7 Capital asset pricing model
If we regard Markowitz’s mean-variance framework as a normative theory of how an ideal- ized investor should behave, the capital asset pricing model (CAPM) (developed by Sharpe, Treynor and Lintner) describes what happens in equilibrium if all investors behave in the same way.1 This theory (and the later development of the efficient market hypothesis) strongly motivated the emergence of passive investment and index funds. In this section, we derive the CAPM and compare its predictions with empirical data.
7.1 Mean-variance analysis with a risk-free asset
In the mean-variance optimization problem studied in Section 6.2, we assumed Σ is strictly positive definite, so that all assets are risky. In practice, there is often a risk-free asset such as the treasury bond.2 We first extend the analysis in Section 6.2 to include a risk-free asset.
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Let Rf be a constant which represents the return of the risk-free asset. Let the other n assets be risky (i.e., σi2 = Var(Ri) > 0) and let μ and Σ be respectively the expected mean and covariance of R = (R1, . . . , Rn)⊤. Let w ∈ Rn be the portfolio weights in the risk assets. We still assume that Σ is strictly positive definite. In order to have a fully invested portfolio, we let the portfolio weight in the risky asset be
The return w0Rf + w⊤R of the resulting portfolio has expected value
μw =w0Rf +w⊤1=Rf +w⊤(μ−Rf1)
and variance σw2 = w⊤Σw. The term μ − Rf 1 is often called the risk premium over the risk-free asset. A main goal of asset pricing theory is to provide an explanation of the risk premium.
With this set-up, we consider again the mean-variance optimization problem min1w⊤Σw, suchthat Rf +w⊤(μ−Rf1)=m, (7.1)
where m is the given expected return. The solution, which follows from another application of the Lagrange multiplier method, is given in the following proposition. The derivation is left as an exercise.
w0 =1−1⊤w=1−
wi, 1=(1,…,1)⊤ ∈Rn.
Proposition 7.1. The solution to (7.1) is
w∗ = κΣ−1(μ − Rf 1),
κ = m − Rf . (μ−Rf1)⊤Σ−1(μ−Rf1)
1Sharpe received the Nobel prize in economics in 1990.
2The treasury bond is not totally risk-free (as even the government can default), but is regarded by market participants as a very safe investment.
0.0 0.2 0.4 0.6 0.8 1.0
standard deviation
Figure 7.1: Efficient frontier with a risk-free asset. The risk-free asset, the minimum variance (risky) portfolio, and the tangency portfolio are shown by the red dots.
The corresponding variance is
2 (m−Rf)2 σw∗(m)= (μ−Rf1)⊤Σ−1(μ−Rf1).
Proof. Exercise.
When a risk-free asset exists, it is better to plot the efficient frontier in terms of the standard deviation. Now the efficient frontier becomes a straight line; it is the upper part
portfolio is given by
1 Σ−1(μ − Rf 1). 1⊤Σ−1(μ − Rf 1)
(σ∗(m),m):σw∗(m)= |m−Rf| . (7.4)
of the set
We illustrate this with an example.
Example 7.2. Consider the efficient frontier (without the risk-free asset) depicted in Figure 6.1. Here, we plot it in terms of (σ,m) rather than (σ2,m), and add the efficient frontier (with the risk-free asset, where Rf is set to be 0.01) given by (7.3).
We show the result in Figure 7.1. In this diagram, the new efficient frontier is the blue line which provides portfolios that are more mean-variance efficient than those provided by the hyperbola. It intersects the y-axis at the risk-free asset which has return Rf . We also observe that the two frontiers intersect at a portfolio. This portfolio, which exists wheneverμGMV ̸=Rf,iscalledthetangencyportfolioandplaysanimportantroleinthe CAPM.
For completeness, we provide an expression of the tangency portfolio. Note that when Rf > μGMV , the tangency portfolio is on the lower side of the frontier.
(μ−Rf1)⊤Σ−1(μ−Rf1)
Proposition 7.3. Suppose μGM V = wG⊤M V μ is different from Rf .
Then the tangency
expected return
−0.2 0.0 0.2 0.4 0.6 0.8
Let A,B,C be given by (6.10), and define D = (μ−Rf1)⊤Σ−1(μ−Rf1). Then the mean and variance of the tangency portfolio is given by
μ = C − BRf , σ2 = D . (7.6) tan B − ARf tan (B − ARf )2
When μGMV > Rf, the tangency portfolio is also the maximizer of the Sharpe ratio. The Sharpe ratio of a portfolio with return R (with respect to the risk-free asset) is defined
E[R−Rf] SR = Var(R) .
On the mean-standard deviation plane, the line m=Rf +μtan−Rfσ
is called the capital market line. In Figure 7.1, it is shown by the blue line.
7.2 Derivation of the CAPM
Now we give a derivation of the basic version of the capital asset pricing model. Consider a one-period market and the following (rather unrealistic) assumptions:
All investors are mean-variance optimizers.
