The X Factor

THE X FACTOR: GROUPING SECURITIES,
DEFINING “SIMILAR” AND FORMING
ESTIMATES
Nick Wade
Northfield Information Services
2011
1

WHAT’S THE MAIN POINT?
 Most of the models we use feature1:
 a fixed factor structure that does not allow for change or evolution in the way that
markets convert information into price change
 and are estimated using one of many techniques that assume all the data in the
sample follow some nice, clean, simple rules that bear no resemblance to real life
 We contend that risk models:
 Need some kind of adaptive structure that allows them to change as new or transient
effects appear
 Need adjustment for “regimes” in the data – this impacts asset allocation and
“insurance”
 Should harness current or forward-looking information as a conditioning input
 Should be estimated using a flexible technique that allows for evolution in the
underlying data
 We also offer some thoughts about emergent techniques, new candidates
for factors, and directions for future research
1 There are a lot more problems, but the focus for today is JUST the idea that things change over time
2

ROADMAP
 Introduction to Factor Models
 Choices of Factors
 Pros and Cons
 A Hybrid Approach
 Better Factors
 Choices of Estimation Methodology
 Data Regimes
 Better Estimation Methodology
 Conclusions
3

WHAT IS A FACTOR MODEL?
 The main purpose of a factor model is to find a set of
common themes that explain the variability in security
prices that is shared across securities.
 Having defined a set of factors, we then look to
estimate the return associated with those factors and
the individual security exposures/sensitivities/betas to
those factors
 The end result is a model of how the portfolio will
behave – how much the securities will move together
and how much they will behave uniquely
 That can lead us to various useful risk characteristics
4

THE LINEAR MODEL
 Relationship between R and F is linear ∀F
 There are N common factor sources of return
 Relationship between R and H is linear ∀H
 There is no correlation between F and H ∀ F,H
 The distribution of F is stationary, Normal, i.i.d. ∀F
 There are M stock-specific sources of return
 There is no correlation between H across stocks
 The distribution of H is stationary, Normal, i.i.d. ∀H
 (Implicitly also the volatility of R and F is stationary)
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5

ESTIMATING A RISK MODEL
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The Variance of a portfolio is given by the double sum over the
factors contributing systematic or common factor risk, plus a
weighted sum of the stock-specific or residual risks.
6

HOW MANY FACTORS…?
 The academic consensus seems to be that
there is not much difference going from 5 to
10 to 15 factors. In other words, 5 do the job.
 Lehmann & Modest (1988)
 Connor & Korajczyk (1988)
 Roll & Ross (1980)
 If somebody is suggesting a 90 factor model
to you, tell them to try harder
8

ENDOGENOUS MODEL
 The Endogenous or Fundamental Model
seeks to estimate Fit assuming Eit by
regression.
 Typical factors include E/P, D/E, Industry
membership, Country membership…
 King (1966)
 Rosenberg and Guy (1975) etc.
 Model is pre-specified
 These models are hugely popular, partly
because we’ve been building them for so
long we’ve become habituated
9

EXOGENOUS MODEL
 The Exogenous or Macro model seeks to
estimate Ei from Fit.
 Typical factors include Market, Sector, Oil,
Interest-Rates…
 Ross (1976)
 Chen (1986)
 Model is pre-specified
10

STATISTICAL MODEL
 Assume N
 Use Factor Analysis or Principle Components
to estimate Ei
 Use Regression to estimate Fit
 Errors in Variables
11

PROS AND CONS OF EACH APPROACH
Approach Pros Cons
Fundamental (micro)
model
• Suitable for concentrated portfolios • Number of factors is fixed thus unchanging
• Dependent on accounting statement
accuracy
• Dependent on accounting standards
comparability
•Membership factors for
industry/country/sector
• Errors will be in factor returns, hence in
covariance matrix, and hence not diversifiable
Macro-economic
model
• No dependence on accounting data
• The response of each security to changes in
market/sector/industry/ whatever to be different
across securities
• Errors will be in loadings (exposures), thus
diversifiable
• Factor number fixed and unchanging
• Exposure to factors is stationary over time
Statistical model • All correlation is information
• Captures new, or transient effects
• Adaptive to the market
• Great for a short-term model
• Attribution of risk is difficult
• Issues with noise in data
• Errors in variables
• Number of factors is either pre-specified or
sample-dependent
12

