CAPM: Beta, the Market Premium & the Security Market Line

You’ve climbed a long ladder. Diversification killed the risk you don’t get paid for, the efficient frontier drew the best-possible menu, and the Capital Market Line bolted the risk-free asset onto the tangency portfolio to give every investor one straight road. There’s one job left: pricing a single asset. Not a portfolio — one stock, one bond, one anything. How much return should the market hand you for holding it?

That’s the question the Capital Asset Pricing Model (CAPM) answers, and it does it with one wonderfully blunt claim: the only thing you get paid for is how much an asset swings with the market. Everything else is noise the market refuses to compensate. This lesson turns that claim into a single equation, a single line, and a single number — beta — that you can compute and read off a chart.

Only systematic risk is priced

Quick recap, because the entire model hinges on it. Any asset’s risk splits into two buckets:

• Idiosyncratic (specific) risk — the CEO resigns, a factory burns, a drug trial flops. These shocks are uncorrelated across companies, so in a diversified portfolio they cancel out. You can erase this risk for free just by owning enough names.
• Systematic (market) risk — recessions, rate shocks, wars. These hit everything at once, so no amount of diversification removes them. This is the risk that genuinely sticks.

Here’s the economic punchline. The market is a haggling crowd of rational investors, and it refuses to pay a premium for any risk you could have deleted yourself at zero cost. Why reward you for taking a risk you didn’t have to take? So idiosyncratic risk earns nothing. The only risk left standing — the only thing that deserves compensation — is systematic risk.

That’s the bridge from the Capital Market Line to CAPM. The CML priced efficient portfolios by their total volatility (because an efficient portfolio is already fully diversified, so all its risk is systematic). CAPM zooms in to price an individual asset, where we must be careful to charge only for its systematic slice. We need a clean ruler for “how much systematic risk does this one asset carry?” That ruler is beta.

Beta — the asset’s market sensitivity

Analogy. Think of the market as the tide and each asset as a boat. Beta measures how high your boat rides when the tide comes in. A beta of 1 means your boat rises exactly with the tide. A beta of 2 means you’re a flimsy dinghy that lurches twice as far. A beta of 0.5 is a heavy barge barely nudged. A beta of 0 floats in a separate pond entirely, and a negative beta is the weird boat that sinks when the tide rises — a natural hedge.

Precisely, beta is the covariance of the asset with the market, scaled by the market’s own variance:

$\beta_i=\frac{\mathrm{Cov}(R_i,R_M)}{\mathrm{Var}(R_M)}.$

Equivalently — and this is the version you can actually see — beta is the slope of the line you get when you regress the asset’s returns on the market’s returns. Run a scatter of “asset return this month” against “market return this month,” fit the best straight line through the cloud, and its steepness is beta. A steeper line means the asset amplifies market moves more.

Here’s how to read the number:

Worked example — computing beta from the formula

Suppose over many months you measure the covariance of a stock with the market as $\mathrm{Cov}(R_i,R_M)=0.012$ and the market’s variance as $\mathrm{Var}(R_M)=0.010$. Then:

$\beta_i=\frac{0.012}{0.010}=1.2.$

So this stock is mildly aggressive: when the market rises 10%, you’d expect it to rise about 12%, and when the market drops 10%, expect roughly a 12% fall. The covariance carried the direction and co-movement; dividing by the market’s own variance turned it into a clean, unitless sensitivity.

When it matters. Beta is the single input that turns “the market” into a price for your specific holding. The moment you want a required return, a discount rate for a project, or a hurdle rate for an acquisition, beta is the dial you reach for first.

The CAPM equation

Now we cash beta in for a return. CAPM says the fair expected return of any asset is the risk-free rate plus a reward proportional to its beta:

$\mathbb{E}[R_i]=r_f+\beta_i\big(\mathbb{E}[R_M]-r_f\big).$

Read it left to right like a price tag built in two pieces:

• $r_f$ — the risk-free rate: what you’d earn for taking no risk at all (think short government bills). This is the floor; every asset starts here.
• $\beta_i\big(\mathbb{E}[R_M]-r_f\big)$ — your risk reward: how many units of systematic risk you carry ($\beta_i$) times the price the market charges per unit.

That second factor, $\big(\mathbb{E}[R_M]-r_f\big)$, has a name worth memorizing: the market risk premium — the extra return the whole market earns over the risk-free rate for being risky at all. It’s the “going rate” for one unit of systematic risk. Beta scales it up or down for your specific asset.

