More Related Content Similar to Factor Analysis: Understanding Portfolio Risk (20) More from Windham Labs (12) Factor Analysis: Understanding Portfolio Risk1. 1© 2018 Windham Capital Management, LLC. All rights reserved.
November 2018
1
Factor Analysis
Cel Kulasekaran
Research Director
Understanding Portfolio Risk
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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
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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.
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The Single Factor Model
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We Studied In School
𝑦 = 𝑚𝑥 + 𝑐
Solve Solve
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Linear Regression Example
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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
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The Multi-Factor Model
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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.
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Multi-Factor Regression Example
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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
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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
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Factor Models in Practice
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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
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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).
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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.
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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
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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
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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
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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.
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Single-Factor Regression: Factor Loadings
Factor Manager A Manager B Manager C
Global Equity 0.59 0.64 0.38
Stocks
US 0.12 0.16 0.12
Europe -0.17 -0.10 -0.27
Asia 0.06 -0.03 0.06
Emerging Markets 0.37 0.30 0.31
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
Commodities 0.16 0.19 0.21
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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.
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Stepwise Multi-Factor Regression: Factor Loadings
Factor Manager A Manager B Manager C
Global Equity 0.58 0.62 0.36
Stocks
US Stocks 0.16 0.13 0.07
Europe -0.16 -0.11 -0.01
Asia 0.05 -0.02 -0.01
Emerging Markets 0.23 0.12 0.17
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
Commodities 0.02 0.04 0.14
Intercept 0.00 0.00 0.00
R2 0.66 0.86 0.73
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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
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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%
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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%
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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).
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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
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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.
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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
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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.
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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.