4.6.3 The capital asset pricing model

Beta

Beta measures how strongly an asset moves with a portfolio M, usually the whole market. It is the asset's covariance with M divided by M's variance. It has no units, since both are in squared returns.

It is the hedge ratio of case study 3.5.5. Selling $\beta_i$ dollars of M per dollar of asset $i$ removes as much of its risk as M can remove. That case study's stock had a beta of 1.1518 to the index.

With a covariance matrix, case study 3.2.5 gives both pieces. The covariance of asset $i$ with M is $(\Sigma\mathbf{w}_M)_i$ , entry $i$ of $\Sigma\mathbf{w}_M$ , and M's variance is $\sigma_M^2 = \mathbf{w}_M^\top\Sigma\mathbf{w}_M$ .

Rule: Beta

\begin{gathered} \beta_i = \frac{\operatorname{Cov}(R_i, R_M)}{\operatorname{Var}(R_M)} \\[4pt] \boldsymbol{\beta} = \frac{\Sigma\mathbf{w}_M}{\mathbf{w}_M^\top\Sigma\mathbf{w}_M} \end{gathered}

$R_i$ and $R_M$ are the returns of asset $i$ and of M. A portfolio's beta is the weighted average of its assets' betas, so M's own beta is $\mathbf{w}_M^\top\boldsymbol{\beta} = 1$ .

$\beta = 1$ : the asset moves one for one with M, on average.
$\beta > 1$ : it amplifies M's moves; $0 < \beta < 1$ damps them.
$\beta = 0$ : no covariance with M. The risk-free bill has $\beta = 0$ .
$\beta < 0$ : it tends to rise when M falls, like insurance (case study 1.2.4).

Take M to be lesson 4.6.2's tangency portfolio, $\mathbf{w}_T = (0.5185, 0.1601, 0.3214)$ . Then $\Sigma\mathbf{w}_T = (0.009468, 0.015779, 0.025247)$ and $\sigma_T^2 = 0.01555$ , so

\boldsymbol{\beta} = (0.6088, 1.0147, 1.6236)

C moves about 1.62 times as much as T, A only 0.61 times. Weighted by $\mathbf{w}_T$ , the betas average to exactly 1.

\Sigma

import numpy as np mu = np.array([0.06, 0.08, 0.11]) sigma = np.array([0.12, 0.18, 0.25]) R = np.array([[1.0, 0.3, 0.1], [0.3, 1.0, 0.5], [0.1, 0.5, 1.0]]) Sigma = np.diag(sigma) @ R @ np.diag(sigma) rf, ones = 0.03, np.ones(3) raw = np.linalg.solve(Sigma, mu - rf * ones) w_T = raw / raw.sum() var_T = w_T @ Sigma @ w_T beta = Sigma @ w_T / var_T print(np.round(Sigma @ w_T, 6), round(var_T, 6)) print(np.round(beta, 4), round(w_T @ beta, 6))

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The security market line

From the tangency condition

Lesson 4.6.2 found $\mathbf{w}_T = \frac{1}{k}\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})$ , with $k$ the sum of the entries of $\Sigma^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})$ . Three steps turn this into a statement about expected returns.

1
$\Sigma\mathbf{w}_T = \frac{1}{k}(\boldsymbol{\mu} - r_f\mathbf{1})$
2
$\sigma_T^2 = \frac{1}{k}(\mu_T - r_f)$
3
$\beta_i = \frac{\mu_i - r_f}{\mu_T - r_f}$

The first step says that each asset's covariance with T is proportional to its excess return $\mu_i - r_f$ . The last says the same with beta: excess return equals beta times T's excess return.

Rule: Security market line

\begin{gathered} \mu_i = r_f + \beta_i\,(\mu_T - r_f) \\[4pt] \text{with } \beta_i \text{ measured against } T \end{gathered}

Every asset's expected return $\mu_i = \mathbb{E}[R_i]$ is a straight line in its beta, with intercept $r_f$ and slope $\mu_T - r_f$ . It holds for portfolios too, and only when the betas are measured against the tangency portfolio.

