FM for Actuaries

294 CHAPTER 9
from which we obtain
r R = 1+r N
1+r I
− 1= r N − r I
1+r I
. (9.2)
If the rate of inflation r I is low, we can write (9.2) as
r R ≈ r N − r I , (9.3)
which says that the real rate of interest is approximately equal to the nominal rate
of interest minus the rate of inflation.
Note that (9.1) and (9.3) are ex post relationships. They hold empirically and
do not refer to any theoretical relationship between the three quantities r N , r I and
r R . On the other hand, the economist Irving Fisher (1867 – 1947) argued that the
nominal rate of interest ought to increase one for one with the expected rate of
inflation. The so-called Fisher equation states that
r N = r R + E(r I ), (9.4)
where E(r I ) is the expected rate of inflation over the term of the financial asset,
also called the inflation premium. This equation implies that if the real rate of
interest is stable, a higher nominal rate of interest predicts a higher expected rate of
inflation. In contrast to (9.3), the Fisher equation is an ex ante prediction. Applying
the Fisher equation to short-term Treasury bill, we have
Nominal risk-free rate of interest = real risk-free rate of interest +
inflation premium. (9.5)
It should be emphasized that inflation premium is time varying. It depends on
the market participants’ expectation of the future rate of inflation. Typically this
premium rises when the experienced rate of inflation is on the rise.
An asset which is not subject to inflation risk is the U.S. inflation-indexed Treasury
bond. This bond adjusts the principal based on the rate of inflation as measured
by the CPI. As the principal rises with the rate of inflation, so does the coupon payments.
Investors are thus protected against inflation. Hence, the nominal rate of
interest of the indexed bond is also its real rate of interest.
Example 9.1: A stock will pay a dividend of $0.50 two months from now and
the annual dividend is expected to grow at a rate of 4% per annum indefinitely. If
the stock is traded at $7.5 and inflation is expected to be 2% per annum, what is
the expected effective annual real rate of return for an investor who purchases the
stock?