Factor Analysis: Understanding Portfolio Risk

1© 2018 Windham Capital Management, LLC. All rights reserved.
November 2018
1
Factor Analysis
Cel Kulasekaran
Research Director
Understanding Portfolio Risk

Agenda
■ Overview of various factor models in investment management
►Single Factor Model
►Multi-Factor Models
■ Issues in implementation of factor models
■ Factors in Practice
■ Case Study
■ Questions / Feedback

What is Factor Analysis?
■ A powerful technique that can identify and measure sources of risk and return
►Managers
►Asset Classes
►Portfolios
■ The single factor Capital Asset Pricing Model (CAPM) is an early example of
factor analysis.
■ Various other applications
►Explain differences in returns across a universe of financial assets.
►Forecasting the expected value of asset returns.
►Explaining systematic variations and co-movements in returns.
►Stress testing asset class returns.

The Single Factor Model

We Studied In School
𝑦 = 𝑚𝑥 + 𝑐
Solve Solve

Linear Regression Example

Single-Factor Example (CAPM)
■ Treynor, Sharpe, and Lintner introduced CAPM in the early 60s.
■ CAPM specifies that an asset’s expected return in excess of the risk free rate is
proportional to asset’s sensitivity to systematic risk (non-diversifiable risk of the
market)
■ The sensitivity term is commonly referred to as beta
𝛽 =
Covariance 𝑅𝑖, 𝑅 𝑚
Variance 𝑅 𝑚
■ The CAPM expected return is
E 𝑅 = 𝑅𝑓 + 𝛽 ∙ 𝑅 𝑚 − 𝑅𝑓

The Multi-Factor Model

Multi-Factor Models
■ Then came Stephen Ross, extended and derived an alternative to CAPM.
■ An asset pricing model based purely on arbitrage arguments, Arbitrage Pricing Theory (APT).
■ APT says that an asset’s expected return is influenced by a variety of risk factors, not just
market risk.
■ Return on an asset is linearly related to some number of factors.
E 𝑅 = 𝛼 + 𝛽1 𝐹1 + 𝛽2 𝐹2 + ⋯ + 𝛽 𝑘 𝐹𝑘 + 𝜀
■ It’s similar to CAPM, the asset’s sensitivity to each factor is quantified by beta.

Multi-Factor Regression Example

What Factors To Use?
■ APT does not define which factors to use, but does offer some guidelines
►Impact of factors on asset prices should be explained by the unexpected
movements of the factors and not the expected movements.
►The factors should represent non-diversifiable sources of risk.
►The relationship between the factors and asset price movement should be
justifiable on economic grounds.
■ Some examples
►Inflation, Credit Risk, Term Structure, Change in Oil Prices, Market Returns
►Fama-French three factor model
E 𝑅 = 𝑅𝑓 + 𝛽 𝑚 ∙ 𝐾 𝑚 − 𝑅𝑓 + 𝛽𝑠 ∙ 𝑆𝑀𝐵 + 𝛽ℎ ∙ 𝐻𝑀𝐿 + 𝜀
 SMB = Small Minus Big (size factor)
 HML = High Minus Low Book (value factor)
►Momentum

Problems of Multi-Factor Regression
■ Overfitting
►The more variables → the higher the amount of variance you can explain
►The more variables → statistical power goes down
■ Correlated Factors
►The explanatory variables are not independent
►Distorts the interpretation of the model (coefficients / results)
 But it does not invalidate the model

Factor Models in Practice

Hierarchy of Factors
Factors
Observed
Market Macro
Unobserved
Security
Specific
Technical Sector Fundamental
Statistical
Source: Modern Investment Management: An Equilibrium Approach, Goldman Sachs Asset Management, Wiley Finance, 2003

Factor Models in Practice
■ Statistical Factor Models
►Historical and cross-sectional data on asset returns are used.
►The goal of these models are to best explain observed returns with factors that are linear
combinations and uncorrelated.
►E.g., Principal Component Analysis.
■ Macroeconomic Factor Models
►Historical asset returns and observable macroeconomic variables are used.
►The goal is to determine which macroeconomic variables are persistent in explaining the
historical asset returns.
■ Fundamental Factor Models
►Most well-known fundamental factor model is Fama-French three-factor model.
►Uses company, industry attributes, market data (and more recently technical indicators
such as momentum).

Practical Issues
■ All factors used in a model should have an economic rationale.
■ Estimation errors and the complexity of the model typically increase with the
number of factors used.

