Further Explorations of the Capital Asset Pricing Model

I. A Mathematical Model of Expected Returns

How much return should a security earn — not more, not less, but what is fair given its risk? The Capital Asset Pricing Model gives a startlingly simple answer, and it is the single most-used equation in practical finance. It says the expected return on any security is the riskless rate plus a reward for one — and only one — kind of risk: the security’s exposure to the market as a whole.

The whole model relies on a single statistic, beta. Formally, beta is the security’s covariance with the market divided by the market’s variance:

Which Model, and When?

It is worth being precise about when each of our two models applies. The Markowitz mean–variance model of Chapter 2 answers a question about an investor’s entire wealth: given all the assets available, what is the single best overall allocation? The CAPM answers a different, marginal question — what happens to an already-diversified portfolio when one more security is added to it. Ross’s argument, expanded below, is exactly that marginal calculation: the effect of a small purchase $dx$ of a single asset.

This marginal view lets us re-read a picture from the beginning of the course. Recall the diversification curve: as securities are added to a portfolio — one, two, ten, thirty and beyond — its standard deviation falls quickly at first and then flattens toward a floor it never crosses. We can now name the two pieces.

The risk that melts away as the portfolio grows is idiosyncratic risk — the residual $e_i$ from the equation above. In a large, well-diversified portfolio it becomes, in the pure model, irrelevant: it averages out, and the investor is paid nothing to bear it. The floor that remains — the risk that cannot be diversified away — is beta risk, the security’s co-movement with the market. Ross’s theorem, below, is the formal proof that only this second piece is priced.

One caveat lies in the fine print of that marginal argument. It sets aside the asset’s own-variance term, $(dx)^2\,\text{var}(A)$, as negligible next to its covariance term — entirely safe for ordinary assets. But for a security with extremely high variance — a lottery-like payoff, or fat, near-undefined tails — that own-variance term need not vanish, and the marginal conclusion can fail. Idiosyncratic risk stops being negligible, and such an asset can matter to a portfolio in its own right. The CAPM’s clean verdict that only beta is priced is a statement about the well-behaved, marginal case.

Where does this come from? Ross’s derivation (optional)

The intuition behind beta — that a well-diversified investor cares about a security’s contribution to portfolio risk, not its standalone volatility — can be made precise. An asset with a low correlation to the tangency portfolio is desirable because it shifts the efficient frontier to the left.

Figure 5.2. Adding a low-correlation asset shifts the efficient frontier to the left, reducing portfolio risk for a given return level.

Stephen Ross formalized this in an article titled “Finance,” published in The New Palgrave. Suppose you hold the market portfolio $m$ and consider buying a quantity $dx$ of asset $A$, financed by borrowing at the riskless rate. This augments the market portfolio’s expected return by

Change in Expected Return

$$dE_m = [E_A - R_f]\,dx$$

and augments its variance. After adding the asset the variance is $v + dv = v + 2\,dx\,\text{cov}(A,m) + (dx)^2\,\text{var}(A)$, so the change is $dv = 2\,dx\,\text{cov}(A,m) + (dx)^2\,\text{var}(A)$, which for small $dx$ is approximately $dv \approx 2\,dx\,\text{cov}(A,m)$. The risk–return tradeoff from investing a little in $A$ is therefore

Risk–Return Tradeoff for A

$$\frac{dE_m}{dv} = \frac{[E_A - R_f]\,dx}{2\,dx\,\text{cov}(A,m)} = \frac{E_A - R_f}{2\,\text{cov}(A,m)}$$

In equilibrium an investor must be indifferent between buying a little more $A$ and simply levering up the market portfolio itself. The same tradeoff for buying $dx$ of the market portfolio is $\dfrac{E_m - R_f}{2\,\text{var}(m)}$ — identical in form, but with $\text{cov}(A,m)$ replaced by $\text{var}(m)$ (the covariance of the market with itself). Setting the two equal,

$$\frac{E_A - R_f}{2\,\text{cov}(A,m)} = \frac{E_m - R_f}{2\,\text{var}(m)} \quad\Longrightarrow\quad E_A - R_f = \frac{\text{cov}(A,m)}{\text{var}(m)}\left[E_m - R_f\right]$$

and the ratio $\text{cov}(A,m)/\text{var}(m)$ is exactly the beta above. The pricing equation of the CAPM falls straight out of the no-free-lunch condition.