OTHER APPROACHES
 Combined Models:
 Northfield Hybrid Model
 Stroyny (2001)
 Simultaneous Estimation
 Black et al (1972)
 Heston and Rouwenhorst (1994, 1995)
 Satchell and Scowcroft (2001)
 GMM Hansen (1982)
 McElroy and Burmeister (1988) using NLSUR (which is
assymptotically equivalent to ML)
 Bayesian Approach:
 Pohlson and Tew (2000)
 Ericsson and Karlsson (2002)
13

SIMULTANEOUS ESTIMATION
 Removing the limitation of binary or membership variables (such as
industry, country, sector, region etc).
 Marsh and Pfleiderer (1997)
 Scowcroft and Satchell (2001)
 Start with an estimate of the exposures (e.g. 1.00 for all companies)
use that estimate to solve for the factor return, then use that factor
return in turn to re-solve for a revised set of exposures, thus
converging iteratively on a better solution for both Eit and Fit.
 Black et al (1972)
 Scowcroft and Satchell (2001)
 Given various limiting restrictions we can ensure that the model
converges and that it is unique.
 This is the idea behind the UBS range of models and it’s good. But
it’s not adaptive.
14

HYBRID MODEL (NORTHFIELD 1998)
 Combine macro, micro, and statistical factors
 An observable factor core, and statistical factors
trawling the residual return to find new factors
 Gain the advantages of each, whilst mitigating the
limitations of each
 Intuitive, explainable, justifiable observable factors
 Minimal dependence on accounting information
 Rapid inclusion of new or transient factors
15

THOUGHTS
 Notice we are already trying to allow for an adaptive set of
factors – we will pick this theme up later
 In a minute we’ll be allowing for time-dependent volatility
and correlations as well
 But are all price movements based on fundamentals?
 Apple?
 What about news?
 What about liquidity impacts?
 What about index membership or common ownership?
 Exposed to “have to” sales as index weight changes for
example
 Some securities are held in many funds, high “centrality”
16

A FEW SUGGESTIONS FOR BETTER FACTORS
 A factor can be any shared behavior
 HISTORICAL: Semantic clustering (text mining)
 Dig into everything published on a universe of companies and
look for similarities by phrase comparison etc
 PREDICTIVE: News flows
 Look at instances of occurrence in news, sentiment
 Inference from other asset classes
 What does a bond spread change tell us about equity vol?
 What about a change in option implied volatility/implied
correlation?
 Social network analysis
 Apply emergent techniques to look at influence within groups,
measures of asset centrality, flow of information, diversification?
 Influence: types of network shape
18

NEWS AS A FACTOR
 Mitra, Mitra, diBartolomeo (2008)
 Since Dan’s going into depth later I will
restrain myself – just note that you can use
news incidence/sentiment to condition risk
forecasts
19

OWNERSHIP DATA
 Not a well-explored source of information
 Starmine: [see Dirk Renick “Research on
sentiment based smart money”]
 Use ownership data to reverse out factor preferences
 Funds are attracted to companies that are “like” ones they
already own
 Funds exhibit biases toward companies with certain
fundamental characteristics, and these biases change over
time
20
1999 2001 2003 2004 2005 2007
StarMine PriceMo EPS_CAGR3 ROE ROE ROE ROE
LTG ROE Profit Margin Interest Coverage F12m E/P F12m E/P
G5 EPS Profit Margin Interest Coverage F12m E/P Interest Coverage Interest Coverage
Debt/Assets Debt/Assets LTG Profit Margin Profit Margin Profit Margin
Interest Coverage LTG F12m E/P LTG StarMine EQ StarMine PriceMo

