Using and testing CAPM

RiskAtoms Bank plc's board asks a simple question: what return do our shareholders expect? The answer sets the hurdle every new business line must clear. Most firms answer it with CAPM, which needs a beta, and a beta is an estimate with an error bar.

L3.1.09: CAPM, beta and the security market line gave the security market line: $\mathbb{E}[R_i] = r_f + \beta_i(\mathbb{E}[R_m] - r_f)$. Two of its three inputs are judgements about the future (the risk-free rate and the equity risk premiumThe expected return on the whole equity market above the risk-free rate.). The third, beta, is estimated from the past.

Estimating beta. Collect monthly returns for the share and for a market index, subtract the risk-free rate from both, and fit a straight line by least squares (M-D03: Linear regression). The slope is beta. In L2.5.04: Sensitivities beta was the covariance with the market divided by the market's variance; the least-squares slope is exactly that number.

The bank's estimate is 1.46 although the data were made with 1.20. Nothing went wrong: the share's own noise happened to line up with the market a little more than usual. The 95% confidence intervalA range built so that, in 95% of samples, it contains the true value., 1.08 to 1.84, does contain the true value. Its width comes from the standard errorThe standard deviation of an estimate across the samples you might have drawn: how far it typically misses the true value. (M-D04: Confidence intervals and hypothesis tests).

Window and frequency. More months shrink the standard error, but a firm's beta drifts as its business and debt change. Common choices are five years of monthly returns or two years of weekly returns. For the bank:

Adjusted betas. Marshall Blume found in 1971 that betas estimated in one period drift towards 1 in the next: a high estimate is partly bad luck that will not repeat. The common adjusted betaA raw regression beta pulled towards 1, because extreme estimates tend to be partly noise. takes two-thirds of the raw estimate plus one-third of 1. Oldrich Vasicek's 1973 version weights the raw estimate by how precise it is: a noisy estimate is pulled harder towards the average beta.

Cost of equity. Put the beta into the security market line and you have the cost of equityThe return shareholders require for holding a company's shares; the discount rate for its dividends.. It is the $r$ in the dividend discount model of L1.5.03: Valuing equities: the dividend discount model, $P_0 = D_1/(r - g)$. Because $r - g$ is small, a modest error in beta moves the value a lot. Try it below with RiskAtoms Bank plc.

Let $y_t = R_{i,t} - r_f$ and $x_t = R_{m,t} - r_f$ for months $t = 1, \dots, n$. The model and its least-squares estimates are:

Where the standard error comes from. Substitute the model into $\hat\beta$. Since $\sum_t (x_t - \bar x) = 0$, the $\alpha$ and $\bar y$ terms drop out:

The estimate is unbiased (its error averages zero), and its spread falls as the market moves more ($S_{xx}$ larger) and as the share's own noise falls. With normal errors, $(\hat\beta - \beta)/\operatorname{SE}(\hat\beta)$ follows a t distribution with $n - 2$ degrees of freedom, which gives the interval and tests.

Adjusted betas, on a few lines. Blume's rule and Vasicek's precision-weighted average, with a prior (the spread of betas across all shares) of mean $\bar\beta$ and standard deviation $\sigma_{\text{prior}}$:

Cost of equity and value: $r_E = r_f + \beta\,\text{ERP}$, then $P_0 = D_1/(r_E - g)$. Going the other way, a market price implies a return $r = D_1/P_0 + g$, which can be compared with CAPM's.

Start with RiskAtoms Bank plc over 60 months. Shorten the window to 24 months and watch the interval widen. Switch to the software company: it is noisier, so its interval is the widest. Then reveal the true beta and see whether the interval caught it.

The 60-month regression of the bank's excess returns on the market's gives these sums (monthly returns as decimals): $\bar x = 0.005000$, $\bar y = 0.003470$, $S_{xx} = 0.119475$, $S_{xy} = 0.174457$, and a residual sum of squares of $0.247575$. Try each step before revealing it.

CAPM makes a sharp prediction: across assets, average excess returns should lie on a straight line through the origin with slope equal to the market's excess return. Tests run in two passes. First, estimate each asset's beta by time-series regression. Second, regress average returns on those betas across assets: a second-pass regressionA regression across assets or portfolios of average return on estimated beta, used to test the security market line.. Using portfolios sorted on beta, rather than single shares, cuts the error in the betas.

The illustrative fit has intercept 2.59% and slope 2.73% a year, against CAPM's 0% and 6%. Low-beta portfolios earn more than predicted, high-beta ones less. That is the shape the published tests found:

• Black, Jensen and Scholes (1972) sorted US shares into portfolios by beta. Average return rose with beta in a straight line, but the line was flatter than CAPM predicts and its intercept was above the risk-free rate.
• Fama and MacBeth (1973) ran the second-pass regression month by month and averaged the slopes, a method still used today. They found a positive, linear relation between beta and return.
• Roll (1977) pointed out that CAPM is really a statement about the market portfolio of all risky assets, which no one observes. A test with a stock index proxyThe index used in place of the true market portfolio, which would include every risky asset. tests whether that index is efficient, not CAPM itself.
• Fama and French (1992) found that, for US shares from 1963 to 1990, beta explained almost none of the differences in average returns once firm size and book-to-market were taken into account. Their 2004 review concluded that its empirical record is “poor enough to invalidate the way it is used in applications”.

The classic anomalies. Each anomalyA pattern in average returns that a pricing model, here CAPM, does not explain. is a way of sorting shares that produced average returns CAPM's betas could not explain. Open each card.

CAPM sets real prices. UK economic regulators for water, energy and telecoms set the allowed return on the equity of the companies they regulate largely with CAPM, and the UK Regulators Network published joint guidance on the method in 2023. Each review argues over the same three inputs as this lesson: the risk-free rate, the equity risk premium and beta, including how to measure it. For a bank, the cost of equity is the hurdle in risk-adjusted performance measures such as RAROC, covered in M7.1: Enterprise risk and economic capital.

• Beta is the slope of a regression of the share's excess returns on the market's: cov ÷ var.
• Its standard error is s ÷ √Sxx, and the 95% interval is β̂ ± t × SE; five years of monthly data still leave an interval of roughly ±0.3 to ±0.5.
• Adjusted betas pull estimates towards 1: Blume uses ⅔ raw + ⅓, Vasicek weights by precision.
• Cost of equity = rf + β × ERP feeds the dividend discount model; small changes in β move the value a lot.
• The evidence: the SML is flatter than CAPM predicts, and size, value, momentum and low-beta sorts earn returns beta does not explain.

Standard error

The standard deviation of an estimate across the samples you might have drawn: how far it typically misses the true value.

Confidence interval

A range built so that, in 95% of samples, it contains the true value.

Cost of equity

The return shareholders require for holding a company's shares; the discount rate for its dividends.

Equity risk premium

The expected return on the whole equity market above the risk-free rate.

Adjusted beta

A raw regression beta pulled towards 1, because extreme estimates tend to be partly noise.

Market proxy

The index used in place of the true market portfolio, which would include every risky asset.

Second-pass regression

A regression across assets or portfolios of average return on estimated beta, used to test the security market line.

Anomaly

A pattern in average returns that a pricing model, here CAPM, does not explain.