# Net present value

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.

1. 8% interest rate on deposit makes your $$1$$ today equal $$1.08$$ dollars in 1 year.