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FINS5513 Lecture 5
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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❑ 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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Introduction to Multi-Factor
Models
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❑ 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?
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❑ 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
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❑ 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
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Arbitrage Pricing Theory
(APT)
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❑ 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
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❑ 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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❑ From BKM Figure 10.3 – Security Characteristic Lines for 2 well diversified portfolios A and B
❑ A and B have the same and therefore the same systematic risk
❑ We can create a portfolio that eliminates systematic risk by buying A and shorting B. We
have locked in a 2% spread () regardless of how the systematic factor moves
Arbitrage Opportunity Example 1
Security
Characteristic
Line
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❑ From BKM Figure 10.4 – now consider arbitrage based on the Security Market Line (SML)
➢ C is plot below the SML (i.e., < 0) - so C is overpriced
➢ Arbitrage opportunity: Short C and Long D – lock in 1% spread (i.e., 7% − 6% = 1%)
❑ What if D is not directly available in the market? We can construct D from A and F:
➢ 50% in A + 50% in F creates a portfolio with = 7% = and = 0.5 = =
➢ Overall arbitrage strategy: Long 50% in A and 50% in F and Short 100% in C
Arbitrage Opportunity Example 2
Security
Market Line
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❑ Widely used hedge fund strategy
❑ Step #1: Construct an Active portfolio :
➢ Hold long positions in stocks with positive alphas (under-priced)
➢ Hold short positions in stocks with negative alphas (over-priced)
➢ This is a concentrated portfolio attempting to maximise alpha ≠
❑ Step #2: Construct a Tracking portfolio :
➢ Alpha of Tracking portfolio is zero: =
➢ Beta of Tracking portfolio equals the beta of the Active portfolio =
❑ If > buy and sell
If < sell and buy
➢ This will hedge out the systematic risk in the Active portfolio
➢ Overall combined portfolio is market neutral – earning “Absolute Returns” that do not
move with the market
Alpha Betting and Tracking Portfolio
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❑ You have constructed a well-diversified Active portfolio that exhibits the following security
characteristic line on excess returns:
= 0.04 + 1.4[ − ] = return on the S&P 500
Describe the Tracking portfolio you would construct, and the trade you would execute.
➢ #1: = 4% = 1.4
> 0 so we buy and sell
➢ #2: Construct T: − × % in S&P500 / −( − ) × % in T-Bills
−140% in the S&P 500 / +40% in T-bills
➢ = = 1.4 and = 0%
− = = 4%
➢ For example: if we were to arbitrage $100m: Short $140m in S&P500, invest $40m in T-bills
(i.e., sell ) , invest $100m in (i.e., buy )– locks in capital-free $4m return ()
Alpha Betting Example 1
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❑ You have constructed a well-diversified Active portfolio that exhibits the following security
characteristic line on excess returns:
= −0.06 + 0.9[ − ]
Describe the Tracking portfolio you would construct, and the trade you would execute.
➢ #1: = -6% = 0.9
< 0 so we sell and buy
➢ #2: Construct T: + × % in S&P500 / +( − ) × % in T-Bills
+90% in the S&P 500 / +10% in T-bills
➢ = = 0.9 and = 0%
− = −= − −6% = +6%
➢ For example: if we were to arbitrage $100m: Short $100m in (i.e., sell ), use proceeds to
buy $90m in S&P500 and $10m in T-bills (i.e., buy ) – locks in capital-free $6m return
Alpha Betting Example 2
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APT and CAPM Compared
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❑ So, what did those arbitrage examples prove?