COUNTRY, INDUSTRY, SECTOR, REGION…
 A useful (I hope) digression into the world of factor selection.
 It is pretty much standard practice to take note of
membership in, or exposure to, one or more countries or
regions, and one or more industries or sectors
 Problems: multinational firms, globalization, index
domination
 Scowcroft and Sefton (2001)
 Diermeier and Solnik (2000)
 MacQueen and Satchell (2001)
 Suggestions:
 Estimate a different kind of index FTSE (1999), Bacon and Woodrow (1999)
 Split into “global” market and “domestic” market either by some cut off on a variable like foreign sales (Diermeier
and Solnik 2000) or by some statistical process (MacQueen and Satchell 2001)
 Solve Model iteratively using Heston and Rouwenhorst (1994, 1995) approach
 Or extensions to that: Scowcroft and Sefton (2001).
 The real problem is that something as bland as “industry” is one-dimensional
and does not pick up enough nuance about company relationships
21

SEMANTIC CLUSTERING
 Let’s get rid of Industry Classifications
 Why? Are all banks the same? Nope.
 Semantic clustering, or text mining can be
used to:
 Help predict ratings changes [Starmine]
 Help update risk/return estimates [Ravenpack
etc]
 Measure distance between companies based on
published information [Quid.com]
22

SOCIAL NETWORK ANALYSIS
 A million followers, and still not winning… [thanks Charlie]
 Recent developments in SNA look at the different kinds of
groups and influence between/within groups
 Solis (2009) stocks form a “small world” network
 “Six degrees of separation” etc.
 Measures of “centrality”
 Degree centrality
 Reach centrality
 Flow centrality
 Betweeness centrality
 Kritzman: Asset Centrality
 Google: page rank algorithm
 Intuitive groups: index membership, ownership in mutual
funds
23

IN CONTEXT
 For our purposes, how is a group
connected?
 Think of localized version of CAPM – which
asset best represents “the market”
 Hubs? What degree of connectedness?
 Ownership commonality across mutual funds
 Why do we care?
 Investment strategy, asset allocation
 Co-movement (risk), diversification, crowding
 Information flow
24

ASSET CENTRALITY
 We can take this idea from Social Network
Analysis and apply it to a variety of contexts:
 Was Lehman too big to fail? Can we quantify it’s
centrality?
 Is BHP more influential than Rio, or less with the
Resources sector?
 Which sectors are the most/least “democratic”?
 Note that this links again with James’ work on
diversification
25

ABSORPTION RATIO (KRITZMAN 2011)
 On a related note – how tightly connected is the market,
or a particular sector?
 Lo (2008)
 Yenilmez and Saltoglu (2011)
 Absorption ratio quantifies this by looking at the
proportion of variance explained by common themes.
 As this number rises, the level of “systemic” risk rises, since
assets are more tightly connected.
 This is one requirement for a crash – just add panic
 A signal for when to apply costly insurance – e.g. zero-cost
collar
 You could use Dispersion and get a similar result (but without
allowing for idiosyncratic vol to move indep of syst vol)
 You could use Implied Correlation and get a similar forward-
looking result
26

ARTIFICIAL IMMUNE SYSTEMS
 At a high level, our immune system consists of
two pieces:
 Innate immunity
 Learned immunity
 In our context
 The factors we believe to be useful at t=0
 Plus the factors the model learns along the way
 Tune the model
 Criteria for accepting a new factor
 Criteria for archiving / forgetting factors
 Memory length for previously useful factors
27

SINGLE-PASS CLUSTERING AND RELATED
METHODS
 Concept: high frequency data in high
volumes presents a storage problem, so
need techniques that can analyze data as it
arrives rather than data-mine.
 Issues: it’s a goldfish. No memory.
28

