Tutorials › VEE Accounting and Finance › Diagnosing CAPM Input Inconsistencies

Home  /  Tutorials  /  VEE Accounting and Finance / Cost of Equity Estimation

VEE Accounting and Finance · Cost of Equity Estimation · Tutorial 632 of 996

Diagnosing CAPM Input Inconsistencies

Check that CAPM inputs describe the same return environment, then quantify how inconsistent assumptions affect the estimated cost of equity.

Intermediate 12 min read

What You'll Learn

  • Identify the currency, return period, market definition, and nominal or real basis required for comparable CAPM inputs.
  • Match a beta to the market return series and currency used to estimate it.
  • Annualize periodic market premiums using an explicit arithmetic convention.
  • Convert nominal CAPM inputs to real terms using consistent inflation assumptions.
  • Explain and calculate how mismatched inputs change a CAPM estimate.

Why CAPM Inputs Must Be Comparable

The CAPM cost of equity is only interpretable when its inputs describe the same investment environment. A risk-free rate in one currency, a market risk premium in another, and a beta estimated against a third market do not become compatible merely because each is expressed as a percentage. The resulting arithmetic may be correct while the estimate has no coherent interpretation.

A useful diagnostic is to label every input before calculating: currency, period, market definition, and nominal or real basis. Then check that the risk-free rate, market risk premium, and beta can all be interpreted together. This is especially important when market data come from different sources or are quoted using different conventions.

Definition: A CAPM input set is internally consistent when the risk-free return and market risk premium refer to the same currency and return period, the beta measures exposure to the market return underlying that premium in the same return framework, and all rates use the same nominal or real basis.

For a one-period CAPM estimate, write the cost of equity as \(k_e\), the risk-free rate as \(R_f\), the market risk premium as \(E[R_m]-R_f\), and equity beta as \(\beta\). The basic estimate is:

$$ k_e=R_f+\beta\big(E[R_m]-R_f\big) $$

Here, \(E[R_m]\) is the expected return on the market portfolio used in the model. CAPM uses the market premium—not the market return by itself—as the amount multiplied by beta. The result is a required expected return for equity under the CAPM assumptions; it is not a guarantee of the return that will be realized.

The formula presumes that the inputs are comparable and that beta is appropriate to the selected market portfolio. The standard CAPM interpretation also depends on model assumptions, including investors evaluating returns over a common period and using a common market opportunity set. If a practical estimate uses a proxy index, the analyst should make that market definition explicit rather than imply the proxy is identical to the theoretical market portfolio.

A Four-Part Consistency Check

1
Match the currency.
Use a risk-free rate, market premium, and beta framework for returns stated in the same currency. If returns are converted between currencies, exchange-rate movements can affect both returns and beta.
2
Match the period.
A monthly premium and an annual risk-free rate cannot be added as if they refer to one period. Convert or annualize using a stated convention, and do not confuse an arithmetic annualization with an exact compounded return.
3
Match the market definition.
The premium must describe the market return against which the beta measures sensitivity. A beta estimated against a domestic broad-market index is not automatically compatible with a premium for a different country or market segment.
4
Match nominal or real terms.
Nominal returns include expected inflation; real returns measure purchasing-power change. Convert all relevant returns consistently before applying CAPM.

Beta is defined relative to a particular market return series. In the usual return-based formulation, it is the covariance of the security return with the market return divided by the variance of the market return. Thus changing the market series can change beta, and changing currency can change the return series used to estimate it.

$$ \beta=\frac{\operatorname{Cov}(R_e,R_m)}{\operatorname{Var}(R_m)} $$

This definition explains why a beta is not a context-free multiplier. It identifies the security’s sensitivity to a specified market return series over a specified return framework. If a USD investor’s security and market returns include currency-conversion effects, the resulting beta can differ from a beta estimated using local-currency returns.

Worked Examples

Worked Example: Reconciling Monthly and Annual Inputs

A valuation team has an equity beta of 1.20 estimated from monthly returns. Its market risk premium estimate is 0.45% per month, stated as an arithmetic expected-return premium. The team’s annual risk-free rate is 4.00%, also on an arithmetic annual expected-return basis. Estimate the annualized arithmetic CAPM cost of equity. State the convention and identify the tempting error.

Method. Annualize the monthly arithmetic premium by multiplying by 12, then apply CAPM using the annual risk-free rate and the same beta framework. This produces an annualized arithmetic estimate, not an exact compounded one-year return.

