Consistent CAPM Estimates in Nominal and Real Terms

Why Nominal and Real CAPM Estimates Must Be Consistent

A nominal return measures the change in dollars. A real return measures the change in purchasing power after accounting for inflation. CAPM can be expressed in either terms, but every return input in a calculation must use the same terms. Combining a real risk-free rate with a nominal market risk premium, for example, generally produces neither a valid nominal estimate nor a valid real estimate.

This matters when a required return is used to assess a project or asset whose cash flows are stated in nominal dollars or in constant purchasing-power dollars. The required return and the cash flows must be expressed on a consistent basis. This tutorial focuses on converting CAPM estimates and diagnosing the assumptions behind that conversion; it is aligned with the SOA VEE Accounting and Finance syllabus.

The Exact Conversion Under Known Inflation

Let \(R_N\) be a one-period nominal return and let \(\pi\) be the effective inflation rate over that same period. The real return is not, in general, obtained by simply subtracting the inflation rate. The exact conversion follows from comparing end-of-period purchasing power with beginning-of-period purchasing power.

Here \(R_R\) is the real return. This relation assumes the nominal return and inflation rate refer to the same period and that the price-level change is known for that period. The familiar approximation \(R_R\approx R_N-\pi\) can be useful for small rates, but it is not the exact Fisher conversion.

For example, if an investment earns 8% nominally while prices rise by 3%, its real return is \(1.08/1.03-1\), not exactly 5%. The difference arises because the nominal ending value must be divided by the increased price level.

CAPM Inputs in Nominal and Real Terms

In nominal terms, the CAPM required return is the nominal risk-free rate plus beta times the nominal market risk premium. Beta measures an asset's covariance with the market relative to the market's variance. Under the standard CAPM assumptions, beta is the measure of systematic risk used in the required-return relation.

In this expression, \(k_N\) is the nominal required return, \(r_{f,N}\) is the nominal risk-free rate, and \(E[R_{M,N}]\) is the expected nominal market return. To express the relation in real terms, suppose inflation is known and is the same for the risk-free investment, the market, and the asset over the period. Then the real risk-free rate and real market premium are:

The nominal market risk premium is therefore divided by \(1+\pi\), not left unchanged. Under this common, known inflation adjustment, the real CAPM relation is:

Why the Market Premium and Beta Transform This Way

The premium conversion follows by applying the same Fisher deflation to the market and risk-free returns and subtracting. For any nominal return \(R_N\), rearranging the Fisher relation gives \(R_R=(R_N-\pi)/(1+\pi)\). Thus, when the same known inflation rate applies to both returns:

The common inflation term cancels in the difference, but the difference is still scaled by \(1/(1+\pi)\). Under the same assumptions, real beta equals nominal beta. To see why, each real return is its nominal return divided by the same constant \(1+\pi\), less the same constant \(\pi/(1+\pi)\). Subtracting a constant does not change covariance or variance, and scaling both market and asset returns by \(1/(1+\pi)\) scales covariance and market variance by the same squared factor. Their ratio is unchanged.

These conclusions depend on the assumptions. In particular, inflation must be a common known rate over the return period. If inflation is uncertain, it is not generally valid to replace it with expected inflation and then use the same formulas as though inflation were known. Real returns are ratios involving random inflation; their expected values and covariances can depend on how inflation co-moves with nominal returns. A separate model of real returns or inflation risk may be needed.

Converting a Real CAPM Estimate to Nominal Terms

The same relationships work in reverse. If the CAPM inputs are supplied in real terms, first calculate the real required return. Then multiply its gross return by \(1+\pi\) to obtain the nominal required return. The nominal market premium is also the real market premium multiplied by \(1+\pi\), provided the common known-inflation assumption holds.

Diagnosing a Mixed-Rate Calculation

A CAPM equation can look arithmetically correct while being economically inconsistent. Before using a calculated required return, label every input as nominal or real. Check that the risk-free rate, expected market return or market premium, and resulting required return all share the same basis. If inflation is used, check that the rate and return periods match.

When the Conversion Is Not Enough

Known, common inflation makes the conversions above straightforward. With uncertain inflation, a real return is determined by both the nominal return and the realized price-level change. Using expected inflation in the deterministic formulas can be an approximation, but it does not by itself establish that a real CAPM relation holds with unchanged beta. In particular, inflation-related variation and its association with market returns may affect real risk and covariance.

A further practical distinction is between an inflation forecast and inflation that is known over the investment period. A forecast is an estimate, not a realized price-level change. If an analysis uses expected inflation as though it were certain, it should state that simplifying assumption and treat the resulting real rate as an estimate. Do not imply that exact Fisher conversion removes inflation uncertainty.

The basis of the cash flows matters as well. Nominal cash flows should be evaluated with a nominal required return; cash flows stated in constant purchasing-power terms should be evaluated with a real required return. This consistency requirement does not depend on whether the rate was calculated from CAPM or obtained by another method.