Last week, you drew a critical distinction: systematic risk is the risk that remains in a well-diversified portfolio — the economy-wide shocks that affect every firm — while unsystematic risk is the firm-specific noise that disappears as you add more stocks. The market doesn't reward you for bearing unsystematic risk because you can eliminate it for free by diversifying. The logical next question is: if only systematic risk matters, how do we measure it?
The answer is beta (β). Beta measures how sensitive a particular asset's returns are to movements in the overall market. A stock with a beta of 1.0 tends to move in lockstep with the market: when the market rises 10%, the stock rises roughly 10%. A stock with a beta of 1.5 is more volatile — it amplifies market movements by 50%, so a 10% market gain translates to roughly a 15% gain for the stock, and a 10% market loss hits the stock with a 15% loss. Conversely, a stock with a beta of 0.6 is more muted, moving only about 60% as much as the market in either direction.
Think of beta as a leverage multiplier on market-wide news. High-beta stocks (technology firms, luxury goods producers, cyclical manufacturers) are highly sensitive to the business cycle. When the economy expands, their earnings surge; when it contracts, they get hit hard. Low-beta stocks (utilities, consumer staples, healthcare) are more insulated — people still need electricity and groceries regardless of whether GDP is growing. A beta of zero means the asset has no systematic risk at all; U.S. Treasury bills are the classic example, since their returns don't fluctuate with the stock market.
A few key benchmarks to keep in mind: the market portfolio itself, by definition, has a beta of exactly 1.0. Risk-free assets have a beta of 0. Most individual stocks fall somewhere between 0.5 and 2.0, though extreme values are possible.
Notice the symmetry: a high beta is a double-edged sword. Red Acre's shareholders enjoy amplified gains in good times, but they suffer amplified losses when the market turns. That's exactly why high-beta stocks need to offer higher expected returns — investors demand compensation for bearing that extra systematic risk.
One of the most useful properties of beta is that it combines linearly across a portfolio. The beta of a portfolio is simply the weighted average of the betas of the individual assets, where the weights are the fraction of portfolio value invested in each asset. This makes portfolio risk easy to compute — you don't need to worry about correlations the way you do when calculating portfolio standard deviation.
The portfolio beta formula has a powerful practical implication: you can dial your systematic risk exposure up or down by adjusting the mix. Want more risk (and more expected return)? Shift weight toward high-beta stocks. Want to reduce your exposure to market swings? Add low-beta stocks or risk-free assets. This is a much more targeted tool than simply "diversifying" — it lets you control the specific dimension of risk the market actually rewards.
This example illustrates a general principle: you can engineer any portfolio beta between the lowest and highest component betas by adjusting the weights. If you could borrow money (leverage), you could even push the portfolio beta above that of your highest-beta holding — but that's a topic for more advanced courses.
We now know that beta measures systematic risk, and we know that only systematic risk is rewarded by the market. The Capital Asset Pricing Model (CAPM) pulls these ideas together into a single equation that tells you exactly how much return you should expect for any given level of systematic risk. It's one of the most important formulas in all of finance.
The logic of the CAPM is straightforward. Start with the risk-free rate — that's the return you can earn with zero risk (Treasury bills). Now consider the extra return the market earns above the risk-free rate; this is the market risk premium, the reward investors collectively demand for bearing the average level of systematic risk (a beta of 1.0). The CAPM says: your asset's expected return is the risk-free rate plus a scaled portion of the market risk premium, where the scaling factor is your asset's beta.
Unpack the formula piece by piece. The term Rf is your baseline — the time value of money compensation you'd earn even with no risk. The term βi × [E(RM) − Rf] is your risk premium — the extra return above the risk-free rate that compensates you for bearing systematic risk. If β = 0, you earn only the risk-free rate. If β = 1, you earn the full market risk premium. If β > 1, you earn more than the market risk premium because you're taking on more systematic risk than the average stock.
It's worth pausing to see how powerfully beta drives expected returns. Using the same risk-free rate (4%) and market return (11%), look at how expected returns escalate across different betas:
That linearity is worth emphasizing. The CAPM doesn't say "more risk, more return" in some vague way — it gives you a precise exchange rate. Every additional unit of beta adds exactly one market risk premium to your expected return. This is the relationship depicted by the Security Market Line.
The Security Market Line is simply the graphical representation of the CAPM equation. On the horizontal axis, you plot beta; on the vertical axis, expected return. The SML is a straight line that starts at the risk-free rate (where β = 0) and passes through the market portfolio (where β = 1.0 and expected return = E(RM)). Every asset that is fairly priced should lie exactly on this line.
The slope of the SML is the market risk premium, E(RM) − Rf. When the market risk premium increases — because investors become more risk-averse, for instance during a recession — the SML steepens, and all risky assets must offer higher expected returns. When the risk-free rate changes, the entire SML shifts up or down.
The Security Market Line plots expected return against beta. Every fairly priced asset lies on the line. The y-intercept is the risk-free rate (4%); the slope is the market risk premium (7%). Each additional unit of beta adds exactly 7% to the required return.
Here's where the SML becomes a practical tool. If you have an estimate of what an asset's return will be (from analyst forecasts, your own analysis, etc.), you can compare that to what the CAPM says the return should be given the asset's beta. The difference between the two is sometimes called alpha (α):
If an asset plots above the SML — meaning its expected return exceeds the CAPM prediction — it's offering more return than its systematic risk warrants. This suggests it's undervalued: you're getting a bargain, a positive alpha. If an asset plots below the SML, the opposite is true: the expected return is too low for the risk, and the asset appears overvalued.
A word of caution: the mispricing analysis is only as good as both your beta estimate and your expected return forecast. In practice, small alphas (like Stock Y's −0.20%) could easily be noise. The CAPM gives you a framework for thinking about fair compensation for risk, but it's not an oracle.
