CAPM: the capital asset pricing model and the security market line

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The CAPM (capital asset pricing model) is the foundational model of capital market theory: the expected return of an investment depends only on its market risk, measured as beta. If you bear more market risk, you are rewarded with more return; risk that can be diversified away, on the other hand, earns nothing. The model is elegant and used everywhere, but it explains the returns in the data surprisingly poorly.

The CAPM formula

The formula says that every investment lies on a straight line, the security market line. A beta of 1 means the market return, a beta of 0 the risk-free rate, and a beta of 1.5 one and a half times the market risk premium on top of the risk-free rate. Any further fluctuation of an investment (unsystematic risk) plays no part in the expected return, because it evens out in a broad portfolio.

If a fund's actual return is above what its beta would lead you to expect, this is called alpha (Jensen (1968) introduced it as a measure of fund performance). If the stock with a beta of 1.2 in the example returned 9% instead of 8%, its alpha would be 1 percentage point.

Capital market line and security market line

The CAPM builds on the portfolio theory of Markowitz (1952). Tobin (1958) showed that, with a risk-free asset, all investors hold the same risky portfolio and only vary the share held at the safe interest rate (separation theorem). In equilibrium, this common portfolio must be the market itself. This gives two straight lines:

• Capital market line: return against volatility for efficient portfolios made up of the market and the safe asset. Its slope is the Sharpe ratio of the market.
• Security market line: return against beta for every individual investment. Inefficient individual stocks also lie on it, because their residual risk is precisely not rewarded.

A single stock with the same volatility of 10.5% but a beta of only 0.3, by contrast, would lie far below the capital market line: its expected return under the CAPM would be 2 + 0.3 × 5 = 3.5%. Most of its risk is unsystematic and is not rewarded. That is exactly why a broadly diversified portfolio is the starting point for any sensible investment.

The model was derived almost simultaneously by Sharpe (1964), Lintner (1965) and Mossin (1966). Sharpe received the Nobel Memorial Prize in Economic Sciences for it in 1990.

The assumptions

1. Investors judge portfolios only by expected return and variance over a single period.
2. Everyone has the same expectations about returns, volatilities and correlations.
3. Everyone can lend and borrow without limit at the risk-free rate.
4. No taxes, no transaction costs, infinitely divisible assets, no short-selling restrictions.
5. The market portfolio contains all the risky assets in the world.

None of these assumptions holds exactly. Black (1972) showed what happens if you cannot borrow at the safe interest rate: the core of the model survives, but the risk-free rate is replaced by the return of a portfolio with zero beta (zero-beta CAPM). This makes the security market line flatter: low-beta investments return more, and high-beta investments less, than the simple CAPM predicts.

What the research shows

The empirical record is sobering. Fama and French (1992) found for US stocks that company size and the ratio of book value to market value capture the differences in average returns well, while the relationship between beta and return is flat once beta is separated from size. In their review article, Fama and French (2004) conclude that the CAPM remains popular because of its simplicity but performs poorly in the data; extended factor models explain returns better.

Frazzini and Pedersen (2014) demonstrated the flat line in many markets: high beta goes hand in hand with low alpha, in US stocks, in 20 international stock markets, in government and corporate bonds and in futures. Their explanation: many investors are not allowed to borrow or do not want to, and therefore buy high-beta assets to achieve more return; this drives up their prices and lowers their subsequent returns. A strategy that buys low-beta assets with leverage and sells high-beta assets short (betting against beta) earned clearly positive risk-adjusted returns. More on this in the article low-volatility anomaly.

The consequence was not a farewell to the CAPM but its extension: alongside the market came further factors such as size and value, later profitability, investment and momentum. The Fama-French model is now the standard for explaining fund returns; the CAPM remains its special case with only one factor.

More fundamental is the criticism by Roll (1977): the true market portfolio contains all risky assets (stocks, bonds, real estate, human capital) and cannot be observed. Every test using a stock index therefore only checks whether this index is efficient, not the CAPM itself.

In practice, the problem shows up in the choice of benchmark index: a US company has a different beta against a US stock index than against a world equity index. For an investor with a globally diversified portfolio, a global index is the more suitable yardstick; a beta against a single-country index then says little about the risk in your own portfolio.

For advanced readers: estimating and testing beta

Beta is estimated with a regression of the investment's excess returns on the market's excess returns. The intercept is Jensen's alpha, and the coefficient of determination R² shows what share of the fluctuation comes from the market.

• Estimation error: the betas of individual stocks are uncertain and change over time. Many providers therefore shrink them toward 1; portfolios of many stocks have more stable betas.
• Period and frequency: with daily data, the beta of thinly traded stocks is underestimated, because their prices lag behind the market. 60 monthly returns are customary.
• Cross-sectional test: Fama and MacBeth (1973) estimate a regression of the returns of all stocks on their betas every month and average the slopes. Under the CAPM, the average slope should equal the market risk premium and the intercept the safe interest rate.
• Multiple factors: the arbitrage pricing theory of Ross (1976) allows several risk factors instead of a single market factor. It is the basis of today's factor models.

What the CAPM is still good for

In theory, the CAPM is a special case of general asset pricing theory: the price of every asset follows from its covariance with a stochastic discount factor, which in the CAPM depends linearly on the market return. The textbook Cochrane (2005) develops this framework, which also includes multi-factor models.

• Cost of capital: companies and valuers often estimate the cost of equity with the CAPM, because it is simple and transparent. A company with a beta of 1.3 would have a cost of equity of 2 + 1.3 × 5 = 8.5% with a 2% interest rate and a 5% market risk premium; investments must earn at least that much.
• Fund evaluation: alpha and beta against an index are the first question to ask of any active fund: does the return come from market risk or from skill?
• Diversification: the central message holds regardless of the tests. Unsystematic risk is not rewarded, so you should not bear it.
• Reference point: deviations from the CAPM (size, value, momentum, low beta) only became visible through the comparison with the model.
• Leverage instead of individual risk: according to the CAPM, if you want more return from the market portfolio, you increase your market allocation or leverage it, rather than switching to risky individual high-beta stocks. The fact that many investors cannot or will not do this is the core of the explanation for the flat security market line.

How Portfolio Lab calculates it

In the Backtest with a benchmark, Portfolio Lab calculates beta from the covariance of the period returns of the portfolio and the benchmark, divided by the variance of the benchmark. Alpha p.a. is the mean of the portfolio's excess returns over beta times the benchmark's excess return, multiplied by 12 (or 52 or 252), i.e. Jensen's alpha using the time-varying money market rate. In addition, there are R², the Treynor ratio and M². In the Factors analysis, the market is the first of three or five factors. Details are in the metrics methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.