Extension: When Does Idiosyncratic Risk Still Matter?

The argument above dropped the own-variance term $(\Delta x)^2\,\text{var}(i)$ as small next to the covariance term. If we keep it, we can say exactly when adding an asset still helps a diversified portfolio — and when a high-variance asset stops obeying the tidy $\beta<1$ rule.

Adding $\Delta x$ of asset $i$ raises portfolio variance by $\Delta v_i = 2\,\Delta x\,\text{cov}(i,p) + (\Delta x)^2\,\text{var}(i)$; adding the same amount of the portfolio itself raises it by $\Delta v_p = 2\,\Delta x\,\text{var}(p) + (\Delta x)^2\,\text{var}(p)$. Asset $i$ is diversifying when $\Delta v_i < \Delta v_p$:

$$2\,\Delta x\,\text{cov}(i,p) + (\Delta x)^2\,\text{var}(i) \;<\; 2\,\Delta x\,\text{var}(p) + (\Delta x)^2\,\text{var}(p)$$

Dividing by $\text{var}(p)$ and using $\beta_i = \text{cov}(i,p)/\text{var}(p)$ gives $\;2\,\Delta x\,\beta_i + (\Delta x)^2\,\dfrac{\text{var}(i)}{\text{var}(p)} < 2\,\Delta x + (\Delta x)^2$, which rearranges to the diversifying condition:

Diversifying condition (own-variance retained)

$$\beta_i \;<\; 1 - \frac{\Delta x}{2}\left(\frac{\text{var}(i)}{\text{var}(p)} - 1\right)$$

For an infinitesimal trade ($\Delta x \to 0$) this is just the familiar $\beta_i < 1$. But the own-variance term reasserts itself for larger positions. Setting $\Delta x = \tfrac{1}{2}$ gives $\beta_i + \tfrac{1}{4}\,\dfrac{\text{var}(i)}{\text{var}(p)} < \tfrac{5}{4}$; with $\sigma_i = 40\%$ and $\sigma_p = 20\%$ (a variance ratio of 4), the threshold tightens all the way to $\beta_i < \tfrac{1}{4}$. A very high-variance asset must have a much lower beta before it earns a place — the CAPM’s “only beta matters” verdict is really the small-trade limit.

Volatility ratio σ_i/σ_p: 2.0× Trade size Δx: 0.50

Figure 5.3. The diversifying-beta threshold once the own-variance term is retained. The curve is $\beta^\ast(\Delta x) = 1 - \tfrac{\Delta x}{2}(k-1)$, where $k=\text{var}(i)/\text{var}(p)$; an asset diversifies only if its beta lies in the shaded region below it. At $\Delta x\to 0$ the threshold is the textbook $\beta<1$; as the position grows, a high-variance asset’s threshold falls sharply. After W. Goetzmann, note on diversification (2022).

II. The Security Market Line

The CAPM equation describes a linear relationship between risk and return. Risk, in this case, is measured by beta. We may plot this line in mean and $\beta$ space:

One remarkable fact that comes from the linearity of this equation is that we can obtain the beta of a portfolio of assets by simply multiplying the betas of the assets by their portfolio weights. For instance the beta of a 50/50 portfolio of two assets, one with a beta of 0.8 and the other with a beta of 1, is 0.9. Easy!

The line also extends out infinitely to the right, implying that you can borrow infinite amounts to lever up your portfolio.

Why is the Line Straight?

Well, suppose it curved, as the blue line does in the figure below. An investor could borrow at the riskless rate and invest in the market portfolio. Any investment of this type would provide a higher expected return than a security which lies on the curved line below. In other words, the investor could receive a higher expected return for the same level of systematic risk. In fact, if the security on the curve could be sold short, then the investor could take the proceeds from the short sale and enter into the levered market position -- generating an arbitrage in expectation.

