# Net Present Value

Honestly, this equation doesn’t tickle me like others do. It’s practical and it’s good to know, but it’s a little too cold-hearted for my tastes. There’s just no love in it. 😦 However I still really want to talk about as a lead up to my next post which will be about the Bellman Equation.

Before we dive deeper into the Bellman Equation let’s look at how you would evaluate some investment $s$, where the best return on alternative investment is given by $1+i$: $NPV(s) = \sum_{t=0}^{\infty} \frac{R_t}{(1+i) ^{t}}$ $NPV(s)$ is the present value of some investment $s$.

What this is trying to tell us is that we shouldn’t just look at the nominal returns of some asset, but we should discount any future returns by how far out in the future we receive it. Think of it this way: A dollar today is more valuable than a dollar tomorrow. Why? We could invest that dollar right now and receive some (small) return. Or the dollar might not be as valuable tomorrow because of inflation. Or we could just straight up get hit by a comet by the time we get to enjoy the fruits of our newfound fortune! That’s what the discount rate $\frac{1}{1+i}$ is trying to factor in: Our value of the present over the value of the future.

Now I’m going to ask you a question to help that discount rate sink in: How much would you pay for a million bucks in your bank account right now? That sounds kind of stupid, right? You’d probably pay any value up to a million bucks. How about a year down the road? Or 50 years down the road? Try to derive a value $\frac{1}{1+i}$ based on how much you would pay to receive that money in the future.