Case Study

Proposed Factors
Factor Proxy Time Series
Global Equity MSCI All Country World
Stocks
US MSCI USA – MSCI All Country World
Europe MSCI Europe – MSCI All Country World
Asia MSCI Asia – MSCI All Country World
Emerging Markets MSCI Emerging Markets – MSCI All Country World
Value-Growth MSCI US Prime Market Value – MSCI US Prime Market Growth
Small-Large MSCI US Small Cap 1750 – MSCI US Large Cap 300
Bonds
US Bonds Barclays US Aggregate
Term Structure Barclays Long Treasury – Barclays Short Treasury
Credit Barclays US Aggregate Long Credit BAA – Barclays US Government Long
High Yield BOA US High Yield Master II – Barclays US Aggregate Long Credit BAA
Other
Inflation US CPI All Urban: All Items Seasonally Adjusted
US Dollar Dollar Trade-Weighted Exchange Index (TWEX)
Volatility CBOE SPX Volatility VIX
Commodities Goldman Sachs Commodity Index Total Return

A B C D E F G H I J K L M N O
A: Global Equity 1.00
B: US Stocks 0.08 1.00
C: Europe 0.07 -0.53 1.00
D: Asia -0.12 -0.57 -0.30 1.00
E: Emerging Markets 0.17 -0.35 -0.19 0.42 1.00
F: Value-Growth -0.31 -0.02 0.06 0.04 -0.13 1.00
G: Small-Large 0.20 -0.23 0.04 0.19 0.28 -0.05 1.00
H: US Bonds -0.08 0.15 -0.17 -0.01 0.01 0.08 -0.08 1.00
I: Term Structure -0.28 0.08 -0.11 -0.01 -0.05 0.11 -0.15 0.84 1.00
J: Credit 0.64 -0.06 0.02 0.07 0.22 -0.16 0.30 -0.16 -0.53 1.00
K: High Yield 0.42 -0.03 0.08 -0.01 0.09 -0.13 0.26 -0.67 -0.80 0.49 1.00
L: Inflation 0.02 0.00 -0.14 0.08 0.04 -0.03 0.06 -0.16 -0.24 0.11 0.27 1.00
M: US Dollar -0.22 -0.19 0.35 -0.06 -0.22 0.06 -0.07 -0.20 -0.02 -0.27 -0.08 -0.29 1.00
N: Volatility -0.70 -0.08 0.00 0.08 -0.14 0.22 -0.22 0.06 0.24 -0.54 -0.28 0.07 0.14 1.00
O: Commodities 0.28 -0.02 -0.20 0.19 0.24 -0.10 0.23 0.01 -0.12 0.28 0.16 0.32 -0.30 -0.22 1.00
Correlations of Factors
Different components of pricing fixed
income tend to move together
* November 1996 – February 2014

Multi-Factor Regression: Factor Loadings
Factor Manager A Manager B Manager C
Global Equity 0.56 0.61 0.32
Stocks
US 0.67 -0.22 0.26
Europe 0.28 -0.26 0.15
Asia 0.23 -0.11 0.06
Emerging Markets 0.24 0.11 0.20
Value-Growth 0.11 0.16 0.05
Small-Large 0.10 0.09 0.01
Bonds
US Bonds -0.04 0.30 0.62
Term Structure 0.15 -0.05 0.30
Credit -0.07 0.01 0.01
High Yield 0.11 0.01 -0.03
Other
Inflation -0.1 -0.41 -0.13
US Dollar -0.18 -0.15 -0.06
Volatility -0.01 0.00 -0.01
Commodities 0.01 0.04 0.14
Intercept 0.00 0.00 0.00
R2 0.68 0.87 0.75

Multi-Factor vs. Single-Factor
■ Because multi co-linearity can distort the significance of individual factors in a
multi-factor regression, we can examine a single factor regression.
■ The multi-factor regression provides the best decomposition of risk when
considering the set of factors
■ The single-factor regression provides the best descriptor of the exposure of
manager to a specific risk factor in isolation.

Single-Factor Regression: Factor Loadings
Stocks
US 0.12 0.16 0.12
Europe -0.17 -0.10 -0.27
Asia 0.06 -0.03 0.06
Value-Growth -0.12 -0.10 -0.09
Small-Large 0.32 0.33 0.18
Bonds
US Bonds 0.10 0.08 1.33
Term Structure -0.16 -0.23 0.30
Credit 0.65 0.78 0.34
High Yield 0.45 0.48 -0.15
Other
Inflation 0.34 0.41 -0.11
US Dollar -0.63 -0.65 -0.58
Volatility -0.10 -0.11 -0.07

Stepwise Multi-Factor Model
■ A method of choosing factors of a particular dependent variable (manager) on
the basis of statistical criteria.
■ The statistical procedure decides which factors is the best predictor, the second
best predictor, etc.
■ The procedure chooses a small subset of predictors from the larger set to
simplify the regression model while preserving good predictive ability.
■ Good choice for modeling all factors simultaneously.