SOME PROBLEMS
 We are dealing with an evolving data set, not a static one
 Explore how this impacts our common techniques
 Look at more advanced / better techniques to fit evolving data
sets
 We are (potentially) dealing with different regimes in the
data, not one uniform set
 Look at models that explicitly allow for regime change (not in a
George Bush sense)
 We are dealing with complex behavior within groups
 For example, some groups play follow the leader
 Some groups herd. There is no leader
 THOUGHTS: sefton: beta compression, social network
analysis, influence
30

THE WORLD IS VERY OBVIOUSLY TIME-VARYING
 Non-stationary volatility (ARCH, GARCH, etc)
 We spend an heroic amount of time trying to forecast
non-stationary volatility
 But we often just ignore it when we calculate correlation,
or perform regression analysis, or run factor analysis (or
PCA)
 Non-stationary mean (Trend)
 We often build models to capture the alpha in
momentum, reversals, and other manifestations of a non-
stationary mean
 But we often ignore those when we calculate correlation,
or perform regression analysis, or run factor analysis
 Read the fine print…
31

SIMPLE ADJUSTMENTS (1)
 Non-stationary factor
return series will lead
to the model
underestimating
portfolio risk
 Adjust by changing
variance calculation
to include trend
component of return
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Adjust Model for the influence of non-stationary factor returns
32

SIMPLE ADJUSTMENTS (2)
 What about security volatility?
 We observe:
 Serial correlation (not i.i.d.)
 Bid-ask bounce
 Non-Normal distributions
 Parkinson volatility
Adjust Model for the influence of non-stationary security returns
33

CONTEMPORANEOUS OR FORWARD-LOOKING
SIGNALS
 You could make the argument that these are “factors” rather than an
estimation approach, but we use them to condition existing factors.
 Take a model that has been estimated on purely historical data
 Find true forward-looking signals
 E.g. option-implied volatility
 Find other contemporaneous signals
 E.g. dispersion measures, range measures, volume
 Adjust the parameters of the “historical” model so that the forecasts of
the model match the signals from “now” and the “future”
 Update it daily so that it stays “current”
 The Advantage: we have kept the same factor structure but removed
the sole dependency on the past.
34

GENERALIZE THE IDEA:
NORTHFIELD “ADAPTIVE NEAR-HORIZON RISK MODELS”
(ANISH SHAH, 2008)
The richest source of information about the future is not the past – increasingly it is consensus
estimates about the future from e.g. option markets, prediction markets…
 Take any risk model. e.g. one of our models estimated monthly
 Add “flexibility points” and fit to information about current conditions
 Adjust for statistical differences between short and long term returns
 Many benefits of this approach
 Avoid statistical complexities of high frequency data
 Keep familiar factor structure
 Common factor structure for long and short horizons permits interpolating
any horizon in between
 Works with any factor model
35

36
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
Northfield Asia ex. Japan Risk Models
Tracking Error LH Tracking Error SH Linear (Tracking Error SH)

KEY QUESTIONS
 Before we get carried away…
 What evidence of detectable regimes?
 How many regimes?
 What kind of model can fit multiple regimes?
 Can any of these fit multiple regimes on evolving
data?
 i.e. learn new regimes as they appear
38

EVIDENCE FOR REGIMES:
TIME-VARYING CORRELATIONS
 Increasing attention is being paid to the issue of correlations varying over time:
 (stocks) De Santis, G. and B. Gerard (1997), International asset pricing and portfolio
diversification with time-varying risk, Journal of Finance, 52, 1881-1912.
 (stocks) Longin, F. and B. Solnik (2001), Extreme correlation of international equity markets,
Journal of Finance, LVI(2), 646-676.
 (bonds) Hunter, D.M. and D.P. Simon (2005), A conditional assessment of the relationships
between the major world bond markets, European Financial Management, 11(4), 463-482.
 (bonds) Solnik, B., C. Boucrelle and Y.L. Fur (1996), International market correlation and
volatility, Financial Analysts Journal, 52(5), 17-34.
 Markov switching model: Chesnay, F and Jondeau, E “Does Correlation
Between Stock Returns really increase during turbulent periods?” Bank of
France research paper.
 To date little explored – however, Implied Correlation also seems useful, and
more powerful than historical correlation in forecasting (we saw the same result
with volatility):
 Campa, J.M. and P.H.K. Chang (1998), The forecasting ability of correlations implied in
foreign exchange options, Journal of International Money and Finance, 17, 855-880.
39