❑ The law of one price requires that assets with the same systematic risk should have the
same expected return
➢ Therefore, alpha should be arbitraged away in efficient markets
➢ Arbitrage will keep assets fairly priced based on their level of systematic risk
➢ Assets with similar systematic risk (measured by ) will have the same expected return
❑ This is essentially the same conclusion as the CAPM
➢ If the market return was the only systematic factor, then the APT and CAPM equation would
basically be the same:
=
➢ Applies to well diversified portfolios and implies = 0
➢ The portfolio would be expected to move with the systematic factor (market return)
according to its
Arbitrage Eliminates Alpha
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❑ APT gets to an − linear relationship without the CAPM’s limiting assumptions
➢ Uses a market index rather than an unobservable market portfolio containing all risky assets
➢ The actions of a few arbitrageurs restore equilibrium (not all investors need to be rational)
➢ APT applies to well diversified portfolios and not necessarily to individual stocks
APT and CAPM Compared
APT CAPM
❑ Uses an observable market index and is
therefore testable
❑ Based on an unobservable “market” portfolio
including all risky assets, and therefore untestable
❑ Does not assume all investors are mean-
variance optimisers (and therefore does not
require derivation of a common efficient frontier)
➢ APT equilibrium is restored even if only a few
sophisticated arbitrageurs (with large capital
outlays) recognise an arbitrage opportunity
❑ All investors have the same information,
homogeneous expectations and are mean-
variance optimisers (resulting in a common
efficient frontier, CAL, ∗ etc)
➢ CAPM is restored by the actions of all investors
❑ Assumes a well-diversified portfolio, but
unsystematic risk may remain
❑ CAPM describes equilibrium for all assets
5.3 Application of
Multi-Factor Models
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Factor Portfolios
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❑ Multi-factor models under APT:
➢ Assumes there is more than a single source of systematic risk
➢ However, APT does not define what the risk factors are
➢ Empirical research using multi-factor models attempts to identify these risk factors
➢ The loading to each factor (i.e., beta) depends on its covariance with the factor
❑ We can input the variables directly into the model or we can construct factor portfolios
❑ What is a factor portfolio?
➢ A factor portfolio is a well-diversified portfolio of stocks which serve as a proxy for a
particular source of systematic risk
➢ A factor portfolio has = 1 for one of the factors and = 0 for all other factors
➢ It tracks a particular source of systematic risk but is uncorrelated with other sources of risk
➢ Regression can be run against factor portfolio risk premiums rather than the factor directly
Multi-Factor APT
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❑ The multi-factor APT predicts that each source of systematic risk (i.e., each factor) has a risk
premium
❑ Portfolio A’s total risk premium is therefore equal to the sum of its factor beta to each
particular risk factor (unique to Portfolio A) times the risk premium of the factor portfolio
tracking that risk factor (common to all assets)
❑ APT Fund has determined that three factors impact Portfolio A’s return. These factors are
Growth, Leverage and Liquidity. It forms 3 factor portfolios with = 1 against the Growth,
Leverage and the Liquidity factors respectively (and = 0 against all other factors in the
model). The risk premiums are λ = 3%, λ = 2% and λ = 5%. Portfolio A’s betas (or
factor loadings) against each of these factors are: = 0.7, =1.2 and = 0.3. The
risk-free rate is = 3.0%. Calculate Portfolio A’s expected return.
