Calculating the Equity Risk Premium from Nominal Returns

Summary

The note explains how to calculate the equity risk premium when both equity and Treasury bill returns are reported in nominal terms. It separates nominal returns from real returns and expresses each as a combination of the real risk free return, inflation, and, for equities, the risk premium. Since inflation affects both nominal return expressions, comparing equities with Treasury bills removes the shared inflation component.

Using the stated historic geometric returns, the answer computes the premium as the relative return of equities over Treasury bills, yielding about 5.4%. The inflation observation is therefore not needed for this calculation. This is a compact illustration of the distinction between a return premium and a real return; it relies on the question's nominal return inputs and stated relationship between rates. The document does not discuss uncertainty in historical estimates, alternative premium definitions, or whether geometric averages are appropriate for other applications.

Key ideas

• The equity risk premium compares equity returns with a risk free return on a consistent nominal or real basis.
• Inflation is common to both nominal returns in the stated setup and cancels in their comparison.
• The premium can be calculated from the nominal equity and Treasury bill returns without separately using inflation.
• The example uses historical geometric returns and does not address estimation uncertainty.

Tags

# Equity Risk Premium calculation


# Equity Risk Premium calculation












Can someone help me understand the intuition of the following?

The formula for calculating the nominal return is:

I have the following problem for which I know the answer is A (5.4%). I was told it is because since "both equity and t-bill returns are nominal, the inflation cancels out". How does it actually cancel out? I am just having a little trouble actually expressing that into the formulas.

```
An analyst observes the following historic geometric returns:

Equities 8.0%

Corporate Bonds 6.5%

Treasury Bills 2.5%

Inflation 2.1%

The risk premium for equities is closest to:

A) 5.4%

B) 5.5%

C) 5.6%"
```

## Answer by nbbo2 (score 0, accepted)

https://quant.stackexchange.com/a/55828

You have to be very clear about what is real and what is nominal. For example let us put a * for real rates and no * for nominal.

Let us call $r^*_f$ the real risk free rate. Then we have the nominal return on tbills is $r_f$ where $$(1+r_f)=(1+r^*_f)(1+\pi)$$

and the nominal return on equities is $R_n$ where $$(1+R_n)=(1+r^*_f)(1+\pi)(1+RP)$$

You can write this in a way that shows the "inflation cancels out": $$(1+R_n)=(1+r_f)(1+RP)$$

Applying $r_f$ = 0.025 and $R_n$ = 0.08 we get RP = 0.0536 or approximately 5.4%, without need to use the inflation rate $\pi$ = 0.021.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.