CORRELATION STABILITY
 One of the first… Kaplanis (1988): STABLE
 Tang (1995), Ratner (1992), Sheedy (1997): STABLE
– although crash of 1987 regarded as an “anomaly”
 Bertero and Mayer (1989), King and Wadwhani
(1990) and Lee and Kim (1993): correlation has
increased, but STABLE
 Not quite so stable? Erb et al (1994) – increases in
bear markets
 Longin and Solnick (1995) – increases in periods of
high volatility
 Longin and Solnick (2001) – increases in bear
markets
40

DETECTING REGIMES
 Use Viterbi’s algorithm (Viterbi 1967) to
detect states.
 Use Jennrich tests (Jennrich 1970) to decide
whether correlation differences between
states are significant
 Mahalanobis Distance (Mahalanobis 1939,
Kritzman 2009)
41

MAHALANOBIS DISTANCE
 MD is one example of a “Bregman divergence” , a group of distance measures.
 Clustering: classifies the test point as belonging to that class for which the Mahalanobis
distance is minimal. This is equivalent to selecting the class with the maximum likelihood.
 Regression: Mahalanobis distance and leverage are often used to detect outliers, especially in
the development of linear regression models. A point that has a greater Mahalanobis distance
from the rest of the sample population of points is said to have higher leverage since it has a
greater influence on the slope or coefficients of the regression equation. Specifically,
Mahalanobis distance is also used to determine multivariate outliers. A point can be an
multivariate outlier even if it is not a univariate outlier on any variable.
 Factor Analysis: recent research couples Mahalanobis distance with Factor Analysis and use
MD to determine whether a new observation is an outlier or a member of the existing factor set.
[Zhang 2003]
 MD depends on covariance (S^-1 is the inverse of the covariance matrix), so is exposed to the
same stationarity issues that affect correlation, however as described above it can help us
reduce correlation’s outlier dependence.
42

MAHALANOBIS IN ACTION
Borrowed from Kritzman: Skulls, financial turbulence, and the
implications for risk management. July 2009 43

TURBULENCE IN THE MARKET
 Kritzman (2009):
 Correlation of US and foreign stocks when both
markets’ returns are one standard deviation
above their mean: -17%
 Correlation of US and foreign stocks when both
markets’ returns are one standard deviation
below their mean: +76%
 “Conditional correlations are essential for
constructing properly diversified portfolios”
44

CORRELATION REGIMES
 If it’s not stable, how about Markov switching
models?
 Ramchand and Susmel (1998), Chesnay and
Jondreau (2001) – correlation, conditioned on
market regime, increases in periods with high
volatility
 Ang and Bekaert (1999) – evidence for two regimes;
a high vol/high corr, and a low vol/low corr.
 Don’t forget this needs to adapt as our world
changes:
 Hidden Markov Experts on evolving data
45

FLEXIBLE ESTIMATION TECHNIQUES
46

TECHNIQUES FOR EVOLVING DATA
 Most of our favorite tools are designed to fit static
data sets where behaviors are mostly unchanged
 Neural network, Kalman filter, OLS/GLS regression,
PCA, ICA, factor analysis, variance, correlation… just
about all of them
 Recent developments in cluster analysis are
encouraging
 Artificial Immune Systems
 Single-pass clustering
 Regime-switching models e.g. HME etc
 [recent] EPCIA
 [very recent] HME on evolving data
47

REGRESSION WITH NONSTATIONARY DATA
 Techniques have been developed specifically
to allow time-varying sensitivities
 FLS (flexible least-squares)
 FLS is primarily a descriptive tool that allows us
to gauge the potential for time-evolution of
exposures
      
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(coefficient estimates)
48

FLS EXAMPLE
 An example from Clayton and MacKinnon (2001)
 The coefficient apparently exhibits structural shift in 1992
49

