In this section, we are going to discuss the Net Present Value rule. So imagine you are running a corporation, how are you going to decide which projects are worth investing in? That's what the net present value rule is designed to do. What this section we'll discuss is why the net present value rule should be the criterion for choosing which projects a corporation should undertake. So I'm not asking you to take on faith that we should use the net present value rule. Hopefully by the end of this section, you will be convinced that the net present value rule is the rule that increases the value of the corporation. I will discuss how to compute net present value. So first we're going to talk about future value. So we deposit a $100 in a bank account that pays 10 percent interest. The future value then is my deposit plus what I get an interest, in this case, 10 percent on a $100. What's that equal to? That's equal to a $110. In general, for some interest rate r, the future value is going to be 100 times 1 plus r. With future value, note that were moving the cash flow forward in time. That's future value. What about present value? Suppose you need a $100 in one year, what do you put aside today? So now a $100 is the future value and you're looking for present value. So you're looking to solve a 100 equals present value times 1 plus r. You're putting aside present value today and by the previous formula, it grows up at 1 plus r to become a $100. Well, let's just solve this equation, present value is $100 over 1 plus r. So if r, for example, is 10 percent, my present value is $90.91, more or less. Note that we're bringing the a $100 backward in time. So what's our general formula for present value? So in general, if we have a cash-flow C in one year, what's my present value? It's simply C over 1 plus r. Now, one thing to note is this is sometimes called the present discounted value. Why is that? Well, if r is greater than 0, then note the present value of C is going to be less than C, that makes sense. So we call this present discounted value because, normally, we think of r being greater than 0. Well, recently, r hasn't always been greater than 0 sometimes we see it actually equal to 0. Well, no big deal. Present value of C is less than or equal to C. So just as an aside, why do we think that r should be greater than or equal to 0? So suppose the bank offers r of minus 8 percent, so let's say if we deposit a $100, next year, we get $92. I think most of us, most of the time would say no thanks, we'll just keep the money in the cookie jar. This example is why we believe that r greater than or equal to 0 is a safe assumption, whereas r negative can't happen. Let's just remember though that the reason why we're assuming r is greater than or equal to 0 is because we can put that money in the cookie jar, we're not afraid, say, of somebody taking the money, of something happening to the money. So this may not hold at all times in all places. There might be certain times where we might not see r greater than or equal to 0 where storing money might become very expensive. So one thing to remember about the 2008 financial crisis is briefly, r was equal to 0 and impossibly less than 0. Other things that happened around that time is that the price of safes and sale of safes spiked, and so did the sale of guns. So sometimes storing money can become expensive, and so r greater than or equal to 0 might not hold all of the time, but most of the time it holds, and that's an assumption that we're going to use. So that was present value. Let's talk about net present value. Net present value is equal to C0, plus C1 over one plus r. This C0, that's the negative of the cost of the investment. This C1, that's the payoff in one year. So most of the time, C0 is going to be a negative number and C1 is going to be positive number. So you can see from this example that net present value is really just present value. The net just emphasizes that we've included a first term, that's the initial investment. So let's do an example of net present value. So let's say we are a software developer and there's a cost or required investment of 0.5 million to develop the software. Then next year, what are we going to get? The payoff is going to be 0.54 million. So my NPV in millions is minus 0.5, that cost is an outflow, that C0 is negative plus we take that 0.54 and discount it back to the present, just as we discussed with present value. So by the way, if we put in r a 5 percent, my NPV is 0.0143 million. So that's net present value.