Stock X (β = 0.8) plots above the SML — its forecast return of 12% exceeds the CAPM-required 9.60%, giving a positive alpha of +2.40%. Stock Y (β = 1.6) plots just below the SML — its forecast return of 15% falls slightly short of the required 15.20%, a negative alpha of −0.20%.
You can solve the CAPM for any one of its four variables — expected return, risk-free rate, beta, or expected market return — as long as you know the other three. This flexibility makes it a versatile analytical tool.
So far, we've used the CAPM as a tool for understanding expected returns on investments. Now we're going to pivot and look at it from the firm's perspective. When a company raises money from shareholders, those shareholders expect a return — they have alternatives, and they won't invest in your stock unless they expect to earn at least as much as other investments of comparable risk. That minimum required return is the firm's cost of equity.
The cost of equity is a critical input in corporate finance. It's one of the key components of a firm's weighted average cost of capital (WACC), which we'll formalize in Week 10. The WACC, in turn, is the discount rate firms use to evaluate new projects — so getting the cost of equity right directly affects which investments a company pursues.
There are two primary methods for estimating the cost of equity: the CAPM approach and the dividend discount model (DDM) approach. Neither is perfect, and in practice analysts often use both and compare.
The logic is immediate. If the CAPM tells you what return investors require for a given level of systematic risk, then that required return is exactly what the firm must deliver to its equity investors. The formula is the same CAPM equation you already know — you're just interpreting it from the issuer's side rather than the investor's side.
To use this method, you need three inputs: a risk-free rate (typically the yield on long-term Treasury bonds), the firm's beta (estimated from historical stock return data, or obtained from a financial data provider), and the market risk premium (either the historical average or a forward-looking estimate). Each of these introduces estimation uncertainty, but the method has the advantage of being directly grounded in the risk-return framework you've been building all quarter.
Recall from Week 4 that the constant-growth dividend discount model prices a stock as:
P0 = D1 ÷ (rE − g)
If we rearrange this equation to solve for rE instead of P0, we get the DDM cost of equity estimate:
The first term, D1 ÷ P0, is called the dividend yield — the immediate cash return you receive from holding the stock. The second term, g, is the capital gains yield — the expected rate of price appreciation if dividends grow at rate g forever. Together, they represent the total return shareholders expect.
This method is appealingly intuitive — you're using market data (the stock price and dividend) to infer what return investors are demanding. But it has real limitations. It only works for firms that pay dividends and are expected to grow at a roughly constant rate. And the result is highly sensitive to your estimate of g: change the growth assumption by even 1%, and the cost of equity shifts significantly.
In a perfect world, both methods would give you the same number. In practice, they almost never do — and the gap can be substantial.
Why might the methods disagree? The CAPM is built on a model of how risk and return relate in equilibrium. It doesn't use any information about the firm's dividends or stock price — it's purely about systematic risk. The DDM, by contrast, is anchored entirely in the firm's dividend policy and stock price, and doesn't consider beta at all. When they diverge, it typically means that one or more of the underlying assumptions — the beta estimate, the market risk premium, or the growth rate — is off. A large divergence should prompt you to reexamine your inputs rather than simply picking whichever number you prefer.
We'll put the cost of equity to work next week when we combine it with the cost of debt to compute the firm's overall weighted average cost of capital.
| Concept | Formula | Notes |
|---|---|---|
| Portfolio Beta | βP = Σ wi × βi | Weighted average of component betas |
| CAPM / SML | E(Ri) = Rf + βi × [E(RM) − Rf] | Expected return given systematic risk |
| Alpha (mispricing) | α = E(R)forecast − E(R)CAPM | Positive α → undervalued; Negative α → overvalued |
| Cost of Equity (CAPM) | rE = Rf + β × MRP | MRP = market risk premium |
| Cost of Equity (DDM) | rE = D1 ÷ P0 + g | Requires constant growth assumption |
You invest 40% of your portfolio in Orange Acre Logistics (β = 0.9), 35% in White Acre Pharmaceuticals (β = 1.7), and the remaining 25% in Treasury bills. What is your portfolio's beta?
A stock has a beta of 1.25. The risk-free rate is 3.5% and the expected market return is 10.5%. According to the CAPM, what is the stock's expected return?
A stock has a beta of 1.4, an expected return of 14%, and the expected market return is 11%. Using the CAPM, solve for the risk-free rate.
Assume the risk-free rate is 4% and the expected market return is 10%. Your analyst provides these forecasts: Stock A has a beta of 0.7 and an expected return of 10%. Stock B has a beta of 1.3 and an expected return of 13%. Determine whether each stock is overvalued, undervalued, or fairly priced according to the CAPM. Which stock would you buy?
Violet Acre Energy has a beta of 0.95. The risk-free rate is 2.5% and the market risk premium is 8%. What is Violet Acre's cost of equity using the CAPM approach?
Violet Acre Energy's stock currently trades at $38.00 per share. The company just paid a dividend of $1.50, and dividends are expected to grow at 5.5% per year indefinitely. What is Violet Acre's cost of equity using the DDM approach?
Silver Acre Technologies has a beta of 1.1. The risk-free rate is 3% and the market risk premium is 6.5%. Silver Acre's stock trades at $45.00, and the company expects to pay a dividend of $2.70 next year. Dividends are expected to grow at 3% per year. Compute the cost of equity using both the CAPM and DDM approaches. What would you recommend as a reasonable estimate of the cost of equity?
In two or three sentences, explain why a stock with a high standard deviation of returns does not necessarily have a high expected return. Specifically, what type of risk is rewarded by the market, and how does the CAPM capture this distinction? (No calculation required.)