, in the presence of model and estimation risk this relationship does not
necessarily remain valid and sacrificing in-sample optimality (by choosing the confidence
parameter ) thereby increasing the entropy might actually increase the out-of-sample
utility.
4
Trivially, if reliable informations on the distribution exist, the maximum diversified portfolio will in general differ from the maximum entropy
portfolio. For example, if in a portfolio two of the assets are linearly dependent (with high confidence), the equally weighted portfolio would
assign a combined weight to these assets that is too high when compared to the weights of the maximal diversified portfolio.
5
Note that, in the language of appendix 5, this is the maximum-entropy estimate such that the corresponding utility
does not deviate too much (as controlled by a) from the in-sample optimum utility given by
6
By using entropy the neutral benchmark is the equally weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
Entropy instead of Entropy. See Appendix 5 for a detailed discussion of the entropy concept.
7
In the mean-variance case the sample-estimators are given by the average returns and the sample covariance matrix of the in-sample
period.
and the
corresponding in-sample maximal utility
4/21
. Any
performance.
subject to
and is the entropy6
sample-estimators7
utility.
4
5
6
7
period.
, the constraint
maximum entropy weights
mer
4/21
s the maximum out-of-sample diversification 4
. Any
o a potentially smaller diversification and should be
stimates.
ticularly simple and straightforward method to employ
imal Portfolio Diversification Using Maximum Entropy
key finding is that, sacrificing in-sample-optimality in
eight’s space) potentially increases the out-of-sample
ptimization problem:
hts and the corresponding in-
nstraint maximum entropy weights are given by
number between 0 and 15
and is the entropy6
The parameter indicates the confidence
the closer to one, the less is the portfolio optimizer
e optimum. An of zero would be appropriate if there
sample estimates, while a value of one would be
in the in-sample estimates.
ons and estimation errors can be demonstrated by
nd out-of-sample optimized portfolios using the same
were no model or estimation errors, the optimal in-
so optimal for the out-of-sample period. This implies
increase with increasing values of the parameter .
del and estimation risk this relationship does not
ing in-sample optimality (by choosing the confidence
the entropy might actually increase the out-of-sample
the maximum diversified portfolio will in general differ from the maximum entropy
are linearly dependent (with high confidence), the equally weighted portfolio would
when compared to the weights of the maximal diversified portfolio.
is the maximum-entropy estimate such that the corresponding utility
by a) from the in-sample optimum utility given by
weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
d discussion of the entropy concept.
given by the average returns and the sample covariance matrix of the in-sample
are given by:

agement of Systematic Return Strategies – A Primer
4/21
ually weighted portfolio promises the maximum out-of-sample diversification 4
. Any
viation from this portfolio leads to a potentially smaller diversification and should be
tified by the existence of reliable estimates.
he following, we will discuss a particularly simple and straightforward method to employ
s notion, which is inspired by “Optimal Portfolio Diversification Using Maximum Entropy
nciple” by Bera/Park (2008). The key finding is that, sacrificing in-sample-optimality in
or of increased entropy (in the weight’s space) potentially increases the out-of-sample
rformance.
e consider the following portfolio optimization problem:
ven the in-sample optimal weights and the corresponding in-
mple maximal utility the constraint maximum entropy weights are given by
subject to
ere the confidence parameter is a number between 0 and 15
and is the entropy6
responding to the weights The parameter indicates the confidence
the estimates of the risk-model: the closer to one, the less is the portfolio optimizer
owed to deviate from the in-sample optimum. An of zero would be appropriate if there
a complete disbelief in the in-sample estimates, while a value of one would be
propriate if there is full confidence in the in-sample estimates.
e impact of model misspecifications and estimation errors can be demonstrated by
mparing the utility of in-sample and out-of-sample optimized portfolios using the same
mple-estimators7
mple weights should trivially be also optimal for the out-of-sample period. This implies
t the out-of-sample utility should increase with increasing values of the parameter .
wever, in the presence of model and estimation risk this relationship does not
cessarily remain valid and sacrificing in-sample optimality (by choosing the confidence
rameter ) thereby increasing the entropy might actually increase the out-of-sample
ity.
ially, if reliable informations on the distribution exist, the maximum diversified portfolio will in general differ from the maximum entropy
olio. For example, if in a portfolio two of the assets are linearly dependent (with high confidence), the equally weighted portfolio would
gn a combined weight to these assets that is too high when compared to the weights of the maximal diversified portfolio.
te that, in the language of appendix 5, this is the maximum-entropy estimate such that the corresponding utility
using entropy the neutral benchmark is the equally weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
opy instead of Entropy. See Appendix 5 for a detailed discussion of the entropy concept.
he mean-variance case the sample-estimators are given by the average returns and the sample covariance matrix of the in-sample
d.
subject to
ent of Systematic Return Strategies – A Primer
4/21
y weighted portfolio promises the maximum out-of-sample diversification 4
. Any
on from this portfolio leads to a potentially smaller diversification and should be
d by the existence of reliable estimates.
following, we will discuss a particularly simple and straightforward method to employ
tion, which is inspired by “Optimal Portfolio Diversification Using Maximum Entropy
ple” by Bera/Park (2008). The key finding is that, sacrificing in-sample-optimality in
of increased entropy (in the weight’s space) potentially increases the out-of-sample
mance.
nsider the following portfolio optimization problem:
the in-sample optimal weights and the corresponding in-
e maximal utility the constraint maximum entropy weights are given by
subject to
the confidence parameter is a number between 0 and 15
and is the entropy6
ponding to the weights The parameter indicates the confidence
estimates of the risk-model: the closer to one, the less is the portfolio optimizer
d to deviate from the in-sample optimum. An of zero would be appropriate if there
omplete disbelief in the in-sample estimates, while a value of one would be
priate if there is full confidence in the in-sample estimates.
mpact of model misspecifications and estimation errors can be demonstrated by
ring the utility of in-sample and out-of-sample optimized portfolios using the same
e-estimators7
e weights should trivially be also optimal for the out-of-sample period. This implies
e out-of-sample utility should increase with increasing values of the parameter .
ver, in the presence of model and estimation risk this relationship does not
sarily remain valid and sacrificing in-sample optimality (by choosing the confidence
eter ) thereby increasing the entropy might actually increase the out-of-sample
if reliable informations on the distribution exist, the maximum diversified portfolio will in general differ from the maximum entropy
For example, if in a portfolio two of the assets are linearly dependent (with high confidence), the equally weighted portfolio would
ombined weight to these assets that is too high when compared to the weights of the maximal diversified portfolio.
at, in the language of appendix 5, this is the maximum-entropy estimate such that the corresponding utility
entropy the neutral benchmark is the equally weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
stead of Entropy. See Appendix 5 for a detailed discussion of the entropy concept.
ean-variance case the sample-estimators are given by the average returns and the sample covariance matrix of the in-sample
,
where the confidence parameter
of Systematic Return Strategies – A Primer
4/21
eighted portfolio promises the maximum out-of-sample diversification 4
. Any
from this portfolio leads to a potentially smaller diversification and should be
y the existence of reliable estimates.
owing, we will discuss a particularly simple and straightforward method to employ
n, which is inspired by “Optimal Portfolio Diversification Using Maximum Entropy
by Bera/Park (2008). The key finding is that, sacrificing in-sample-optimality in
ncreased entropy (in the weight’s space) potentially increases the out-of-sample
nce.
der the following portfolio optimization problem:
e in-sample optimal weights and the corresponding in-
aximal utility the constraint maximum entropy weights are given by
subject to
confidence parameter is a number between 0 and 15
and is the entropy6
ding to the weights The parameter indicates the confidence
imates of the risk-model: the closer to one, the less is the portfolio optimizer
o deviate from the in-sample optimum. An of zero would be appropriate if there
plete disbelief in the in-sample estimates, while a value of one would be
te if there is full confidence in the in-sample estimates.
ct of model misspecifications and estimation errors can be demonstrated by
g the utility of in-sample and out-of-sample optimized portfolios using the same
stimators7
eights should trivially be also optimal for the out-of-sample period. This implies
ut-of-sample utility should increase with increasing values of the parameter .
in the presence of model and estimation risk this relationship does not
ly remain valid and sacrificing in-sample optimality (by choosing the confidence
r ) thereby increasing the entropy might actually increase the out-of-sample
able informations on the distribution exist, the maximum diversified portfolio will in general differ from the maximum entropy
xample, if in a portfolio two of the assets are linearly dependent (with high confidence), the equally weighted portfolio would
ned weight to these assets that is too high when compared to the weights of the maximal diversified portfolio.
n the language of appendix 5, this is the maximum-entropy estimate such that the corresponding utility
not deviate too much (as controlled by a) from the in-sample optimum utility given by
opy the neutral benchmark is the equally weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
of Entropy. See Appendix 5 for a detailed discussion of the entropy concept.
variance case the sample-estimators are given by the average returns and the sample covariance matrix of the in-sample
is a number between 0 and
143
and
um out-of-sample diversification 4
. Any
y smaller diversification and should be
le and straightforward method to employ
Diversification Using Maximum Entropy
s that, sacrificing in-sample-optimality in
) potentially increases the out-of-sample
oblem:
and the corresponding in-
mum entropy weights are given by
ween 0 and 15
and is the entropy6
he parameter indicates the confidence
one, the less is the portfolio optimizer
n of zero would be appropriate if there
ates, while a value of one would be
ple estimates.
mation errors can be demonstrated by
mple optimized portfolios using the same
del or estimation errors, the optimal in-
or the out-of-sample period. This implies
h increasing values of the parameter .
mation risk this relationship does not
e optimality (by choosing the confidence
might actually increase the out-of-sample
sified portfolio will in general differ from the maximum entropy
nt (with high confidence), the equally weighted portfolio would
the weights of the maximal diversified portfolio.
entropy estimate such that the corresponding utility
-sample optimum utility given by
ny other neutral benchmark can be chosen by using the Cross-
ntropy concept.
e returns and the sample covariance matrix of the in-sample
is the entropy44
corresponding to the weights
c Return Strategies – A Primer
4/21
portfolio promises the maximum out-of-sample diversification 4
. Any
s portfolio leads to a potentially smaller diversification and should be
stence of reliable estimates.
e will discuss a particularly simple and straightforward method to employ
is inspired by “Optimal Portfolio Diversification Using Maximum Entropy
/Park (2008). The key finding is that, sacrificing in-sample-optimality in
entropy (in the weight’s space) potentially increases the out-of-sample
ollowing portfolio optimization problem:
mple optimal weights and the corresponding in-
tility the constraint maximum entropy weights are given by
subject to
ce parameter is a number between 0 and 15
and is the entropy6
the weights The parameter indicates the confidence
of the risk-model: the closer to one, the less is the portfolio optimizer
from the in-sample optimum. An of zero would be appropriate if there
sbelief in the in-sample estimates, while a value of one would be
e is full confidence in the in-sample estimates.
odel misspecifications and estimation errors can be demonstrated by
ity of in-sample and out-of-sample optimized portfolios using the same
7
hould trivially be also optimal for the out-of-sample period. This implies
mple utility should increase with increasing values of the parameter .
presence of model and estimation risk this relationship does not
n valid and sacrificing in-sample optimality (by choosing the confidence
thereby increasing the entropy might actually increase the out-of-sample
ons on the distribution exist, the maximum diversified portfolio will in general differ from the maximum entropy
a portfolio two of the assets are linearly dependent (with high confidence), the equally weighted portfolio would
these assets that is too high when compared to the weights of the maximal diversified portfolio.
age of appendix 5, this is the maximum-entropy estimate such that the corresponding utility
e too much (as controlled by a) from the in-sample optimum utility given by
al benchmark is the equally weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
See Appendix 5 for a detailed discussion of the entropy concept.
e the sample-estimators are given by the average returns and the sample covariance matrix of the in-sample
. The parameter
nt of Systematic Return Strategies – A Primer
4/21
weighted portfolio promises the maximum out-of-sample diversification 4
. Any
n from this portfolio leads to a potentially smaller diversification and should be
by the existence of reliable estimates.
ollowing, we will discuss a particularly simple and straightforward method to employ
ion, which is inspired by “Optimal Portfolio Diversification Using Maximum Entropy
e” by Bera/Park (2008). The key finding is that, sacrificing in-sample-optimality in
increased entropy (in the weight’s space) potentially increases the out-of-sample
ance.
sider the following portfolio optimization problem:
the in-sample optimal weights and the corresponding in-
maximal utility the constraint maximum entropy weights are given by
subject to
he confidence parameter is a number between 0 and 15
and is the entropy6
onding to the weights The parameter indicates the confidence
estimates of the risk-model: the closer to one, the less is the portfolio optimizer
to deviate from the in-sample optimum. An of zero would be appropriate if there
omplete disbelief in the in-sample estimates, while a value of one would be
iate if there is full confidence in the in-sample estimates.
pact of model misspecifications and estimation errors can be demonstrated by
ing the utility of in-sample and out-of-sample optimized portfolios using the same
-estimators7
weights should trivially be also optimal for the out-of-sample period. This implies
e out-of-sample utility should increase with increasing values of the parameter .
er, in the presence of model and estimation risk this relationship does not
arily remain valid and sacrificing in-sample optimality (by choosing the confidence
ter ) thereby increasing the entropy might actually increase the out-of-sample
reliable informations on the distribution exist, the maximum diversified portfolio will in general differ from the maximum entropy
r example, if in a portfolio two of the assets are linearly dependent (with high confidence), the equally weighted portfolio would
mbined weight to these assets that is too high when compared to the weights of the maximal diversified portfolio.
t, in the language of appendix 5, this is the maximum-entropy estimate such that the corresponding utility
oes not deviate too much (as controlled by a) from the in-sample optimum utility given by
entropy the neutral benchmark is the equally weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
ead of Entropy. See Appendix 5 for a detailed discussion of the entropy concept.
an-variance case the sample-estimators are given by the average returns and the sample covariance matrix of the in-sample
indicates the confidence
in the estimates of the risk model: the closer to one, the less
the portfolio optimizer is allowed to deviate from the in-sample
optimum. An
tegies – A Primer
4/21
promises the maximum out-of-sample diversification 4
. Any
o leads to a potentially smaller diversification and should be
reliable estimates.
cuss a particularly simple and straightforward method to employ
ed by “Optimal Portfolio Diversification Using Maximum Entropy
008). The key finding is that, sacrificing in-sample-optimality in
(in the weight’s space) potentially increases the out-of-sample
portfolio optimization problem:
mal weights and the corresponding in-
the constraint maximum entropy weights are given by
t to
eter is a number between 0 and 15
and is the entropy6
hts The parameter indicates the confidence
k-model: the closer to one, the less is the portfolio optimizer
in-sample optimum. An of zero would be appropriate if there
n the in-sample estimates, while a value of one would be
onfidence in the in-sample estimates.
specifications and estimation errors can be demonstrated by
sample and out-of-sample optimized portfolios using the same
, if there were no model or estimation errors, the optimal in-
ially be also optimal for the out-of-sample period. This implies
ty should increase with increasing values of the parameter .
e of model and estimation risk this relationship does not
nd sacrificing in-sample optimality (by choosing the confidence
ncreasing the entropy might actually increase the out-of-sample
stribution exist, the maximum diversified portfolio will in general differ from the maximum entropy
o of the assets are linearly dependent (with high confidence), the equally weighted portfolio would
that is too high when compared to the weights of the maximal diversified portfolio.
endix 5, this is the maximum-entropy estimate such that the corresponding utility
(as controlled by a) from the in-sample optimum utility given by
is the equally weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
5 for a detailed discussion of the entropy concept.
estimators are given by the average returns and the sample covariance matrix of the in-sample
of zero would be appropriate if there were a
complete disbelief in the in-sample estimates, while a value of
one would be appropriate if there were full confidence in the
in-sample estimates.
The impact of model misspecifications and estimation errors
can be demonstrated by comparing the utility of in-sample and
out-of-sample optimized portfolios using the same sample
estimators.45
In fact, if there were no model or estimation errors,
the optimal in-sample weights should also be optimal for the
out-of-sample period. This implies that the out-of-sample utility
should increase with increasing values of the parameter
4/21
. Any
performance.
subject to
and is the entropy6
sample-estimators7
utility.
4
5
6
7
period.
.
However, in the presence of model and estimation risk, this
relationship does not necessarily remain valid, and sacrificing
in-sample optimality (by choosing the confidence parameter
he maximum out-of-sample diversification 4
. Any
a potentially smaller diversification and should be
mates.
ularly simple and straightforward method to employ
al Portfolio Diversification Using Maximum Entropy
y finding is that, sacrificing in-sample-optimality in
ht’s space) potentially increases the out-of-sample
mization problem:
and the corresponding in-
raint maximum entropy weights are given by
umber between 0 and 15
and is the entropy6
The parameter indicates the confidence
closer to one, the less is the portfolio optimizer
ptimum. An of zero would be appropriate if there
mple estimates, while a value of one would be
he in-sample estimates.
s and estimation errors can be demonstrated by
out-of-sample optimized portfolios using the same
re no model or estimation errors, the optimal in-
optimal for the out-of-sample period. This implies
crease with increasing values of the parameter .
and estimation risk this relationship does not
in-sample optimality (by choosing the confidence
e entropy might actually increase the out-of-sample
maximum diversified portfolio will in general differ from the maximum entropy
inearly dependent (with high confidence), the equally weighted portfolio would
en compared to the weights of the maximal diversified portfolio.
he maximum-entropy estimate such that the corresponding utility
a) from the in-sample optimum utility given by
ted portfolio. Any other neutral benchmark can be chosen by using the Cross-
cussion of the entropy concept.
n by the average returns and the sample covariance matrix of the in-sample
1) and thereby increasing the entropy might actually
increase the out-of-sample utility.
Now, let us investigate a concrete example using the systematic
return strategies introduced in section 4. Here the utility
function is chosen to be the Sharpe ratio:46