For the three assets the slope is $\mu_T - r_f = 0.049274$ :

asset	$\beta_i$	$0.03 + 0.049274\,\beta_i$	$\mu_i$
A	0.6088	0.06	0.06
B	1.0147	0.08	0.08
C	1.6236	0.11	0.11

The line gives back each expected return exactly. This is not luck: the betas were measured against the one portfolio that makes it true.

\Sigma\mathbf{w}_T

import numpy as np mu = np.array([0.06, 0.08, 0.11]) sigma = np.array([0.12, 0.18, 0.25]) R = np.array([[1.0, 0.3, 0.1], [0.3, 1.0, 0.5], [0.1, 0.5, 1.0]]) Sigma = np.diag(sigma) @ R @ np.diag(sigma) rf, ones = 0.03, np.ones(3) raw = np.linalg.solve(Sigma, mu - rf * ones) k = raw.sum() w_T = raw / k mu_T, var_T = w_T @ mu, w_T @ Sigma @ w_T beta = Sigma @ w_T / var_T print(np.round(Sigma @ w_T, 6), np.round((mu - rf) / k, 6)) print(round(var_T, 6), round((mu_T - rf) / k, 6)) print(round(mu_T - rf, 6), np.round(rf + beta * (mu_T - rf), 6)) print([round(0.03 + 0.05 * b, 4) for b in [-0.5, 0, 0.8, 1, 1.4]])

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From the tangency portfolio to the market

The capital asset pricing model (CAPM) adds an argument about everyone at once. Suppose every investor chooses by mean–variance utility (lesson 4.6.1) with the same $\boldsymbol{\mu}$ and $\Sigma$ . Suppose too that all can borrow and lend at $r_f$ , with no costs or taxes, over one period.

Then every investor holds the bill and $\mathbf{w}_T$ , in amounts set by their own $\gamma$ (two-fund separation, lesson 4.6.2).

Add up everyone's risky holdings: each investor's risky part has the same proportions, so the total has them too. That total is the market portfolio, every asset in proportion to its total market value. So the market portfolio is the tangency portfolio.

Rule: The capital asset pricing model

\mathbb{E}[R_i] = r_f + \beta_i\,\big(\mathbb{E}[R_m] - r_f\big)

$R_m$ is the return of the market portfolio M, and $\beta_i$ is measured against it. This is the security market line of case study 1.2.4: a stock's expected return is a straight line in its beta.

Case study 1.2.4 drew the line with an illustrative market return of 8%, giving $0.03 + 0.05\beta$ : 0.5% at $\beta = -0.5$ , 7% at 0.8 and 10% at 1.4. For the three assets the market is $\mathbf{w}_T$ , which expects 7.93%, so the slope is 4.93%.

Expected return against beta, measured two ways. Against the tangency portfolio every asset sits on the line. Against the 50/30/20 portfolio they miss it, and each gap is an alpha.

Only beta is paid

Split each asset's variance into the part that moves with the market and the part that is its own. As in case study 3.5.5, the best hedge with the market removes the share $\rho_{iM}^2$ , where $\rho_{iM}$ is the asset's correlation with the market.

Rule: Systematic and specific risk

\begin{gathered} \sigma_i^2 = \beta_i^2\,\sigma_M^2 + \sigma_{\varepsilon,i}^2 \\[4pt] \frac{\beta_i^2\sigma_M^2}{\sigma_i^2} = \rho_{iM}^2 \end{gathered}

$\beta_i^2\sigma_M^2$ is the systematic risk, which moves with the market. $\sigma_{\varepsilon,i}^2$ (epsilon) is the specific risk, the asset's own, which spreading money across many assets averages away.

asset	$\sigma_i^2$	$\beta_i^2\sigma_M^2$	systematic share	specific volatility
A	0.0144	0.005764	0.4003	0.0929
B	0.0324	0.016012	0.4942	0.128
C	0.0625	0.040991	0.6559	0.1467

The security market line pays for $\beta$ alone, not for $\sigma_i$ . An asset with C's 25% volatility but a beta of only 0.5 would be expected to return $0.03 + 0.5 \times 0.049274 = 5.46\%$ , less than A. Most of its risk would be specific, which investors can diversify away at no cost, so the market pays nothing for it.

