Risk premiums using geometric averages

cfageist · February 16, 2018, 2:06am

Hi everyone,

In one of the EOC questions in the portfolio management book, they are asking us to calculate the risk premium with geometric averages. They are giving equity return as 8% and treasury bills at 2 %. I calculated the risk premium as 6% but they calculated this way:

(1.08)/(1.02) - 1 = 5.88%

I understand why we need to do it this way when we want to calculate real returns using the inflation rate, but why do we use this way to calculate the risk premium ? I am sure it has to do with geometric returns. If arithmetic returns were used, would the risk premium be 6%?

Thank you!

Nukular · February 19, 2018, 12:42pm

I can only guess what EOC question you are after (it would be beneficial to state the reading and number of the question). So I assume you are looking at EOC question 13 of reading 42. It is asked to calculate the risk premium of equities when the goemtric return has been 8.0% for equities, 2.5% for T-Bills, and 2.1% for inflation.

In general any nominal return is a function of risk premium, inflation and risk-free return:

(1+r_nominal) = (1+r_{risk premium}) * (1+r_risk-free) * (1+r_inflation)

In the case of treasury bills it is assumed that r_{risk premium}= 0 and thus for treasury bills

(1+r_{nominal, T-Bills}) = (1+r_risk-free) * (1+r_inflation)

This equation can then be inserted in the first equation above to obtains:

(1+r_{nominal, equities}) = (1+r_{risk premium, equities}) * (1+r_risk-free) * (1+r_inflation) = (1+r_{risk premium}) * (1+r_{nominal, T-Bills})

and thus

r_{risk premium, equities}= (1+r_{nominal, equities}) / (1+r_{nominal, T-Bills}) - 1

cfageist · February 19, 2018, 3:49pm

Got it, thanks!