How much return should an investor demand for holding a risky share? The Capital Asset Pricing Model (CAPM) gives a remarkably simple answer: the risk-free rate plus a reward for the share's exposure to market-wide risk. Developed in the early 1960s, mainly by William Sharpe, with independent work by John Lintner, Jan Mossin and Jack Treynor, CAPM is still the most common way companies estimate their cost of equity and the starting point for nearly every finance course.
The Big Idea
Portfolio theory shows that diversification removes unsystematic risk, the part of a company's risk that is unique to it. What remains is systematic risk, caused by forces that hit the entire market, such as interest rate changes, inflation or political uncertainty.
CAPM's key insight follows: since investors can eliminate unsystematic risk for free by diversifying, the market will not reward them for bearing it. Only systematic risk earns a risk premium. And systematic risk is measured by a single number, beta.
The CAPM Formula
$$ E(R_i) = R_f + \beta_i\left[E(R_m) - R_f\right] $$
where:
- \( E(R_i) \) is the expected (required) return on asset \( i \);
- \( R_f \) is the risk-free rate, usually the yield on short-term government securities such as Pakistan's Market Treasury Bills or US Treasury bills;
- \( E(R_m) \) is the expected return on the market portfolio, often proxied by an index such as the KSE-100 or the S&P 500;
- \( E(R_m) - R_f \) is the market risk premium;
- \( \beta_i \) is the asset's beta.
Understanding Beta
Beta measures how strongly a stock's return moves with the market:
$$ \beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)} $$
- \( \beta = 1 \): the stock moves in line with the market.
- \( \beta > 1 \): the stock is aggressive; it amplifies market movements. Cyclical industries such as cement, steel and automobiles often fall in this group.
- \( \beta < 1 \): the stock is defensive. Utilities, food producers and pharmaceuticals often have low betas.
- \( \beta = 0 \): no systematic risk, like the risk-free asset.
Beta is also the slope of a regression line of the stock's returns on the market's returns.
Worked Example 1: Calculating Beta from Data
A student has five years of annual returns for a stock and the market index:
| Year | Market return (%) | Stock return (%) |
|---|---|---|
| 1 | 10 | 14 |
| 2 | −4 | −8 |
| 3 | 15 | 20 |
| 4 | 6 | 5 |
| 5 | 8 | 9 |
Step 1: Means.
$$ \bar{R}_m = \frac{10 - 4 + 15 + 6 + 8}{5} = \frac{35}{5} = 7\% $$
$$ \bar{R}_i = \frac{14 - 8 + 20 + 5 + 9}{5} = \frac{40}{5} = 8\% $$
Step 2: Deviations and products.
- Year 1: market deviation 3, stock deviation 6, product 18, squared market deviation 9.
- Year 2: \( -11 \) and \( -16 \), product 176, square 121.
- Year 3: 8 and 12, product 96, square 64.
- Year 4: \( -1 \) and \( -3 \), product 3, square 1.
- Year 5: 1 and 1, product 1, square 1.
Step 3: Sample covariance and variance (dividing by \( n-1 = 4 \)).
$$ \text{Cov}(R_i,R_m) = \frac{18 + 176 + 96 + 3 + 1}{4} = \frac{294}{4} = 73.5 $$
$$ \text{Var}(R_m) = \frac{9 + 121 + 64 + 1 + 1}{4} = \frac{196}{4} = 49 $$
Step 4: Beta.
$$ \beta = \frac{73.5}{49} = 1.5 $$
The stock is aggressive: when the market rises or falls by 1 percentage point, this stock tends to move by about 1.5 points. (The divisor cancels, so using \( n \) instead of \( n-1 \) gives the same beta.)
In practice analysts use 36 to 60 monthly returns rather than five annual ones, but the method is identical. In Excel, =SLOPE(stock_range, market_range) returns beta directly; in R, coef(lm(stock ~ market))[2] does the same.