. Any
performance.
subject to
and is the entropy6
sample-estimators7
utility.
subject to
ystematic Return Strategies – A Primer
s investigate a concrete example using the systematic return strategies
n chapter 4. Here the utility function is chosen to be the Sharpe ratio8
:
subject to
e determined the maximum entropy weights, such that the corresponding
o does not deviate too much (as controlled by the confidence parameter )
mal in-sample Sharpe ratio.
matically visualized in Figure 13.
.
In words, we determined the maximum entropy weights in such
a way that the corresponding Sharpe ratio does not deviate too
much (as controlled by the confidence parameter
es – A Primer
concrete example using the systematic return strategies
the utility function is chosen to be the Sharpe ratio8
:
o
e maximum entropy weights, such that the corresponding
te too much (as controlled by the confidence parameter )
harpe ratio.
d in Figure 13.
from the
optimal in-sample Sharpe ratio. This is schematically visualized
in Figure 13.
1/1
Figure 1: Sharpe ratio as a function of estimation error intensity, , compared to a
naively diversified portfolio
Formeln für Figure 13
√
√
Figure 13: Illustration of the Constrained Entropy
Maximization Method
The blue line is the in-sample efficient frontier. With this method, the portfolio
is chosen within the shaded area (which is determined by the parameter
4/21
equally weighted portfolio promises the maximum out-of-sample diversifica
deviation from this portfolio leads to a potentially smaller diversification and
In the following, we will discuss a particularly simple and straightforward method
this notion, which is inspired by “Optimal Portfolio Diversification Using Maximu
Principle” by Bera/Park (2008). The key finding is that, sacrificing in-sample-o
favor of increased entropy (in the weight’s space) potentially increases the out
performance.
Given the in-sample optimal weights and the corresp
sample maximal utility the constraint maximum entropy weights a
subject to
and is th
corresponding to the weights The parameter indicates the
in the estimates of the risk-model: the closer to one, the less is the portfolio
allowed to deviate from the in-sample optimum. An of zero would be appropri
is a complete disbelief in the in-sample estimates, while a value of one
The impact of model misspecifications and estimation errors can be demon
comparing the utility of in-sample and out-of-sample optimized portfolios using
sample-estimators7
. In fact, if there were no model or estimation errors, the
sample weights should trivially be also optimal for the out-of-sample period. T
that the out-of-sample utility should increase with increasing values of the pa
However, in the presence of model and estimation risk this relationship
necessarily remain valid and sacrificing in-sample optimality (by choosing the
parameter ) thereby increasing the entropy might actually increase the out
utility.
4
Trivially, if reliable informations on the distribution exist, the maximum diversified portfolio will in general differ from the m
portfolio. For example, if in a portfolio two of the assets are linearly dependent (with high confidence), the equally weighte
assign a combined weight to these assets that is too high when compared to the weights of the maximal diversified portfolio
5
Note that, in the language of appendix 5, this is the maximum-entropy estimate such that the corres
6
By using entropy the neutral benchmark is the equally weighted portfolio. Any other neutral benchmark can be chosen by
7
In the mean-variance case the sample-estimators are given by the average returns and the sample covariance matrix
period.
that
has maximal entropy. In the presence of estimation errors, this can improve the
robustness of the portfolio. Source: own calculations.
1/1
√
√
1/1
Figure 1: Sharpe ratio as a function of estimation error intensity, , comp
√
√
1/1
√
√
1/1
√
√
1/1
√
√
We considered monthly return data of the selected strategies
and used a rolling 12-month period for the estimation of the
following one-month returns and covariance matrices.
We measured the in-sample performance by applying the
resulting weights to the estimation period and out-of-sample
performance, where we applied the weights to the month
following the estimation period. We set the confidence level at
5/21
ow, let us investigate a concrete example using the systematic return strategies
roduced in chapter 4. Here the utility function is chosen to be the Sharpe ratio8
:
subject to
words, we determined the maximum entropy weights, such that the corresponding
harpe Ratio does not deviate too much (as controlled by the confidence parameter )
om the optimal in-sample Sharpe ratio.
his is schematically visualized in Figure 13.
ure 1: Illustration of the constrained entropy maximization method The blue line is the in-
mple efficient frontier. With this method, the portfolio is chosen within the shaded area
hich is determined by the parameter a), that has maximal entropy. In the presence of
timation errors, this can improve the robustness of the portfolio.
e considered monthly return data of the selected strategies and used a rolling 12 month
riod for the estimation of the following 1-month returns and covariance matrices.
e measured the in-sample performance by applying the resulting weights to the
timation period, and out-of-sample performance, where we applied the weights to the
lowing month of the estimation period. We set the confidence level with the
erpretation that our confidence that the sample estimators are valid for the out-of-
ny other utility function or risk measure (after a slight modification) is possible as well. For example the popular VaR or
aR would have been equally suitable.
, with the interpretation that our confidence that the
sample estimators are valid for the out-of-sample period is
70%. For both cases, in- and out-of-sample, we compared the
following three portfolios: mean-variance optimized, constrained
maximum entropy and (unconstrained) maximum entropy. In
Figure 14 we show the resulting portfolio values, and in Table 8
we show some sample statistics for the in-sample periods.
43
Note that in the language of Appendix 5, this is the maximum-entropy estimate
such that the corresponding utility
4/21
. Any
performance.
subject to
and is the entropy6
sample-estimators7
utility.
4
5
6
7
period.
does not deviate too much (as
controlled by
matic Return Strategies – A Primer
ed portfolio promises the maximum out-of-sample diversification 4
. Any
this portfolio leads to a potentially smaller diversification and should be
existence of reliable estimates.
, we will discuss a particularly simple and straightforward method to employ
ch is inspired by “Optimal Portfolio Diversification Using Maximum Entropy
era/Park (2008). The key finding is that, sacrificing in-sample-optimality in
sed entropy (in the weight’s space) potentially increases the out-of-sample
e following portfolio optimization problem:
sample optimal weights and the corresponding in-
al utility the constraint maximum entropy weights are given by
subject to
dence parameter is a number between 0 and 15
and is the entropy6
to the weights The parameter indicates the confidence
s of the risk-model: the closer to one, the less is the portfolio optimizer
ate from the in-sample optimum. An of zero would be appropriate if there
disbelief in the in-sample estimates, while a value of one would be
here is full confidence in the in-sample estimates.
model misspecifications and estimation errors can be demonstrated by
utility of in-sample and out-of-sample optimized portfolios using the same
ors7
s should trivially be also optimal for the out-of-sample period. This implies
sample utility should increase with increasing values of the parameter .
he presence of model and estimation risk this relationship does not
main valid and sacrificing in-sample optimality (by choosing the confidence
) thereby increasing the entropy might actually increase the out-of-sample
) from the in-sample optimum utility given by
c Return Strategies – A Primer
4/21
portfolio promises the maximum out-of-sample diversification 4
. Any
s portfolio leads to a potentially smaller diversification and should be
stence of reliable estimates.
e will discuss a particularly simple and straightforward method to employ
is inspired by “Optimal Portfolio Diversification Using Maximum Entropy
a/Park (2008). The key finding is that, sacrificing in-sample-optimality in
d entropy (in the weight’s space) potentially increases the out-of-sample
ollowing portfolio optimization problem:
mple optimal weights and the corresponding in-
tility the constraint maximum entropy weights are given by
subject to
ce parameter is a number between 0 and 15
and is the entropy6
the weights The parameter indicates the confidence
of the risk-model: the closer to one, the less is the portfolio optimizer
from the in-sample optimum. An of zero would be appropriate if there
sbelief in the in-sample estimates, while a value of one would be
e is full confidence in the in-sample estimates.
odel misspecifications and estimation errors can be demonstrated by
lity of in-sample and out-of-sample optimized portfolios using the same
s7
hould trivially be also optimal for the out-of-sample period. This implies
mple utility should increase with increasing values of the parameter .
presence of model and estimation risk this relationship does not
n valid and sacrificing in-sample optimality (by choosing the confidence
thereby increasing the entropy might actually increase the out-of-sample
tions on the distribution exist, the maximum diversified portfolio will in general differ from the maximum entropy
a portfolio two of the assets are linearly dependent (with high confidence), the equally weighted portfolio would
o these assets that is too high when compared to the weights of the maximal diversified portfolio.
uage of appendix 5, this is the maximum-entropy estimate such that the corresponding utility
e too much (as controlled by a) from the in-sample optimum utility given by
al benchmark is the equally weighted portfolio. Any other neutral benchmark can be chosen by using the Cross-
See Appendix 5 for a detailed discussion of the entropy concept.
e the sample-estimators are given by the average returns and the sample covariance matrix of the in-sample
.
44
By using entropy, the neutral benchmark is the equally weighted portfolio. Any
other neutral benchmark can be chosen by using the cross-entropy instead of
entropy. See Appendix 5 for a detailed discussion of the entropy concept.
45
In the mean-variance case, the sample-estimators are given by the average
returns and the sample covariance matrix of the in-sample period.
46
Any other utility function or risk measure (after a slight modification) is possible
as well. For example, the popular VaR or CVaR risk measures would have been
equally suitable.

