The CAPM

The CAPM#

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The CAPM#

The most famous Linear Factor Pricing Model is the Capital Asset Pricing Model (CAPM).

\[ \mathbb{E}[\tilde{r}^i] = \beta^{i,m} \mathbb{E}[\tilde{r}^m] \]
\[ \beta_{i,m} \equiv \frac{\operatorname{Cov}(\tilde{r}^i, \tilde{r}^m)}{ \operatorname{Var}(\tilde{r}^m)} \]

where \(\tilde{r}^m\) denotes the return on the entire market portfolio, meaning a portfolio that is value-weighted to every asset in the market.

The market portfolio#

The CAPM identifies the market portfolio as the tangency portfolio.

  • The market portfolio is the value-weighted portfolio of all available assets.

  • It should include every type of asset, including non-traded assets.

  • In practice, a broad equity index is typically used.

Explaining expected returns#

The CAPM is about expected returns:

  • The expected return of any asset is given as a function of two market statistics: the risk-free rate and the market risk premium.

  • The coefficient is determined by a regression. If \(\beta\) were a free parameter, then this theory would be vacuous.

  • In this form, the theory does not say anything about how the risk-free rate or market risk premium are given.

  • Thus, it is a relative pricing formula.

Deriving the CAPM#

If returns have a joint normal distribution…

  1. The mean and variance of returns are sufficient statistics for the return distribution.

  2. Thus, every investor holds a portfolio on the MV frontier.

  3. Everyone holds a combination of the tangency portfolio and the risk-free rate.

  4. Then aggregating across investors, the market portfolio of all investments is equal to the tangency portfolio.

Deriving CAPM by investor preferences#

Even if returns are not normally distributed, the CAPM would hold if investors only care about mean and variance of return.

  • This is another way of assuming all investors choose MV portfolios.

  • But now it is not because mean and variance are sufficient statistics of the return distribution, but rather that they are sufficient statistics of investor objectives.

  • So one derivation of the CAPM is about return distribution, while the other is about investor behavior.

CAPM assumptions and asset classes#

But if we assume normally distributed and iid. returns…

  • Application is almost exclusively for equities.

  • The CAPM is often not even tried on derivative securities, or even debt securities.

The CAPM decomposition of risk premium#

The CAPM says that the risk premium of any asset is proportional to the market risk premium.

\[ \mathbb{E}[\tilde{r}^i] = \beta^{i,m} \mathbb{E}[\tilde{r}^m] \]

The risk premium of an asset is defined as the expected excess return of that asset.

  • The scale of proportionality is given by a measure of risk—the market \(\beta\) of asset i.

  • What would a negative \(\beta\) indicate?

Beta as the only priced risk#

Equation (2) says that market \(\beta\) is the only risk associated with higher average returns.

  • No other characteristics of asset returns command a higher risk premium from investors.

  • Beyond how it affects market \(\beta\), CAPM says volatility, skewness, other covariances do not matter for determining risk premia.

Return variance decomposition#

The CAPM implies a clear relation between volatility of returns and risk premia.

\[ \tilde{r}^i_t =\beta^{i,m}\tilde{r}^m_t + \varepsilon_t \]

Take the variance of both sides of the equation to get

\[ \sigma^2_i = \underbrace{\beta_{i,m}^2 \sigma^2_m}_{\text{systematic}} + \underbrace{\sigma^2_{\varepsilon}}_{\text{idiosyncratic}} \]

So CAPM implies…

  • The variance of an asset’s return is made up of a systematic (or market) portion and an idiosyncratic portion.

  • Only the former risk is priced.

