Why the Risk-Free Rate Must Be Chosen Carefully
In the Capital Asset Pricing Model (CAPM), the risk-free rate is the baseline return used to estimate an investor’s required return for bearing equity risk. It may look like a single market input, but it is a choice of benchmark: the currency, horizon and return convention of the benchmark should fit the equity-return estimate. A rate that does not fit can make the CAPM result inconsistent even when the arithmetic is correct.
In practice, analysts often use a yield on a government security as a proxy for the risk-free rate. That is a proxy, not a guarantee that the security has no risk. The analyst must consider what the quoted yield represents, including its currency, maturity and any material sovereign credit risk.
For an annual CAPM estimate, the standard relationship is:
Here \(E(R_i)\) is the expected return required on equity \(i\), \(\beta_i\) measures its market risk, and \(E(R_m)-R_f\) is the market risk premium. The earlier material on deriving the CAPM cost of equity establishes this relationship. This tutorial focuses on selecting \(R_f\); the examples treat the beta and market risk premium as given rather than estimating them.
Matching the Benchmark to the Estimate
Begin by stating what the equity-return estimate represents. Is it a nominal or real return? Which currency is used for the investment’s cash flows and expected returns? What period does the estimate represent? Choose a government yield with those features in mind. When a term structure is available, a yield that corresponds to the relevant horizon is generally a better starting point than a rate chosen merely because it is familiar.
Currency matching matters because nominal yields reflect the expected inflation and other conditions associated with their currency. A U.S. dollar government yield is not automatically an appropriate baseline for a Canadian-dollar nominal return, for example. A foreign-currency yield does not become the right benchmark simply because it is observable or higher. Any currency conversion or hedging would need to be reflected consistently in the return estimate; applying a foreign yield directly does not do that.
Horizon matching is also important. A short-term government yield and a long-term government yield can differ because market rates vary by term. If the CAPM estimate is an annual required return used for an investment horizon of several years, a government yield near that horizon may be more relevant than a short-term yield. For an estimate over a single year, a short-term benchmark may be more suitable. The company’s total expected life does not, by itself, mean that every CAPM estimate should use a government yield with that same maturity. The relevant horizon is the period represented by the return being estimated.
A coupon bond’s yield to maturity is not necessarily the same thing as a zero-coupon yield for a particular horizon. The quoted yield summarizes returns from the bond’s coupon and principal cash flows under the bond-yield convention; a spot yield refers to a specific maturity cash flow. When a suitable maturity-specific government spot rate is available, it can be a closer match to a single horizon. Whatever benchmark is selected, document its source, observation date, currency, maturity and yield convention.
Worked Example: Choosing a Dollar Benchmark for a Five-Year Estimate
A hypothetical firm wants an annual nominal U.S. dollar cost of equity estimate for an investment analysis focused on a five-year horizon. The available government yields, observed on the same date and quoted on a consistent annual basis, are 3.2% for one year, 3.8% for five years and 4.1% for ten years. For this example, assume the five-year government yield is an appropriate proxy for the relevant risk-free rate. Given \(\beta_i=1.20\) and a market risk premium of 5.5%, estimate the cost of equity. Also calculate what the estimate would be using the one-year yield, holding the premium fixed.
Method. Match the risk-free benchmark to the dollar currency and five-year horizon, then apply CAPM. The given premium and beta are inputs; they are not being re-estimated.
Step 1: Select the benchmark. The estimate is nominal, in U.S. dollars, and focused on five years. Of the supplied yields, the five-year U.S. dollar government yield is the closest horizon match. Thus \(R_f=3.8\%\).
Step 2: Calculate the market risk contribution. The given market risk premium contribution is \(1.20\times 5.5\%=6.6\%\).
Step 3: Calculate the cost of equity. CAPM gives \(3.8\%+6.6\%=10.4\%\).
Step 4: Compare with the one-year benchmark. Using \(R_f=3.2\%\) with the same beta and fixed premium gives \(3.2\%+6.6\%=9.8\%\). The difference is \(10.4\%-9.8\%=0.6\) percentage points, or 60 basis points.
Check. The selected yield is the five-year yield, not the highest yield or the shortest yield. The difference in the two CAPM estimates is exactly the 0.6 percentage-point difference between their risk-free rates because the premium was held fixed. The common mistake avoided is selecting a maturity without regard to the estimate’s horizon.
