Reading: Treating Inflation Consistently

When valuing projects, we have to make sure to treat inflation consistently. The general rule is very simple:
 

• Discount nominal cash flows at the nominal discount rate
   
• Discount real cash flows at the real discount rate.

If we follow this procedure, both methods will produce the same NPV!

The corresponding math...

We can see this mathematically. With in nominal terms, we discount the nominal net cash flow (NCF) with the nominal rate of return (R):

\(\text{Present Value}_{\text{nominal}} = \frac{NCF}{(1+R)}\)

In contrast, in real terms, we discount the real (deflated) net cash flow (NCF*) with the real rate of return (R*):
 

\(\text{Present Value}_{\text{real}} = \frac{NCF^*}{(1+R^*)}\)

From the considerations from before, remember that the deflated (real) cash flow corresponds to the nominal cash flow divided by one plus the expected rate of inflation:

\( NCF^* = \frac{NCF}{(1+\pi)} \)

Also, we know from before that the relation between the nominal and the real rate of return is:

\((1+R^*) = \frac{(1+R)}{(1+\pi)}\)

If we plug these two terms into the valuation formula for the real NPV, we get:

\(\text{Present Value}_{\text{real}} = \frac{NCF^*}{(1+R^*)}\) \(\frac{\frac{NCF}{(1+\pi)}}{\frac{(1+R)}{(1+\pi)}} = \frac{NCF}{(1+R)} =\text{Present Value}_{\text{nominal}}\)

Example

You have estimated the following (nominal) net cash flows for a project that last's 5 years (see also the accompanying Excel file):

You also know that the project's nominal cost of capital (R) is 12% per year and that the annual expected rate of inflation (\(\pi\)) is 5%. Based on this information, what's the project's NPV in nominal as well as in real terms?

To value the project in nominal terms, we can proceed "as usual" and capitalize the nominal NCFs from the table above at the nominal cost of capital of 12%. The result is a NPV of 7'028 (see also the Excel file):

\(NPV_{nominal}=\sum_{t=1}^5 \frac{NCF_t}{(1+R)^t}=-15'000+\frac{3'000}{1.12}+...+\frac{4'000}{1.12^5}=7'028\)

In contrast, if we want to value the project in real terms, we first have to...:

1. … compute the real cost of capital (R*) 
2. … deflate the the nominal NCFs to get the real NCFs (NCF*)
3. … capitalize the real NCFs (NCF*) at the real cost of capital (R*).

Using the equations from before, the real cost of capital (R*) is 6.67% per year:

\(R^* = \frac{1+R}{1+\pi} - 1 = \frac{1.12}{1.05}-1 = 0.0667 = 6.67\%\)

To deflate the nominal net cash flows, we divide them by one plus the cumulative inflation. With an annual inflation (\(\pi\)) of 5%, the cumulative inflation is 5% after 1 year, 10.3% after 2 years \((=(1+\pi)^2-1 = 1.05^2-1=0.103)\), 15.8% after 3 years \((=(1+\pi)^3-1 = 1.05^3-1=0.158)\), etc. The following table shows the computation of the deflated NCFs (again, see the Excel file for details):

Now we can capitalize the real NCFs at the real cost of capital. Not surprisingly, the resulting NPV is the same as above, namely 7'028:

\(NPV_{real}=\sum_{t=1}^5 \frac{NCF^*_t}{(1+R^*)^t}=-15'000+\frac{2'857}{1.0667}+...+\frac{3'134}{1.0667^5}=7'028\)

This example has shown how to value projects in nominal terms and in real terms. If we treat inflation consistently, both approaches yield the same result!