Proportional risk premium#

To appreciate how idiosyncratic risk does not increase return, consider the following calculations for expected returns.

\[ \mathbb{E}[\tilde{r}^i] = \beta^{i,m} \mathbb{E}[\tilde{r}^m] \]

Using the definition of \(\beta_{i,m}\),

\[\frac{\mathbb{E}[\tilde{r}^i]}{\sigma_i} = \rho_{i,m} \frac{\mathbb{E}[\tilde{r}^m]}{\sigma_m} \]

where \(\rho_{i,m}\) denotes \(\operatorname{Corr}(\tilde{r}^m, \tilde{r}^i).\)

The CAPM and Sharpe-Ratios#

Using the definition of the Sharpe ratio in (3), we have

\[\mathrm{SR}_i = \rho_{i,m}\mathrm{SR}_m\]

  • The Sharpe ratio earned on an asset depends only on the correlation between the asset return and the market.

  • A security with large idiosyncratic risk, \(\sigma^2_{\varepsilon}\), will have lower \(\rho_{i,m}\) which implies a lower Sharpe Ratio.

  • Thus, risk premia are determined only by systematic risk.

Treynor’s Ratio#

If CAPM does not hold, then Treynor’s Measure is not capturing all priced risk.

\[ \text{Treynor Ratio} = \frac{\mathbb{E}[\tilde{r}^i]}{\beta_{i,m}} \]

If the CAPM does hold, then what do we know about Treynor Ratios?

Testing#

CAPM and realized returns#

The CAPM implies that expected returns for any security are

\[ \mathbb{E}[\tilde{r}^i] = \beta^{i,m} \mathbb{E}[\tilde{r}^m] \]

This implies that realized returns can be written as

\[ \tilde{r}^i_t = \beta^{i,m} \tilde{r}^m_t + \varepsilon_t \]

where \varepsilont is not assumed to be normal, but

\[ \mathbb{E}[\varepsilon] = 0 \]

Of course, taking expectations of both sides we arrive back at the expected-return formulation.

Testing the CAPM on an asset#

Using any asset return \(i\), we can test the CAPM.

  • Run a time-series regression of excess returns i on the excess market return.

  • Regression for asset \(i\), across multiple data points \(t\):

\[ \tilde{r}^i_t = \alpha^i + \beta^{i,m} \tilde{r}^m_t + \varepsilon^i_t \]

Estimate \(\alpha\) and \(\beta\).

  • The CAPM implies \(\alpha^i = 0\).

Testing the CAPM on a group of assets#

Can run a CAPM regression on various assets, to get various estimates \(\alpha^i\).

  • CAPM claims every single \(\alpha^i\) should be zero.

  • A joint-test on the \(\alpha^i\) should not be able to reject that all \(\alpha^i\) are jointly zero.

CAPM and realized returns#

CAPM explains variation in \(\mathbb{E}[\tilde{r}^i]\) across assets—NOT variation in \(\tilde{r}^i\) across time!

\[ \tilde{r}^i_t = \tilde{r}^i + \beta^{i,m} \tilde{r}^m_t + \varepsilon_t \]

  • The CAPM does not say anything about the size of \varepsilont.

  • Even if the CAPM were exactly true, it would not imply anything about the R-squared of the above regression, because \(\sigma_{\varepsilon}\) may be large.

CAPM as practical model#

For many years, the CAPM was the primary model in finance.

  • In many early tests, it performed quite well.

  • Some statistical error could be attributed to difficulties in testing.

  • For instance, the market return in the CAPM refers to the return on all assets—not just an equity index. (Roll critique.)

  • Further, working with short series of volatile returns leads to considerable statistical uncertainty.

Industry portfolios#

A famous test for the CAPM is a collection of industry portfolios.

  • Stocks are sorted into portfolios such as manufacturing, telecom, healthcare, etc.

  • Again, variation in mean returns is fine if it is accompanied by variation in market \(\beta\).

Industry portfolios: beta and returns#

Figure: Data Source: Ken French. Monthly 1926-2011.

Evidence for CAPM?#

The plot of industry portfolios shows monthly risk premia from about 0.5% to 0.8%.

  • Still, there is substantial spread in betas, and the correlation seems to be positive.

  • Note that the risk-free rate and market index are both plotted (black diamonds.)

