I have a textbook which contains a question which is as follows. Conveniently, the textbook doesn't have an answer key:

Calculate the net present value of the following cash flows: You invest \$2000 today and receive \$200 one year from now, \$800 two years from now, and \$1000 a year for 10 years starting four years from now. Assume that the interest rate is 8%.

I think I am okay regarding the present value of the inflows. If you let $i=0.08$ and $u=(1+i)^{-1}$, then ${PV}_{in}=200u+800u^2+1000(u^4+u^5+...+u^{13})=6197.74$, right?

But regarding the present value of the outflows, is it correct to think of them as ${PV}_{out}=2000$, or do I need to somehow take into account the interest payments I would have been receiving on the \$2000, had I not invested in the project?

On the one hand, doing this by thinking of discounting the FV of the (principal * interest) seems to suggest that the PV is just 2000. The FV of (principal * interest) after $k$ periods is $2000(1+i)^k$, but the discounting factor after $k$ periods is ${(1+i)}^{-k}$, for any $k$, so the PV is just 2000.

On the other hand, when I think of discounting the interest rate payments I would be receiving in each period, I get a different result? In period $k$, the marginal interest rate payment you receive on principal $P$ is $P(1+i)^k-P{(1+i)}^{k-1}=Pi{(1+i)}^{k-1}$, so its PV is $Pi{(1+i)}^{k-1}{(1+i)}^{-k}=Pi{(1+i)}^{-1}$. So on this basis ${PV}_{out} = P+nPi{(1+i)}^{-1}=2000+\frac{13*2000*0.08}{1.08}=3925.93$.

Whence the NPV is either $4197.74$ or $2271.81$, depending on how you look at it.

Could someone please help?


I'm afraid you got confused a bit about economic meaning of discounting. It actually takes into account the interest payments. The economic logic is following:

  1. 8% interest rate on deposit makes your $1$ today equal $1.08$ dollars in 1 year.
  2. When you calculate NPV, you basically convert future dollars into today dollars. You know rate of conversion because you have an alternative (a deposit)
  3. The result is how much the investment better or worse than an alternative (the deposit)

So the correct approach is the the first one (among the two you used), because discounting already accounts for interest payments


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