Table 8: Summary of In-Sample Statistics
Source: Bloomberg L.P., own calculations. Monthly data as from 30.11.2000 to 29.08.2014. Historical performance indications and financial market scenarios are not reliable indicators of
current or future performance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription and/or redemption.
Statistics Mean-Variance Constrained Maximum
Entropy
Maximum Entropy
Total Return 365.2% 278.7% 142.0%
Return p.a. 11.8% 10.2% 6.6%
Volatility 4.4% 4.3% 6.4%
Skewness 0.69 0.34 -1.22
Figure 14: Cumulative In-Sample Performance Chart
Source: Bloomberg L.P., own calculations. Monthly data as from 30.11.2000 to 29.08.2014. Indices are not directly investable. Historical performance indications and financial market
subscription and/or redemption. In-sample performance indications and financial market scenarios are not reliable indicators of current or future performance.
Mean-variance Constrained entropy Maximum entropy
500
400
300
200
100
0
11. 2000 08.2003 05.2006 02.2009 11.2011 08.2014
Figure 15: Cumulative Out-of-Sample Performance Chart
Source: Bloomberg L.P., own calculations. Monthly data as from 30.11.2000 to 29.08.2014. Indices are not directly investable. Historical performance indications and financial market
Mean-variance Constrained entropy Maximum entropy
11.2000 08.2003 05.2006 02.2009 11.2011 08.2014
300
200
100
0

38 / 54
As expected, in-sample the mean-variance portfolio showed the
best results when measured by the Sharpe ratio. The maximum
entropy portfolio was the weakest, and the constrained maximum
entropy portfolio lies in the middle, both in terms of total return and
Sharpe ratio. This changes significantly when considering the out-
of-sample performances shown in Figure 15 and Table 9.
We see that if the optimal in-sample weights are applied to out-
of-sample data,47
the mean-variance portfolio hardly beats the
maximum entropy portfolio and is significantly inferior to the
constrained maximum entropy portfolio in terms of total return,
volatility and, accordingly, also in terms of the Sharpe ratio.
There are several important lessons to learn here. First, the
in-sample mean-variance optimized portfolio performance points
to a high overall cumulative return, whereas out-of-sample this
methodology gives a drastically lower number. Second, the
in-sample mean-variance optimization overestimates the
diversification effect and underestimates the drawdown risk
because estimation errors are ignored.
Third, the constrained maximum entropy represents a balance
between in-sample optimality and portfolio entropy, depending
on one’s confidence in the in-sample estimates. The constrained
maximum entropy strategy dominates both the mean-variance
portfolio and the naïvely maximum entropy diversified portfolio in
key performance indicators in Table 8.
Summing up, the constrained maximum entropy method is
straightforward to implement and can potentially significantly
improve the risk-adjusted out-of-sample performance of a
systematic return portfolio.48
It is evident that the benefits of this
method will be particularly prominent if model and estimation
risk is high, which is often the case, not only for systematic
return strategies.
47
We applied the in-sample weights, which were based on an estimation period
of 12 months subsequent to the next month.
48
An attractive side effect is the lower portfolio turnover, which helps to lower
transaction costs and might require less market liquidity for rebalancing.
Table 9: Summary of Out-of-Sample Statistics
Source: Bloomberg L.P., own calculations. Monthly data as from 30.11.2000 to 29.08.2014. Historical performance indications and financial market scenarios are not reliable indicators of current
Statistics Mean-Variance Constrained Maximum
Entropy
Maximum Entropy
Total Return 94.5% 122.4% 142.0%
Return p.a. 5.0% 6.0% 6.6%
Volatility 4.8% 4.5% 6.4%
Skewness -1.13 -0.64 -1.22

2.5 Case Study 2: The Effect of Adding Systematic
Return Strategies to a Balanced Portfolio
In this second case study, we illustrate the effect of adding
systematic return strategies to a classic balanced portfolio
consisting of 60% equities and 40% bonds.49
As for systematic
return strategies, we are using the out-of-sample constrained
entropy portfolio of the last section. We analyze the total return,
the Sharpe ratio and the maximum drawdown of the combined
portfolio consisting of the constrained entropy and balanced
portfolios. The weights of the former range from 0% to
100% – in other words, from purely balanced to purely
systematic. In the upper half of Figure 16, we see the
cumulated performances of balanced, constrained entropy and
the combination of both in equal weights. While the total return
is about the same, the superior diversification properties
of the systematic return manifest themselves in the evidently
smoother returns.
In the lower half of Figure 16, we show the combination of total
return and maximum drawdown for each combination of
balanced and constrained entropy portfolios, with highlighted
cases given by the 100%, 50% and 0% weights of the
systematic return portfolio. It is evident that the inclusion of
systematic return strategies primarily reduces the risk while
keeping the total return virtually unchanged. In particular, an
inclusion of 50% reduces the maximum drawdown by almost
50% while the total return essentially stays the same.
49
We took 60% of the MSCI Daily TR Net World USD (NDDUWI) and 40% of
the Barclays Global Aggregate Total Return Index (LEGATRUU) with monthly
rebalancing.
Source: own calculations. As from 30.11.2000 to 29.08.2014. Strategies are not directly investable. Historical performance indications and financial market scenarios are not
reliable indicators of current or future performance. Performance indications do not consider commissions, fees and other charges, including commissions levied at subscription
and/or redemption.
Only systematic Only balanced 50 : 50 Mixed portfolios (shown as the connecting line)
124%
122%
120%
118%
116%
114%
112%
110%
108%
35%5% 15%10% 20% 25% 30%
TotalreturnIndex30.11.2000=100
Maximum drawdown
Figure 16: Performance Comparison
Performance comparison between classical balanced portfolio, constrained entropy portfolio of systematic return strategies and 50 : 50 mix of both. Source: own calculations.
As from 30.11.2000 to 29.08.2014. Strategies are not directly investable. Historical performance indications and financial market scenarios are not reliable indicators of current
Only systematic Only balanced 50 : 50
2000 2003 2005 2008 2010 2013
220
200
180
160
140
120
100
80

40 / 54
Figure 17 summarizes the isolated effect on the portfolio’s
Sharpe ratio and maximum drawdown for each added weight of
the systematic return portfolio. We have highlighted the effect
at a weight of 20%. We see that the Sharpe ratio can be
doubled and the maximum drawdown can be reduced by three-
quarters. In particular, by adding 20% of systematic strategies,
we see that the Sharpe ratio could be increased from 0.57 to
0.67 and the maximum drawdown could be reduced from 36%
to 30% as the total return remains virtually unchanged (moving
from 101.35% to 101.21%).
We therefore conclude that adding systematic return strategies
to classic balanced portfolios can substantially increase their
diversification potential, with a significantly reduced maximum
drawdown and an increased Sharpe ratio.
40%
35%
30%
25%
20%
15%
10%
5%
0%
Maximumdrawdown
100%0% 20% 40% 60% 80%
Figure 17: Sharpe Ratio (left) and Maximum Drawdown (right) of a Combined Position Consisting of a Traditional
Balanced Portfolio and a Portfolio of Systematic Return Strategies
The percentage weight of the systematic return strategies is denoted on the x-axis. The highlighted points correspond to the combination of 20% systematic return strategies
and 80% balanced portfolio. The effect is an increase in the Sharpe ratio from 0.57 to 0.67 and a reduction in the maximum drawdown from 36% to 30%.
Source: own calculations. As from 30.11.2000 to 29.08.2014. Historical performance indications and financial market scenarios are not reliable indicators of current or future
performance.
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
Sharperatio
Percentage allocation to systematic return strategies
100%0% 20% 40% 60% 80%
Percentage allocation to systematic return strategies

3. Implications for Investors
Systematic return strategies are nondiscretionary investment rules that aim to
monetize the performance potential from established and well-documented risk
premia in asset classes. Adding them to a balanced portfolio can lead to higher
return persistency and thus portfolio robustness.
In this section, we show how investors can overcome the common pitfalls when
dealing with systematic return strategies. We highlight again how important it is to
have not only an effective classification scheme, but also a deep understanding of
how risk premia harmonize with each other to unlock the full potential from these
interesting investment solutions. In the conclusion, we underline our firm belief
that the future for systematic return strategies looks bright, as more and more
investors start to realize how these strategies can help to diversify their portfolios
and particularly their core fixed-income holdings to achieve their investment
targets in both falling and rising interest-rate environments.

44 / 54
3.1 Overview
We would like to summarize the key findings before moving to
the conclusion:
ƁƁ By using systematic return strategies, investors can access
additional diversification benefits. At the same time, they can
enjoy sufficient liquidity since most of the strategies are based
on liquid traditional asset classes. We have seen that due to
the convex or concave nature of systematic return strategies,
linear concepts such as Sharpe ratios are not sufficient to
explain the risk/return profile. A more appropriate risk-
adjusted return comparison would at least need to include
the convexity.50
ƁƁ Our simple classification scheme shows that if only carry
strategies are chosen, this might have limited diversification
potential during crisis periods. However, the comovement
of carry strategies across different asset classes is one of
our central critical points and suggests that they are mainly
influenced by general risk aversion in crisis periods.
Therefore, it is important that absolute-return investors
combine concave (carry) strategies with convex (trend-
following) strategies. Investors can use these strategies as
building blocks to remodel the return distribution of their
fixed-income or balanced portfolios, introducing positive
skewness to mitigate downside risks.
ƁƁ Positive convexity in itself is not a “free lunch.” The investor
should carefully analyze the upside and downside properties
of the relevant strategies. Trend-following strategies need
trending markets. However, this means that they are not
consistently profitable. As with positive gamma or positive
convexity trades in general, many trend-following strategies
are not profitable. They usually lose a little, but can earn
a lot in strong-trending markets. The performance of
individual trend-following strategies may vary substantially.
Hence, performance dispersion among strategies may
be very wide because they depend on choosing the right
markets. In addition, the investor is still exposed to gap risk
– the risk that the strategy is not fast and nimble enough to
adapt to sudden changes in markets.
ƁƁ In a world of growing complexities, we firmly believe that
“simplicity is key.” Our guiding principles can help investors
to navigate the universe of systematic return strategies.
We stressed that costs, liquidity and capacity constraints
play an important role. However, most importantly, we have
shown that strategies need to be based on sound economic
rationales. Active management can provide an edge in this
respect.
ƁƁ We demonstrated that systematic strategies exhibit
lower pair-wise correlations compared to traditional
asset classes (please also see Figure 2). This means
that increasing the number of different systematic return
strategies in a portfolio can substantially reduce portfolio
risk. Portfolio theory describes how to determine their
respective weights. However, model and estimation errors
are serious challenges in practical portfolio optimization.
For this reason, optimized portfolios are not necessarily
optimal portfolios. In Appendix 4, we show that portfolio
optimization with large enough estimation errors can even
be expected to underperform naïvely diversified portfolios.
Additionally, positive properties like low correlations, short
time series, skewed return distributions and nonlinear
relationships to traditional asset classes compound model
and estimation errors. Therefore, particular steps have to
be taken to mitigate some of the adverse effects of model
and estimation errors. This is especially true for systematic
return strategies. So we presented the constrained
maximum entropy method and showed how it can help to
build more stable and robust systematic return portfolios.
ƁƁ Path dependency requires special attention. Many investors
have experienced that even simple stop-loss rules from risk
management can introduce the risk of path dependency in
the return-generating process of the portfolio. This means
that initial decisions in the risk management framework may
limit future portfolio choices even though the circumstances
under which those decisions were based may no longer be
relevant. In general, the more successful a particular risk
management strategy has been in the past, the more likely
it will lead to inflexibility. When decisions are made based
on past experience, other options are ruled out and a path
emerges that often becomes irreversible as the investor
gets locked into the path.
50
Taleb (1997) provides a good overview on this.

3.2 Conclusion
In this report, we first examined the reasons why a typical
balanced portfolio has done so well over the past three decades.
Two drivers were identified. First, bonds and equities exhibited
very reliable diversification, particularly in times of economic
shocks. Second, interest rates have been declining globally over
the last 30 years. These are also the reasons why many market
professionals question the future return potential of such
investments and fear higher drawdowns going forward.
Systematic return strategies may be a valuable alternative in this
situation. After providing a definition of systematic return
strategies and outlining some well-documented examples, we
argued that risk premia investing is not new. Many investors
already have direct or indirect exposure to this form of investing.
However, indirect exposure – a portfolio of value stocks or
small-cap stocks, for example – is still dominated by market
beta and cannot take full advantage of the low correlation
between value and small-cap risk premia. We pointed out that
only direct exposure allows investors to take full advantage of
the diversification benefits.
We introduced the concept of nonlinear payouts from the option
theory and investigated the empirical nonlinear relationships to
the market of several systematic return strategies. We argued
that a simple classification scheme (carry and trend-following)
based on the positive or negative convexity of the strategy can
help investors to avoid overlaps in terms of tail-risk exposure
and to construct more robust portfolios. We specifically argued
that trend-following strategies can help to preserve capital in
bear markets due to their notable risk-mitigating properties.
Furthermore, we presented principles that can be used as a
guide for selecting systematic return strategies. We emphasized
that strategies should be based on sound economic rationales
and that they should be simple in order to avoid falling into the
most common pitfalls.
In our first case study, we showed one way how investors can
mitigate some of the effects of estimation errors using the
constrained maximum entropy method. It became evident how
sacrificing in-sample optimality in favor of portfolio entropy can
help to construct more stable and robust portfolios.
In the second case study, we investigated how adding
systematic return strategies to a balanced portfolio can help
investors to earn higher risk-adjusted returns and reduce
drawdown risks. Our approach offers an innovative framework
for generating persistent returns from a portfolio of systematic
return strategies that is effectively diversified.
A diversified basket of systematic return strategies across asset
classes can deliver more stable portfolio returns due to its
inherent correlation characteristics. Since the returns are not
directly exposed to interest rates, a systematic return solution
can be a viable way for both relative-return and absolute-return
investors to diversify their portfolios. This is particularly true for
their core fixed-income holdings.