  • Note that the markers for the “Health” and “Tech” portfolio cover up most of the markers for “Energy” and “Durables”.

CAPM-implied relation between beta and returns#

Figure: Data Source: Ken French. Monthly 1926-2011.

CAPM and risk premium#

CAPM can be separated into two statements:

  • Risk premia are proportional to market \(\beta\):

\[ \mathbb{E}[\tilde{r}^i] = \beta^{i,m}\lambda_m \]

  • The proportionality is equal to market risk premium:

\[ \lambda_m = \mathbb{E}[\tilde{r}^m] \]

The risk-return tradeoff#

The parameter \(\lambda_m\) is particularly important.

  • It represents the amount of risk premium an asset gets per unit of market \(\beta\).

  • Thus, can divide risk premium, into quantity of risk, \(\beta_{i,m}\), multiplied by price of risk, \(\lambda_m\) .

  • \(\lambda_m\) is also the slope of the Security Market Line (SML), which is the line plotted above.

Cross-sectional test of the CAPM#

We can run a cross-sectional regression to test implications (5) and (6).

\[ \underbrace{\mathbb{E}\left[\tilde{r}^{i}\right]}_{n\times 1\text{ data}} = \textcolor{ForestGreen}{\underbrace{\eta}_{\text{regression intercept}}} + \underbrace{{\beta}^{i,\text{mkt}};}_{n\times 1\text{ data}}~ \textcolor{ForestGreen}{\underbrace{\lambda_{\text{mkt}}}_{\text{regression estimate}}} + \textcolor{ForestGreen}{\underbrace{\upsilon}_{n\times 1\text{ residuals}}} \]

  • The data on the left side is a list of mean returns on assets, \(\mathbb{E}[\tilde{r}^i]\).

  • The data on the right side is a list of asset betas: \(\beta_{i,m}\) for each asset \(i\).

  • The regression parameters are \(\eta\) and \(\lambda_m\).

  • The regression errors are \(\upsilon\).

CAPM implications in the cross-section#

\[ \mathbb{E}[\tilde{r}^i] = \eta + \beta^{i,m}\lambda_m + \upsilon^i \]

  • CAPM statement (5) implies the R-squared of the cross-sectional regression is 100%.

  • That is to say, the CAPM implies \(\upsilon^i=0\) for each \(i\).

  • CAPM statement (6) implies the cross-sectional regression parameters are:

\[ \eta = 0,\; \lambda_m = \mathbb{E}[\tilde{r}^m] \]

  • That is, the SML goes through zero and the market return. (See slide 24.)

Estimating the cross-sectional CAPM equation#

Estimation of the cross-sectional equation on industry portfolios shows:

  • The estimated slope, $\lambda_m$ is too small relative to the full CAPM theory.

  • The SML line doesn’t start at zero, \(\eta\) > 0. This is a well-known fact. (But only a puzzle if you really believe the CAPM!)

Unrestricted SML for industry portfolios;#

Figure: Data Source: Ken French. Monthly 1926-2011.

Risk-reward tradeoff is too flat relative to CAPM#

Figure: Data Source: Ken French. Monthly 1926-2011.

Trading on the security market line#

Suppose one believes the CAPM: market \(\beta\) completely describes (priced) risk.

  • Relatively small \(\lambda_m\) in estimation implies that there is little difference in mean excess returns even as risk, as seen in \(\beta_{i,m}\), varies.

  • A trading strategy would then be to bet against \(\beta\): go long small-\(\beta\) assets and short large-\(\beta\) assets.

  • Frazzini and Pedersen (2011) have an interesting paper on this.

References#

  • Back, Kerry. Asset Pricing and Portfolio Choice Theory. 2010. Chapter 6.

  • Bodie, Kane, and Marcus. Investments. 2011. Chapters 9 and 10.

  • Cochrane. Discount Rates. Journal of Finance. August 2011.

  • Frazzini, Adrea and Lasse Pedersen. Betting Against Beta. Working Paper. October 2011.

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