First convert the monthly premium:

$$ 12(0.45\%)=5.40\% $$

Apply the CAPM formula:

$$ k_e=4.00\%+1.20(5.40\%)=4.00\%+6.48\%=10.48\% $$

The estimated annualized arithmetic cost of equity is 10.48%. The calculation matches the annual risk-free rate with an annualized arithmetic premium and uses beta from the monthly return framework, assuming that beta is appropriate for the market underlying that premium.

Common mistake and correction. Using the monthly premium directly gives \(4.00\%+1.20(0.45\%)=4.54\%\). That combines an annual risk-free rate with a monthly premium and is not a valid annual CAPM estimate. The correction is to express the premium on the annual arithmetic basis before applying the formula. Multiplying by 12 is an arithmetic annualization convention; it does not claim that the exact compounded annual premium must equal twelve times the monthly premium.

Worked Example: Detecting a Currency and Market Mismatch

An analyst is estimating the required return for an equity investment. A local-currency analysis has a local risk-free rate of 2.50%, a local market risk premium of 6.00%, and a local-market beta of 1.30. A separate USD analysis has a USD risk-free rate of 4.00%, a USD market risk premium of 5.00%, and a beta of 1.10 estimated against the USD market series. Calculate both internally consistent estimates and diagnose the result of combining selected inputs across the two analyses.

Method. Apply CAPM separately within each complete currency and market framework. Then calculate the mixed-input result only to show why it is not a valid estimate under either framework.

For the local-currency framework:

$$ k_{e,\text{local}}=2.50\%+1.30(6.00\%)=2.50\%+7.80\%=10.30\% $$

For the USD framework:

$$ k_{e,\text{USD}}=4.00\%+1.10(5.00\%)=4.00\%+5.50\%=9.50\% $$

These are two distinct estimates: 10.30% in the local-currency framework and 9.50% in the USD framework. Now consider the tempting but inconsistent calculation that uses the USD risk-free rate with the local beta and local premium:

$$ 4.00\%+1.30(6.00\%)=11.80\% $$

The arithmetic is correct, but the result combines a USD rate with a local-currency market premium and a beta measured against a local-market return series. It is not justified as either a USD or local-currency CAPM estimate by the supplied inputs.

Common mistake and correction. Choosing a risk-free rate in the investor’s preferred currency does not, by itself, make the other inputs compatible. Use a complete set of inputs in the target currency and matching market framework, or obtain the information needed to convert and re-estimate them consistently. Do not treat the 11.80% mixed result as a meaningful midpoint or adjustment.

Worked Example: Converting a Nominal CAPM Estimate to Real Terms

A nominal annual CAPM analysis has a nominal risk-free rate of 4.00%, a nominal market risk premium of 5.00%, and beta of 1.20. Expected inflation is 2.50% annually. Assume the same expected inflation factor applies when converting the risk-free, market, and equity returns. Calculate the nominal and real costs of equity, and check the real result in two ways.

Method. First calculate the nominal CAPM return. Then convert the relevant nominal gross returns to real gross returns using the exact Fisher relationship. The conversion is \(1+r_{\text{real}}=(1+r_{\text{nominal}})/(1+\pi)\), where \(\pi\) is expected inflation over the same period.

The nominal expected market return is the nominal risk-free rate plus the nominal premium:

$$ E[R_m]=4.00\%+5.00\%=9.00\% $$

The nominal CAPM cost of equity is:

$$ k_{e,\text{nominal}}=4.00\%+1.20(5.00\%)=10.00\% $$

Convert the nominal risk-free rate to a real rate:

$$ R_{f,\text{real}}=\frac{1.04}{1.025}-1=0.0146341\ldots=1.4634\% $$

Convert the expected market return, then find the real market premium:

$$ E[R_{m,\text{real}}]=\frac{1.09}{1.025}-1=0.0634146\ldots=6.3415\% $$
$$ E[R_{m,\text{real}}]-R_{f,\text{real}}=6.3415\%-1.4634\%=4.8781\% $$

Applying CAPM in real terms gives:

$$ k_{e,\text{real}}=1.4634\%+1.20(4.8781\%)=7.3171\% $$

Check directly by converting the nominal equity return:

$$ \frac{1.10}{1.025}-1=0.0731707\ldots=7.3171\% $$

The two calculations agree, to four decimal places: the real cost of equity is 7.3171%, while the nominal cost is 10.00%.