46 / 54
Appendix
Appendix 1: Drawdowns and Drawups
Drawdown
13/21
option for both relative and absolute return oriented investors to diversify especially their
core fixed income holdings.
3 Appendix
The drawdown (DD) measures the current cumulative loss from the previous maximum
price level within a given time-span, i.e., given a price process and the considered time
span [ ] the drawdown at time can formally be defined as
The drawup (DU) is just the opposite, as it reflects the cumulative gain from the previous
minimum, that is, with the notation from above, it can be defined as:
The maximum drawdown and maximal drawup within a time span [ ] are accordingly
given by
| |
and
Appendix 3: Systematic Return Strategies and the Efficient Market Hypothesis
Many investment consultants rightly argue that systematic return strategies are xzy
The Efficient Market Hypothesis (EMH) (Fama, 1970) in its strong form asserts that all
relevant information is publicly available and immediately reflected in the prices. If
investors are assumed to be risk averse and agree on a single risk-measure, this implies
the existence of some “optimal” market portfolio that no single investment strategy should
be able to persistently outperform on a risk-adjusted basis.
However, every now and then researchers and practitioners choose a particular market,
for example an equity index, and claim to have spotted such an investment strategy and
measures the current cumulative loss from the
previous maximum price level within a given time span, i.e.
given a price process
– A Primer
13/21
solute return oriented investors to diversify especially their
Drawups
s the current cumulative loss from the previous maximum
span, i.e., given a price process and the considered time
e can formally be defined as
posite, as it reflects the cumulative gain from the previous
on from above, it can be defined as:
maximal drawup within a time span [ ] are accordingly
| |
rn Strategies and the Efficient Market Hypothesis
ghtly argue that systematic return strategies are xzy
is (EMH) (Fama, 1970) in its strong form asserts that all
y available and immediately reflected in the prices. If
sk averse and agree on a single risk-measure, this implies
” market portfolio that no single investment strategy should
m on a risk-adjusted basis.
researchers and practitioners choose a particular market,
nd claim to have spotted such an investment strategy and
and the considered time span
13/21
option for both relative and absolute return oriented investors to diversify especia
3 Appendix
The drawdown (DD) measures the current cumulative loss from the previous ma
price level within a given time-span, i.e., given a price process and the consider
The drawup (DU) is just the opposite, as it reflects the cumulative gain from the p
The maximum drawdown and maximal drawup within a time span [ ] are acco
given by
| |
and
The Efficient Market Hypothesis (EMH) (Fama, 1970) in its strong form asserts
relevant information is publicly available and immediately reflected in the pr
investors are assumed to be risk averse and agree on a single risk-measure, this
the existence of some “optimal” market portfolio that no single investment strategy
However, every now and then researchers and practitioners choose a particular
for example an equity index, and claim to have spotted such an investment strate
,
the drawdown at time
anagement of Systematic Return Strategies – A Primer
13/21
ption for both relative and absolute return oriented investors to diversify especially their
ore fixed income holdings.
3 Appendix
he drawdown (DD) measures the current cumulative loss from the previous maximum
rice level within a given time-span, i.e., given a price process and the considered time
pan [ ] the drawdown at time can formally be defined as
he drawup (DU) is just the opposite, as it reflects the cumulative gain from the previous
he maximum drawdown and maximal drawup within a time span [ ] are accordingly
iven by
| |
nd
he Efficient Market Hypothesis (EMH) (Fama, 1970) in its strong form asserts that all
elevant information is publicly available and immediately reflected in the prices. If
nvestors are assumed to be risk averse and agree on a single risk-measure, this implies
he existence of some “optimal” market portfolio that no single investment strategy should
e able to persistently outperform on a risk-adjusted basis.
or example an equity index, and claim to have spotted such an investment strategy and
can formally be defined as:

13/21
wdowns and Drawups
D) measures the current cumulative loss from the previous maximum
a given time-span, i.e., given a price process and the considered time
wdown at time can formally be defined as
is just the opposite, as it reflects the cumulative gain from the previous
with the notation from above, it can be defined as:
awdown and maximal drawup within a time span [ ] are accordingly
| |
ematic Return Strategies and the Efficient Market Hypothesis
consultants rightly argue that systematic return strategies are xzy
ket Hypothesis (EMH) (Fama, 1970) in its strong form asserts that all
on is publicly available and immediately reflected in the prices. If
umed to be risk averse and agree on a single risk-measure, this implies
ome “optimal” market portfolio that no single investment strategy should
ntly outperform on a risk-adjusted basis.
ow and then researchers and practitioners choose a particular market,
quity index, and claim to have spotted such an investment strategy and
ement of Systematic Return Strategies – A Primer
13/21
n for both relative and absolute return oriented investors to diversify especially their
fixed income holdings.
Appendix
endix 1: Drawdowns and Drawups
drawdown (DD) measures the current cumulative loss from the previous maximum
level within a given time-span, i.e., given a price process and the considered time
[ ] the drawdown at time can formally be defined as
drawup (DU) is just the opposite, as it reflects the cumulative gain from the previous
mum, that is, with the notation from above, it can be defined as:
maximum drawdown and maximal drawup within a time span [ ] are accordingly
by
| |
endix 3: Systematic Return Strategies and the Efficient Market Hypothesis
y investment consultants rightly argue that systematic return strategies are xzy
Efficient Market Hypothesis (EMH) (Fama, 1970) in its strong form asserts that all
ant information is publicly available and immediately reflected in the prices. If
tors are assumed to be risk averse and agree on a single risk-measure, this implies
xistence of some “optimal” market portfolio that no single investment strategy should
ble to persistently outperform on a risk-adjusted basis.
ever, every now and then researchers and practitioners choose a particular market,
xample an equity index, and claim to have spotted such an investment strategy and
gement of Systematic Return Strategies – A Primer
13/21
on for both relative and absolute return oriented investors to diversify especially their
e fixed income holdings.
Appendix
pendix 1: Drawdowns and Drawups
e level within a given time-span, i.e., given a price process and the considered time
n [ ] the drawdown at time can formally be defined as
n by
| |
pendix 3: Systematic Return Strategies and the Efficient Market Hypothesis
ny investment consultants rightly argue that systematic return strategies are xzy
vant information is publicly available and immediately reflected in the prices. If
stors are assumed to be risk averse and agree on a single risk-measure, this implies
existence of some “optimal” market portfolio that no single investment strategy should
able to persistently outperform on a risk-adjusted basis.
wever, every now and then researchers and practitioners choose a particular market,
example an equity index, and claim to have spotted such an investment strategy and
Drawup
13/21
option for both relative and absolute return oriented investors to diversify especially their
3 Appendix
given by
| |
and
is just the opposite since it reflects the cumulative
gain from the previous minimum. With the notation from above,
it can be defined as:

ment of Systematic Return Strategies – A Primer
13/21
ixed income holdings.
ppendix
ndix 1: Drawdowns and Drawups
by
| |
ndix 3: Systematic Return Strategies and the Efficient Market Hypothesis
investment consultants rightly argue that systematic return strategies are xzy
tors are assumed to be risk averse and agree on a single risk-measure, this implies
xistence of some “optimal” market portfolio that no single investment strategy should
le to persistently outperform on a risk-adjusted basis.
The maximum drawdown and maximum drawup within a time
span
d investors to diversify especially their
ative loss from the previous maximum
rice process and the considered time
defined as
the cumulative gain from the previous
n be defined as:
hin a time span [ ] are accordingly
| |
he Efficient Market Hypothesis
ematic return strategies are xzy
970) in its strong form asserts that all
mmediately reflected in the prices. If
e on a single risk-measure, this implies
at no single investment strategy should
d basis.
actitioners choose a particular market,
otted such an investment strategy and
are accordingly given by:

ement of Systematic Return Strategies – A Primer
13/21
fixed income holdings.
Appendix
endix 1: Drawdowns and Drawups
n by
| |
endix 3: Systematic Return Strategies and the Efficient Market Hypothesis
y investment consultants rightly argue that systematic return strategies are xzy
stors are assumed to be risk averse and agree on a single risk-measure, this implies
existence of some “optimal” market portfolio that no single investment strategy should
ble to persistently outperform on a risk-adjusted basis.
ent of Systematic Return Strategies – A Primer
13/21
for both relative and absolute return oriented investors to diversify especially their
xed income holdings.
ppendix
dix 1: Drawdowns and Drawups
awdown (DD) measures the current cumulative loss from the previous maximum
evel within a given time-span, i.e., given a price process and the considered time
] the drawdown at time can formally be defined as
awup (DU) is just the opposite, as it reflects the cumulative gain from the previous
um, that is, with the notation from above, it can be defined as:
aximum drawdown and maximal drawup within a time span [ ] are accordingly
by
| |
dix 3: Systematic Return Strategies and the Efficient Market Hypothesis
nvestment consultants rightly argue that systematic return strategies are xzy
ficient Market Hypothesis (EMH) (Fama, 1970) in its strong form asserts that all
nt information is publicly available and immediately reflected in the prices. If
rs are assumed to be risk averse and agree on a single risk-measure, this implies
stence of some “optimal” market portfolio that no single investment strategy should
e to persistently outperform on a risk-adjusted basis.
er, every now and then researchers and practitioners choose a particular market,
mple an equity index, and claim to have spotted such an investment strategy and
and:

13/21
for both relative and absolute return oriented investors to diversify especially their
xed income holdings.
ppendix
ndix 1: Drawdowns and Drawups
rawdown (DD) measures the current cumulative loss from the previous maximum
evel within a given time-span, i.e., given a price process and the considered time
] the drawdown at time can formally be defined as
rawup (DU) is just the opposite, as it reflects the cumulative gain from the previous
um, that is, with the notation from above, it can be defined as:
by
| |
ndix 3: Systematic Return Strategies and the Efficient Market Hypothesis
investment consultants rightly argue that systematic return strategies are xzy
fficient Market Hypothesis (EMH) (Fama, 1970) in its strong form asserts that all
nt information is publicly available and immediately reflected in the prices. If
ors are assumed to be risk averse and agree on a single risk-measure, this implies
istence of some “optimal” market portfolio that no single investment strategy should
e to persistently outperform on a risk-adjusted basis.
ver, every now and then researchers and practitioners choose a particular market,
ample an equity index, and claim to have spotted such an investment strategy and
Appendix 2: Convexity of Systematic Return Strategies
The convexity of a strategy with respect to its (main) underlying
describes a certain functional relationship.
Typical examples are shown in Figure 18.
Strategy’s return
Underlying’s return
Strategy’s return
Strategy’s return
Underlying’s returnUnderlying’s return
Strategy’s return
More formally, a function
1/3
More formally, a function of one variable is called convex, if
connecting any two points of its graph lies above the graph (left figu
concave (negative convexity), if the line lies below (right figure).
In the seminal paper Perold/Sharpe (1988) convexity of strategy values w
value of risky asset is investigated. Strictly speaking, the authors conside
the strategy’s value to certain price scenarios in time of the underlying
instance a monotone rise/fall in the stock’s price within a certain per
simulated fluctuations are considered. In this setting, they arrive at the follo
strategy’s return strateg
x
f(x) f(x)
of one variable
1/3
More formally, a function of one variable is called convex, if the
connecting any two points of its graph lies above the graph (left figure).
In the seminal paper Perold/Sharpe (1988) convexity of strategy values with re
value of risky asset is investigated. Strictly speaking, the authors consider th
the strategy’s value to certain price scenarios in time of the underlying stoc
instance a monotone rise/fall in the stock’s price within a certain period
simulated fluctuations are considered. In this setting, they arrive at the followin
strategy’s return strategy’s re
Un
x
f(x) f(x)
is called convex
if the line connecting any two points of its graph lies above the
graph (left figure). It is called concave (negative convexity) if the
line lies below the graph (right figure).
xx
1/3
Convexity of a strategy with respect to its (main) underlying describes a certain functional
relationship. Typical examples are given in Figure 1.
Figure 1: Convexity of systematic return strategies, source: Credit Suisse AG
In the first case, an increase in the underlying’s return yields an increase of the strategy’s
return at an increasing rate. In the second case, an increase in the underlying’s return
yields a decrease of the derivative’s returns at a decreasing rate. The corresponding
examples for “negative convexity” are shown below.
More formally, a function of one variable is called convex, if the straight line
connecting any two points of its graph lies above the graph (left figure). It is called
In the seminal paper Perold/Sharpe (1988) convexity of strategy values with respect to the
value of risky asset is investigated. Strictly speaking, the authors consider the reaction of
the strategy’s value to certain price scenarios in time of the underlying stock price. For
instance a monotone rise/fall in the stock’s price within a certain period of time and
simulated fluctuations are considered. In this setting, they arrive at the following findings:
strategy’s return strategy’s return
x
f(x) f(x)
x
1/3
Convexity of a strategy with respect to its (main) underlying describes a certain f
relationship. Typical examples are given in Figure 1.
Figure 1: Convexity of systematic return strategies, source: Credit Suisse AG
In the first case, an increase in the underlying’s return yields an increase of the s
return at an increasing rate. In the second case, an increase in the underlying
yields a decrease of the derivative’s returns at a decreasing rate. The corre
examples for “negative convexity” are shown below.
More formally, a function of one variable is called convex, if the stra
connecting any two points of its graph lies above the graph (left figure). It
In the seminal paper Perold/Sharpe (1988) convexity of strategy values with resp
value of risky asset is investigated. Strictly speaking, the authors consider the re
the strategy’s value to certain price scenarios in time of the underlying stock p
instance a monotone rise/fall in the stock’s price within a certain period of
simulated fluctuations are considered. In this setting, they arrive at the following fi
Underly
x
f(x) f(x)
Figure 18: Convexity of Systematic Return Strategies. Source: Credit Suisse AG
In the first case, an increase in the underlying’s return yields an
increase in the strategy’s return at an increasing rate. In the
second case, an increase in the underlying’s return yields a
decrease in the derivative’s return at a decreasing rate. The
corresponding examples for negative convexity are shown
below.
The seminal study by Perold and Sharpe (1988) investigates
the convexity of strategy values with respect to the value of
risky assets (stocks). Strictly speaking, the authors consider
the reaction of the strategy’s value to certain price scenarios of
the underlying stock price over time. For instance, a monotone
rise/fall in the stock’s price within a certain time period and
simulated fluctuations are considered. In this setting, they arrive
at the following findings:
ƁƁ Strategies that purchase stocks as they fall in value and
sell stocks as they rise in value lead to concave strategies.
In periods of monotonically rising or falling stock prices,
this type of strategy is inferior to linear (“buy-and-hold”)
strategies. However, these strategies are superior in flat
but oscillating markets. So, they are favorable in markets
with small absolute movements but with high volatility in the
underlying’s price.
ƁƁ Strategies that sell stocks as they fall and buy stocks
as they rise lead to convex strategies. Their qualitative
behavior is just the opposite of concave strategies.
In particular, this type of strategy loses against linear
strategies in the absence of big price trends.
13/21
given by
| |
and