➢ = + λ + λ + λ
= 0.03 + 0.7 × 0.03 + 1.2 × 0.02 + 0.3 × 0.05
= 0.09 = 9.0%
Factor Portfolios Example
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Fama-French 3 Factor Model
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❑ Widely known model that uses factor portfolios
➢ Factors are based on firm/stock characteristics (fundamental factors)
➢ Each factor portfolio is self-financing, i.e., it takes long (buy) and short (sell) positions that
offset each other so that the net portfolio cost is zero
➢ Factor portfolio returns are stated as risk premiums (excess returns)
❑ Size and value are priced risk factors impacting returns, in addition to the market risk premium
➢ These characteristics (size, value) are assumed to be correlated with systematic risk factors
➢ SMB = Small Minus Big (firm size)
• The risk premium of small cap stocks over large cap stocks
➢ HML = High Minus Low (book-to-market ratio)
• The risk premium of “value” stocks over “growth” stocks
❑ Fama-French three-factor model:
= + + + +
Fama-French 3 Factor Model
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❑ SMB = Small-Minus-Big = risk premium of small cap stocks over large cap stocks
❑ Fama-French quantifies the size risk premium:
➢ The SMB portfolio consists of a long position in the market’s smallest 50% of firms ranked
by market cap (small) and a short position in the market’s largest 50% of firms ranked by
market cap (big)
❑ Smaller firms experience higher returns:
➢ Probably due to a risk premium for small firms
➢ Size effect may be related to liquidity (small caps have less liquidity)
Fama-French Factors: Size
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❑ HML = High-Minus-Low = risk premium of “value” stocks over “growth” stocks
❑ Fama-French quantifies the value risk premium:
➢ Rank each stock based on the ratio: Book Value of equity / Market Value of equity (B/M)
➢ Stocks with higher B/M ratios are considered “value” stocks
➢ Stocks with lower B/M ratios are considered “growth” stocks
➢ The HML portfolio consists of a long position in market stocks with the highest 30% ranked
B/M ratio (value stocks) and a short position in market stocks with the lowest 30% ranked
B/M ratio (growth stocks)
❑ Why the risk premium for high B/M stocks (value stocks)?
➢ High B/M stocks may have discounted share prices or are closer to financial distress leading
to higher returns
Fama-French Factors: Value
39
❑ From BKM Figure 13.1
❑ Fama-French appears to fit the data better than CAPM
Empirical Results of Fama-French
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Carhart 4 Factor Model
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❑ Carhart expanded the Fama-French model with a fourth factor, Momentum
➢ WML = Winners Minus Losers
• The risk premium of uptrend stocks over downtrend stocks
❑ Carhart quantifies the momentum risk premium:
➢ The WML portfolio consists of a long position in market stocks with the highest 30% ranked
returns in the preceding year (winners) and a short position in market stocks with the lowest
30% ranked returns in the preceding year (losers)
❑ The Fama-French-Carhart four-factor model equation to derive an asset’s risk premium is
given by:
= + + + + +
Carhart Additional Factor: Momentum
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❑ Refer to the data contained in the spreadsheet “Multi-factor Regression”, Worksheet: Data
Sheet:
a) Using the returns contained in the “Data Sheet” worksheet for the period 1976-2017, run
the simple SIM model (i.e., derive the SCL), the Fama-French 3-Factor model, and the
Carhart 4-factor model.
➢ SIM Model: From Menu Choose Data>>>Data Analysis>>>Regression (OK)
Input Y Range: $G$14:$G$499 (Stock Excess Returns)
Input X Range: $H$14:$H$499 (Market Excess Returns)
➢ Fama-French Model: From Menu Choose Data>>>Data Analysis>>>Regression (OK)
Input Y Range: $G$14:$G$499 (Stock Excess Returns)
Input X Range: $H$14:$J$499 (Market, SMB and HML Excess Returns)
➢ Carhart Model: From Menu Choose Data>>>Data Analysis>>>Regression (OK)
Input Y Range: $G$14:$G$499 (Stock Excess Returns)
Input X Range: $H$14:$K$499 (Market, SMB, HML and WML Excess Returns)
Carhart 4 Factor Model Example
Excel: “Multi-factor Regression”
43
b) Analyse the coefficients derived from the Carhart 4-factor model. What can we conclude
from the statistical properties of these coefficients?