Currency and Return-Basis Consistency
All CAPM inputs must use a consistent return basis. If the equity return is nominal, the risk-free rate and market risk premium should also be nominal. If the estimate is expressed in real terms, use real inputs. Do not combine a nominal government yield with a real market risk premium simply because both rates are available.
When converting a nominal rate to a real rate, the exact relationship depends on the inflation assumption. If \(i_{\text{nominal}}\) is the effective nominal rate and \(\pi\) is the effective inflation rate over the same period, the exact real rate is:
This relationship assumes that the nominal return and inflation rate refer to the same period and that the stated inflation rate is the applicable measure for the conversion. Subtracting inflation from the nominal rate is only an approximation, not the exact conversion.
Worked Example: Converting a Nominal Benchmark to a Real Basis
A hypothetical analyst needs a real annual cost of equity estimate. The available nominal government rate is 4.04% per year, and the assumed annual inflation rate is 2.00%. For this example, assume both are effective annual rates for the same currency and horizon. The real market risk premium is given as 5.0%, and \(\beta_i=0.90\). Find the real cost of equity.
Method. Convert the risk-free rate to a real rate using the exact nominal-to-real relationship, then use real CAPM inputs throughout.
Step 1: Convert the rates to decimal form. The nominal rate is \(0.0404\), and inflation is \(0.0200\).
Step 2: Calculate the real rate. The relationship gives \(1+i_{\text{real}}=1.0404/1.0200=1.0200\), so \(i_{\text{real}}=0.0200=2.00\%\).
Step 3: Calculate the market risk contribution. The given contribution is \(0.90\times5.0\%=4.5\%\).
Step 4: Calculate the real cost of equity. The estimate is \(2.0\%+4.5\%=6.5\%\).
Check. The calculation uses a real risk-free rate and a real market risk premium. The common mistake avoided is inserting the 4.04% nominal yield directly into a real-return estimate.
Government Yields and Sovereign Credit Risk
A government security is not necessarily default-free. If the government issuing a security has material credit risk, its yield may include compensation for expected default losses as well as other risk and market effects. In that case, treating its full yield as a risk-free rate may overstate the CAPM baseline. The analyst should consider whether a more appropriate government benchmark exists for the same currency, or whether a transparent adjustment is justified by the available information. Subtracting an estimated credit spread is not a universal rule: the estimate must be credible and the adjustment must fit the purpose and basis of the analysis.
Worked Example: Assessing a Government Yield with Credit Risk
A hypothetical firm estimates a local-currency cost of equity. A local government bond yields 6.4% per year, but the analyst has reason to believe that about 1.5 percentage points of the yield reflects sovereign credit risk. For this illustration only, assume that this estimate is a suitable adjustment and that the adjusted rate is an appropriate proxy for the local-currency risk-free rate. Given \(\beta_i=1.10\) and a local-currency market risk premium of 6.0%, calculate the cost of equity using the adjusted rate. Compare it with the result from using the unadjusted government yield.
Method. First make the stated illustrative credit adjustment, then apply CAPM to each candidate rate while holding the stated market risk premium fixed.
Step 1: Calculate the adjusted benchmark. The adjusted rate is \(6.4\%-1.5\%=4.9\%\).
Step 2: Calculate the market risk contribution. The contribution is \(1.10\times6.0\%=6.6\%\).
Step 3: Calculate the adjusted-rate estimate. The cost of equity is \(4.9\%+6.6\%=11.5\%\).
Step 4: Calculate the unadjusted comparison. Using 6.4% as \(R_f\) gives \(6.4\%+6.6\%=13.0\%\). This is 1.5 percentage points higher, exactly the assumed adjustment.
Check. The adjustment is explicitly an assumption in this example, not a general method for all sovereign yields. The common mistake avoided is treating every government yield as automatically risk-free.
How a Rate Choice Changes the CAPM Result
The effect of changing \(R_f\) depends on which other CAPM input is held fixed. This distinction matters when comparing estimates, because the market risk premium is itself defined relative to \(R_f\).
If \(E(R_m)-R_f\) is treated as a fixed input, then \(E(R_i)=R_f+\beta_i\text{(fixed premium)}\). A change in \(R_f\) changes the CAPM estimate by the same amount.
