How to combine compound interest formula with inflation?

Question

Given a compound interest formula for an investment of FV = PV(1 + r/n)^nt, with a different APR for each year and a timeline of 2.5 years, how could this be combined with an inflation formula of R = (1 + r) / (1 + i) - 1? Would the first be plugged into the second, other way round or does it not matter? Would the non-full year affect inflation?

Answer 1

To handle the partial year, one way is to start with rates compounded annually, so if your given rates are, say 4.5, 5 & 4 % annual nominal interest compounded monthly then the rates r compounded annually are . . .

r1 = (1 + 0.045/12)^12 - 1 = 0.0459398
r2 = (1 +  0.05/12)^12 - 1 = 0.0511619
r3 = (1 +  0.04/12)^12 - 1 = 0.0407415

With inflation i at 6% compounded annually the longhand calculation for pv is

pv   = 1000
y1   = pv (1 + r1)/(1 + i)
y2   = y1 (1 + r2)/(1 + i)
y2.5 = y2 (1 + r3)^(1/2)/(1 + i)^(1/2) = 969.579

So, in one step

The same result is achieved using the monthly compounded rates:

pv   = 1000
y1   = pv (1 + 0.045/12)^12/(1 + i)
y2   = y1 (1 +  0.05/12)^12/(1 + i)
y2.5 = y2 (1 +  0.04/12)^6/(1 + i)^(1/2) = 969.579

Either way you get the same result.

Answer 2

Inflation does not affect future value. If you want to see the effect of inflation of that future value, you would just discount it by the inflation rate:

So your overall "real" return would be

FV(real) = PV(1+r1)(1+r2)(1+(r3)/2)
           ------------------------
                  (1+i)^2.5

So in your example, the "r" in the inflation equation would be the total nominal return over those 2.5 years, but you would use an exponent of 2.5 for the inflation effect in the denomitator.

Would the non-full year affect inflation?

Yes. Inflation is a continuous phenomenon, so you would count the half year, which is done by using the exponent 2.5