 So, as we noted, the ethics and morality of charging interest are not really subject to mathematics, but let's at least mention them. One rationale for charging interest on a loan. If you lend money, it won't be available, and you might miss an opportunity. So one way to look at it is interest corresponds to the opportunity cost. To extend this idea, we introduce the concept of present value. Suppose you could loan someone $100 for a year, but if you didn't loan the money, you could invest it and receive $120 at the end of the year. So the $100 now is worth $120 a year from now. We say that the $100 is the present value of $120 received in a year. Note that the present value is the same as the amount a borrower would receive now to pay back the final amount later. So the present value of a dollar corresponds to the discount function, and, if we assume the values are proportional, the present value of an amount L received at time t is Lv of t. So suppose our accumulation function is 1.03 to the power t, where t is in years. Let's find the present value of 500 to be paid back in five years. The discount function gives us. So the present value of 500 received five years from now is, or about $431.30. So why is this useful? Suppose you want to compare two investment opportunities. If they have the same beginning and ending dates, and you invest the same amount, the comparison is straightforward, which opportunity gives you more money at the end. But if they don't have the same ending period, we can compare their present values. To do so, we'll need to introduce an accumulation function as well. So suppose the bank of Alice offers a CD which pays $1,150 after two years on a deposit of $1,000. Meanwhile, Bob's bank offers a CD which will pay $1,200 after three years on a deposit of $1,000. We'll use the accumulation function 1.03 to the power t to the two CDs and determine which is the better investment. So Alice's CD returns $1,150 after two years. We find the present value of $1,150 in two years to be. And so using our accumulation function, we find the present value is about $1,083.99. Now we can interpret this result as follows. If you were to put this amount into an account earning 3% interest, you'd make $1,150 after two years. But you can make the same amount $1,150 by giving Alice $1,000. So Alice's CD provides an additional $83.99 in present value. It's like getting a bargain. But wait, there's more. Bob's CD returns $1,200 after three years. So again, we find the present value of $1,200 in three years, which will be. And again, we can interpret this as follows. You could get $1,200 after three years by investing $1,098.17 today. But you could get the same amount by giving Bob $1,000. So Bob's CD provides an additional $98.17 in present value. So since Alice's CD gives an additional $83.99 in present value, but Bob's CD gives an additional $98.17, Bob's is the better investment. You get more for the same amount under the assumption that our accumulation function is 1.03 to the power of t. And that assumption is important because in some sense it reflects all the other things we could do with the money we have. So what if our accumulation function was 1.05 to the power of t? That's the equivalent of being able to find something that pays 5% interest. So this time, the present value of $1,152 years will be, while the present value of $1,200 in three years will be. So this time, Alice's CD provides an additional $4,308 in present value, while Bob's only provides $3,661, and this makes Alice's CD a better investment.