CAPM — risk and return in one equation

The economic story

The Capital Asset Pricing Model (Sharpe 1964, Lintner 1965, Mossin 1966 — sometimes called Sharpe-Lintner CAPM) gives the simplest "right answer" for what a stock's expected return should be:

$E[r_i] = r_f + \beta_i \cdot (E[r_m] - r_f)$

In words: the expected return on stock $i$ equals the risk-free rate, plus beta times the market risk premium. Concrete: risk-free rate 4%, expected market return 10%, beta 1.5 → $4\% + 1.5 \times 6\% = 13\%$ (the pretest).

Components, read aloud:

• $E[r_i]$ — "the expected return on asset $i$": the return you'd get on average, per A1-00's $E[X]$ notation
• $r_f$ — "r sub f", the risk-free rate: what you earn taking no risk at all (e.g., a 1-year Treasury yield)
• $E[r_m] - r_f$ — the expected market return minus the risk-free rate: the market risk premium (also called the equity premium), the extra return the whole stock market pays for bearing its risk
• $\beta_i$ — "beta sub $i$", the asset's sensitivity to market moves: beta 1.5 means when the market moves 1%, this stock tends to move 1.5%

The logic: you should be compensated only for risk you can't diversify away.

Suppose you hold a well-diversified portfolio — many stocks across many industries. Each stock's idiosyncratic risk (risk specific to that one company, like Apple-specific news) cancels out against the others. Only the market risk — the part correlated with everything else — remains. So in equilibrium, the market only pays a risk premium for the un-diversifiable part. That part is exactly what beta measures.

Why this is beautiful (and why it's wrong)

Beautiful because it gives a clean prediction:

• A stock's expected return depends only on its beta.
• Two stocks with the same beta should have the same expected return, regardless of any other property (industry, size, valuation).
• High-beta stocks should outperform low-beta stocks by the market risk premium times the beta gap.

Wrong in practice:

• Empirically — in the actual data — low-beta stocks outperform what CAPM predicts (the "low-beta anomaly," Frazzini & Pedersen 2014).
• High-beta stocks underperform predictions.
• Stocks with the same beta but different sizes (small vs. large cap — cap is market capitalization, the company's total market value) have systematically different returns. Pure CAPM says this shouldn't happen.

Fama and French (1992) put a fork in single-factor CAPM. They showed that size and book-to-market — a company's accounting value divided by its market price; high means the stock is cheap — explain cross-sectional returns (the differences in return between stocks) better than beta alone. This is the launching point for multi-factor models (next several lessons in D2).

But CAPM is still the right baseline — every more-sophisticated model is a correction to it.

How to use CAPM in practice

Three practical applications:

1. Hurdle rate for evaluating a stock

A hurdle rate is the minimum return that makes holding a stock worthwhile. What return is reasonable to expect?

$E[r] = r_f + \beta \cdot \text{MRP}$

In words: expected return equals the risk-free rate plus beta times the market risk premium (MRP — the same $E[r_m] - r_f$ from above, just abbreviated). With $r_f = 4\%$, MRP = 6%, $\beta = 1.2$: expected return $= 4\% + 1.2 \cdot 6\% = 11.2\%$.

If you believe the stock will return more than this, it's underpriced (positive alpha — return above what its market exposure explains). If less, overpriced.

2. Required return for capital budgeting

Companies use CAPM to set the discount rate — the rate used to shrink future cash into today's money — when computing a project's NPV (net present value, the total worth of a project's future cash flows in today's terms):

$\text{NPV} = \sum_t \frac{\text{cash flow}_t}{(1 + r_\text{CAPM})^t}$

In words: for each future period $t$, take that period's cash flow and divide it by $(1 + \text{the CAPM rate})$ once for every period you have to wait; the $\sum_t$ is A1-00's for-loop, adding those discounted amounts up. Concrete: 100 dollars arriving in 2 years, discounted at 10% per year, is worth $100 / 1.1^2 \approx 82.6$ dollars today.

Higher business risk means higher $\beta$, which means a higher required return, which means a lower NPV for the same cash flows.

3. Decomposing strategy returns

Take a strategy's returns $r_s$ ("r sub s") and regress them on the market's returns — fit the straight-line regression from A2-06:

$r_s = \alpha + \beta r_m + \varepsilon$

In words: the strategy's return equals alpha ($\alpha$ — the intercept, the return not explained by market exposure; this is your claimed edge), plus beta times the market's return, plus $\varepsilon$ (epsilon — the leftover noise the fit doesn't explain). Then read the fitted numbers:

• $\alpha > 0$ with $|\beta|$ reasonable: real edge (subject to all our hypothesis-testing caveats).
• $\alpha \approx 0$, $\beta = 1$: you're effectively holding the market.
• $\alpha < 0$, $\beta > 1$: you're paying for leveraged market exposure (worse than buying SPY on margin).

The market risk premium — how big is it really?

The historical equity risk premium (US, 1928–present): roughly 5–6% per year. Globally and over longer windows: 3–5%.

This is the biggest single edge available to long-term investors. Compound 5% per year for 30 years and you have ~4.3× your money in real terms — that is, after inflation.

Forward-looking estimates (Damodaran, Asness, others) suggest the future MRP might be lower (3–4%) than historical (5–6%), because current valuations are high. Reasonable people disagree by 2–3 percentage points. The right answer is "use a range and stress-test."

Why CAPM isn't enough

Fama-French (1992) showed that:

1. Size effect — small caps earned more than CAPM predicted in the historical data FF studied (though this premium has weakened substantially since publication).
2. Value effect — high book-to-market ("value" — cheap relative to accounting worth) stocks earn ~5% per year more than CAPM predicts.

This led to the Fama-French 3-Factor model: market + size (SMB, "small minus big") + value (HML, "high minus low"). Later extended to 5 factors (adding profitability and investment quality). We'll meet each in subsequent lessons.

The pattern is: CAPM is a useful baseline. Real-world expected returns are explained by multiple factors. The whole field of "factor investing" / "smart beta" is built on this.

The most important practical takeaway

Always benchmark a strategy's return against the CAPM prediction, not against zero. Example: risk-free rate $r_f = 4\%$ and a 12% market return, so MRP = 8%. A strategy earns 15% per year with $\beta = 1.5$. Its CAPM-required return is $4\% + 1.5 \cdot 8\% = 16\%$ — so its alpha is $15\% - 16\% =$ −1%, despite the impressive-looking headline number. It's not generating edge; it's just leveraging beta inefficiently. Most retail strategies fall into this trap.

We'll see this rigorously in D4 (backtesting methodology) when we discuss benchmark choice.

Try it

Turn the CAPM equation into code:

Predict before the next lesson

Tomorrow we'll look at regression diagnostics — when does OLS (ordinary least squares, the standard line-fitting method from A2-06) lie to you? Predict:

• If two of your predictors are perfectly correlated, what happens to their estimated coefficients?
• What does a "fat-tailed" residual distribution (residuals — the leftover errors after the fit) imply about hypothesis tests on the regression?