CLUSTER ANALYSIS WITH NONSTATIONARY DATA
 Guedalia, London, Werman; “An on-line agglomerative clustering
method for nonstationary data” Neural Computation, February 15, 1999,
Vol. 11, No. 2, Pages 521-54
 C. Aggarwal, J. Han, J. Wang, and P. S. Yu, On Demand Classification of Data
Streams, Proc. 2004 Int. Conf. on Knowledge Discovery and Data Mining
(KDD'04), Seattle, WA, Aug. 2004.
 G. Widmer and M. Kubat, “Learning in the Presence of Concept Drift and Hidden
Contexts”, Machine Learning, Vol. 23, No. 1, pp. 69-101, 1996.
 Again, there are techniques available to
conquer the problem
50

FACTOR ANALYSIS WITH NONSTATIONARY DATA
 Dahlhaus, R. (1997). Fitting Time Series Models to Nonstationary
Processes. Annals of Statistics, Vol. 25, 1-37.
 Del Negro and Otrok (2008): Dynamic Factor Models with Time-
Varying Parameters: Measuring Changes in International
Business Cycles (Federal Reserve Bank New York)
 Eichler, M., Motta, G., and von Sachs, R. (2008). Fitting dynamic
factor models to non-stationary time series. METEOR research
memoranda RM/09/002, Maastricht University.
 Stock and Watson (2007): Forecasting in dynamic factor models
subject to structural instability (Harvard).
 There are techniques available, and they are
being applied to financial series.
51

EVOLVING PRINCIPAL COMPONENT INNOVATION
ANALYSIS (EPCIA)
 You want PCA but your factor structure is
changing
 EFA (evolving factor analysis): keep adding
factors
 EFWFA (evolving fixed-window factor analysis):
keep adding new factors, but forget old ones to
make room!
 EPCIA allows for new factors to emerge
52

LAYER-EMBEDDED NETWORKS
 Within our networks, information and
interactions may flow at multiple levels
 E.g. Pasquel and de Weck (2011 working paper)
 E.g. Luttrell (2010 working paper)
 Enter multi-layer modeling
 At a high level, a layer-embedded network captures
the effects of multiple interacting processes at
different frequencies or across different groups.
 E.g. combining short and long-term alpha signals
 Combining global and local factors
 Combining pervasive and short-term factors
53

CONCLUSIONS
 Our world changes
 This requires an adaptive risk model factor structure
 This requires the ability to accommodate regimes in our risk
models, our portfolio construction, hedging, and asset allocation
by harnessing contemporaneous and forward-looking signals
 The market is not driven solely by fundamentals
 We need to leverage news/perception
 We need to explore nuanced relationships beyond bland
membership
 Techniques exist to address all of these issues, and are
being applied today.
55

TAKE HOME
 Northfield:
 risk models that utilize implied volatility since 1997
 adaptive hybrid risk models since 1998
 risk models utilizing cross-sectional dispersion since
2003
 using implied volatility and dispersion in our entire range
of short-horizon adaptive models since 2009
 If you’re doing some kind of time-series analysis on
financial data you need to keep time-dependence,
regimes, and evolving data in mind
 There are techniques to conquer all of these
challenges, but they’re not the easy ones that come
as part of Excel! 56

REFERENCES
 Black F., Jensen M., Scholes M. “The Capital Asset Pricing Model: some empirical tests” In Jensen M.C., editor,
“Studies in the Theory of Capital Markets” Praeger, New York, 1972.
 Bulsing M., Scowcroft A., and Sefton J., “Understanding Forecasting: A Unified framework for combining both
analyst and strategy forecasts” UBS Working Paper, 2003.
 Chen N.F. Roll R. Ross S.A. “Economic Forces and the Stock Market” Journal of Business 59, 1986.
 Connor G and Korajczyck R.A. “Risk and Return in an equilibrium APT: application of a new test methodology”
Journal of Financial Economics 21, 1988.
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63

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