\beta_i^2\sigma_M^2

import numpy as np mu = np.array([0.06, 0.08, 0.11]) sigma = np.array([0.12, 0.18, 0.25]) R = np.array([[1.0, 0.3, 0.1], [0.3, 1.0, 0.5], [0.1, 0.5, 1.0]]) Sigma = np.diag(sigma) @ R @ np.diag(sigma) rf, ones = 0.03, np.ones(3) raw = np.linalg.solve(Sigma, mu - rf * ones) w_T = raw / raw.sum() mu_T, var_T = w_T @ mu, w_T @ Sigma @ w_T beta = Sigma @ w_T / var_T systematic = beta**2 * var_T print(np.round(sigma**2, 4), np.round(systematic, 6), np.round(systematic / sigma**2, 4)) print(np.round(np.sqrt(sigma**2 - systematic), 4), np.round((Sigma @ w_T) / (sigma * np.sqrt(var_T)), 4)) print(round(rf + 0.5 * (mu_T - rf), 4))

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Alpha

An asset's alpha is how far its expected return sits above the line, as in case study 1.2.4. Against a benchmark portfolio B, with return $\mu_B$ :

Rule: Alpha

\alpha_i = \mu_i - r_f - \beta_i\,(\mu_B - r_f)

Here $\beta_i$ is measured against B. Weighted by B's own weights, the alphas add up to 0. Every alpha is 0 exactly when B is the tangency portfolio.

Against $\mathbf{w}_T$ every alpha is 0: that is the security market line. Now take case study 3.2.5's 50/30/20 mix as the benchmark, as if it were an index. Its betas are $(0.6833, 1.2244, 1.4551)$ and the alphas are

\boldsymbol{\alpha} = (-0.0014, -0.0063, 0.0131)

C sits 1.31 points above the benchmark's line, B 0.63 below. A positive alpha does not make C a bargain in any absolute sense. It says the benchmark is not mean–variance efficient: holding a little more C would raise its Sharpe ratio.

Moving 1% of the benchmark from B to C lifts its Sharpe ratio from 0.3852 to 0.3868, and from A to C to 0.3863. Moving all the way to $\mathbf{w}_T$ reaches 0.3951, where every alpha is 0. So an alpha is always relative to a benchmark: against a poor one, many assets seem to have it.

\mathbf{w}_T

import numpy as np mu = np.array([0.06, 0.08, 0.11]) sigma = np.array([0.12, 0.18, 0.25]) R = np.array([[1.0, 0.3, 0.1], [0.3, 1.0, 0.5], [0.1, 0.5, 1.0]]) Sigma = np.diag(sigma) @ R @ np.diag(sigma) rf, ones = 0.03, np.ones(3) def sharpe(w): return (w @ mu - rf) / np.sqrt(w @ Sigma @ w) def alphas(w): beta = Sigma @ w / (w @ Sigma @ w) return beta, mu - rf - beta * (w @ mu - rf) w_b = np.array([0.5, 0.3, 0.2]) beta_b, alpha_b = alphas(w_b) print(np.round(beta_b, 4), np.round(alpha_b, 4), abs(w_b @ alpha_b) < 1e-12) raw = np.linalg.solve(Sigma, mu - rf * ones) w_T = raw / raw.sum() print(round(sharpe(w_b), 4), round(sharpe(w_b + np.array([0, -0.01, 0.01])), 4), round(sharpe(w_b + np.array([-0.01, 0, 0.01])), 4), round(sharpe(w_T), 4)) print(np.abs(alphas(w_T)[1]).max() < 1e-12)

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On real data, beta is estimated from returns, as the hedge ratio was in case study 3.5.5. Fit the least-squares line of the asset's excess returns against the market's (lesson 3.5.3): its slope estimates beta and its intercept alpha.

