What the Market Risk Premium Measures
The market risk premium is a central input to the Capital Asset Pricing Model (CAPM). It represents the expected return investors require for bearing the risk of a diversified market portfolio rather than earning a risk-free return over the same horizon. Estimating it is not just a matter of choosing a number from a table: the result depends on the return measure, the historical period or forecast method, and the market proxy.
The word matched matters. Market and risk-free returns must use the same currency, return interval, and general timing convention. If one return is measured over a year and the other over a month, subtracting them directly does not produce a meaningful annual premium. Market returns should generally include both price changes and distributions such as dividends; using price changes alone can omit an important part of the investor’s return.
Once an estimate of the premium is available, CAPM relates it to a security’s required return. For this calculation, the risk-free rate and beta are treated as given inputs; estimating beta is a separate task.
Here \(r_e\) is the CAPM cost of equity and \(\beta\) measures the security’s sensitivity to market risk. A premium estimate is multiplied by beta, so a change in the premium changes the CAPM cost of equity by beta times that change.
Estimating a Premium from Historical Returns
A historical estimate begins with a series of market returns and corresponding risk-free returns for the same intervals. For each interval, calculate the market’s excess return by subtraction. The arithmetic historical premium is the average of those matched excess returns.
This estimator uses the arithmetic mean because the target is an expected one-period excess return. It does not claim that each future year will earn the estimated premium, nor that an investor’s multi-year compound growth will equal it.
Worked Example: Calculating a Matched Historical Premium
A hypothetical diversified market proxy returns 4%, 10%, −2%, and 8% over four successive years. The matched annual risk-free returns are 1%, 2%, 1%, and 3%. Estimate the arithmetic historical premium. State the method and identify a common calculation error.
Method. Subtract the risk-free return from the market return in each year, then average the four annual excess returns.
Year 1 excess return: \(4\%-1\%=3\%\). Year 2: \(10\%-2\%=8\%\). Year 3: \(-2\%-1\%=-3\%\). Year 4: \(8\%-3\%=5\%\).
The sum is \(3\%+8\%-3\%+5\%=13\%\). Divide by four years:
The estimated arithmetic historical premium is 3.25% per year. As a check, the annual excess returns average to a value between their minimum of −3% and maximum of 8%, and their sum is 13 percentage points.
Common mistake avoided: Do not average the market returns and risk-free returns separately and then mis-add or mis-subtract the results. Here the market-return average is \(20\%/4=5\%\), and the risk-free average is \(7\%/4=1.75\%\); their difference is also 3.25%. The direct matched calculation makes the underlying observations and signs easier to verify.
Why the Arithmetic Average Is Used
The reason for averaging matched annual excess returns follows from the definition of an expectation. Suppose the annual excess returns \(X_t=R_{M,t}-r_{f,t}\) are observations from a stable process with a common expected value \(\mu\) and finite mean. The sample arithmetic average is \(\bar X=(X_1+\cdots+X_n)/n\). By linearity of expectation:
Thus, under this common-mean assumption, the arithmetic sample average is unbiased for the expected one-period excess return. Independence is not needed for this expectation calculation, although dependence across years affects how much information the sample provides and therefore the uncertainty of the estimate. If the market environment changes over time, treating all observations as having one stable expected premium may be inappropriate.
Arithmetic and Geometric Estimates Answer Different Questions
The geometric mean summarizes the constant annual rate that would produce the same compounded growth over a historical period. It is not the same quantity as the arithmetic mean of annual excess returns. Compounding makes the order and size of gains and losses relevant to accumulated wealth, whereas the arithmetic mean is designed to estimate a one-period expected return.
Worked Example: Comparing an Arithmetic Premium with a Geometric-Return Spread
A hypothetical market proxy has annual returns of 10%, −10%, and 10%. The risk-free return is 2% each year. Calculate (1) the arithmetic average of the matched excess returns and (2) the market’s geometric average return minus the geometric average risk-free return. Explain why the answers differ.
