So we've made an important simplifying assumption, and that is that compounding occurs once a year. Suppose that interest is paid and reinvested more frequently, let's say, two times a year. Example, so a bank offers a stated annual interest rate of eight percent compounded semi-annually. If we invest $100, what do we have after one year? So what does this mean? So if we have a stated annual interest rate of something R, what happens is that we take that R and we adjust it to create a period rate. So in this case, if we have a stated annual interest rate of eight percent compounded semi-annually, my future value of $100 is going to be a 100 times 1.04 squared. So this 0.04, that's what I call my period rate, and this two that corresponds to my frequency. We're going to use the terminology, stated annual interest rate. We'll be consistent about that. So in consumer credit, sometimes this is called the APR, the annual percentage rate. So what's the general formula? The general formula is that the future value is equal to, well, let's just say it's $100 that we start with times 1 plus, so our period rate is the stated annual interest rate divided by our compounding frequency, m, and then we compound this m times. That's our future value. We can also have our present value. So what's the present value of $100? The present value is $100. Now we're going to divide by this. Well, that would be a mess to put this in the denominator. So we'll just use minus m. If we want to have this going on for t years, you just stick a t over here. This is for t years. So let's do some examples. Remember, future value is 100 times 1 plus stated annual interest rate divided by m to the mth power. In our example, we'll take the stated annual interest rate to be eight percent per year. Well, If m equals 1, we're back to annual compounding. So our future value is just simply 108. You can see how that works in this formula. When m equals 2, now we're doing semi-annual compounding. The future value is equal to 100 times 1.04 squared, which is $108.16. Well, there's other possibilities, you could do quarterly, for example, but let's skip ahead to monthly. So for monthly, we get 100 times 1 plus 0.08 over 12 to the 12th power. That's $108.30. If we want, we can compound every single day, including holidays, 365 days a year. So this m is a frequency per year, so that's 365 times per year. What's that? Well, that's $108, and we'll do some rounding, 33 cents. If we really wanted to, we could compound every second. That would be 31,536,000 times a year. In this case, what would we get? Well, we would get $108.32871. By the way, that's a little more than what we got here. This was really 108.32867. So yeah, you could compound as often as you want. Notice that even though these ms are getting pretty high, it's making less and less of a difference. So we can think about a mathematical construct, which is the limit as m goes to infinity. So m equals infinity, that's continuous compounding. It is physically impossible to compound something continuously. So we can think of m equals infinity as the mathematical limit of compounding more and more frequently. You can see in the previous example that for all intents and purposes, you can reach that limit. It's not too hard. So we ask ourselves, what is the limit as m goes to infinity of our future value formula? 1 plus the stated annual interest rate, which holds constant, divided by m to the power m. What is this? Well, it has a name, we can calculate it, it is the exponential. We take whatever is here and we raise e to this power. E, by the way is a number on your calculator, it's 2.718, and so on and so forth. So now, using this formula, we can calculate future value under continuous compounding. So let's just say that r_a, we use this notation, is the stated annual interest rate. So if we want to calculate the future value of a $100 compounded continuously for t years, we get 100 times e to the r_a multiplied by t. Or we could calculate the present value of a $100, that's $100 divided by e to the r_a t or e to the minus r_a t. So it might interest you to know that this e, this exponential was actually invented due to the continuous compounding problem. Leibnitz created e to answer this very question of what happens when you compound continuously. So people sometimes think that calculus was invented to deal with physics, but actually that's not true, finance is the much more fundamental subject. So now we have a potential problem, because we have a bunch of different compounding frequencies. Let's just put them all on the same footing, and the way we're going to do that is we're going to define an effective annual interest rate. An effective annual interest rate is just what you think. It's the rate that, when compounded annually, produces the same return as the stated annual interest rate, compounded m times year. So it's very simple, E AR solves 1 plus E AR equals 1 plus, remember r_a is our SAIR, divided by m to the mth power, this is r_a. So that's just the effective annual rate, so we can always solve for the effective annual rate. E AR is 1 plus r_a over m to the m minus 1. So for example, if r_a is six percent, if m equals 2, the E AR is 1 plus 0.03 squared minus 1, or 6.09 percent. If m equals 4, the E AR is 1 plus 0.015 to the fourth minus 1, or 6.14 percent. If m equals to infinity, the E AR is e_0.06 minus 1, which is equal to 6.18 percent. So let's do one more example, this one based on consumer credit. So credit card statements say something like the following, they'll tell you the period rate as 1.5 percent per month, then they'll say something like APR, annual percentage rate is 1.5 percent times 12 equals 18 percent. Now, notice this 18 percent is the interest rate you would pay if the credit card company charged you simple interest over the year. This corresponds to a stated annual interest rate, that's the concept that we defined before. Yes, you would pay 18 percent if you just were charged 1.5 percent per month and there was no compounding. But in reality, of course, the credit card company charges you on interest. So let's say you spend a $100 and you never pay your bill for a year. So after one month, your statement reads a $100 plus interest or 101.50. After two months, your statement reads a 101 times 1.015 squared. After so on and so forth 12 months, your statement reads 100 times 1.015 to the power 12. Now, that's not a $118, that is a $119.56. So really your effective annual rate is equal to 1 plus the stated annual rate or the APR, divided by 12 to the 12th power or 19.56 percent. That's the interest rate you're actually being charged, and that's what the credit card companies should report. So now we've talked about different ways of doing compounding and calculating present value, so now we're going to go on to actually value some securities.