Stepwise Multi-Factor Regression: Factor Loadings
Stocks
US Stocks 0.16 0.13 0.07
Europe -0.16 -0.11 -0.01
Asia 0.05 -0.02 -0.01
Value-Growth 0.12 0.15 0.05
Small-Large 0.09 0.09 0.01
Bonds
US Bonds 0.19 0.20 0.78
Term Structure 0.09 -0.05 0.27
Credit -0.10 0.03 0.02
High Yield -0.03 0.02 -0.03
Other
Inflation -0.24 -0.35 -0.25
US Dollar -0.22 -0.18 -0.04
Volatility -0.01 -0.01 -0.01
Intercept 0.00 0.00 0.00
R2 0.66 0.86 0.73

Risk Decomposition by Factors
■ Analysis so far describes only beta exposures.
■ Does not reveal if we should be concerned with a particular factor exposure.
►For example, we could have high beta to a low risk factor and low beta to high
risk factor
■ Betas tell us direction, but it does not reveal risk.
■ We can calculate asset class risk decomposition using
►Weighted averages of the manager’s multi-factor loadings
►Factors’ standard deviations
►Factors’ correlation matrix

Standard Deviation of Factors
Factor Standard Deviation
Global Equity 16.30%
Stocks
US 4.44%
Europe 5.70%
Asia 9.70%
Emerging Markets 11.88%
Value-Growth 12.83%
Small-Large 10.64%
Bonds
US Bonds 3.70%
Term Structure 10.65%
Credit 8.93%
High Yield 8.02%
Other
Inflation 1.04%
US Dollar 5.86%
Volatility 86.75%
Commodities 24.33%

Risk Decomposition by Factors
Manager A Manager B Manager C
Global Equity
54.55%
Global Equity
72.91%
Global Equity
26.79%
Residual
31.78%
Residual
13.32%
Residual
25.39%
Emerging Markets
8.38%
Emerging Markets
3.55%
Commodities
15.69%
Dollar
2.84%
Commodities
3.46%
US Bonds
9.93%
Credit
-2.71%
Dollar
2.71%
Term Structure
9.13%
Size
2.43%
Size
2.66%
Emerging Markets
7.90%

Managers
■ GMO Global Asset Allocation
►World Allocation, benchmarked against (65% MSCI ACWI and 35% Barclays
U.S. Aggregate Index).
■ BlackRock Global Allocation
►World Allocation (at least 40% of assets in non-U.S. stocks or bonds)
■ Salient Risk Parity Index
►Short track-records of existing managers, index as proxy
►Equal risk exposure to four major asset classes (equities, commodities, rates,
and credit).

Questions and Comments
info@windhamlabs.com
Next Webinar: Windham Software Overview
Asset Allocation and Risk Management
TOMORROW, November 15th, 2PM EST
www.windhamlabs.com/webinars

Appendix: R2 and Statistical Significance
■ The R2 value indicates how well variance is explained by the set of factors
►A value of 100% indicates the variance is entirely explained by the multi-factor
model.
■ Statistical significance of the loadings are measured using the t-statistic
►A t-statistic value above 2 of below -2 suggests that a factor is statistically
significant.
►Statistically significant results imply that the results have an acceptable amount
of error.

Appendix: Factor Loadings Notes
■ Regression coefficients are also known as factor loadings (or betas).
■ Three types of regressions to estimate sensitivity of managers to each factor.
►Multi-factor regression
 regress each manager’s returns against the entire set of fourteen factors.
 describes manager factor exposures but does not help with cross-sectional comparisons to other
managers.
►Stepwise multi-factor regression
 mitigates correlated factors which dampens key factors loadings from the multi-factor regression.
►Single-factor regression
 regress each manager’s returns against each individual factor.
 allows comparison of a factor exposure across managers

Appendix: Stress-Testing
■ How do factor exposures change under times of duress for these managers?
■ What are these managers most vulnerable to?
■ Consider partitioning data into normal market conditions and those associated
with market turbulence.
►Examine each vector of multivariate distance
𝑑 = 𝑅 − 𝜇 Σ−1
𝑅 − 𝜇 ′
𝑰 𝑁
►If exceeds a chi-square score threshold, then consider turbulent.
►Reevaluate factor loadings and variance decomposition.

Appendix: Further Reading
■ Chow, G., E. Jacquier, M. Kritzman, and K. Lowry, “Optimal Portfolios in Good
Times and Bad,” Financial Analysts Journal, May / June 1999.
■ Fabozzi, F., et al, Financial Modeling of the Equity Market: From CAPM to
Cointegration, Wiley Finance, 2006.
■ Kritzman, M., The Portable Financial Analyst, Wiley Finance, 2003.
■ Litterman, B., et al, Modern Investment Management, Wiley Finance, 2003.

Factor Analysis: Understanding Portfolio Risk

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