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FINS5513 Multi-Factor Models
创建日期：2024-06-06 16:21:48
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Multi-Factor Models

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Multi-Factor Models
2❑ 5.1 Empirical Testing of Single Index Models
➢ Methodology For Testing the SML Relationship
➢ Overview of Single Factor Model Empirical Tests
❑ 5.2 Multi-Factor Models
➢ Introduction to Multi-Factor Models
➢ Arbitrage Pricing Theory (APT)
➢ APT and CAPM Compared
❑ 5.3 Application of Multi-Factor Models
➢ Factor Portfolios
➢ Fama-French 3 Factor Model
➢ Carhart 4 Factor Model
➢ Smart beta
Lecture Outline
5.1 Empirical Testing of
Single Index Models
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4Methodology For Testing the
SML Relationship
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5❑ As we know, the CAPM is untestable because there is no observable “true” market portfolio
➢ The CAPM is an equilibrium model applying to all assets in all markets
❑ However, one type of empirical test which is often conducted is whether the predicted linear
relationship between asset excess return and asset beta (i.e., the “SML relationship”) holds in
a single market – using a single index as a proxy for the market (a single factor model)
➢ For example, for the U.S. stock market, we could test whether the excess return on stocks
in the S&P500 increase linearly with their beta x the market risk premium. In other words,
we could test whether:
= &500 , , … … ℎ &500
❑ Many empirical tests have used a broad index as a proxy for to test the SML relationship
Testing the SML Relationship
6There is a general method for testing the SML for a single market, using a single index:
❑ First Pass Regression (Step 1 – Time-Series)
➢ Derive n Security Characteristic Lines (SCLs) over a reasonable sample period for n
stocks in the index
➢ Estimate for each stock in the sample:
a) Beta ,…. – the slope of the SCL over the sample period; and
b) Average excess return ,….
❑ Second Pass Regression (Step 2 – Cross-Sectional)
➢ Derive ONE Security Market Line (SML) using the estimates from the first pass regression
➢ The second pass regression equation is tested with the beta estimates ,…. from the first
pass regression as the independent variable
➢ Test whether the sample of stocks follow the predicted SML relationship – (that is, on
average, stock excess return changes linearly with beta × market risk premium)
Testing the SML Relationship
7❑ What is the hypothesis we are testing in the second pass regression?
❑ Say we were testing all 500 stocks in the S&P500 over 5 years using monthly data (60
observations). The specific second pass regression equation we are testing is:
− = 0 + 1 = 1,2, … … 500 &500
− = sample average (over 60 months) of the excess return of each S&P500 stock
= sample estimate of the beta coefficients of each of the S&P500 stocks
❑ If the SML relationship holds, then 0 and 1 should satisfy:
0 = 0 and 1 = − (1 ≠ 0)
− = sample average of the excess return on the S&P500 index
➢ The second pass regression tests whether the SML relationship holds. If the intercept 0
is statistically equal to zero (i.e., no alpha) and the slope 1 is statistically equal to the
market excess return, the relationship is consistent with the SML
Second Pass Regression Tests
8❑ Open the spreadsheet “First and Second Pass Regression”.
a) Run first pass regressions on the 9 stock returns contained in the “Excess Returns”
spreadsheet.
b) Using the results from your first pass regression, and using the Excel regression tools, run
a second pass regression to test whether the 9-stock sample conforms to the predicted
Security Market Line relationship. Interpret your results.
❑ First Pass Regression
➢ For each of the 9 sample stocks, run a regression of the stock’s excess returns against the
market excess returns
➢ Derive a regression estimate of for each stock using Excel’s “SLOPE” function – see
“First Pass” worksheet e.g., for BRK Slope(Returns!C9:C128,B9:B128)=0.6242
➢ Calculate the average excess return for each stock using Excel’s “AVERAGE” function –
see “First Pass” worksheet e.g., for BRK: Average(Returns!C9:C128)=1.04% per month
(12.46% pa)
Example: First and Second Pass Regression
Excel: “First and Second Pass Regression”
9Example: First and Second Pass Regression
❑ Second Pass Regression
➢ At the top of the “Second Pass” worksheet (A16:C24), we have a sample of 9 average
excess returns and 9 beta estimates (9 individual SCLs) derived from the first pass
➢ Using Excel regression tools, run ONE second pass regression:
• Regress 9 average excess returns (dependent (y) variable) against the 9 beta estimates
(independent (x) variable)
• The test is to determine if the data conforms to the SML (i.e., 0 = 0 and 1 = − )
❑ Remember: if we have n stocks in our sample, we run n time series (1st pass) regressions (n
SCLs) to derive n betas (1,2…n ) and n average excess returns (R1,2,…n). Then we run ONE (2
nd
pass) regression to determine if the cross section of stocks conforms to the SML
10
❑ Results do appear to be broadly consistent with
the predicted SML relationship (linear relationship
between stock excess return and stock beta)
➢ y-intercept is not significantly different to zero
➢ The slope of the SML is positive, and relatively
consistent with the market excess return
➢ The general direction of the sample is higher
beta/higher excess return
➢ R-Squared is high – 85% of the variability in a
stock’s excess return is explained by its beta
❑ Of course, we cannot draw any definitive
conclusions from such a small sample (n = 9)
Example: Testing the SML Relationship
11
Overview of Single Factor
Model Empirical Tests
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12
❑ Early results by Lintner (1965) and Miller and Scholes (1972) for NYSE stocks were
inconsistent with the predicted SML relationship
➢ The SML was “too flat”
• In other words, the slope of the SML significantly underestimated the market risk premium
and its intercept was significantly higher than zero
❑ However, a number of issues were identified with these early empirical tests:
➢ The market index used was not the true market portfolio (Roll’s critique)