The premium must then change when \(R_f\) changes. Expanding CAPM gives \(E(R_i)=\beta_i E(R_m)+(1-\beta_i)R_f\), so the effect depends on \(1-\beta_i\).
The second result follows directly from rearranging the CAPM equation:
Thus, if \(E(R_m)\) is fixed, increasing \(R_f\) raises the cost of equity when \(\beta_i<1\), has no effect when \(\beta_i=1\), and lowers the cost of equity when \(\beta_i>1\). This does not contradict the one-for-one effect when the premium is fixed: those are different assumptions about what stays unchanged.
Worked Example: Changing the Rate with the Market Return Fixed
A hypothetical equity estimate uses \(\beta_i=1.40\) and a fixed expected market return of 9.0%. Compare the CAPM estimate when the risk-free rate is 3.5% with the estimate when it is 4.0%. Assume both rates refer to the same currency, horizon and return basis.
Method. Because the expected market return is fixed, recalculate the market risk premium at each risk-free rate before applying CAPM.
Step 1: Calculate the first premium. At \(R_f=3.5\%\), the premium is \(9.0\%-3.5\%=5.5\%\).
Step 2: Calculate the first estimate. The market risk contribution is \(1.40\times5.5\%=7.7\%\). Therefore the cost of equity is \(3.5\%+7.7\%=11.2\%\).
Step 3: Calculate the second premium. At \(R_f=4.0\%\), the premium is \(9.0\%-4.0\%=5.0\%\).
Step 4: Calculate the second estimate. The market risk contribution is \(1.40\times5.0\%=7.0\%\). Therefore the cost of equity is \(4.0\%+7.0\%=11.0\%\).
Step 5: Check the change. The risk-free rate rose by 0.5 percentage points, while the CAPM estimate fell by \(11.0\%-11.2\%=-0.2\) percentage points. The rearranged formula predicts this: \((1-1.40)\times0.5\%=-0.2\%\).
Check. The premium decreased when the risk-free rate increased because the expected market return was held fixed. The common mistake avoided is holding both the expected market return and its implied premium fixed at once.
Practical Selection Checklist
Before using a government yield in CAPM, record the following decisions. Doing so makes the estimate easier to review and reduces the chance that a numerical input is used outside its intended context.
- Currency: Is the government yield in the currency of the equity-return estimate?
- Horizon: Does the yield maturity reasonably correspond to the return period being estimated?
- Return basis: Are the risk-free rate and market risk premium both nominal or both real?
- Yield convention: Is the quoted figure a spot yield, yield to maturity or another rate measure, and is that measure suitable for the estimate?
- Credit risk and date: Could sovereign credit risk materially affect the yield, and is the rate observed on a date consistent with the other market inputs?
- Comparison assumption: When testing sensitivity, is the market risk premium fixed, or is the expected market return fixed?
A well-chosen risk-free rate does not remove uncertainty from CAPM. It makes the baseline consistent with the estimate being made. These selection and consistency checks are aligned with the SOA VEE Accounting and Finance syllabus.
Check Your Understanding
For each problem, identify the appropriate input treatment before calculating.
- A hypothetical investment has an annual nominal U.S. dollar return estimate with a five-year horizon. Same-date government yields are 3.1% for one year and 3.7% for five years. Given \(\beta_i=0.90\) and a market risk premium of 5.2%, calculate the estimate using the better horizon match. (Answer: 8.38%)
- A hypothetical Canadian-dollar estimate has a matching Canadian government rate of 3.7%. Given \(\beta_i=0.70\) and a Canadian-dollar market risk premium of 6.0%, calculate the CAPM estimate. (Answer: 7.9%)
- A hypothetical estimate has a fixed expected market return of 9.2%, \(\beta_i=1.30\), and \(R_f=3.2\%\). Calculate the market risk premium and the cost of equity. (Answer: Market risk premium = 6.0%; cost of equity = 11.0%)
- With expected market return held fixed, \(R_f\) rises by 0.40 percentage points and \(\beta_i=1.25\). By how much does the CAPM estimate change? (Answer: Decreases by 0.10 percentage points, or 10 basis points)
- A real-return estimate uses a nominal effective annual government rate of 5.06% and same-period inflation of 3.00%. Given a real market risk premium of 4.5% and \(\beta_i=1.10\), calculate the real risk-free rate and real cost of equity. (Answer: Real risk-free rate = 2.00%; real cost of equity = 6.95%)