The convexity of a trading strategy is also strongly related to the
gamma of a European option, i.e. the second derivative of the
option’s price with respect to the underlying’s price. To highlight
this relationship, we regard a systematic return strategy as a
dynamic portfolio strategy known from the mathematical
finance literature. We assume that the price process of the risky
asset
2/3
es are superior in flat but oscillating markets. Thus, this strategy is favorable
ets with small absolute moves but high volatility in the underlying’s price.
ies that sell stocks as they fall and buy stocks as they rise lead to convex
es. Its qualitative behavior is just the opposite of concave strategies. In
ar, this strategy loses against linear strategies in the absence of big price
of a trading strategy is also strongly related to the gamma of an European
the second derivative of the option’s price with respect to the underlying’s
ight this relationship we regard a systematic return strategy as a dynamic
gy known from the mathematical finance literature. We assume that the
of the risky asset is given by the solution of
are positive deterministic processes and is a standard Brownian
he sake of simplicity we assume that interest rates are zero. A dynamic
is stochastic process (adapted to the filtration generated by , that
end on current time and the price path of up to time ) that assigns at
e number of shares in the risky asset and the money market account.
iven some initial wealth , the strategy value at each time is given by
sume that the strategy is self-financing, that is, we have that
∫
his assumption is standard, and means that besides the initial investment
e strategy’s value are solely due to price changes of the risky asset given the
European option claim (meeting standard regularity assumptions) on the
with maturity , we have the following well-known decomposition
∫
denotes initial price of the option and the option’s delta at time , that is,
This relationship reflects the well-known paradigm (given for example in
oles setting) that under suitable assumptions, any European option payoff
ted by an initial investment and a delta-hedging portfolio. Thus, European
e decomposed into an initial investment and a systematic return strategy.
(defined on some filtered probability space) is given by
the (strong) solution of
2/3
ype of strategy is inferior to linear (“buy-and-hold”) strategies. However, these
trategies are superior in flat but oscillating markets. Thus, this strategy is favorable
n markets with small absolute moves but high volatility in the underlying’s price.
trategies that sell stocks as they fall and buy stocks as they rise lead to convex
trategies. Its qualitative behavior is just the opposite of concave strategies. In
articular, this strategy loses against linear strategies in the absence of big price
ends.
vexity of a trading strategy is also strongly related to the gamma of an European
hat is, the second derivative of the option’s price with respect to the underlying’s
o highlight this relationship we regard a systematic return strategy as a dynamic
strategy known from the mathematical finance literature. We assume that the
ocess of the risky asset is given by the solution of
and are positive deterministic processes and is a standard Brownian
For the sake of simplicity we assume that interest rates are zero. A dynamic
n depend on current time and the price path of up to time ) that assigns at
me the number of shares in the risky asset and the money market account.
ngly, given some initial wealth , the strategy value at each time is given by
her assume that the strategy is self-financing, that is, we have that
∫
This assumption is standard, and means that besides the initial investment
s in the strategy’s value are solely due to price changes of the risky asset given the
n .
ven an European option claim (meeting standard regularity assumptions) on the
set with maturity , we have the following well-known decomposition
∫
k-Scholes setting) that under suitable assumptions, any European option payoff
eplicated by an initial investment and a delta-hedging portfolio. Thus, European
can be decomposed into an initial investment and a systematic return strategy.
,
where
2/3
type of strategy is inferior to linear (“buy-and-hold”) strategies. However, these
strategies are superior in flat but oscillating markets. Thus, this strategy is favorable
in markets with small absolute moves but high volatility in the underlying’s price.
- Strategies that sell stocks as they fall and buy stocks as they rise lead to convex
strategies. Its qualitative behavior is just the opposite of concave strategies. In
particular, this strategy loses against linear strategies in the absence of big price
trends.
The convexity of a trading strategy is also strongly related to the gamma of an European
option, that is, the second derivative of the option’s price with respect to the underlying’s
price. To highlight this relationship we regard a systematic return strategy as a dynamic
portfolio strategy known from the mathematical finance literature. We assume that the
price process of the risky asset is given by the solution of
where and are positive deterministic processes and is a standard Brownian
motion. For the sake of simplicity we assume that interest rates are zero. A dynamic
portfolio is stochastic process (adapted to the filtration generated by , that
is, can depend on current time and the price path of up to time ) that assigns at
each time the number of shares in the risky asset and the money market account.
Accordingly, given some initial wealth , the strategy value at each time is given by
We further assume that the strategy is self-financing, that is, we have that
∫
for all This assumption is standard, and means that besides the initial investment
changes in the strategy’s value are solely due to price changes of the risky asset given the
allocation .
Now, given an European option claim (meeting standard regularity assumptions) on the
risky asset with maturity , we have the following well-known decomposition
∫
where denotes initial price of the option and the option’s delta at time , that is,
the Black-Scholes setting) that under suitable assumptions, any European option payoff
can be replicated by an initial investment and a delta-hedging portfolio. Thus, European
options can be decomposed into an initial investment and a systematic return strategy.
and
2/3
trends.
∫
allocation .
∫
are positive deterministic processes and
2/3
trends.
∫
allocation .
∫
is
a standard Brownian motion. For the sake of simplicity, we
assume that interest rates are zero. A dynamic portfolio
2/3
value lead to concave strategies. In monotonously rising or falling stock prices this
trends.
∫
allocation .
∫
is a stochastic process (adapted to the filtration
generated by
and sell stocks as they rise in
sing or falling stock prices this
strategies. However, these
Thus, this strategy is favorable
lity in the underlying’s price.
s as they rise lead to convex
e of concave strategies. In
s in the absence of big price
o the gamma of an European
th respect to the underlying’s
return strategy as a dynamic
ature. We assume that the
nd is a standard Brownian
est rates are zero. A dynamic
e filtration generated by , that
up to time ) that assigns at
nd the money market account.
at each time is given by
s, we have that
besides the initial investment
nges of the risky asset given the
regularity assumptions) on the
own decomposition
option’s delta at time , that is,
aradigm (given for example in
, any European option payoff
ng portfolio. Thus, European
a systematic return strategy.
, i.e.
2/3
trends.
∫
allocation .
∫
can depend on current time
2/3
trends.
∫
allocation .
∫
and the
price path of
2/3
Strategies that purchase stocks as they fall in value and sell stocks as they rise in
Strategies that sell stocks as they fall and buy stocks as they rise lead to convex
trends.
onvexity of a trading strategy is also strongly related to the gamma of an European
, that is, the second derivative of the option’s price with respect to the underlying’s
To highlight this relationship we regard a systematic return strategy as a dynamic
io strategy known from the mathematical finance literature. We assume that the
process of the risky asset is given by the solution of
n. For the sake of simplicity we assume that interest rates are zero. A dynamic
io is stochastic process (adapted to the filtration generated by , that
can depend on current time and the price path of up to time ) that assigns at
time the number of shares in the risky asset and the money market account.
dingly, given some initial wealth , the strategy value at each time is given by
rther assume that the strategy is self-financing, that is, we have that
∫
es in the strategy’s value are solely due to price changes of the risky asset given the
ion .
given an European option claim (meeting standard regularity assumptions) on the
sset with maturity , we have the following well-known decomposition
∫
ack-Scholes setting) that under suitable assumptions, any European option payoff
e replicated by an initial investment and a delta-hedging portfolio. Thus, European
s can be decomposed into an initial investment and a systematic return strategy.
up to time
2/3
ocks as they fall in value and sell stocks as they rise in
egies. In monotonously rising or falling stock prices this
o linear (“buy-and-hold”) strategies. However, these
at but oscillating markets. Thus, this strategy is favorable
ute moves but high volatility in the underlying’s price.
as they fall and buy stocks as they rise lead to convex
havior is just the opposite of concave strategies. In
es against linear strategies in the absence of big price
y is also strongly related to the gamma of an European
ve of the option’s price with respect to the underlying’s
we regard a systematic return strategy as a dynamic
mathematical finance literature. We assume that the
s given by the solution of
terministic processes and is a standard Brownian
y we assume that interest rates are zero. A dynamic
c process (adapted to the filtration generated by , that
e and the price path of up to time ) that assigns at
s in the risky asset and the money market account.
alth , the strategy value at each time is given by
gy is self-financing, that is, we have that
∫
andard, and means that besides the initial investment
e solely due to price changes of the risky asset given the
aim (meeting standard regularity assumptions) on the
ave the following well-known decomposition
∫
of the option and the option’s delta at time , that is,
eflects the well-known paradigm (given for example in
der suitable assumptions, any European option payoff
stment and a delta-hedging portfolio. Thus, European
an initial investment and a systematic return strategy.
that assigns at each time the
number of shares in the risky asset
2/3
- Strategies that purchase stocks as they fall in value and sell stocks as they rise in
trends.
∫
allocation .
∫
and the money market
account. Accordingly, given some initial wealth
2/3
trends.
∫
allocation .
∫
, the strategy
value
2/3
they fall in value and sell stocks as they rise in
n monotonously rising or falling stock prices this
“buy-and-hold”) strategies. However, these
scillating markets. Thus, this strategy is favorable
es but high volatility in the underlying’s price.
all and buy stocks as they rise lead to convex
s just the opposite of concave strategies. In
st linear strategies in the absence of big price
strongly related to the gamma of an European
e option’s price with respect to the underlying’s
ard a systematic return strategy as a dynamic
atical finance literature. We assume that the
by the solution of
tic processes and is a standard Brownian
ssume that interest rates are zero. A dynamic
s (adapted to the filtration generated by , that
the price path of up to time ) that assigns at
risky asset and the money market account.
he strategy value at each time is given by
f-financing, that is, we have that
∫
and means that besides the initial investment
due to price changes of the risky asset given the
meeting standard regularity assumptions) on the
following well-known decomposition
∫
option and the option’s delta at time , that is,
he well-known paradigm (given for example in
able assumptions, any European option payoff
and a delta-hedging portfolio. Thus, European
investment and a systematic return strategy.
at each time
2/3
hey fall in value and sell stocks as they rise in
monotonously rising or falling stock prices this
buy-and-hold”) strategies. However, these
cillating markets. Thus, this strategy is favorable
s but high volatility in the underlying’s price.
ll and buy stocks as they rise lead to convex
just the opposite of concave strategies. In
t linear strategies in the absence of big price
strongly related to the gamma of an European
option’s price with respect to the underlying’s
rd a systematic return strategy as a dynamic
tical finance literature. We assume that the
y the solution of
c processes and is a standard Brownian
sume that interest rates are zero. A dynamic
s (adapted to the filtration generated by , that
he price path of up to time ) that assigns at
risky asset and the money market account.
e strategy value at each time is given by
financing, that is, we have that
∫
nd means that besides the initial investment
due to price changes of the risky asset given the
eeting standard regularity assumptions) on the
ollowing well-known decomposition
∫
ption and the option’s delta at time , that is,
e well-known paradigm (given for example in
ble assumptions, any European option payoff
nd a delta-hedging portfolio. Thus, European
nvestment and a systematic return strategy.
is given by

Systematic Return Strategies – A Primer
2/3
tegies that purchase stocks as they fall in value and sell stocks as they rise in
e lead to concave strategies. In monotonously rising or falling stock prices this
of strategy is inferior to linear (“buy-and-hold”) strategies. However, these
egies are superior in flat but oscillating markets. Thus, this strategy is favorable
arkets with small absolute moves but high volatility in the underlying’s price.
tegies that sell stocks as they fall and buy stocks as they rise lead to convex
egies. Its qualitative behavior is just the opposite of concave strategies. In
cular, this strategy loses against linear strategies in the absence of big price
ds.
xity of a trading strategy is also strongly related to the gamma of an European
is, the second derivative of the option’s price with respect to the underlying’s
ghlight this relationship we regard a systematic return strategy as a dynamic
ategy known from the mathematical finance literature. We assume that the
ss of the risky asset is given by the solution of
r the sake of simplicity we assume that interest rates are zero. A dynamic
depend on current time and the price path of up to time ) that assigns at
the number of shares in the risky asset and the money market account.
y, given some initial wealth , the strategy value at each time is given by
assume that the strategy is self-financing, that is, we have that
∫
the strategy’s value are solely due to price changes of the risky asset given the
.
an European option claim (meeting standard regularity assumptions) on the
∫
Scholes setting) that under suitable assumptions, any European option payoff
icated by an initial investment and a delta-hedging portfolio. Thus, European
n be decomposed into an initial investment and a systematic return strategy.
We further assume that the strategy is self-financing, meaning

ystematic Return Strategies – A Primer
2/3
egies that purchase stocks as they fall in value and sell stocks as they rise in
lead to concave strategies. In monotonously rising or falling stock prices this
of strategy is inferior to linear (“buy-and-hold”) strategies. However, these
gies are superior in flat but oscillating markets. Thus, this strategy is favorable
rkets with small absolute moves but high volatility in the underlying’s price.
egies that sell stocks as they fall and buy stocks as they rise lead to convex
gies. Its qualitative behavior is just the opposite of concave strategies. In
ular, this strategy loses against linear strategies in the absence of big price
s.
ty of a trading strategy is also strongly related to the gamma of an European
s, the second derivative of the option’s price with respect to the underlying’s
hlight this relationship we regard a systematic return strategy as a dynamic
tegy known from the mathematical finance literature. We assume that the
s of the risky asset is given by the solution of
nd are positive deterministic processes and is a standard Brownian
the sake of simplicity we assume that interest rates are zero. A dynamic
epend on current time and the price path of up to time ) that assigns at
he number of shares in the risky asset and the money market account.
given some initial wealth , the strategy value at each time is given by
ssume that the strategy is self-financing, that is, we have that
∫
he strategy’s value are solely due to price changes of the risky asset given the
.
an European option claim (meeting standard regularity assumptions) on the
∫
choles setting) that under suitable assumptions, any European option payoff
cated by an initial investment and a delta-hedging portfolio. Thus, European
be decomposed into an initial investment and a systematic return strategy.
for all
2/3
trends.
∫
allocation .
∫
. This assumption is standard and means that
besides the initial investment
2/3
trends.
∫
allocation .
∫
, changes in the strategy’s value
are solely due to price changes in the risky asset given the
allocation
2/3
trends.
∫
allocation .
∫
.
Now, given a European option claim
2/3
trends.
∫
allocation .
∫
(meeting standard
regularity assumptions) on the risky asset
trends.
∫
allocation .
∫
with maturity
2/3
trends.
∫
allocation .
∫
, we
have the following well-known decomposition