Worked Example 2: Required Return
Suppose the risk-free rate is 12% and the expected market return is 17%. What return should investors require from the stock in Example 1?
Step 1: Market risk premium.
$$ E(R_m) - R_f = 17\% - 12\% = 5\% $$
Step 2: Apply CAPM.
$$ E(R_i) = 12\% + 1.5 \times 5\% = 12\% + 7.5\% = 19.5\% $$
Investors should demand 19.5% from this share. A company with this beta would use 19.5% as its cost of equity when evaluating projects or computing its weighted average cost of capital.
Sanity check. The stock is riskier than the market (beta above 1), so its required return should exceed the market's 17%. It does.
The Security Market Line
Plotting CAPM with beta on the horizontal axis and expected return on the vertical axis gives a straight line called the security market line (SML). It crosses the vertical axis at \( R_f \) and has slope equal to the market risk premium. Every fairly priced asset should lie on this line.
If an analyst's forecast return for a stock lies above the SML, the stock offers more than CAPM requires, so it appears undervalued. If it lies below, it appears overvalued. The gap is called Jensen's alpha:
$$ \alpha = \text{Forecast return} - \text{CAPM required return} $$
Worked Example 3: Spotting Mispriced Stocks
With \( R_f = 12\% \) and \( E(R_m) = 17\% \), an analyst forecasts a return of 17% for both Stock X (\( \beta = 0.8 \)) and Stock Y (\( \beta = 1.2 \)). Which, if either, should she buy?
Stock X:
$$ E(R_X) = 12\% + 0.8 \times 5\% = 16\%, \qquad \alpha_X = 17\% - 16\% = +1\% $$
Stock Y:
$$ E(R_Y) = 12\% + 1.2 \times 5\% = 18\%, \qquad \alpha_Y = 17\% - 18\% = -1\% $$
Stock X lies above the SML and looks undervalued: buy. Stock Y lies below it and looks overvalued: avoid or sell. Both are forecast to return the same 17%, but Y carries more systematic risk, so it should be paying more.
Portfolio Beta
The beta of a portfolio is the weighted average of the individual betas. A portfolio of 40% X and 60% Y has
$$ \beta_P = 0.4(0.8) + 0.6(1.2) = 0.32 + 0.72 = 1.04 $$
and a CAPM required return of \( 12\% + 1.04 \times 5\% = 17.2\% \).
Assumptions and Criticisms
CAPM rests on strong assumptions: investors hold diversified portfolios, can borrow and lend at the risk-free rate, share the same expectations, and face no taxes or transaction costs. None of these is exactly true.
Empirical tests have found that beta alone does not fully explain average returns. In 1992 Eugene Fama and Kenneth French showed that company size and book-to-market ratio also matter, leading to their three-factor model. Arbitrage Pricing Theory, developed by Stephen Ross, allows several risk factors too. Even so, CAPM remains the standard first estimate of the cost of equity because it is simple and grounded in clear logic.
Applying CAPM in Pakistan
Students writing theses on the Pakistan Stock Exchange often estimate betas for KSE-100 companies using monthly returns, with the T-bill yield as \( R_f \). Because Pakistani interest rates have changed sharply in recent years, results can be very sensitive to the period chosen, so it is good practice to state the sample period and test the robustness of your estimates.
Did You Know?
- William Sharpe shared the 1990 Nobel Memorial Prize in Economic Sciences with Harry Markowitz and Merton Miller.
- A stock can in principle have a negative beta; gold-mining shares and gold itself have sometimes behaved this way during market crashes.
- The market portfolio itself has a beta of exactly 1 by definition, since the covariance of the market with itself is its variance.
Key Takeaways
- CAPM: \( E(R_i) = R_f + \beta_i[E(R_m) - R_f] \).
- Beta is the covariance of a stock with the market divided by the market's variance.
- Only systematic risk is rewarded, because unsystematic risk can be diversified away.
- Stocks above the security market line have positive alpha and appear undervalued.
- Portfolio beta is the weighted average of the individual betas.
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