Inflation and NPV

An important aspect of project valuation is the discounting of future cash flows using an appropriate opportunity cost of capital.
When cash flows and the opportunity cost of capital are given, obtaining the net present value is straightforward using either a financial calculator or a spreadsheet.
However, under inflationary economy, good care must be given to both cash flows and the opportunity cost of capital to obtain the same net present value in either nominal or real terms.
Only today I realize why my students were finding it difficult to deal with inflation in discounting real cash flows. The exercise involving a constant inflation rate was well understood, but I got only 1 right over 42 students on the exercise involving varying inflation rates.
I thought it may be good to demonstrate for the unknown student how to deal with the term structure of inflation rates. But as for every thing else in life, let start with the easiest case and end with the most involved case.
Suppose the following:
Investment (I(t=0)) = $800, real cash flow at time 1 (RCF(t=1)) = $500, RCF(t=2) = $700, and RCF(t=3) = $800. Assume an opportunity cost of capital of 25%.

What is the net present value in nominal terms when the expected inflation rate is 10% and constant over the three years?

Step 1: Convert real cash flows into nominal cash flows using
RCF(t)(1+i(t))^t,
where i(t) is the expected inflation rate at time t.
Step 2: Discount nominal cash flows using the nominal discount rate (the opportunity cost of capital).

Applying the two steps at once we obtain:

NPV = -$800 + $500(1+0.1)/(1+0.25)  + $700(1+0.1)^2/(1+0.25)^2 + $800(1+0.1)^3/(1+0.25)^3 = $727.26

What is the net present value in real terms?

Step 1: Adjust the nominal opportunity cost of capital for inflation rate by using
r(t) = [(1 + R(t))/(1 + i(t))] - 1,
where r(t) is the real rate at t, R(t) is the nominal rate at t, and i(t) is the expected inflation rate at t.

Doing so, we obtain, [(1+0.25)/(1+0.1)] - 1 = 0.1364 or 13.64%, which is constant over t.
Step 2: Discount real cash flows in terms of real opportunity cost of capital
-$800 + $500/1.1364 + $700/1.1364^2 + $800/1.1364^3 = $727.26

Conclusion: We obtain the same NPV in both nominal and real terms.

The following part is the more involved, and you will see why.

What if inflation rate is not constant over the years? How can we be consistent?
Let now assume that the expected inflation rate is 10%, 12%, and 15% at time 1, 2, and 3, respectively. We keep the rest of the given unchanged.

In nominal terms, we get:
-$800 + $500*1.1/1.25 + $700*1.1*1.12/1.25^2 + $800*1.1*1.12*1.15/1.25^3 = $772.26
So the formula is: RCF(t)(1+i(1))*(1+i(2))*...*(1+i(n)/(1+R)^t)

In real terms
step 1: Get the nominal rates at each t using R(t)(1+i(t))^t
step 2: Adjust R(t) for inflation using r(t) = [R(t)/((1+i(1))*(1+i(2))*...*(1+i(n)))^(1/n)]-1
So, in our cases
r(1) = [(1+0.25)/(1+0.1)] - 1 = 0.1364 or 13.64% as before
r(2) = [1.25^2/(1.1*1.12)^(1/2)] - 1 = 0.1262 or 12.62%
r(3) = [1.25^3/(1.1*1.12*1.15)^(1/3)] -1 = 0.1130 or 11.30%

We now use these real discount rates to discount our real cash flows to obtain the following:
-$800+$500/1.136 + $700/1.1262^2 + $800/1.1130^3 = $772.26

Conclusion: We are consistent. Both methods agree.

I hope this helps.