nt of Systematic Return Strategies – A Primer
2/3
trategies that purchase stocks as they fall in value and sell stocks as they rise in
alue lead to concave strategies. In monotonously rising or falling stock prices this
ype of strategy is inferior to linear (“buy-and-hold”) strategies. However, these
trategies are superior in flat but oscillating markets. Thus, this strategy is favorable
n markets with small absolute moves but high volatility in the underlying’s price.
trategies that sell stocks as they fall and buy stocks as they rise lead to convex
trategies. Its qualitative behavior is just the opposite of concave strategies. In
articular, this strategy loses against linear strategies in the absence of big price
ends.
vexity of a trading strategy is also strongly related to the gamma of an European
hat is, the second derivative of the option’s price with respect to the underlying’s
o highlight this relationship we regard a systematic return strategy as a dynamic
strategy known from the mathematical finance literature. We assume that the
ocess of the risky asset is given by the solution of
For the sake of simplicity we assume that interest rates are zero. A dynamic
n depend on current time and the price path of up to time ) that assigns at
me the number of shares in the risky asset and the money market account.
ngly, given some initial wealth , the strategy value at each time is given by
her assume that the strategy is self-financing, that is, we have that
∫
s in the strategy’s value are solely due to price changes of the risky asset given the
n .
ven an European option claim (meeting standard regularity assumptions) on the
set with maturity , we have the following well-known decomposition
∫
k-Scholes setting) that under suitable assumptions, any European option payoff
eplicated by an initial investment and a delta-hedging portfolio. Thus, European
can be decomposed into an initial investment and a systematic return strategy.
,
where
2/3
trends.
∫
allocation .
∫
denotes the initial price of the option and
2/3
trends.
∫
allocation .
∫
the option’s delta at time
trends.
∫
allocation .
∫
, that is,
2/3
trends.
∫
allocation .
∫
. This
relationship reflects the well-known paradigm (given, for
example, in the Black-Scholes setting) that under suitable
assumptions (e.g. no volatility risk), any European option payoff
can be replicated by an initial investment and a delta-hedging
portfolio. European options therefore can be decomposed into
an initial investment and a systematic return strategy.
In certain cases this point of view can be reversed. Formally,
given a fixed maturity time
2/3
∫
for all This assumption is standard, and means that besides the in
changes in the strategy’s value are solely due to price changes of the ri
allocation .
Now, given an European option claim (meeting standard regularity as
risky asset with maturity , we have the following well-known decomp
∫
where denotes initial price of the option and the option’s delta
This relationship reflects the well-known paradigm (give
the Black-Scholes setting) that under suitable assumptions, any Europe
can be replicated by an initial investment and a delta-hedging portfolio.
options can be decomposed into an initial investment and a systematic
and a systematic return strategy
In certain cases this point of view can be r
a systematic return strategy on a price p
where the initial investment is understoo
corresponds to the option’s delta. In this ca
and accordingly we could say that the syst
and concave if for all . Th
the convexity of the strategy is equivalent
corresponding option.
A slightly different decomposition is sugge
introduce in the following. This decomposi
similar qualitative findings as in Perold/Sha
They define the option profile by
formula yields
As above, we see that the corresponding o
since . Another conseq
return strategies corresponding to convex
and the reverse is true for systematic retur
In particular, in flat markets we have
impact dominates the stratetgy’s value wh
Bruder, Benjamin, and Nicolas Gaussel. Risk-
Available at SSRN 2465623 (2011).
Perold, Andre F., and William F. Sharpe. Dyna
Analysts Journal (1988): 16-27.
Option pro
on a price process
2/3
particular, this strategy loses against linear strategies in the absence of big p
trends.
The convexity of a trading strategy is also strongly related to the gamma of an Euro
option, that is, the second derivative of the option’s price with respect to the underly
price. To highlight this relationship we regard a systematic return strategy as a dyna
portfolio strategy known from the mathematical finance literature. We assume that
where and are positive deterministic processes and is a standard B
motion. For the sake of simplicity we assume that interest rates are zero. A d
portfolio is stochastic process (adapted to the filtration generated by
is, can depend on current time and the price path of up to time ) that as
each time the number of shares in the risky asset and the money market a
Accordingly, given some initial wealth , the strategy value at each time is give
∫
for all This assumption is standard, and means that besides the initial investm
changes in the strategy’s value are solely due to price changes of the risky asset g
allocation .
Now, given an European option claim (meeting standard regularity assumptions)
∫
where denotes initial price of the option and the option’s delta at time , t
This relationship reflects the well-known paradigm (given for examp
the Black-Scholes setting) that under suitable assumptions, any European option p
can be replicated by an initial investment and a delta-hedging portfolio. Thus, Europ
options can be decomposed into an initial investment and a systematic return strate
, we can define the “option”
2/3
We further assume that the strategy is self-financing, that is, w
∫
for all This assumption is standard, and means that bes
changes in the strategy’s value are solely due to price change
allocation .
Now, given an European option claim (meeting standard reg
risky asset with maturity , we have the following well-know
∫
where denotes initial price of the option and the opt
This relationship reflects the well-known parad
the Black-Scholes setting) that under suitable assumptions, an
can be replicated by an initial investment and a delta-hedging
options can be decomposed into an initial investment and a sy
(note that
2/3
∫
allocation .
∫
is not necessarily nonnegative) by

3/3
In certain cases this point of view can be reversed. Formally, given a fixed maturity time ,
a systematic return strategy on a price process we can define the “option” by
∫
where the initial investment is understood as the “price” of the option and naturally
corresponds to the option’s delta. In this case the option’s gamma is given by
and accordingly we could say that the systematic return strategy is convex if for all
and concave if for all . Thus for this class of systematic return strategies
the convexity of the strategy is equivalent to the non-negative gamma of the
A slightly different decomposition is suggested by Bruder/Gaussel (2011), which we will
introduce in the following. This decomposition is less straight-forward but leads to very
similar qualitative findings as in Perold/Sharpe (1988), which were summarized above.
They define the option profile by ∫ . Then, a simple application of Ito’s
formula yields
∫
As above, we see that the corresponding option profile is convex if and only if
since . Another consequence of this decomposition is that systematic
return strategies corresponding to convex option profiles have a negative trading impact
and the reverse is true for systematic return strategies leading to concave option profiles.
In particular, in flat markets we have , and accordingly the trading
impact dominates the stratetgy’s value which increases with volatility.
Bruder, Benjamin, and Nicolas Gaussel. Risk-Return Analysis of Dynamic Investment Strategies.
Perold, Andre F., and William F. Sharpe. Dynamic strategies for asset allocation. Financial
Option profile Trading impact
,
where the initial investment
3/3
In certain cases this point of view can be reversed. Formally, given a fixe
a systematic return strategy on a price process we can define the “o
∫
where the initial investment is understood as the “price” of the option
corresponds to the option’s delta. In this case the option’s gamma is give
and accordingly we could say that the systematic return strategy is conv
and concave if for all . Thus for this class of systematic
the convexity of the strategy is equivalent to the non-negative gamma of
A slightly different decomposition is suggested by Bruder/Gaussel (2011
introduce in the following. This decomposition is less straight-forward bu
similar qualitative findings as in Perold/Sharpe (1988), which were summ
They define the option profile by ∫ . Then, a simple ap
formula yields
∫
As above, we see that the corresponding option profile is convex if and o
since . Another consequence of this decomposition is
return strategies corresponding to convex option profiles have a negative
and the reverse is true for systematic return strategies leading to concav
In particular, in flat markets we have , and accordingly
Bruder, Benjamin, and Nicolas Gaussel. Risk-Return Analysis of Dynamic Inves
Perold, Andre F., and William F. Sharpe. Dynamic strategies for asset allocatio
is understood as the “price” of the
option
2/3
∫
allocation .
∫
and
3/3
∫
formula yields
∫
naturally corresponds to the option’s delta. In
this case, the option’s gamma is given by
3/3
∫
formula yields
∫
, and
accordingly we could say that the systematic return strategy
is convex if
3/3
∫
formula yields
∫
for all
3
In certain cases this point of view can be rev
a systematic return strategy on a price pro
where the initial investment is understood a
corresponds to the option’s delta. In this case
and accordingly we could say that the system
and concave if for all . Thus
the convexity of the strategy is equivalent to
A slightly different decomposition is suggeste
introduce in the following. This decompositio
similar qualitative findings as in Perold/Sharp
They define the option profile by ∫
formula yields
As above, we see that the corresponding opt
since . Another conseque
return strategies corresponding to convex op
and the reverse is true for systematic return s
impact dominates the stratetgy’s value which
Bruder, Benjamin, and Nicolas Gaussel. Risk-Re
Perold, Andre F., and William F. Sharpe. Dynam
Option profile
and concave if
3
In certain cases this point of view can be reve
a systematic return strategy on a price proc
where the initial investment is understood a
corresponds to the option’s delta. In this case
and accordingly we could say that the system
and concave if for all . Thus
the convexity of the strategy is equivalent to t
A slightly different decomposition is suggeste
introduce in the following. This decomposition
similar qualitative findings as in Perold/Sharp
They define the option profile by ∫
formula yields
As above, we see that the corresponding opti
since . Another consequen
return strategies corresponding to convex opt
and the reverse is true for systematic return s
impact dominates the stratetgy’s value which
Bruder, Benjamin, and Nicolas Gaussel. Risk-Ret
Perold, Andre F., and William F. Sharpe. Dynami
Option profile
for all
3/3
In certain cases this point of view can be reversed. Formally, given a
a systematic return strategy on a price process we can define the
∫
where the initial investment is understood as the “price” of the optio
corresponds to the option’s delta. In this case the option’s gamma is g
and accordingly we could say that the systematic return strategy is co
and concave if for all . Thus for this class of system
the convexity of the strategy is equivalent to the non-negative gamma
A slightly different decomposition is suggested by Bruder/Gaussel (20
introduce in the following. This decomposition is less straight-forward
similar qualitative findings as in Perold/Sharpe (1988), which were su
They define the option profile by ∫ . Then, a simple
formula yields
∫
As above, we see that the corresponding option profile is convex if an
since . Another consequence of this decompositio
return strategies corresponding to convex option profiles have a nega
and the reverse is true for systematic return strategies leading to con
In particular, in flat markets we have , and accordin
Bruder, Benjamin, and Nicolas Gaussel. Risk-Return Analysis of Dynamic In
Perold, Andre F., and William F. Sharpe. Dynamic strategies for asset alloca
. Thus, for this class of systematic return strategies, the
convexity of the strategy is equivalent to the nonnegative
gamma of the corresponding option.
A slightly different decomposition is suggested by Bruder and
Gaussel (2011), which we will introduce next. This decomposition
is less straightforward, but leads to very similar qualitative
findings as those by Perold and Sharpe (1988) summarized
above. They define the option profile by
3/3
In certain cases this point of view can be reversed. Formally, give
a systematic return strategy on a price process we can define
∫
where the initial investment is understood as the “price” of the o
corresponds to the option’s delta. In this case the option’s gamma
and accordingly we could say that the systematic return strategy
and concave if for all . Thus for this class of sys
the convexity of the strategy is equivalent to the non-negative gam
A slightly different decomposition is suggested by Bruder/Gausse
introduce in the following. This decomposition is less straight-forw
similar qualitative findings as in Perold/Sharpe (1988), which wer
They define the option profile by ∫ . Then, a sim
formula yields
∫
As above, we see that the corresponding option profile is convex
since . Another consequence of this decompo
return strategies corresponding to convex option profiles have a n
and the reverse is true for systematic return strategies leading to
In particular, in flat markets we have , and acco
impact dominates the stratetgy’s value which increases with volat
Bruder, Benjamin, and Nicolas Gaussel. Risk-Return Analysis of Dynam
Perold, Andre F., and William F. Sharpe. Dynamic strategies for asset a
(note that
2/3
trends.
∫
allocation .
∫
is not necessarily nonnegative). Then, a simple
application of Itô’s formula yields

3/3
In certain cases this point of view can be reversed. Formally, given a fixed maturity tim
∫
where the initial investment is understood as the “price” of the option and natura
and accordingly we could say that the systematic return strategy is convex if fo
and concave if for all . Thus for this class of systematic return strateg
A slightly different decomposition is suggested by Bruder/Gaussel (2011), which we w
similar qualitative findings as in Perold/Sharpe (1988), which were summarized above
They define the option profile by ∫ . Then, a simple application of Ito
formula yields
∫
since . Another consequence of this decomposition is that systema
return strategies corresponding to convex option profiles have a negative trading impa
and the reverse is true for systematic return strategies leading to concave option profi
Bruder, Benjamin, and Nicolas Gaussel. Risk-Return Analysis of Dynamic Investment Strategie
.
As above, we see that the corresponding option profile
is convex if and only if
3/3
∫
formula yields
∫
since
Management of Systematic Return Strategies – A Prim
In certain cases this point of view can
a systematic return strategy on a pr
where the initial investment is under
corresponds to the option’s delta. In th
and accordingly we could say that the
and concave if for all
the convexity of the strategy is equiva
A slightly different decomposition is su
introduce in the following. This decom
similar qualitative findings as in Perold
They define the option profile by
formula yields
As above, we see that the correspond
since . Another con
return strategies corresponding to con
and the reverse is true for systematic
impact dominates the stratetgy’s valu
Bruder, Benjamin, and Nicolas Gaussel.
Perold, Andre F., and William F. Sharpe.
Opti
.
Another consequence of this decomposition is that systematic
return strategies corresponding to convex option profiles have a
negative trading impact, and the reverse is true for systematic
return strategies leading to concave option profiles. In particular,
in flat markets we have
3/3
In certain cases this point of view can be reversed. Formally, given a fixed maturity
∫
where the initial investment is understood as the “price” of the option and nat
and accordingly we could say that the systematic return strategy is convex if
and concave if for all . Thus for this class of systematic return stra
A slightly different decomposition is suggested by Bruder/Gaussel (2011), which we
introduce in the following. This decomposition is less straight-forward but leads to v
similar qualitative findings as in Perold/Sharpe (1988), which were summarized abo
They define the option profile by ∫ . Then, a simple application of
formula yields
∫
since . Another consequence of this decomposition is that system
return strategies corresponding to convex option profiles have a negative trading im
and the reverse is true for systematic return strategies leading to concave option pr
Bruder, Benjamin, and Nicolas Gaussel. Risk-Return Analysis of Dynamic Investment Strate
, and accordingly
the trading impact dominates the strategy’s value, which
increases with volatility.