All investors agree on the mean vector μ and the covariance matrix Σ of the risky
There is a risk-free asset with return Rf , and investors can borrow and lend at this rate. We assume Rf > μGMV (as otherwise the capital market line has a negative slope).
All assets are tradable and perfectly divisible.
There are no market frictions (such as transaction costs).
Consider the situation of a fixed investor. From the analysis in Section 7.1, the investor chooses to hold a portfolio on the capital market line (the blue line in Figure 7.1). The exact location on the line depends on the risk aversion of the investor. But any portfolio on the capital market line is a linear combination of the risk-free asset and the tangency portfolio wtan. This observation is a version of the two-fund separation theorem. Since all investors hold the same risky portfolio, namely the tangency portfolio, in market equi- librium this portfolio is also equal to the market portfolio. We repeat this important conclusion:
In market equilibrium, the tangency portfolio is the market portfolio. In particular, the market portfolio is mean-variance efficient and lies on the efficient frontier.
This conclusion strongly suggests investing in the market portfolio. (Note that in practice most investors do not hold the market portfolio (which is often approximated by a market index).)
The famous CAPM equation ((7.11) below) is a consequence of this observation. Let
wM = wtan be the market or tangency portfolio, and let its return be RM. Fix a risky
asset i and let e(i) = (0, …, 1, …, 0) be the portfolio which invests everything in asset i (so
e(i) =1ifj=iandis0otherwise).
For t ∈ (−ε, ε), where ε > 0, consider the portfolio
w(t) = (1 − t)wtan + te(i) = (1 − t)wM + te(i).
This portfolio has expected return
μw(t) = (1 − t)E[RM ] + tE[Ri]
and variance
σw(t) = (1 − t)2Var(RM ) + 2t(1 − t)Cov(Ri, RM ) + t2Var(Ri).
Consider the curve (μw(t),σw(t)) in the mean-stand deviation plane. The slope of this curve, whenever it is not vertical, is given by dμw(t) /dσw(t) . Since the curve is tangent to
the capital market line when t = 0, we have
dμw(t) E[RM]−Rf
dμw(t) = E[Ri] − E[RM ] dt t=0
dσw(t) Var(RM )
It remains to compute the left hand side. We have
dσw(t) dt
−2(1 − t)Var(RM ) + 2(1 − 2t)Cov(Ri, RM ) + 2tVar(Ri) = 2(1−t)2Var(R )+2t(1−t)Cov(R ,R )+t2Var(R )
M i M i t=0 −Var(RM ) + Cov(Ri, RM )
= Var(RM) . Plugging these into (7.10) and simplifying, we have
E[Ri]=Rf +Cov(Ri,RM)(E[RM]−Rf)=Rf +βi(E[RM]−Rf), Var(RM )
βi = Cov(Ri,RM) Var(RM )
is called the beta of the stock.
Here is the financial meaning of (7.11). Since
E[Ri]−Rf =βi(E[RM]−Rf),
the risk premium of an asset is directly proportional to that of the market portfolio.
Moreover, the constant of proportionality is the beta. Since β is proportional to the 54
covariance between the asset and market returns, only the part of the return which is correlated with the market is rewarded with extra expected return. To understand this, it is helpful to think of the total risk of an asset to be the sum of a systematic part and an idiosyncratic part. Idiosyncratic risk may be practically eliminated by sufficient diversification, while the systematic risk (which is the part which is correlated to the market) cannot be diversified away. As such, risk premium is awarded in equilibrium for investors to be willing to bear this risk.
7.3 Empirical results
In the literature, there are many empirical tests of the CAPM.3 Since the basic CAPM is a one-period model, it cannot be directly applied to a multi-period market.4 While it may be assumed that it holds period by period, different investors usually have different investment horizons. Another major difficult is that testing the CAPM requires knowing the market portfolio; in principle, the market portfolio may contains non-financial assets such as real estates. Many empirical tests restrict the market portfolio to be a market index. Nevertheless, one can still ask whether asset returns are adequately described by (7.11).
7.3.1 The single index model
The CAPM equation (7.11) may be written in the form
Ri −Rf =βi(RM −Rf)+εi, (7.13)
where εi is a random variable with E[εi] = 0. We may express this as the constraint αi = 0 in the single index model:
Ri −Rf =αi +βi(RM −Rf)+εi. (7.14)
The coefficient αi, the alpha of the asset, may be regarded as a measure of abnormal return (with respect to CAPM).
Suppose we observe time series data Ri,t, Rf,t and RM,t, t = 1, . . . , T , of the respective returns. Then we may entertain the model
Ri,t − Rf,t = αi + βi(RM,t − Rf,t) + εi,t, (7.15) and estimate αi and βi using ordinary least squares (OLS). If the risk-free return is not
included, we have the alternative model
Ri,t = αi + βiRM,t + εi,t. (7.16)
Note that in both (7.15) and (7.16) we have βi = Cov(Ri,RM). When Rf is close to zero Var(RM )
(which is approximately the case for the US market in recent years), the models (7.16) and (7.15) are essentially equivalent.
3See Fama and French (2004) for a survey of empirical tests.
