# Is this formula for net present value correct?

I have been presented with this formula for the net present value: $$NPV = I_o + \sum \frac{F_t}{(1+r+p_t)^t},$$ where $$F_t =$$ net cash flow for period $$t$$; $$R=$$ required ret of return; $$I_o=$$ Initial cash investment, and $$P_t=$$ inflation rate during period $$t$$.

Although I've been told it is an industry standard, I don't get it.

1. Why does rate of return increase with time? I assume it is constant - if I want 8% return each year, I assume it comes from inflation adjusted value, so the term should be out of the brackets
2. "Initial cash investment" pharsing suggest the value will be positive, but I assume it has to be negative, otherwise it wouldn't make sense to sum initial investments with inflows.

Any comment will be appreciated, thank you!

• Shouldn't $I_0$ enter the equation with a negative sign? Jun 20 '16 at 7:51

Q2 : Indeed the correct formulation is for initial investment to appear with a minus sign, since it is an outflow. This reflects the position of somebody who contemplates making the investment. Note also that you shuold check whether there exists some residual value of assets at the end of the operational period examined - this should be added (and discounted) together with the net cash flow of the last future period examined.

Q1 : the logic of the net present value calculation is to notionally compare the returns of the specific investment with some abstract alternative that pays a constant $r$ per period, as a percentage of value left to produce returns. It is as though we have a bank account from which we draw and to which we deposit.

Now think in reverse : in order to "have" amount $F_3$ (three periods in the future), you only need to deposit now in this account $F_3/(1+r)^3$, and leave it there to accrue interest.

Now, if the cash flows are calculated in nominal terms, you would want to deflate them, so as to express them in today's value/purchasing power. The real value of $F_3$ is $F_3/[(1+\pi_1)(1+\pi_2)(1+\pi_3)]$, where $\pi$ is the rate of inflation. So over all we have

$$NPV (F_3) = \frac {F_3}{(1+r)^3 \cdot (1+\pi_1)(1+\pi_2)(1+\pi_3)}$$

Now, if you assume a constant inflation rate (or an average - after all, $r$ is also essentially an average), then you can write

$$NPV (F_3) = \frac {F_3}{(1+r)^3 \cdot (1+\pi)^3} = \frac {F_3}{[(1+r) \cdot (1+\pi)]^3}$$

$$=\frac {F_3}{(1+r +\pi +r\pi)^3}$$

Usually the term $r\pi$ is considered negligible (when we look at "usual" required rate of returns, and when inflation is low, think something like $0.1\cdot 0.02 = 0.002$). So it is common practice to be dropped (both in industry and in academia), arriving at

$$NPV (F_3) = \frac {F_3}{(1+r +\pi)^3}$$

What you were given keeps the inflation rate time variant. This comes from the approximation

$$i_t \approx r_t + \pi_t$$

i.e. that the nominal interest rate equals approximately the real interest rate/rate of return plus inflation. So the nominal rate of return may increase with time solely due to inflation.

• Thank you very much! This has explained everything to me. Indeed, missing r*π was confusing. Also, the rate of return makes sense now if thinking of it as a bank deposit alternative.
– Ivan
Jun 21 '16 at 9:33