➢ See worksheet “Carhart Model” for the estimated Carhart model alphas and betas for BRK
➢ T=486 and K=5; (T-K)=481 degrees of freedom
➢ The adjusted 2 is relatively low at 0.257
➢ = 10.8% (0.9% per month) with -stat = 3.16, compared to the single-factor model
alpha of 13.4% - the 3 additional factors explain some of Buffet’s alpha, but it’s still significant
➢ = 0.842 with -stat = 12.64 (significant at 99%), BRK moves with the market
➢ = −0.275 with –stat = −2.87 (significant at 95%), BRK buys large stocks, so
performance is lower when small stocks outperform
➢ = 0.500 with -stat = 4.83 (significant at 99%), BRK buys value stocks (measured by
high B/M), so performance is higher when value stocks outperform
➢ = 0.06 with -stat = 0.95 (insignificant), BRK does not chase momentum. Momentum
explains little variation in BRK’s performance
Carhart 4 Factor Model Example
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Smart Beta
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❑ “Smart beta” investment strategies arose as a result of the identification of alternative factors
which may impact on security returns
❑ Smart beta attempts to capture defined factors in a rules based and transparent way by
constructing an alternative index to the traditional value-weighted indices
➢ A Smart Beta ETF will then “track” this alternative index
➢ Smart beta strategies are therefore a mix of passive and active investing
• They are passive in the sense that a smart beta investor follows an index, but also active
in selecting the stock characteristics that comprise the index
❑ Smart Beta ETFs might choose companies that only exhibit certain behaviours or metrics
such as: earnings growth, value, momentum, low (market) beta or volatility, quality (for
example measured by return on equity) etc
❑ Smart beta ETFs allows investors to tailor portfolio exposure either away from, or toward,
additional systematic factors
Smart Beta
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❑ We have seen BRK’s alpha was 13.4% from 1976 to 2017
❑ The study “Buffett’s Alpha” (Frazzini and Kabiller, 2018) attempts to explain this alpha in
terms of additional factors which correspond to Buffett’s investing style
➢ Based on the 4-factor model, we have already seen that Buffett’s style is large-cap value
and isn’t concerned with momentum
➢ The authors add two further factors which attempt to explain Buffett’s alpha – investing in
low beta stocks and focusing on quality stocks
❑ Based on the results of “Buffet’s Alpha” Table 4 which are summarised below, derive the risk
premium on the BAB and QMJ factors
Using Multi-factor Models to Explain Alpha
Factors BRK.A Market SMB HML MOM BAB QMJ
Risk Premiums 18.6% 7.5% 2.7% 3.5% 7.6% ??? ???
5 Factor Model ( and ) = 8.5% = 0.83 = -0.30 = 0.31 = -0.02 = 0.33 ---
6 Factor Model ( and ) = 5.4% = 0.95 = -0.13 = 0.40 = -0.05 = 0.27 = 0.47
47
➢ Hint: Calculate the BAB Risk Premium first, based on the information in the 5-factor model,
then use this risk premium to derive the QMJ Risk Premium in the 6 Factor model.
➢ We can use the betas and risk premiums from the 4-factor model to calculate the BAB risk
premium:
(18.6% - 8.5% - 0.83 x 7.5% + 0.30 x 2.7% - 0.31 x 3.5% + 0.02 x 7.6%) / .33
= 11.3%
➢ Similarly, we can use the betas and risk premiums from the 5-factor model to calculate the
QMJ risk premium:
(18.6% - 5.4% - 0.95 x 7.5% + 0.13 x 2.7% - 0.40 x 3.5% + .05 x 7.6% - 0.27 x 11.3%) / 0.47
= 5.0%
➢ Addition of BAB and QMJ factors reduces alpha to 5.4% (from ~11% in the 4-factor model)
Using Multi-factor Models to Explain Alpha
RBRK BRK RM SMB HML MOM BAB
RBRK BRK RM SMB HML MOM QMJBAB
Mini Case Study: “Can Buffett’s Alpha Be Explained?”