48 / 54
Appendix 3: Systematic Return Strategies and the
Efficient Market Hypothesis
Some market participants have expressed criticism about risk
premia investing. This criticism mostly falls into two categories.
First, they argue that investing in traditional assets already gives
some exposure to these risk premia. We addressed this
concern in section 2. We argued that by directly investing in
these risk premia – as opposed to an indirect investment via
traditional long-only portfolios – the investment manager’s
degree of freedom can be increased and he or she accordingly
is more likely to achieve an “optimal” allocation. Also, many risk
premia cannot be harvested with such simple portfolios. The
second pillar of concern rests on the hypothesis that risk premia
should not be regarded as “new” asset classes or risk factors.
In conjunction with traditional equilibrium asset pricing models
like the CAPM, this would imply – in an efficient market – that
exposure to these risk premia is not expected to be rewarded
beyond the indirect exposure to the traditional market factors.
Here, we address this second concern by showing that it is
indeed common sense that some of these risk premia should
be regarded as additional risk factors that promise a premium,
which is why they are called “risk premia.”
The Efficient Market Hypothesis (EMH) (Fama 1970) in its
strong form asserts that all relevant information is publicly
available and immediately reflected in the prices of financial
investments. If investors are assumed to be risk-averse and
agree on a single risk measure, this implies the existence of
some “optimal” market portfolio that no single investment
strategy should be able to persistently outperform on a risk-
adjusted basis.
However, every now and then researchers and practitioners
choose a particular market, for example an equity index, and
claim to have spotted such an investment strategy and conclude
by contradiction that markets cannot be efficient.51
Advocates of the EMH point to the joint hypothesis problem
(Fama 1991), which states that the EMH cannot be rejected on
the basis of a single market model. That is, an investment
strategy might be superior to the market with respect to one risk
measure, but inferior with respect to another. This problem is
particularly evident in the earlier criticisms of the EMH, which
were based on the (single-factor) CAPM.52
They found that
certain investment strategies consistently outperformed others
without having a higher beta (some investment managers called
these the “manager’s alpha”) and concluded the invalidity of the
EMH.53
The standard approach to support the EMH against such
contradictions is to enlarge the corresponding market model.
One of the earliest of these attempts came from Fama (1991),
who showed that by extending the standard CAPM by two
additional factors, many of the strategies that seem to be
inconsistent with the EMH disappeared. Parts of the returns
that could not be explained by the beta and were falsely
attributed to alpha could now be assigned to the additional risk
factors. Some investment managers call these additional factor
exposures “alternative beta” or “exotic beta.”
According to the joint hypothesis problem, the EMH can be
rejected only by an investment strategy that produces superior
risk-adjusted returns in every possible market model, meaning
for every possible (reasonable) risk measure. This is what
Jarrow and his colleagues claimed to have done in a series of
papers (see Hogan et al. 2004 and Jarrow et al. 2005). They
introduced the concept of “statistical arbitrage” strategies,
which are strategies that asymptotically generate riskless
profits. Here, “riskless” means that the variance of the gains
process vanishes as time passes, i.e. the randomness of the
gains disappears when considering (very) long time horizons.
This is obviously independent of any market model and cannot
be explained by the introduction of further risk factors and,
accordingly, is incompatible with the EMH. Using a rigorous54
methodology, they find that many of the investigated carry and
trend-following strategiescan be classified as statistical arbitrage
with high confidence.
In summary, even if it is assumed that the EMH holds, unless
the market portfolio includes all sources of risk with appropriate
weights, it will be possible to find strategies that are exposed to
a wider opportunity set that persistently outperform this
particular market.
51
The existence of such seemingly inefficient markets is backed by the scholars
of behavioral finance theory; see for example Shleifer (2000). Here, some
concerns are formulated regarding the assumption of the “rational investor.”
52
Here, the risk premia of each single asset should correspond to its exposure
to the market risk, the beta. Expected returns that are not backed by an
appropriate beta are attributed to alpha. In equilibrium, according to the theory,
this should equal the risk-free rate of return.
53
Note that when restricting to the narrow CAPM, the PutWrite strategy as well
as the trend-following strategy we discussed in the previous section seem to
violate the EMH as well.
54
In particular, they include most market frictions in their tests.

Appendix 4: Portfolio Optimization in the Presence of
Estimation Errors
Optimization results can be far from optimal in the presence of
estimation errors. Under certain circumstances, even naïve
portfolio construction schemes, i.e. equal weights, can be
expected to lead to better results.
Let us start with 30 German large-cap stocks55
and estimate
the returns and the covariance matrix based on the weekly
returns between 30.12.2005 and 29.08.2014. Next, we
perform a standard mean-variance portfolio optimization to
optimize56
the in-sample Sharpe ratio57
, which we find to be
1.57. In practice, it is unrealistic to achieve such an attractive
Sharpe ratio because here the optimizer was allowed to
determine the portfolio weights based on known risks and
returns of the investment period. Weights are fixed over the
entire period. This reflects the hypothetical situation where risks
and returns could be estimated without any errors. In real life,
investors will have to estimate risks and returns employing
whichever techniques58
he or she prefers.
Whichever estimation technique is employed, chances are that
its estimation involves estimation errors. Let us take a
conservative approach and ignore estimation errors in estimating
the covariance matrix and model estimation errors of the returns
in the following form:
1/2
which we find to be 1.573
. In practice, it is unrealistic to achiev
ratio because the optimizer was allowed to determine the
known risks and returns of the investment period. Weights are
real life, nobody has this luxury but will have to estimate ri
whatever technique4
of his or her preference.
Whatever the estimation technique employed chances are
estimation errors. Let us take a conservative approach and
estimating the covariance matrix and model estimation er
following from: ̂ where ( ) is standard no
realized expected return, is the realized volatility of the a
which determines the intensity of the estimation error. The
assumes that the estimation error of the return linearly increa
asset without any directional bias. The exact form of the m
̂ leads to qualitatively similar results.
Next we perform a Monte Carlo simulation of 250 realizations
error. Figure 1 shows the Sharpe ratio as a function as the est
expected, the ensemble average of the Sharpe ratio declin
error. The ensemble maximum and minimum observed Sh
envelope around the average. At around the mean S
optimization drops below the one of a naively diversified (equa
be interpreted that in this setting one cannot expect to outpe
portfolio unless the standard deviation of one’s estimation erro
volatility of respective asset.
1
We use the constituents of the Deutsche Aktienindex (DAX) as of March 2014.
2
Optimization constrains are 1) sum of all weights equal to 100% ( ∑ )
3
In-sample sharpe ratios are naturally quite high since the risk and return paramete
4
For example, some people rely on analysts looking at company balance sheets an
rely on quantitative methods or technical analysis.
where
1/2
. In practice, it is unrealistic to achiev
ratio because the optimizer was allowed to determine the
known risks and returns of the investment period. Weights are
real life, nobody has this luxury but will have to estimate r
whatever technique4
Whatever the estimation technique employed chances are
estimation errors. Let us take a conservative approach and
estimating the covariance matrix and model estimation e
following from: ̂ where ( ) is standard n
realized expected return, is the realized volatility of the a
which determines the intensity of the estimation error. The
assumes that the estimation error of the return linearly incre
asset without any directional bias. The exact form of the m
Next we perform a Monte Carlo simulation of 250 realizations
error. Figure 1 shows the Sharpe ratio as a function as the est
expected, the ensemble average of the Sharpe ratio declin
error. The ensemble maximum and minimum observed Sh
envelope around the average. At around the mean
optimization drops below the one of a naively diversified (equ
be interpreted that in this setting one cannot expect to outpe
portfolio unless the standard deviation of one’s estimation erro
1
2
3
In-sample sharpe ratios are naturally quite high since the risk and return paramete
4
For example, some people rely on analysts looking at company balance sheets a
is
standardly normally distributed,
1/2
standard mean variance portfolio optimization to optimize2
the in-sample Sharpe ratio
. In practice, it is unrealistic to achieve such an attractive Sharpe
ratio because the optimizer was allowed to determine the portfolio weights based on
known risks and returns of the investment period. Weights are fix over the entire period. In
real life, nobody has this luxury but will have to estimate risks and returns employing
whatever technique4
Whatever the estimation technique employed chances are that its estimation involves
estimation errors. Let us take a conservative approach and ignore estimation errors in
estimating the covariance matrix and model estimation errors of the returns in the
following from: ̂ where ( ) is standard normally distributed is the
realized expected return, is the realized volatility of the asset, and is a parameter
which determines the intensity of the estimation error. The model of estimation errors
assumes that the estimation error of the return linearly increases with the volatility of an
asset without any directional bias. The exact form of the model does not matter, e.g.,
Next we perform a Monte Carlo simulation of 250 realizations for each level of estimation
error. Figure 1 shows the Sharpe ratio as a function as the estimation error intensity, . As
expected, the ensemble average of the Sharpe ratio declines with growing estimation
error. The ensemble maximum and minimum observed Sharpe ratio is shown as an
envelope around the average. At around the mean Sharpe ratio of the portfolio
optimization drops below the one of a naively diversified (equal weight) portfolio. This can
be interpreted that in this setting one cannot expect to outperform the naively diversified
portfolio unless the standard deviation of one’s estimation error is less than half their of the
1
2
3
In-sample sharpe ratios are naturally quite high since the risk and return parameters are known in advance.
4
For example, some people rely on analysts looking at company balance sheets and talking to the management. Others
is the realized expected
return,
1/2
the in-sample Sharpe ra
. In practice, it is unrealistic to achieve such an attractive Shar
ratio because the optimizer was allowed to determine the portfolio weights based o
known risks and returns of the investment period. Weights are fix over the entire period.
real life, nobody has this luxury but will have to estimate risks and returns employi
whatever technique4
Whatever the estimation technique employed chances are that its estimation involv
estimation errors. Let us take a conservative approach and ignore estimation errors
estimating the covariance matrix and model estimation errors of the returns in t
following from: ̂ where ( ) is standard normally distributed is t
realized expected return, is the realized volatility of the asset, and is a paramet
which determines the intensity of the estimation error. The model of estimation erro
assumes that the estimation error of the return linearly increases with the volatility of
asset without any directional bias. The exact form of the model does not matter, e.g
Next we perform a Monte Carlo simulation of 250 realizations for each level of estimati
error. Figure 1 shows the Sharpe ratio as a function as the estimation error intensity, . A
expected, the ensemble average of the Sharpe ratio declines with growing estimatio
error. The ensemble maximum and minimum observed Sharpe ratio is shown as a
envelope around the average. At around the mean Sharpe ratio of the portfo
optimization drops below the one of a naively diversified (equal weight) portfolio. This ca
be interpreted that in this setting one cannot expect to outperform the naively diversifie
portfolio unless the standard deviation of one’s estimation error is less than half their of t
1
2
3
4
For example, some people rely on analysts looking at company balance sheets and talking to the management. Oth
is the realized volatility of the asset and
1/2
whatever technique4
1
2
3
4
is a
parameter that determines the intensity of the estimation error.
The model of estimation errors assumes that the estimation
error of the return linearly increases with the volatility of an asset
without any directional bias. The exact form of the model for the
estimation error does not matter, for example
Let us start with 30 Germ
matrix based on the wee
standard mean varianc
ratio because the optim
known risks and returns
real life, nobody has th
whatever technique4
of h
Whatever the estimatio
estimation errors. Let u
estimating the covarian
following from: ̂
realized expected retur
which determines the i
assumes that the estim
asset without any direc
̂ leads to qua
Next we perform a Mon
error. Figure 1 shows the
expected, the ensemble
error. The ensemble m
envelope around the av
optimization drops below
be interpreted that in th
portfolio unless the stan
volatility of respective as
1
We use the constituents of the
2
Optimization constrains are 1)
3
In-sample sharpe ratios are na
4
For example, some people rely
rely on quantitative methods or t
leads to qualitatively similar results.
Next, we perform a Monte Carlo simulation of 250 realizations
for each level of estimation error. Figure 19 shows the Sharpe
ratio as a function of the estimation error intensity,
1/2
Appendix 4: Portfolio Optimization in the Presence of Estimation Errors
Optimization results can be far from optimal in the presence of estimation errors. Under
certain circumstances even naïve portfolio construction schemes, i.e., equal weights, can
be expected to lead to better results.
Let us start with 30 German large cap stocks1
and estimate the returns and the covariance
matrix based on the weekly returns between 30.12.2005 to 14.3.2014. Next, we perform a
whatever technique4
1
2
3
4
. As
expected, the average of the Sharpe ratio declines with growing
estimation error. The maximum and minimum observed Sharpe
ratios are shown as an envelope around the average. At around
1/2
whatever technique4
1
2
3
4
, the mean Sharpe ratio of the portfolio optimization
drops below the one of a naïvely diversified (equally weighted)
portfolio. This can be interpreted as meaning that in this setting,
one cannot expect to outperform the naïvely diversified portfolio
unless the standard deviation of one’s estimation error for
expected returns is less than half that of the volatility of the
respective asset.
55
We use the constituents of the Deutscher Aktienindex (DAX) as of March 2014.
56
Optimization constraints are 1) the sum of all weights equal to 100%
mer
15/21
n in the Presence of Estimation Errors
m optimal in the presence of estimation errors. Under
ortfolio construction schemes, i.e., equal weights, can
ge cap stocks 18
and estimate the returns and the
ekly returns between 30.12.2005 to 14.3.2014. Next,
nce portfolio optimization to optimize19
the in-sample
.5720
. In practice, it is unrealistic to achieve such an
e optimizer was allowed to determine the portfolio
returns of the investment period. Weights are fix over
y has this luxury but will have to estimate risks and
ue21
employed chances are that its estimation involves
nservative approach and ignore estimation errors in
and model estimation errors of the returns in the
e is standard normally distributed is the
realized volatility of the asset, and is a parameter
he estimation error. The model of estimation errors
f the return linearly increases with the volatility of an
The exact form of the model does not matter, e.g.,
milar results.
ulation of 250 realizations for each level of estimation
o as a function as the estimation error intensity, . As
f the Sharpe ratio declines with growing estimation
d minimum observed Sharpe ratio is shown as an
round the mean Sharpe ratio of the portfolio
a naively diversified (equal weight) portfolio. This can
e cannot expect to outperform the naively diversified
on of one’s estimation error is less than half their of the
nindex (DAX) as of March 2014.
hts equal to 100% ( ∑ )
h since the risk and return parameters are known in advance.
looking at company balance sheets and talking to the management.
analysis.
.
57
In-sample Sharpe ratios are naturally quite high, since the risk and return
parameters are known in advance.
58
For example, some people rely on analysts looking at company balance sheets
and talking to the company management. Others rely on quantitative methods
(e.g. Merton 1980, Stein 1956) or on technical analysis.
Figure 19: Sharpe Ratio as a Function of Estimation Error Intensity,
√
√
, Compared to a Naïvely Diversified Portfolio
Source: Bloomberg L.P., own calculations. Historical performance indications and financial market scenarios are not reliable indicators of current or future performance.
Mean Sharpe ratio Minimum Sharpe ratio Maximum Sharpe ratio Naïve diversification
0 0.5 32.521.51
2
1.5
1
0.5
0
-0.5
-1
Sharperatio
Estimation error
Appendix 4: Portfolio Optimization in the Presence of Estimation Errors
whatever technique4