Method for the arithmetic estimate. The annual excess returns are \(10\%-2\%=8\%\), \(-10\%-2\%=-12\%\), and \(10\%-2\%=8\%\). Their sum is \(8\%-12\%+8\%=4\%\), so the arithmetic average is \(4\%/3=1.3333\%\), or 1.33% to two decimal places.
Method for the geometric-return spread. The market’s cumulative growth factor is \(1.10\times0.90\times1.10=1.089\). Its geometric average annual return is therefore \(1.089^{1/3}-1\), approximately \(2.88\%\). The risk-free cumulative factor is \(1.02^3\), so its geometric average annual return is exactly 2.00%. The geometric-return spread is \(2.88\%-2.00\%=0.88\%\), to two decimal places.
The arithmetic estimate is 1.33%, while the geometric-return spread is 0.88%. The arithmetic figure estimates an average one-year excess return; the geometric spread compares annualized compounded growth rates. They are not interchangeable estimates of the same statistic. The calculation avoids the common mistake of compounding annual excess returns as if doing so produced the arithmetic expected premium.
Choosing the Historical Period and Market Proxy
A historical premium is sensitive to the observations included. A long period gives more observations but may include economic conditions unlike those expected in the forecast period. A short, recent period may seem more relevant but can be dominated by a few unusually strong or weak years. There is no universal period that eliminates this judgment. The choice should be disclosed and justified, and the estimate should be treated as uncertain rather than exact.
The proxy also matters because the true market portfolio is not directly observable. An index or portfolio is used as a practical stand-in. A proxy that differs in geography, asset composition, or return construction can produce a different estimate. For example, a domestic equity index need not represent a globally diversified market portfolio. The analyst should use a proxy that is defensible for the company and valuation context, and should keep the return definition consistent across the full sample.
The following choices can materially affect the result:
- Start and end dates: including or excluding a boom or downturn can shift the average.
- Return definition: total returns, including distributions, are not interchangeable with price-only returns.
- Risk-free series: use rates matched to the market-return intervals, rather than one unrelated current rate for every historical year.
- Market proxy: different indices can represent different market exposures.
- Estimation method: an arithmetic average, a geometric-return spread, and a forward-looking estimate answer different questions.
Worked Example: Seeing the Effect of the Sample Period
A hypothetical broad market proxy has annual returns of 12%, −8%, 10%, 6%, and 4%. The matched risk-free return is 2% in every year. Compare the arithmetic historical premium using all five years with the estimate using only the most recent three years. Then calculate the CAPM cost of equity for a company with a given beta of 1.10 and a current matched risk-free rate of 2%. Treat this as a sensitivity illustration, not evidence that one period is automatically preferable.
Method. First calculate each year’s excess return. The five observations are \(12\%-2\%=10\%\), \(-8\%-2\%=-10\%\), \(10\%-2\%=8\%\), \(6\%-2\%=4\%\), and \(4\%-2\%=2\%\). Their sum is \(10\%-10\%+8\%+4\%+2\%=14\%\), giving a five-year arithmetic premium of \(14\%/5=2.80\%\).
For the most recent three years, the excess returns are 8%, 4%, and 2%. Their sum is 14%, and their average is \(14\%/3=4.6667\%\), or 4.67% to two decimal places.
Using the five-year estimate, the CAPM cost of equity is \(2\%+1.10(2.80\%)=2\%+3.08\%=5.08\%\). Using the three-year estimate, it is \(2\%+1.10(4.6667\%)\), approximately \(2\%+5.1333\%=7.13\%\) to two decimal places.
The two historical periods produce costs of equity of 5.08% and 7.13%. The difference is consistent with the higher recent-period premium. The five-year excess-return sum is 14 percentage points, which also checks the first average. The common trap is to use the premium from one period with a CAPM input set described as coming from another without explaining the mismatch.
Forward-Looking Evidence
Historical observations are backward-looking. A forward-looking estimate instead uses information intended to represent investors’ expectations for future market returns. Evidence may include forecasts of market returns, investor surveys, or estimates inferred from market prices and expected cash flows. These approaches rely on assumptions about the forecast horizon, expected returns, and the relationship between prices and future cash flows. They are not direct observations of the future premium.