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❑ No Lecture Week 6
❑ Mid-term Test this Saturday (end of week 5)
❑ iLab: Week 6 Tuesday and Thursday
❑ Week 7: BKM Chapter 11 and 12
❑ 7.1 The Efficient Market Hypothesis
❑ 7.2 Empirical Testing on Market Efficiency
❑ 7.3 Behavioural Finance
❑ 7.4 Technical Trading
Next Lecture
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
FINS5513
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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❑ 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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Introduction to Multi-Factor
Models
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❑ 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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❑ 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
23
❑ From BKM Figure 10.3 – Security Characteristic Lines for 2 well diversified portfolios A and B
❑ A and B have the same and therefore the same systematic risk
❑ We can create a portfolio that eliminates systematic risk by buying A and shorting B. We
have locked in a 2% spread () regardless of how the systematic factor moves
Arbitrage Opportunity Example 1
Security
Characteristic
Line
24
❑ From BKM Figure 10.4 – now consider arbitrage based on the Security Market Line (SML)
➢ C is plot below the SML (i.e., < 0) - so C is overpriced
➢ Arbitrage opportunity: Short C and Long D – lock in 1% spread (i.e., 7% − 6% = 1%)
❑ What if D is not directly available in the market? We can construct D from A and F:
➢ 50% in A + 50% in F creates a portfolio with = 7% = and = 0.5 = =
➢ Overall arbitrage strategy: Long 50% in A and 50% in F and Short 100% in C
Arbitrage Opportunity Example 2
Security
Market Line
25
❑ Widely used hedge fund strategy
❑ Step #1: Construct an Active portfolio :
➢ Hold long positions in stocks with positive alphas (under-priced)
➢ Hold short positions in stocks with negative alphas (over-priced)
➢ This is a concentrated portfolio attempting to maximise alpha ≠
❑ Step #2: Construct a Tracking portfolio :
➢ Alpha of Tracking portfolio is zero: =
➢ Beta of Tracking portfolio equals the beta of the Active portfolio =
❑ If > buy and sell
If < sell and buy
➢ This will hedge out the systematic risk in the Active portfolio
➢ Overall combined portfolio is market neutral – earning “Absolute Returns” that do not
move with the market
Alpha Betting and Tracking Portfolio
26
❑ You have constructed a well-diversified Active portfolio that exhibits the following security
characteristic line on excess returns:
= 0.04 + 1.4[ − ] = return on the S&P 500
Describe the Tracking portfolio you would construct, and the trade you would execute.
➢ #1: = 4% = 1.4
> 0 so we buy and sell
➢ #2: Construct T: − × % in S&P500 / −( − ) × % in T-Bills
−140% in the S&P 500 / +40% in T-bills
➢ = = 1.4 and = 0%
− = = 4%
➢ For example: if we were to arbitrage $100m: Short $140m in S&P500, invest $40m in T-bills
(i.e., sell ) , invest $100m in (i.e., buy )– locks in capital-free $4m return ()
Alpha Betting Example 1
27
❑ You have constructed a well-diversified Active portfolio that exhibits the following security
characteristic line on excess returns:
= −0.06 + 0.9[ − ]
Describe the Tracking portfolio you would construct, and the trade you would execute.
➢ #1: = -6% = 0.9
< 0 so we sell and buy
➢ #2: Construct T: + × % in S&P500 / +( − ) × % in T-Bills
+90% in the S&P 500 / +10% in T-bills
➢ = = 0.9 and = 0%
− = −= − −6% = +6%
➢ For example: if we were to arbitrage $100m: Short $100m in (i.e., sell ), use proceeds to
buy $90m in S&P500 and $10m in T-bills (i.e., buy ) – locks in capital-free $6m return
Alpha Betting Example 2
28
APT and CAPM Compared
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❑ So, what did those arbitrage examples prove?