III. Expectations vs. Realizations

It is important to stress that the vertical dimension in the security market line picture is expected return. Things rarely turn out the way you expect. However, the CAPM equation also tells us about the realized rate of return. Since the realization is just the expectation plus random error, we can write:

This is useful, because it tells us that when we look at past returns, they will typically deviate from the security market line -- not because the CAPM is wrong, but because random error will push the returns off the line. Notice that the realized $R_m$ does not have to behave as expected, either. So, even the slope of the security market line will deviate from the average equity risk premium. Sometimes it will even be negative!

IV. An Example

The appeal of the CAPM is clear -- it radically simplifies an inherently complex and troublesome problem. The question of the appropriate discount rate becomes virtually a back-of-the-envelope calculation! In fact, if you know a security's beta, estimating the discount rate is a snap: multiply beta times the expected risk premia of the market portfolio over the riskless rate.

For example, suppose you are a banker considering a private equity investment in a company with a new drug process. The process is inherently risky, i.e. the standard deviation of the project is 75% per year. The beta of the project is 0.5. The $R_f = 5\%$ and the $E[R_m] = 13.5\%$. What is the required rate of return on the project?

Theory tells us that the answer does not depend upon the volatility associated with the returns. Instead we use the beta of the project:

This is the required rate of return on the project. The answer would not change if the range of outcome next year broadened or narrowed. The $\beta$ is the only relevant piece of information -- now all that remains is to estimate it!

V. How Do You Estimate Beta?

$\beta$ may be all we need, but it is not immediately clear how it should be estimated. What we really need is a quantitative estimate of how the future return changes in response to future changes in the world market portfolio. Good Luck! It is tough to even guess the empirical composition of the market portfolio, let alone estimate a beta. In practice (although it is not theoretically justified) analysts typically use the S&P 500 equity risk premium in this calculation. To estimate beta, regress the security returns for the past several periods (usually 60 months) on the market returns. The slope in this regression is an estimate of $\beta$.

Notice that this shows concretely the empirical property of $\beta$ as it measures the co-movement of the security with the market. Unfortunately, since the S&P 500 is not the world market portfolio, we are somewhat in the dark about how well this beta measures the true systematic risk.

VI. Assessing the CAPM

The CAPM is a classical model in finance. It is an equilibrium argument that, if true, answers most important investment questions. It tells us where to invest, how to invest and what discount rate to use for project cash flows. Not only that, it is a disarmingly simple model. The expected return of a security depends upon a simple statistic: $\beta$. The relationship between risk and return is linear. Calculation of portfolio risk is trivial.

At the same time, the CAPM is revolutionary. It tells us that the variance of a project is NOT a factor in determining the appropriate, risk-adjusted discount rate. It turns financial research from roll-up-your-sleeves fundamental analysis into a statistics problem. In short, the CAPM turned Wall Street on its head.

VII. Conclusion: Is the CAPM True?

Here comes the bad news. Despite twenty years of attempts to verify or refute the Capital Asset Pricing Model, there is no consensus on its legitimacy. There are a few hints that the model is incorrect. For starters, we all hold different portfolios. Therefore, it cannot be exactly true. Researchers have focused upon the more interesting issue of whether rates of return depend upon $\beta$ and whether the elegant, linear form of the model holds for stocks. What they have found is that real markets typically deviate broadly from the exact model. While there are long periods in U.S. Capital market history when realized returns are positively related to betas, there are also long periods when they are not.

Among the most forceful arguments against the CAPM advanced in recent times is a study by Eugene Fama and Kenneth French. These authors found that "beta did a relatively poor job at explaining differences in the actual returns of portfolios of U.S. stocks." Instead, Fama and French noted that there were other variables besides beta with respect to the market that explained returns. Some of these were "fundamental" ratios long used by financial analysts in the pre-CAPM era such as Book to Market Ratio and Earnings Price Ratio. Another was simply the relative size of the company. The evidence against the CAPM continues to grow and despite its elegance, most researchers have turned to more complex, but more powerful models.