50 / 54
Appendix 5: Maximum Entropy Estimates in Information
Theory
From information theory we get a constructive answer to the
statistical inference problem: “Given a discrete random variable
for which we have only partial information about its probability
distribution, what is the least biased estimate possible?” This is
the maximum entropy estimate. The key for the maximum
entropy estimate is the entropy measure that acts on
probability distributions. For a discrete probability distribution
ntensity, , compared to a
the statistical inference problem:
partial information about its
possible?” This is the maximum-
mate is the entropy measure that
distribution the
2 2.5 3
, the corresponding entropy is given by:

16/21
e ratio as a function of estimation error intensity, , compared to a
rsified portfolio
ntropy estimates in Information Theory
ation theory we get a constructive answer to the statistical inference problem:
rete random variable for which we have only partial information about its
stribution, what is the least biased estimate possible?” This is the maximum-
mate. The key for the maximum-entropy estimate is the entropy measure that
ability distributions. For a discrete probability distribution the
g entropy is given by
∑
0.5 1 1.5 2 2.5 3

Now let
17/21
While this expression seems to be somewhat arbitrary, it can be proved22
to be basically
the unique measure based on three axioms matching our intuition on the “amount of
uncertainty” of a given probability distribution. The three axioms are: 1) The measure
should be higher for broader distributions (attaining its maximum if all states are equally
likely). 2) The measure should be lower for sharply peaked distributions (attaining its
minimum if only one state can be attained with positive probability). 3) The measure shall
be additive for independent sources of uncertainty.
Now let be a discrete random variable, that can take a finite collection of states
and denote by it’s probability distribution. Assume we have only partial
information about the probability distribution , and denote by the subset of all discrete
probability distributions that meet the partial information we have about , then the
maximum-entropy estimate is given by , where
This estimate is the least-biased among all possible estimates that meet the partial
information in the sense, that it is the distribution that has the minimum distance to the
most uncertain, that is, to the maximum entropy distribution. Any other estimate that meets
the partial information we have and deviates from has incorporated more information
then given and hence introduced a bias.
4 Literature
Arthur, W. B.: Increasing Returns and Path Dependence in the Economy; The
University of Michigan Press, 1997
Avramov, D., Cheng, S., Hameed, A., (2014), “Time-Varying Momentum Payoffs and
Illiquidity”, http://ssrn.com/abstract=2289745
Bailey, D., Borwein, J., Lopez de Prado, M., Zhu, Q., (2014), “Pseudo-Mathematics and
Financial Charlatanism: The Effects of Backtest Overfitting on Out-of-Sample
Performance », http://www.ams.org/notices/201405/rnoti-p458.pdf ,
Bender, J., Briand, R., Nielsen, F., (2009), „Portfolio of Risk Premia: A New Approach to
Diversification”, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1331080
22
see Jaynes (1957) for short cut proof and the references therein
be a discrete random variable that can take a finite
collection of states
bitrary, it can be proved22
to be basically
tching our intuition on the “amount of
he three axioms are: 1) The measure
ng its maximum if all states are equally
ply peaked distributions (attaining its
ositive probability). 3) The measure shall
ty.
n take a finite collection of states
ribution. Assume we have only partial
nd denote by the subset of all discrete
mation we have about , then the
re
possible estimates that meet the partial
on that has the minimum distance to the
distribution. Any other estimate that meets
rom has incorporated more information
endence in the Economy; The
, “Time-Varying Momentum Payoffs and
hu, Q., (2014), “Pseudo-Mathematics and
Backtest Overfitting on Out-of-Sample
405/rnoti-p458.pdf ,
tfolio of Risk Premia: A New Approach to
ers.cfm?abstract_id=1331080
in
and denote its probability
distribution by
17/21
to be basically
4 Literature
22
. If we assume that we have only
partial information about the probability distribution
17/21
to be basically
4 Literature
22
and
denote by
17/21
ewhat arbitrary, it can be proved22
to be basically
ioms matching our intuition on the “amount of
ution. The three axioms are: 1) The measure
ns (attaining its maximum if all states are equally
for sharply peaked distributions (attaining its
ed with positive probability). 3) The measure shall
uncertainty.
e, that can take a finite collection of states
bility distribution. Assume we have only partial
ution , and denote by the subset of all discrete
artial information we have about , then the
y , where
ong all possible estimates that meet the partial
distribution that has the minimum distance to the
entropy distribution. Any other estimate that meets
eviates from has incorporated more information
.
ath Dependence in the Economy; The
, (2014), “Time-Varying Momentum Payoffs and
9745
o, M., Zhu, Q., (2014), “Pseudo-Mathematics and
s of Backtest Overfitting on Out-of-Sample
ces/201405/rnoti-p458.pdf ,
09), „Portfolio of Risk Premia: A New Approach to
sol3/papers.cfm?abstract_id=1331080
ences therein
the subset of all discrete probability distributions
that meet the partial information we have about
17/21
to be basically
4 Literature
22
, then the
maximum entropy estimate is given by
17/21
to be basically
4 Literature
22
, where:
17/21
to be basically
4 Literature
22
among all
17/21
to be basically
4 Literature
22
in
17/21
to be basically
4 Literature
22
.
This estimate is the least biased among all possible estimates
that meet the partial information in the sense that it is the
distribution that has the minimum distance to the most
uncertain, i.e. the maximum entropy, distribution. Any other
estimate that meets the partial information we have and that
deviates from
atic Return Strategies – A Primer
ssion seems to be somewhat arbitrary, it can be proved22
to be basically
sure based on three axioms matching our intuition on the “amount of
given probability distribution. The three axioms are: 1) The measure
r for broader distributions (attaining its maximum if all states are equally
easure should be lower for sharply peaked distributions (attaining its
one state can be attained with positive probability). 3) The measure shall
ndependent sources of uncertainty.
iscrete random variable, that can take a finite collection of states
it’s probability distribution. Assume we have only partial
ut the probability distribution , and denote by the subset of all discrete
butions that meet the partial information we have about , then the
opy estimate is given by , where
s the least-biased among all possible estimates that meet the partial
he sense, that it is the distribution that has the minimum distance to the
that is, to the maximum entropy distribution. Any other estimate that meets
mation we have and deviates from has incorporated more information
hence introduced a bias.
creasing Returns and Path Dependence in the Economy; The
chigan Press, 1997
heng, S., Hameed, A., (2014), “Time-Varying Momentum Payoffs and
ssrn.com/abstract=2289745
wein, J., Lopez de Prado, M., Zhu, Q., (2014), “Pseudo-Mathematics and
latanism: The Effects of Backtest Overfitting on Out-of-Sample
http://www.ams.org/notices/201405/rnoti-p458.pdf ,
nd, R., Nielsen, F., (2009), „Portfolio of Risk Premia: A New Approach to
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1331080
for short cut proof and the references therein
has incorporated more information than given
and has thus introduced a bias.
Although this expression seems somewhat arbitrary, it can be
proven59
to basically be the unique measure based on three
axioms matching our intuition regarding the amount of
uncertainty of a given probability distribution. The three axioms
are:
1) The measure should be higher for broader distributions
(attaining its maximum if all states are equally likely).
2) The measure should be lower for sharply peaked distributions
(attaining its minimum if only one state can be attained with
positive probability).
3) The measure shall be additive for independent sources of
uncertainty.
59
See Jaynes (1957) for shortcut proof and the references therein.

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delivered to, or relied on by, any other type of Person. The Fund to which this document relates is not subject to any form of regulation or approval by
the DFSA. The DFSA has no responsibility for reviewing or verifying any Prospectus or other documents in connection with this Fund. Accordingly, the
DFSA has not approved this marketing material or any other associated documents nor taken any steps to verify the information set out in this document
and has no responsibility for it. The Units to which this marketing material relates may be illiquid and/or subject to restrictions on their resale. Prospective
purchasers of the Units offered should conduct their own due diligence on the Units. If you do not understand the contents of this brochure, you should
consult an authorized financial advisor. For investors in Bahrain: This document has not been approved by the Central Bank of Bahrain, which takes
no responsibility for its contents. No offer to the public to purchase the fund units/securities/services will be made in the Kingdom of Bahrain and this
document is intended to be read by the addressee only and must not be passed to, issued to, or shown to the public generally. For investors in
Lebanon: The Fund/Products/Services has not been approved by the local regulator for marketing, promotion, offer or sale (“offered”) of the Fund/
Products/Services in Lebanon and such may not be offered into Lebanon, except through a licensed local entity and on a reverse solicitation basis. For
Residents of the Sultanate of Oman: The information contained in this brochure neither constitutes a public offer of securities in the Sultanate of
Oman as contemplated by the Commercial Companies Law of Oman (Royal Decree 4/74) or the Capital Market Law of Oman (Royal Decree 80/98),
nor does it constitute an offer to sell, or the solicitation of any offer to buy non-Omani securities in the Sultanate of Oman as contemplated by Article
139 of the Executive Regulations to the Capital Market Law (issued by Decision No. 1/2009). Additionally, this private placement memorandum is not
intended to lead to the conclusion of any contract of any nature whatsoever within the territory of the Sultanate of Oman. For investors in Qatar: The
Units/securities/services are only being offered to a limited number of investors who are willing and able to conduct an independent investigation of the
risks involved in an investment in such Units/securities/services. The promotional documentation does not constitute an offer to the public and is for the
use only of the named addressee and should not be given or shown to any other person (other than employees, agents or consultants in connection with
the addressee’s consideration thereof). The fund has not been and will not be registered with the Qatar Central Bank or under any laws of the State of
Qatar. No transaction will be concluded in your jurisdiction and any inquiries regarding the Units/securities/services should be made to Credit Suisse,
Frédéric Schmutz, The Gate, Dubai International Financial Center, 9th
Floor East, P.O. Box 33660, Dubai, United Arab Emirates; +971 4 362 0024;
frederic.schmutz@credit-suisse.com For investors in the UAE/ABU DHABI/DUBAI (NON-DIFC): This document, and the information contained
herein, does not constitute and is not intended to constitute a public offer of securities in the United Arab Emirates and accordingly should not be
construed as such. The Units/securities/services are only being offered to a limited number of sophisticated investors in the UAE (a) who are willing and
able to conduct an independent investigation of the risks involved in an investment in such Units/securities/services, and (b) upon their specific request.
The Units/securities/services have not been approved by or licensed or registered with the UAE Central Bank, the Securities and Commodities Authority
or any other relevant licensing authorities or governmental agencies in the UAE. The document is for the use of the named addressee only and should
not be given or shown to any other person (other than employees, agents or consultants in connection with the addressee’s consideration thereof).
No transaction will be concluded in the UAE and any enquiries regarding the Units/securities/services should be made to Credit Suisse,
Frédéric Schmutz, The Gate, Dubai International Financial Center, 9th
Floor East, P.O. Box 33660, Dubai, United Arab Emirates; +971 4 362 0024;
frederic.schmutz@credit-suisse.com. Copyright © 2015 Credit Suisse Group AG and/or its affiliates. All rights reserved.

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