❑ The law of one price requires that assets with the same systematic risk should have the
same expected return
➢ Therefore, alpha should be arbitraged away in efficient markets
➢ Arbitrage will keep assets fairly priced based on their level of systematic risk
➢ Assets with similar systematic risk (measured by ) will have the same expected return
❑ This is essentially the same conclusion as the CAPM
➢ If the market return was the only systematic factor, then the APT and CAPM equation would
basically be the same:
=
➢ Applies to well diversified portfolios and implies = 0
➢ The portfolio would be expected to move with the systematic factor (market return)
according to its
Arbitrage Eliminates Alpha
30
❑ APT gets to an − linear relationship without the CAPM’s limiting assumptions
➢ Uses a market index rather than an unobservable market portfolio containing all risky assets
➢ The actions of a few arbitrageurs restore equilibrium (not all investors need to be rational)
➢ APT applies to well diversified portfolios and not necessarily to individual stocks
APT and CAPM Compared
APT CAPM
❑ Uses an observable market index and is
therefore testable
❑ Based on an unobservable “market” portfolio
including all risky assets, and therefore untestable
❑ Does not assume all investors are mean-
variance optimisers (and therefore does not
require derivation of a common efficient frontier)
➢ APT equilibrium is restored even if only a few
sophisticated arbitrageurs (with large capital
outlays) recognise an arbitrage opportunity
❑ All investors have the same information,
homogeneous expectations and are mean-
variance optimisers (resulting in a common
efficient frontier, CAL, ∗ etc)
➢ CAPM is restored by the actions of all investors
❑ Assumes a well-diversified portfolio, but
unsystematic risk may remain
❑ CAPM describes equilibrium for all assets
5.3 Application of
Multi-Factor Models
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Factor Portfolios
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❑ Multi-factor models under APT:
➢ Assumes there is more than a single source of systematic risk
➢ However, APT does not define what the risk factors are
➢ Empirical research using multi-factor models attempts to identify these risk factors
➢ The loading to each factor (i.e., beta) depends on its covariance with the factor
❑ We can input the variables directly into the model or we can construct factor portfolios
❑ What is a factor portfolio?
➢ A factor portfolio is a well-diversified portfolio of stocks which serve as a proxy for a
particular source of systematic risk
➢ A factor portfolio has = 1 for one of the factors and = 0 for all other factors
➢ It tracks a particular source of systematic risk but is uncorrelated with other sources of risk
➢ Regression can be run against factor portfolio risk premiums rather than the factor directly
Multi-Factor APT
34
❑ The multi-factor APT predicts that each source of systematic risk (i.e., each factor) has a risk
premium
❑ Portfolio A’s total risk premium is therefore equal to the sum of its factor beta to each
particular risk factor (unique to Portfolio A) times the risk premium of the factor portfolio
tracking that risk factor (common to all assets)
❑ APT Fund has determined that three factors impact Portfolio A’s return. These factors are
Growth, Leverage and Liquidity. It forms 3 factor portfolios with = 1 against the Growth,
Leverage and the Liquidity factors respectively (and = 0 against all other factors in the
model). The risk premiums are λ = 3%, λ = 2% and λ = 5%. Portfolio A’s betas (or
factor loadings) against each of these factors are: = 0.7, =1.2 and = 0.3. The
risk-free rate is = 3.0%. Calculate Portfolio A’s expected return.
