In the previous section, I discussed the Net Present Value rule. In that section, the examples were simple. There was a cost for the project and a payoff at some future time. But usually, examples are a little bit more complicated. There are payoffs spanning multiple years. So we need a way to deal with that. Also, this payoffs may go on for an indefinite length of time. That happens, for instance, if you're valuing a stock. So that's where the perpetuity formulas come in. In this section, I will cover compound versus simple interest. The annuity and perpetuity formulas and finally, compounding within a year. These topics are important not just for valuing bonds and stocks in the next sections but also for Finance 612 and for all finance electives. Suppose we have $100, What can we count on getting if we invest this in a bank at an interest rate r for 2 years? Let's say r is 7% and we're investing for 2 years. Well under something called simple interest, My future value. So what I'd have at the end of 2 years would be my principal + my first interest payment, $7 + my second interest payment, $7 or $114. So we get the interest twice. But there's something a little funny here. So the second interest payment, we're getting interest on $100. Well, why not earn interest on the $7? That leads us naturally to compound interest. Under compound interest, we earn interest on interest. So we get the $100, that's our principal. Then we get $7. Then we get another $7. And finally, we get the interest payment on the first $7. Another $0.49, let's just do it with the letter r. We get $100, then we get our interest payment, then our second interest payment and then interest on interest. If you do the algebra, this is 100 (1 + r). Okay, that's what we get after 1 year. Then after the second year, we get that same amount reinvested, that's 100 (1+r) (1 + r) again, or 100 (1 + r) squared. Okay, so if we have compound interest, T years, then the future value is 100 (1 + r) to the power t. Well, we've just seen this can make a difference. So, for t = 2, and let's stick with the r = 7%. For t = 2 with simple interest, that gave us $114. Compound interest gave us $114.49. Well, who cares if it's just $0.49? Suppose we weren't just investing for 2 years. Suppose we were investing for 100 years. Well in that case, under simple interest, We would get our, $100, And we get $100 interest payments. Each of them would be, $7 for a total of $800, not too bad. What if we had compound interest? We would earn interest on interest for 100 years, That's 100 (1.07) to the 100th power. That's not $800. That's 86,771. So it's a big difference. This is the power of compounding. So what's the correct formula to use? Do we use the simple interest formula or the compound interest formula? Do we get the 800 or the 86,771? Well, it all depends if we can reinvest the proceeds or not. Unless stated otherwise, I will be assuming that the proceeds can be reinvested. Okay, so that's future value. What about present value? So for present value, we ask, what do we need to set aside today to have a $100 in 2 years? So present value solves the equation. PV (1 + r) to the t = 100. So we can see that the present value, the amount that we set aside today is 100 divided by (1 + r) to the t. So this 1 over (1 + r) to the t, as a name, it is sometimes called the discount factor. Sometimes people just call it DF. So that's the simplest present value calculation you can do. You have one cash flow, t years in the future and then you discount it back to the present. The next most complicated cash flow is something called an annuity.