Worked example — fair returns across betas

Let the risk-free rate be $r_f=3\%$ and the market risk premium be $7\%$ (so the market itself returns $\mathbb{E}[R_M]=3\%+7\%=10\%$). Plug a few betas into $\mathbb{E}[R_i]=3\%+\beta_i\times 7\%$:

Sanity checks fall right out: at $\beta=0$ you earn exactly the risk-free rate (no systematic risk, no premium), and at $\beta=1$ you earn exactly the market’s return (you are the market’s sensitivity). Everything in between is a straight-line interpolation, and a high-beta ($\beta=1.6$) asset is rewarded above the market for amplifying its swings.

The Security Market Line

Take the CAPM equation, put beta on the horizontal axis and expected return on the vertical axis, and plot it. Because the equation is linear in beta, you get a straight line — the Security Market Line (SML). It starts at the risk-free rate ($\beta=0$), passes through the market portfolio ($\beta=1,\ \mathbb{E}[R]=\mathbb{E}[R_M]$), and keeps climbing. Its slope is the market risk premium: each extra unit of beta buys you exactly one more premium’s worth of return.

Drag the slider below to walk an asset along its beta and watch the fair return the SML demands. The dots are example assets; the vertical connectors show how far each strays from the line (we’ll name that gap next).

The SML looks a lot like the Capital Market Line, but do not confuse them — they answer different questions:

The crucial difference: the CML only knows how to price fully diversified, efficient portfolios, because it measures risk as total volatility (which only equals systematic risk once you’re diversified). The SML works for anything — a single stock, a lopsided portfolio, your weird crypto bag — because it measures risk as beta, the systematic slice, which is the only thing priced regardless of how undiversified you are.

Alpha — beating (or losing to) the line

The SML tells you what an asset should return given its beta. But what an asset is actually forecast to return can differ. The gap between the two is alpha:

$\alpha_i=\mathbb{E}[R_i]-\big[r_f+\beta_i(\mathbb{E}[R_M]-r_f)\big].$

In words: alpha = actual expected return − CAPM-fair return. It’s the part of your return that beta can’t explain — the “free” excess (or shortfall) after you’ve accounted for systematic risk.

• Above the SML → positive alpha. The asset offers more than its risk deserves. It’s underpriced — a buy. Investors will pile in, bidding the price up and the future return back down toward the line.
• Below the SML → negative alpha. It offers less than its risk deserves. It’s overpriced — a sell (or short). In a perfectly efficient market, alpha is competed away to zero and everything sits on the line.

Worked example — one of each

Keep $r_f=3\%$ and market premium $7\%$, so fair return is $3\%+\beta\times7\%$.

• Stock U has $\beta=0.8$ and you forecast $\mathbb{E}[R_U]=11\%$. Fair return $=3\%+0.8\times7\%=8.6\%$. Alpha $=11\%-8.6\%=\mathbf{+2.4\%}$. It sits above the line — underpriced, a buy.
• Stock O has $\beta=1.6$ and you forecast $\mathbb{E}[R_O]=11.5\%$. Fair return $=3\%+1.6\times7\%=14.2\%$. Alpha $=11.5\%-14.2\%=\mathbf{-2.7\%}$. It sits below the line — you’re taking aggressive risk and getting underpaid for it. Overpriced, a sell.

Notice Stock O has the higher raw return (11.5% vs 11%) yet the worse deal, because its beta demanded far more. Alpha, not raw return, is the scorecard once you adjust for risk.

Assumptions & criticisms

CAPM is elegant, which is a polite way of saying it leans on assumptions no real market obeys. Be honest about them — CAPM is a baseline, not gospel.

The strong assumptions:

• Everyone is a mean-variance optimizer holding well-diversified portfolios (so only beta matters).
• Homogeneous expectations — every investor agrees on the same expected returns, variances, and covariances. (In reality we wildly disagree; that’s what makes a market.)
• Everyone can borrow and lend freely at the same risk-free rate $r_f$, with no limits.
• A single period, frictionless world — no taxes, no transaction costs, no liquidity constraints, all assets infinitely divisible.

Where it fails empirically:

• The low-beta anomaly. When you actually measure it, the relationship between beta and return is too flat — low-beta stocks have historically returned more than CAPM predicts, and high-beta stocks less. The real-world line is gentler than the theoretical SML.
• Size and value effects. Small-cap stocks and cheap “value” stocks have earned returns CAPM can’t explain by beta alone. This is what drove the Fama–French factor models, which add size and value (and later, profitability and investment) as extra priced risks beyond the market.
• Roll’s critique. CAPM is built on “the market portfolio” — all risky assets, everywhere. But that portfolio is unobservable; any index we use (the S&P 500, say) is just a proxy. Roll argued this makes CAPM nearly untestable: a rejection might mean the model is wrong, or just that we picked the wrong proxy.

Key Takeaways

That completes the toolkit — diversification, the frontier, the CML, and CAPM. Next: the final exam.