➢ = + λ + λ + λ
= 0.03 + 0.7 × 0.03 + 1.2 × 0.02 + 0.3 × 0.05
= 0.09 = 9.0%
Factor Portfolios Example
35
Fama-French 3 Factor Model
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❑ Widely known model that uses factor portfolios
➢ Factors are based on firm/stock characteristics (fundamental factors)
➢ Each factor portfolio is self-financing, i.e., it takes long (buy) and short (sell) positions that
offset each other so that the net portfolio cost is zero
➢ Factor portfolio returns are stated as risk premiums (excess returns)
❑ Size and value are priced risk factors impacting returns, in addition to the market risk premium
➢ These characteristics (size, value) are assumed to be correlated with systematic risk factors
➢ SMB = Small Minus Big (firm size)
• The risk premium of small cap stocks over large cap stocks
➢ HML = High Minus Low (book-to-market ratio)
• The risk premium of “value” stocks over “growth” stocks
❑ Fama-French three-factor model:
= + + + +
Fama-French 3 Factor Model
37
❑ SMB = Small-Minus-Big = risk premium of small cap stocks over large cap stocks
❑ Fama-French quantifies the size risk premium:
➢ The SMB portfolio consists of a long position in the market’s smallest 50% of firms ranked
by market cap (small) and a short position in the market’s largest 50% of firms ranked by
market cap (big)
❑ Smaller firms experience higher returns:
➢ Probably due to a risk premium for small firms
➢ Size effect may be related to liquidity (small caps have less liquidity)
Fama-French Factors: Size
38
❑ HML = High-Minus-Low = risk premium of “value” stocks over “growth” stocks
❑ Fama-French quantifies the value risk premium:
➢ Rank each stock based on the ratio: Book Value of equity / Market Value of equity (B/M)
➢ Stocks with higher B/M ratios are considered “value” stocks
➢ Stocks with lower B/M ratios are considered “growth” stocks
➢ The HML portfolio consists of a long position in market stocks with the highest 30% ranked
B/M ratio (value stocks) and a short position in market stocks with the lowest 30% ranked
B/M ratio (growth stocks)
❑ Why the risk premium for high B/M stocks (value stocks)?
➢ High B/M stocks may have discounted share prices or are closer to financial distress leading
to higher returns
Fama-French Factors: Value
39
❑ From BKM Figure 13.1
❑ Fama-French appears to fit the data better than CAPM
Empirical Results of Fama-French
40
Carhart 4 Factor Model
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41
❑ Carhart expanded the Fama-French model with a fourth factor, Momentum
➢ WML = Winners Minus Losers
• The risk premium of uptrend stocks over downtrend stocks
❑ Carhart quantifies the momentum risk premium:
➢ The WML portfolio consists of a long position in market stocks with the highest 30% ranked
returns in the preceding year (winners) and a short position in market stocks with the lowest
30% ranked returns in the preceding year (losers)
❑ The Fama-French-Carhart four-factor model equation to derive an asset’s risk premium is
given by:
= + + + + +
Carhart Additional Factor: Momentum
42
❑ Refer to the data contained in the spreadsheet “Multi-factor Regression”, Worksheet: Data
Sheet:
a) Using the returns contained in the “Data Sheet” worksheet for the period 1976-2017, run
the simple SIM model (i.e., derive the SCL), the Fama-French 3-Factor model, and the
Carhart 4-factor model.
➢ SIM Model: From Menu Choose Data>>>Data Analysis>>>Regression (OK)
Input Y Range: $G$14:$G$499 (Stock Excess Returns)
Input X Range: $H$14:$H$499 (Market Excess Returns)
➢ Fama-French Model: From Menu Choose Data>>>Data Analysis>>>Regression (OK)
Input Y Range: $G$14:$G$499 (Stock Excess Returns)
Input X Range: $H$14:$J$499 (Market, SMB and HML Excess Returns)
➢ Carhart Model: From Menu Choose Data>>>Data Analysis>>>Regression (OK)
Input Y Range: $G$14:$G$499 (Stock Excess Returns)
Input X Range: $H$14:$K$499 (Market, SMB, HML and WML Excess Returns)
Carhart 4 Factor Model Example
Excel: “Multi-factor Regression”
43
b) Analyse the coefficients derived from the Carhart 4-factor model. What can we conclude
from the statistical properties of these coefficients?