Common mistake and correction. Subtracting expected inflation from every nominal rate is only an approximation, not the exact conversion. Also, converting the risk-free rate but leaving the market premium nominal would mix real and nominal inputs. Convert the gross market and risk-free returns consistently, calculate the real premium from those converted returns, and then apply CAPM.

Why Mismatches Change the Estimate

The CAPM formula makes the effect of changed inputs transparent. Holding beta fixed, changing the risk-free rate changes the estimate directly; changing the market premium changes it in proportion to beta. If the inputs used are \(R_f^*\), \(M^*\), and \(\beta^*\), where \(M^*\) denotes the chosen premium, the estimate is:

$$ k_e^*=R_f^*+\beta^*M^* $$

Relative to a consistent estimate \(k_e=R_f+\beta M\), the difference is:

$$ k_e^*-k_e=(R_f^*-R_f)+(\beta^*-\beta)M^*+\beta(M^*-M) $$

This identity follows by expanding the right side:

$$ (R_f^*-R_f)+\beta^*M^*-\beta M^*+\beta M^*-\beta M =R_f^*+\beta^*M^*-R_f-\beta M =k_e^*-k_e $$

It shows why an inconsistent estimate cannot generally be repaired by adjusting only one input. For example, a different market definition can affect both the premium and the beta. A currency conversion can also alter the return series used to estimate beta. A change in basis from nominal to real affects the risk-free return and the market premium as well.

A further trap is to assume that any annualization procedure preserves CAPM exactly. Multiplying a monthly arithmetic premium by 12 is a clear and common estimate convention when an annualized arithmetic premium is required. Compounding returns instead answers a different question about a multi-period return. With uncertain returns, a compounded multi-period expected return is not generally obtained by simply compounding a single-period expected return. State the intended convention and avoid presenting a rough annualization as an exact multi-period CAPM result.

Practical Interpretation

A CAPM estimate is useful only when its interpretation is clear. Before reporting a cost of equity, document which currency the return is expressed in, what period the rates cover, which market return defines beta and the premium, and whether the estimate is nominal or real. These details allow another analyst to reproduce the calculation and tell what kind of required return the result represents.

When sources disagree, first check whether they estimate the same quantity under the same conventions. A difference may reflect a genuinely different risk assessment, but it may instead arise from mismatched markets, currencies, or rate bases. Recalculate only after identifying the source of the difference; changing inputs without preserving their meaning can make the estimate less defensible rather than more precise.

Key takeaway: Apply CAPM only after aligning currency, period, market definition, and nominal or real basis. A numerically correct calculation using incompatible inputs is not a valid cost-of-equity estimate.

Check Your Understanding

For each problem, check both the calculation and the consistency of the stated inputs.

  1. A stock has beta 0.90 against a specified annual market series. The matching annual risk-free rate is 3.60% and the matching annual market risk premium is 4.80%. Calculate the CAPM cost of equity. (Answer: 7.92%)
  2. A monthly-return analysis gives an arithmetic market risk premium of 0.40% per month and beta 1.10. For an annualized arithmetic estimate, use an annual risk-free rate of 4.00% and multiply the monthly premium by 12. Calculate the estimate. (Answer: 9.28%)
  3. A nominal annual risk-free rate is 5.00%, the nominal annual market return is 11.00%, beta is 0.75, and expected inflation is 3.00%. Convert the CAPM estimate to real terms using the exact Fisher relationship. (Answer: 6.3107%)
  4. A local-currency analysis gives a local risk-free rate of 2.50%, a local market premium of 7.00%, and local beta 1.30. A USD analysis gives a USD risk-free rate of 4.20%, a USD market premium of 5.50%, and USD beta 0.90. Calculate each consistent estimate and the result from combining the USD risk-free rate with the local beta and premium; identify which result is mismatched. (Answer: Local 11.60%; USD 9.15%; mixed 13.30%, which is not a consistent estimate.)
  5. A monthly-return analysis gives an arithmetic market premium of 0.45% per month and beta 1.20. An annualized arithmetic estimate uses a matching annual risk-free rate of 3.50%. Calculate the annualized arithmetic CAPM estimate, and state why it should not automatically be called an exact compounded one-year expected return. (Answer: 9.98%; it uses a 12-times arithmetic premium annualization, not an exact multi-period compounding calculation.)

Previous

Next

Back to Tutorials
Clinical Biostats