factor analysis

1© 2014 Windham Capital Management, LLC. All rights reserved.

Confidential

May 2014 1

Factor Analysis

Cel KulasekaranVice President, Research and Development

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: Global Asset Allocation Managers

■ 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

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

� =Covariance � , ��

Variance ��

■ The CAPM expected return is

E � = �� + � ∙ �� − ��


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 � = � + �� + �� +⋯+ �� + �

■ 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 � = �� + �� ∙ �� − �� + �� ∙ � ! + �" ∙ # $ + �

� 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

Global Asset Allocation Funds


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).


Proposed Factors

Factor Proxy Time Series

Global Equity MSCI All Country World

Sto

cks

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

Oth

er

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 GMO BlackRock Risk Parity

Global Equity 0.56 0.61 0.32

Sto

cks

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

Oth

er

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 multicollinearity 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



Sto

cks

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

Oth

er

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.

■ Good choice for modeling all factors simultaneously.


Stepwise Multi-Factor Regression: Factor Loadings



Sto

cks

US Stocks 0.16 0.13 0.07

Europe -0.16 -0.11 -0.01

Asia 0.05 -0.02 -0.01



Small-Large 0.09 0.09 0.01

Bonds

US Bonds 0.19 0.20 0.78


Credit -0.10 0.03 0.02

High Yield -0.03 0.02 -0.03

Oth

er

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%

Sto

cks

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%

Oth

er

Inflation 1.04%

US Dollar 5.86%

Volatility 86.75%

Commodities 24.33%


Risk Decomposition by Factors

GMO BlackRock Risk Parity

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%


May/June Market Sell-Off

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

January February March April May June July August September October

$ m

illi

on

s

Stocks

Bonds

2013


May/June Market Sell-Off

2013

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

January February March April May June July August September October

$ millions

Stocks

Bonds

Fed anticipates ending

QE3 by mid 2013

10-year yields begin increasing


Pre Sell-off: Stepwise Multi-Factor Factor Loadings



Sto

cks

US 0.16 0.13 0.09

Europe -0.16 -0.11 -0.04

Asia 0.05 -0.03 -0.01



Small-Large 0.09 0.09 0.01

Bonds

US Bonds 0.18 0.19 0.73


Credit -0.11 0.03 0.00

High Yield -0.02 0.02 -0.01

Oth

er

Inflation -0.24 -0.38 -0.24

US Dollar -0.22 -0.19 -0.06

Volatility -0.01 -0.01 -0.01


Intercept 0.00 0.00 0.00

R2 0.65 0.86 0.73


Factor Returns During the Sell-Off

Factor May 2013 June 2013 Cumulative

Global Equity 1.44% -2.69% -1.29%

Sto

cks

US Stocks 0.66% 1.36% 2.03%

Europe 1.25% -2.32% -1.10%

Asia -2.32% 0.26% -2.06%

Emerging Markets -0.60% -2.36% -2.94%

Value-Growth -0.13% 1.15% 1.02%

Small-Large 1.62% 0.78% 2.41%

Bonds

US Bonds -1.78% -1.55% -3.30%

Term Structure -6.24% -3.18% -9.22%

Credit 1.12% -2.34% -1.25%

High Yield 4.46% 2.97% 7.56%

Oth

er

Inflation 0.18% 0.32% 0.50%

US Dollar 0.97% -0.95% 0.01%

Volatility 20.56% 3.44% 24.70%

Commodities -1.49% 0.23% -1.26%


What Happened

■ Using information prior to May 2013, we can project portfolio performance.

■ Managers were exposed to duration risk.

May / June 2013 GMO BlackRock Risk Parity

Projected Return (Factors) -2.21% -1.88% -6.03%

Realized Return -2.57% -2.11% -10.55%


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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.


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

% = � − & Σ)� � − & * +,

►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.

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