# Net Present Value in Nominal Terms

I have the following question from my homework:

When calculating the net present value of an investment project, the firm of Henry & Norman expects profit in the first year to be $60,000, and they expect real profits to remain at that level over the next five years. Since they are using nominal discount rate of 11 percent in their net present value calculation, they want to convert future real profits to nominal profits. They expect inflation to be 4 percent per year over the next five years. The nominal profit for year 2 of the investment project is ___________(correct answer, 62,400) If the investment project has an initial cost of 200,000, the net present value in nominal dollars is _________ (my answer 64,474.50 book answer 38,264.90) The difference between the book answer and my answer is that I used 60,000 as year 1 profit with no discount. I realized this was incorrect according to the given methodology in my textbook of$NPV=-C+\pi_1/(1+R)+\pi_2/(1+R)^2+...$What I don't understand is why we discount the profits in year one but don't consider inflation in year one. This seems inconsistent to me. Why do we discount the real profit in the first year and not the nominal profit, or if we assume nominal profits in year one are the same as real profit, why do we discount in year one? • Did I answer your question? Is it still unclear? – Aaron Hall Nov 28 '15 at 13:26 • @AaronHall It still is unclear to me why the difference in treatment. In my mind it is still an inconsistency. At T0 both cost, profit are real, but the profit is treated as occurring during year one and being discounted in year one, in the meantime we pretend inflation doesn't exist in year one. I still don't get it, I'll just follow the algorithm. – Owen Sechrist Nov 28 '15 at 16:26 ## 2 Answers What I don't understand is why we discount the profits in year one but don't consider inflation in year one. This seems inconsistent to me. Why do we discount the real profit in the first year and not the nominal profit, or if we assume nominal profits in year one are the same as real profit, why do we discount in year one? As you said, the firm expects profit in the first year to be$60,000, and they expect real profits to remain at that level over the next five years.

They are expecting both real and nominal profits (because they do not explicitly state only one or the other) to be \$60,000 for year one but they need to calculate NPV for$T_0$, thus they need to discount the \$60,000 for a single year (and the others for an additional year).

We're realizing the profit as income starting in $T_1$, and realizing the initial capital expenditure at time $T_0$.

$T_0$----------$T_1$--------$T_2$--------$T_3$--------$T_4$--------$T_5$
-200000 + 60000 + 60000 + 60000 + 60000 + 60000 Real
-200000 + 60000 + 62400 + 64896 + 67492 + 70192 Nominal
-200000 + 54054 + 50645 + 47451 + 47451 + 41655 Nominal Discounted

Calculated in Python:

>>> -200000 + sum(60000 * (1.04**i) / (1.11**(i+1)) for i in range(5))
38264.90224378838


As to the reason the authors of the question gave these conditions, I presume that they want to ensure that you can quickly arrive at the correct calculations based on seemingly trivial but important details. I recall a similar trick in a corporate finance final, where a careful reading of the question indicated that a speedy reading of the question (one that I initially made and expected most of my peers would make) would result in an off-by-one time period mistake.

• why are you treating Year 1 profit as nominal and at the same time real? – london Nov 29 '15 at 19:18

Because the cash flows are in real dollars, you should first calculate their nominal values using the rate of inflation 4%. Then, discount these cash flows using the nominal discount rate 11%. You should get NPV in nominal dollars. The answer will be the same if you use real discount rate to calculate the NPV in real dollars, but this time, use the original \$60000 for each year as a cash flow as the company expect the real profits remain constant for the next 5 years.

• Yes, I understand that you need to use nominal profits/cash flows if you are using nominal rates to achieve the discount rate. What I'm trying to understand is when you use nominal rates in this example, why do you use 60,000 for the first year's profit but also discount it. Why not use 60,000 for the first year with no discount on the first year, then use 60,000x(1+inflation) for the second year and discount the second year at (1+R)^1 ...? – Owen Sechrist Nov 27 '15 at 22:35
• I think you should understand that the investment outflow of 200,000 occurs in Year 0, that's 'today', in a year's time, exactly 12 months later you expect to receive the first year's profit of 60000. Therefore, you should account for a year's inflation effect and discount it to present. First year is not "now", it is exactly 12 months later from now. – london Nov 28 '15 at 17:19
• You are proposing exactly my point, however, that produces a different answer than the accepted answer for the homework. – Owen Sechrist Nov 29 '15 at 18:31
• @OwenSechrist, there is an issue with the wording of this problem, implicitly, Year 1 profit is nominal for some reason (treated as nominal value, though is real explicitly). I tried to calcluate the NPV, but it does not work and you will not get the 'correct' answer (as supplied by the authors). Aaron Hall provided an answer above, but I think he just matched the book answer. He assumes the real and nominal values in year one are the same, though, this was not noted in the question. I think this issue could be reported in the errata of the book, most probably available from book's authors. – london Nov 29 '15 at 19:16