4In the literature there are multi-period extensions of the CAPM but they are beyond the scope of this course.
−0.06 −0.04
ACF of residuals
−0.02 0.00
S&P500 excess return
0.02 0.04 0.06
Normal Q−Q Plot
0 5 10 15 20 25 30
−3 −2 −1 0 1 2 3
Theoretical Quantiles
Figure 7.2: Top: Scatterplot of the excess returns of Mi- crosoft and S&P 500. The OLS regression line (fitted using the model (7.15)) is shown by the blue line. Bottom: ACF and normal Q-Q plot of the residuals.
Example 7.4. Here we repeat Example 17.3 of Ruppert and Matteson (2015).5 Consider the daily returns of Microsoft from Nov 1, 1993 to Apr 3, 2003. We let RM be the return of the S&P 500 Index. The risk-free rate is approximated by T-bill rates.
In Figure 7.2 we plot the excess return Ri,t −Rf,t of Microsoft against the excess return of S&P 500. We see a positive linear correlation (the R2 is 0.34 whose size is typical for the single index model). The estimated beta (for the model (7.15))is βˆi = 1.25 which is bigger than 1, and αˆ = 0.0009. Under the normal approximation, the p-value of αˆ is 0.0256. For practical purposes we may regard the value of α as zero. In 7.2 we also show the sample ACF and normal Q-Q plot of the residuals. It appears that the noise process
5The data set and codes can be download at https://people.orie.cornell.edu/davidr/SDAFE2/ index.html
−0.05 0.05 0.10 0.15
0.2 0.4 0.6 0.8 1.0
Sample Quantiles
MSFT excess return
−0.15 −0.05 0.05 0.15
Rolling beta estimate
Figure 7.3: Rolling beta estimates of Microsoft with respect to S&P 500. Each beta estimate is computed using 100 weekly returns.
is close to a white noise process but is fat-tailed.
The next example investigates the stability of the beta over time.
Example 7.5 (Rolling beta estimates). We consider instead the weekly returns of Microsoft and S&P 500. For convenience, we consider the model (7.16) which does not include the risk-free return. At time t, we consider the previous 100 weekly returns and estimate β using OLS. We show the result in Figure 7.3. From the figure, it is clear that the (historical) beta is not constant over time. The beta decreased from 2000 to 2010 but increased rather quickly in 2015. The beta also decreased rapidly in early 2020.
We also consider a very crude visual test of the CAPM.
Example 7.6. Consider monthly returns of Dow Jone stocks and the S&P 500 from Jan 2015 to Dec 2021.6 Again for convenience we do not include Rf,t in the model. Since the interest rate is close to zero in this period, the result with Rf,t should be similar. We estimate using OLS the beta for each stock. We also compute the average arithmetic return.
In Figure 7.4 we plot the result. The solid line shows the OLS fit. We see a small but positive association between the average return and the beta. However, the linear correlation is not significant (the R2 is 0.07). Letting Rf,t = 0 for simplicity, if the CAPM holds exactly we should observe the dashed line which represents (7.11). Thus, it appears that the relationship between average return and beta is weaker than what CAPM predicts (consistent with the finding of many published studies).
7.3.2 Additional empirical evidence
Careful tests of CAPM involve a lot of statistical complications (such as errors in the estimation of beta) that are beyond the scope of this course. We end this section by
6One of the stocks is removed because of missing data. 57
Figure 7.4: Average (monthly) return versus beta for the Dow Jow stocks over 2015 − −2021.
reviewing an empirical study reported in Fama and French (2004) (it is a much more careful implementation of Example 7.6).
The idea is that the estimation of beta is more accurate for portfolios than individual assets. We quote Fama and French (2004) for the methodology adopted:
In December of each year, we estimate a preranking beta for every NYSE (1928-–2003), AMEX (1963–2003) and NASDAQ (1972–2003) stock in the CRSP (Center for Research in Security Prices of the University of Chicago) database, using two to five years (as available) of prior monthly returns.5 We then form ten value-weight portfolios based on these preranking betas and compute their returns for the next twelve months. We repeat this process for each year from 1928 to 2003. The result is 912 monthly returns on ten beta-sorted portfolios. Figure 2 [reproduced in Figure 7.5] plots each portfolio’s average return against its postranking beta, estimated by regressing its monthly returns for 1928-–2003 on the return on the CRSP value-weight portfolio of U.S. common stocks.
Once again, we observe that the relationship between average return and beta is weaker than that predicted by CAPM. Overall, it is fair to say that while the CAPM has had huge practical impacts in portfolio management, its empirical performance is not satisfactory. Later models, in particular factor models, improve the CAPM by including extra factors to explain asset returns. We also mention that in the literature there are many improved versions of CAPM to include, among other things, consumption, intertemporal investment, and continuous-time trading.
average return
0.000 0.010 0.020
References
Figure 7.5: Figure 2 of Fama and French (2004).
Fama, E. F., & French, K. R. (2004). The capital asset pricing model: Theory and evidence. Journal of Economic Perspectives, 18(3), 25–46.
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