# Real vs nominal rate of return

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%

Why is the answer is A? Surely we need to obtain the real rate of return for equities which is 5.8% (1.08/1.021) and then divide this by the treasury bill (risk-free) rate of 1.025. Yet the book uses the nominal return in the top half of the fraction (1.08/1.025)?

In the book it says that the REAL RATE of interest is equal to (1+Rf)x(1+Rp) NOT the nominal rate of interest so I’m confused as to why they’ve used the nominal rate to compute the risk premium. Any ideas?

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Both the equity return and the Treasury return will include inflation; it cancels.

Ah right! That makes sense, thanks!

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My pleasure.

Hey, Could you pls explain this? I’m still struggling with it.

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Risk premia compound.

\left(1 + r_f\right)\left(1 + ERP\right) = 1 + r_{eq}
1 + ERP = \frac{1 + r_{eq}}{1 + r_f} = \frac{1.08}{1.025} = 1.0537
ERP = 1.0537 - 1 = 0.0537 = 5.37\%
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what i dnot understand about this question why in 1 we are using treasuary bill as denominator and in the other one the inflation?

What do you mean by, “the other one”?

Which other one?

some question ask for The real rate of return of equity instead of risk premium for equities
i guess the real rate is calculated with the infaltion in the denominator and the risk premiun with the closest one to a risk free investment in the denominator but i’m not sure on why though.

On a \$1 equity investment you would receive an extra 0.055 (1.08-1.025) in premium vs the tbill investment. This premium is 5.366% (0.055/1.025) more than you would receive over and above the tbill.

OR the long way
prem
= (1.08-1.025) / 1.025
= 1.08/1.025 - 1.025/1.025
= 1.05366 - 1
= 0.05366

No formula required. Just use basic math to calculate a % gain. If you purchase a stock for 102.5 and sell for 108. Your 5.5 gain (premium) is 5.366% more than your original purchase price.

108/102.5 -1 = 0.05366

or

5.5/102.5 = 0.05366