Suppose you have an annual yield of 5% on bonds (or a return of 5% on stocks), and annual inflation is running say 7%.
What’s the the “real” yield after adjusting for inflation?
INCORRECT ANSWER:
AdjustedYield = Yield - Inflation
CORRECT ANSWER:
AdjustedYield = (1+Yield)/(1+Inflation) - 1
CORRECT ANSWER DERIVATION:
The definition of ‘yield’ (or ‘return’), evaluated at some date in the future is:
Yield = (FuturePrice - TodaysPrice)/TodaysPrice = FuturePrice/TodaysPrice - 1
When there’s inflation then both prices in the above equation for Yield (or Return) need to be put on equal terms. Let’s adjust TodaysPrice to FuturePrice based on an annual inflation rate, “Inflation”. For simplicity, we’ll assume the future date is one year from today:
TodaysPriceAdjusted = (1+Inflation)*TodaysPrice
We need to use TodaysPriceAdjusted instead of TodaysPrice in the above formula in order to get AdjustedYield:
AdjustedYield = FuturePrice/TodaysPriceAdjusted - 1
Substituing in TodaysPriceAdjusted:
AdjustedYield = FuturePrice/[(1+Inflation)*TodaysPrice] - 1
Rearrange to isolate FuturePrice/TodaysPrice
AdjustedYield = [1/(1+Inflation)] * FuturePrice/TodaysPrice - 1
We know from above that
Yield = FuturePrice/TodaysPrice - 1
so
FuturePrice/TodaysPrice = 1 + Yield
Substitute this into the formula for AdjustedYield to get:
AdjustedYield = [1/(1+Inflation)]*(1 + Yield) - 1
i.e.
AdjustedYield = (1+Yield)/(1+Inflation) - 1
Example 1:
If the nominal yield is 5% and inflation is 8%, then the adjusted yield is
(1+0.05)/(1+0.08) -1 = -0.0277 or a loss of 2.77%
The loss is not 5-8= -3%
Example 2:
If the nominal yield is 10% and inflation is 8%, then the adjusted yield is
(1+0.10)/(1+0.08) -1 = 0.0185 or a gain of 1.85%
The gain is not 10-8= 2%