Case study 5.10.5 does this with the tools of regression.

What the model claims, and what it does not

Under its assumptions, CAPM claims three things:

the market portfolio is the tangency portfolio;
every asset's expected excess return is its beta times the market's;
specific risk earns nothing, however large.

It does not claim:

That high-beta assets do better every year. The line is about expected returns. In a year when the market falls, high-beta assets tend to fall most.
That a stock index is the market. The market portfolio holds every risky asset, bonds and property included. A test against an index tests that index.
That no benchmark has alphas. Only the true market portfolio has none; against any other portfolio, alphas are normal.
That real returns follow it closely. Tests on stock returns have found the line flatter than the model predicts. Other features of firms explain part of the returns. Factor models add more betas, one for each source of systematic risk.

The model stays in daily use: as a first estimate of the return investors require, and to measure a fund's alpha against its benchmark.

Common mistakes

Beta is covariance over the market's variance, not over the asset's own: $\frac{(\Sigma\mathbf{w}_M)_i}{\sigma_M^2}$ . Equivalently $\beta_i = \rho_{iM}\frac{\sigma_i}{\sigma_M}$ , so a volatile asset can still have a small beta.
Measure beta and alpha against the same benchmark. Betas against the 50/30/20 mix are $(0.6833, 1.2244, 1.4551)$ ; against $\mathbf{w}_T$ , $(0.6088, 1.0147, 1.6236)$ .
Subtract $r_f$ on both sides. The line is $\mu_i - r_f = \beta_i(\mu_M - r_f)$ , not $\mu_i = \beta_i\mu_M$ .
Volatility is not what the line pays for. Only the systematic part, $\beta_i^2\sigma_M^2$ , earns a reward.
An alpha is relative. A positive alpha says the benchmark could be improved, not that the asset is cheap.
The line is about expectations. Realised returns scatter around it, widely, year by year.

The capital asset pricing model

0 of 11 correct

1.
An asset has covariance 0.018 with the market, whose variance is 0.0225. Find its beta
2.
With $r_f = 0.03$ and $\mathbb{E}[R_m] = 0.08$ , what expected return does CAPM give an asset with $\beta = 1.2$ ?
3.
On the same line, what does an asset with $\beta = -0.4$ expect?
4.
A portfolio holds 60% in an asset with $\beta = 0.5$ and 40% in one with $\beta = 1.5$ . What is its beta?
5.
An asset has correlation 0.6 with the market, volatility 0.3, and the market's volatility is 0.15. Find its beta
6.
For that asset, what share of its variance is systematic?
7.
Excess returns are $\mu_i - r_f = 0.04$ and $\mu_T - r_f = 0.05$ against the tangency portfolio. What is the asset's beta?
8.
With $r_f = 0.03$ and $\mathbb{E}[R_m] = 0.08$ , an asset with $\beta = 0.8$ expects 6%. What is its alpha?
9.
What expected return makes an asset's beta 1.2 on the line $0.03 + 0.05\beta$ give zero alpha?
10.
A benchmark holds 50% in A and 50% in B. A's alpha against it is 0.004. What is B's alpha?
11.
Two assets each have volatility 0.25. One has $\beta = 1.5$ , the other $\beta = 0.5$ . With $r_f = 0.03$ and $\mathbb{E}[R_m] = 0.08$ , by how much does CAPM say their expected returns differ?

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\beta_i = \frac{\operatorname{Cov}(R_i, R_M)}{\operatorname{Var}(R_M)}

The capital asset pricing model

Beta

Rule: Beta

The security market line

From the tangency condition

Rule: Security market line

From the tangency portfolio to the market

Rule: The capital asset pricing model

Only beta is paid

Rule: Systematic and specific risk

Alpha

Rule: Alpha

What the model claims, and what it does not

Common mistakes

The capital asset pricing model

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