To form a forward-looking premium, estimate the expected market return over a stated horizon and subtract the risk-free rate for the same horizon and currency. If the expected return is supplied as scenarios, calculate its probability-weighted mean first. Probabilities must sum to one, and all scenario returns must refer to the same period.
Worked Example: Estimating a Forward-Looking Premium from Scenarios
An analyst constructs three hypothetical one-year market-return scenarios: a 25% probability of a 15% return, a 50% probability of an 8% return, and a 25% probability of a −5% return. The matched one-year risk-free rate is 2.5%. Estimate the expected market risk premium and the CAPM cost of equity for a company whose beta is given as 1.20.
Method. The scenario probabilities sum to \(0.25+0.50+0.25=1.00\). The probability-weighted expected market return is \(0.25(15\%)+0.50(8\%)+0.25(-5\%)\). The three contributions are 3.75%, 4.00%, and −1.25%, so the expected market return is \(3.75\%+4.00\%-1.25\%=6.50\%\).
Subtract the matched risk-free rate: \(6.50\%-2.50\%=4.00\%\). The forward-looking premium estimate is therefore 4.00%. Applying CAPM gives \(r_e=2.50\%+1.20(4.00\%)=2.50\%+4.80\%=7.30\%\).
The result checks because the expected market return of 6.50% exceeds the risk-free rate by exactly 4.00%, and the premium multiplied by beta contributes 4.80 percentage points to the cost of equity. A common mistake is to subtract the risk-free rate from each scenario before taking the probability-weighted average and then subtract it again. Either subtract once from the expected market return, as done here, or average the scenario excess returns directly—not both.
Using and Communicating an Estimate
No estimation method is automatically correct in every setting. Historical estimates benefit from observed returns but can be unstable across periods and proxies. Forward-looking evidence is more directly aimed at future expectations but depends on forecasts and modeling assumptions. Analysts may compare more than one method to understand the range of plausible inputs, while clearly distinguishing estimates rather than blending them without justification.
A well-supported estimate should state the return horizon, currency, market proxy, period or forecast source, and method. It should also explain why the selected evidence is relevant and acknowledge material limitations. When the estimate feeds into CAPM, the risk-free rate must be consistent with the premium’s horizon and currency. The premium is an input to a required-return model, not a promise that the market will earn that amount.
Check Your Understanding
For each problem, keep the market and risk-free returns matched by horizon and subtract the risk-free rate only once.
- A hypothetical index returns 7%, −4%, and 11% over three years. Matched annual risk-free returns are 2%, 1%, and 3%. Find the arithmetic historical premium. (Answer: 2.67%)
- A market proxy has four annual returns of 9%, 5%, −7%, and 13%. The matched risk-free return is 3% each year. Calculate the four annual excess returns and their arithmetic average. (Answer: 6%, 2%, −10%, and 10%; arithmetic premium 2.00%)
- For a one-year forecast, market returns are 18%, 7%, and −6% with probabilities 0.20, 0.50, and 0.30. The matched risk-free rate is 2.3%. Find the forward-looking premium and, for a given beta of 1.40, the CAPM cost of equity. (Answer: expected market return 5.30%; premium 3.00%; cost of equity 6.50%)
- Proxy A has annual returns of 6%, 8%, and 10%; Proxy B has returns of 10%, 0%, and 8%. The matched risk-free return is 2% in every year. Calculate both arithmetic premiums and the CAPM costs of equity using a current matched risk-free rate of 3% and a given beta of 1.25. (Answer: Proxy A premium 6.00% and cost of equity 10.50%; Proxy B premium 4.00% and cost of equity 8.00%)
- A four-year historical estimate is 3.5%, while a forward-looking estimate is 5.0%. With a matched risk-free rate of 2.8% and a given beta of 0.90, calculate the two CAPM costs of equity and their difference. (Answer: historical-based cost 5.95%; forward-looking-based cost 7.30%; difference 1.35 percentage points)