CAPM and Beta: Why Some Risk Gets Paid and Some Doesn't

CAPM answers a question that sounds simple and isn't: what return should you expect from a risky asset, given how risky it is? The formula looks small, but the argument behind it — that not all risk deserves to be paid for — is one of the more counterintuitive ideas in finance.

The formula

E(R) = Rf + β · (E(Rm) − Rf)

• E(R) — expected return of the asset
• Rf — the risk-free rate (Treasury yield)
• β (beta) — the asset's sensitivity to the overall market
• E(Rm) − Rf — the "market risk premium," how much more the market is expected to return over the risk-free rate

Read left to right: your expected return is the risk-free rate, plus your beta times the extra return the market as a whole is expected to earn over risk-free. A beta of 1 means "moves with the market, gets exactly the market's extra return." A beta of 2 means amplified market exposure, and a proportionally larger expected return. A beta of 0 — no correlation with the market at all — means CAPM predicts you should expect only the risk-free rate, no premium at all, regardless of how volatile that asset is on its own.

That last case is the whole point of the model.

Why total risk isn't what gets paid

It's tempting to assume riskier assets should always pay more, full stop. CAPM says that's wrong, and the argument is diversification. If an asset's volatility is entirely uncorrelated with the market — pure company-specific risk, like a lawsuit or a factory fire — a diversified investor can eliminate that risk almost completely just by holding many uncorrelated assets together; the idiosyncratic wobbles average out across the portfolio. Since that risk is avoidable for free (just diversify), the market doesn't need to pay you extra to bear it — and in an efficient market, it won't.

What diversification can't eliminate is the risk that moves with everything else at once — a recession, a rate shock, a systemic crisis. That's systematic risk, and beta measures exactly how exposed a given asset is to it. CAPM's claim is that systematic risk is the only risk that earns a premium, because it's the only risk a rational, diversified investor is actually forced to bear.

Computing beta

Beta is a regression coefficient — literally the slope you get from regressing an asset's returns against the market's returns:

β = Cov(R_asset, R_market) / Var(R_market)

import statistics

def beta(asset_returns, market_returns):
    n = len(asset_returns)
    mean_a = statistics.mean(asset_returns)
    mean_m = statistics.mean(market_returns)

    cov = sum((a - mean_a) * (m - mean_m) for a, m in zip(asset_returns, market_returns)) / (n - 1)
    var_m = statistics.variance(market_returns)

    return cov / var_m

# Toy weekly returns — asset moves roughly 1.5x the market
market = [0.01, -0.02, 0.015, 0.005, -0.01, 0.02, -0.005]
asset  = [0.018, -0.028, 0.021, 0.009, -0.017, 0.031, -0.006]

b = beta(asset, market)
print(round(b, 3))  # comes out close to 1.5

Once you have beta, plugging it into the CAPM formula gives the expected return — and comparing that expected return against an asset's actual average return is exactly how "alpha" gets defined: alpha is the return an asset earned above and beyond what CAPM says it should have, given its systematic risk. Positive alpha is what every active fund manager claims to generate and what CAPM predicts should, on average, be zero.

What CAPM gets wrong

CAPM assumes markets are efficient, investors are rational and diversified, and that beta is a complete description of risk. Decades of empirical work have chipped at all three — small-cap stocks and value stocks have historically earned returns CAPM doesn't fully explain, which is exactly the gap multi-factor models (Fama-French and its descendants) were built to close. CAPM isn't the final word on asset pricing. It's the starting point every later model is a response to — which is exactly why it's still the first thing taught.