823
Appendix F
Technical Issues in 
Estimating the Market 
Risk Premium
In its simplest form, the historical market risk premium can be measured by 
subtracting the return on government bonds from the return (total return to 
shareholders) on a large sample of companies over some time frame. But this 
requires many choices that will affect the results. For the best measurement 
of the risk premium using historical data, follow the guidelines presented in 
this appendix.
Calculate Premium Relative to Long-Term Government 
Bonds
When calculating the market risk premium, compare historical market returns 
with the return on ten-year government bonds. Long-term government bonds 
match the duration of a company’s cash flows better than short-term bonds.
Use the Longest Period Possible
How far back should you look when using historical observations to predict 
future results? If the market risk premium is stable, a longer history will re-
duce estimation error. Alternatively, if the premium changes and estimation 
error is small, a shorter period is better. To determine the appropriate histori-
cal period, consider any trends in the market risk premium compared with the 
imprecision associated with short-term estimates.

824  Appendix F
To test for the presence of a long-term trend, we regress the U.S. market 
risk premium against time. Over the past 119 years, no statistically significant 
trend is observable.1 Based on regression results, the average excess return 
has fallen by two basis points a year, but this result cannot be statistically 
distinguished from zero. Premiums calculated over shorter periods are too 
volatile to be meaningful. For instance, U.S. stocks outperformed bonds by 
18 percent in the 1950s but offered no premium in the 1970s. Given the lack of 
any discernible trend and the significant volatility of shorter periods, use the 
longest time series possible.
Use an Arithmetic Average of Longer-Dated (e.g., Ten-Year) 
Intervals
When reporting market risk premiums, most data providers report an annual 
number, such as 6.3 percent per year. But how do they convert a century of 
data into an annual number? And is the annualized number even relevant?
Annual returns can be calculated using either an arithmetic average or a 
geometric average. An arithmetic (simple) average sums each year’s observed 
premium and divides by the number of observations:
Arithmetic Average =
+
( )
+
( )
−
=∑
1
1
1
1
1
T
R
t
r
t
m
f
t
T
where 
T
R
t
t
r t
m
f
=
=
=
number of observations
market return in year
risk-
( )
( )
free rate in year t
A geometric average compounds each year’s excess return and takes the root 
of the resulting product:
Geometric Average =
+
( )
+
( )






−
=∏
1
1
1
1
1
R
t
r
t
m
f
t
T
T
/
The choice of averaging methodology will affect the results. For 
instance, between 1900 and 2019, U.S. stocks outperformed long-term govern-
ment bonds by 6.3 percent per year when averaged arithmetically. Using a 
1 Some authors, such as Jonathan Lewellen, argue that the market risk premium does change over 
time—and can be measured using financial ratios, such as the dividend yield. We address these mod-
els separately. J. Lewellen, “Predicting Returns with Financial Ratios,” Journal of Financial Economics, 
74, no. 2 (2004): 209–235.