➢ The daily volatility of stocks significantly reduced the precision in using average return
estimates
• The betas derived from the first pass regression would contain substantial sampling error
and were not reliable inputs to the second pass regressions
➢ Borrowers cannot borrow at the risk-free rate which impacts on the results
Early Tests of the SML Relationship
13
❑ It is a well-known problem in statistics that if the independent variable is measured with
error, then the slope coefficient of the regression equation will be underestimated and the
intercept will be overestimated (the regression line will be too flat)
➢ The independent variable in the second pass regression is the beta
❑ The next wave of tests were designed to overcome this beta measurement error
➢ Black, Jensen and Scholes (1972) and Fama and MacBeth (1973) used diversified
portfolios with minimal unsystematic risk (rather than individual stocks) to increase the
precision of the beta estimates
➢ These tests ranked the portfolios by beta and used a wide dispersion of portfolio betas
➢ This method increased the precision of the SML tests (under APT)
❑ When tested with a value weighted index these results also still seem to produce an SML
which is too flat
Measurement Error in Beta
14
More Recent Tests
❑ More recent tests still seem to indicate the SML is too flat compared to CAPM theory
❑ A graphical representation of Fama-French (2004) using 1928-2003 U.S. data is shown:
❑ Implication: Low beta stocks do better than the CAPM SML predicts, and high beta stocks do
worse than the CAPM SML predicts
Predicted CAPM
SML (based on
actual and ) The actual SML which fits the
data is flatter (higher
intercept/lower slope)
Portfolios Betas
5.2 Multi-Factor Models
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16
Introduction to Multi-Factor
Models
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17
❑ The CAPM / SIM expresses asset expected return as a function of its covariance with
➢ We sometimes refer to the return on as a risk factor, and as a “loading” on that factor
➢ CAPM / SIM are single-factor models. Only one common factor is used – the return on
❑ A factor is a variable or a characteristic which impacts on an asset’s expected return (asset
expected returns are correlated with the factor)
❑ In principle, we could imagine other risk factors with which assets co-vary and add them to
our expected return equation
➢ Multi-factor models use at least two factors
➢ Estimate a beta or factor loading for each factor using regression
❑ The theory behind multi-factor models differs from CAPM, but interpretation of results is similar
❑ Multi-factor models have become one of the most popular tools used in the investment industry
today
What is a Factor?
18
❑ Macroeconomic factors
➢ Unexpected “surprises” in macroeconomic variables that impact on asset returns
➢ Examples include: GDP growth, industrial production, interest rates, inflation, investor
confidence, slope of the yield curve, business cycle (corporate default spreads)
❑ Fundamental factors
➢ Attributes of the assets themselves that impact on asset returns
➢ Examples include: book-to-price ratios, market capitalisation, P/E ratios, profitability,
financial leverage and price trends (momentum)
❑ Statistical factors
➢ Statistical methods are applied to extract factors that can explain the observed returns
➢ Interpretation of statistical factors is generally difficult, and associating a statistical factor
with economic meaning may not be possible
➢ Relies on advanced quantitative methods (e.g., data mining) which are outside our scope
Factor Types
19
❑ In comparison to single-factor models, multi-factor models offer increased explanatory power
➢ Most empirical factor models are not derived formally with clearly specified assumptions
➢ The theoretical foundation for such models is Arbitrage Pricing Theory (APT)
❑ A general macroeconomic factor model expresses the excess return on an asset as follows:
= + 1,1 + 2,2 … , +
= expected excess return of asset
= unexpected (surprise) element of the th systematic factor
, = loading of asset on the th factor
❑ For fundamental factor models, each factor is often expressed as a risk premium:
= + 1,λ1 + ⋯ ,λ +
= the element of asset ’s return not explained by the risk factors
λ = risk premium on the th factor
Multi-Factor Models
20
Arbitrage Pricing Theory
(APT)
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21
❑ 3 key Arbitrage Pricing Theory (APT) assumptions:
➢ Investors can arbitrage in unlimited quantities
➢ Investors form well-diversified portfolios which eliminate unsystematic risk
➢ Factor models describe asset returns
❑ Pure arbitrage occurs when there is a:
I. zero-investment portfolio (no net capital outlay)
II. with a certain (riskless) profit
➢ Since no capital is required, investors will want infinite positions in risk-free arbitrage
opportunities - regardless of wealth or risk aversion
➢ In efficient markets, profitable pure arbitrage opportunities should disappear quickly
❑ Law of one price: Two securities with the same payoff must have the same price
➢ The existence of arbitrage keeps assets in a price equilibrium
➢ If equal-payoff assets have different prices – buy the cheaper one/short the expensive one
Arbitrage
22
❑ Recall from 3.2 that unsystematic risk is eliminated for well diversified portfolios
➢ By grouping stocks into a diversified portfolio, firm-specific movements offset → reducing
unsystematic risk - the observations will hug the SCL (high R2)
➢ Risk is mostly systematic (arising from market movements)
❑ The same principle applies to multi-factor models
➢ Consider a multi-factor model for portfolio :
= + 1,1 + 2,2 + ⋯ + , +
➢ For a well-diversified portfolio, → 0 as the number of portfolio securities increases and
their weights decrease. Therefore, firm-specific risk is diversified away
❑ This assumption of the APT allows that investors form portfolios with factor risk but without
unsystematic risk
➢ Empirical evidence (e.g., Roll and Ross (2001)) suggests that only 1-3% of a diversified
portfolio’s variance comes from unsystematic variance
Well Diversified Portfolios

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