➢ See worksheet “Carhart Model” for the estimated Carhart model alphas and betas for BRK
➢ T=486 and K=5; (T-K)=481 degrees of freedom
➢ The adjusted 2 is relatively low at 0.257
➢ = 10.8% (0.9% per month) with -stat = 3.16, compared to the single-factor model
alpha of 13.4% - the 3 additional factors explain some of Buffet’s alpha, but it’s still significant
➢ = 0.842 with -stat = 12.64 (significant at 99%), BRK moves with the market
➢ = −0.275 with –stat = −2.87 (significant at 95%), BRK buys large stocks, so
performance is lower when small stocks outperform
➢ = 0.500 with -stat = 4.83 (significant at 99%), BRK buys value stocks (measured by
high B/M), so performance is higher when value stocks outperform
➢ = 0.06 with -stat = 0.95 (insignificant), BRK does not chase momentum. Momentum
explains little variation in BRK’s performance
Carhart 4 Factor Model Example
44
Smart Beta
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❑ “Smart beta” investment strategies arose as a result of the identification of alternative factors
which may impact on security returns
❑ Smart beta attempts to capture defined factors in a rules based and transparent way by
constructing an alternative index to the traditional value-weighted indices
➢ A Smart Beta ETF will then “track” this alternative index
➢ Smart beta strategies are therefore a mix of passive and active investing
• They are passive in the sense that a smart beta investor follows an index, but also active
in selecting the stock characteristics that comprise the index
❑ Smart Beta ETFs might choose companies that only exhibit certain behaviours or metrics
such as: earnings growth, value, momentum, low (market) beta or volatility, quality (for
example measured by return on equity) etc
❑ Smart beta ETFs allows investors to tailor portfolio exposure either away from, or toward,
additional systematic factors
Smart Beta
46
❑ We have seen BRK’s alpha was 13.4% from 1976 to 2017
❑ The study “Buffett’s Alpha” (Frazzini and Kabiller, 2018) attempts to explain this alpha in
terms of additional factors which correspond to Buffett’s investing style
➢ Based on the 4-factor model, we have already seen that Buffett’s style is large-cap value
and isn’t concerned with momentum
➢ The authors add two further factors which attempt to explain Buffett’s alpha – investing in
low beta stocks and focusing on quality stocks
❑ Based on the results of “Buffet’s Alpha” Table 4 which are summarised below, derive the risk
premium on the BAB and QMJ factors
Using Multi-factor Models to Explain Alpha
Factors BRK.A Market SMB HML MOM BAB QMJ
Risk Premiums 18.6% 7.5% 2.7% 3.5% 7.6% ??? ???
5 Factor Model ( and ) = 8.5% = 0.83 = -0.30 = 0.31 = -0.02 = 0.33 ---
6 Factor Model ( and ) = 5.4% = 0.95 = -0.13 = 0.40 = -0.05 = 0.27 = 0.47
47
➢ Hint: Calculate the BAB Risk Premium first, based on the information in the 5-factor model,
then use this risk premium to derive the QMJ Risk Premium in the 6 Factor model.
➢ We can use the betas and risk premiums from the 4-factor model to calculate the BAB risk
premium:
(18.6% - 8.5% - 0.83 x 7.5% + 0.30 x 2.7% - 0.31 x 3.5% + 0.02 x 7.6%) / .33
= 11.3%
➢ Similarly, we can use the betas and risk premiums from the 5-factor model to calculate the
QMJ risk premium:
(18.6% - 5.4% - 0.95 x 7.5% + 0.13 x 2.7% - 0.40 x 3.5% + .05 x 7.6% - 0.27 x 11.3%) / 0.47
= 5.0%
➢ Addition of BAB and QMJ factors reduces alpha to 5.4% (from ~11% in the 4-factor model)
Using Multi-factor Models to Explain Alpha
RBRK BRK RM SMB HML MOM BAB
RBRK BRK RM SMB HML MOM QMJBAB
Mini Case Study: “Can Buffett’s Alpha Be Explained?”
48
❑ No Lecture Week 6
❑ Mid-term Test this Saturday (end of week 5)
❑ iLab: Week 6 Tuesday and Thursday
❑ Week 7: BKM Chapter 11 and 12
❑ 7.1 The Efficient Market Hypothesis
❑ 7.2 Empirical Testing on Market Efficiency
❑ 7.3 Behavioural Finance
❑ 7.4 Technical Trading
Next Lecture
