Compound Interest Compound Interest is a way to calculate the quantity of money in a bank account which is compounded regularly on some time scale. For example, you might have a bank account which has an annual interest rate of 2%, compounded monthly. This situation is modeled by the equation: A = P (1 + , r nt ) n
Where A is the amount of money you will have in your account after t units of time, P is the initial amount of money you deposit, r is the annual interest rate, n is the number of times your rate applies in one unit of time, and t is the time. Now, as an example, suppose you invest $500 in a bank with 2% interest compounded monthly. We can already tell that the equation we want to use will have P = 500, and r = .02 (which is 2%). Let’s say that we want our units of time in years. We notice that there are 12 months (from the rate) in one year (from the unit of time), so n = 12. Now our equation reads: A = 500(1 + or if you want to write this as a function, A(t) = 500(1 + .02 12t ) 12 .02 12t ) , 12
. Now remember here that the 2% rate was an annual rate. What would happen if our account only compounded once a year? We would end up in one year with 1.02 ∗ 500, which is $510. We don’t really need to set up an equation here. If the rate is 2%, then we’ll end up with 102% of our money at the end of the year. Now with our monthly compounded interest bank, what do we have at the end of the year? A(1) = 500(1 + .02 12t ) = 510.092, 12
so we end up with $510.09, not much of a difference. Give it more time, however, and we can start to see a difference. Let’s look at the yearly compounded bank in 20 years, and we’ll give it the name B (try to see if you can derive the formula for the yearly compounded bank without looking): B(t) = 500(1 + and so in 20 years, B(20) = 500(1 + .02 1t ) , 1
1
.02 20 ) = 742.974. 1
2
This means bank B (which has yearly interest) will have $742.97 waiting in our bank account after 20 years. How much better will the monthly compounded interest bank do? Well, .02 12∗20 A(20) = 500(1 + ) = 745.664, 12 so the monthly compounded interest bank give us $745.66, a bit better than the bank which yearly (or annually) compounds the interest. Try this with some examples with bigger interest, a larger starting amount, or a longer time, and you’ll see how this difference can really add up. One other factor to keep in mind is what the percent really is. It’s always an annual interest. This means that we can think of the monthly compounded interest situation as you getting 2% per year, but it’s broken up. Every month you get about .16666% in interest 1 (really that number is 6 %, which means that over the course of the year, you ”sort of” get 1 % ∗ 12 = 2% in interest. The benefit, like I said above, is that you get interest on your 6 interest, so you actually make more money. Now let’s look at one more thing. We wrote our equation so that t had units of years, but this doesn’t have to be the case. We can write the equation so that the units are in terms of months, but we’ll need to play around with the interest rate a bit. We were getting 2% per year, but this is the same as .16666% per month. We’ll need to use that. We’ll still have P = 500, we now have r = .0016666, and we need to figure out n. This n needs to relate the units in the r (which is months) to the units of t (which is also months), so n will just be 1. Now, let’s call this situation C(t). .0016666 1∗t C(t) = 500(1 + ) = 500(1.0016666)t, 1 so in 12 months, C(12) = 500(1.0016666)12 = 510.092. This means in 12 months, the bank account which offers 2% annual interest compounded monthly (which is where we got our .16666%) in which we deposited $500, has a total of $510.09. This is exactly what we found for our first equation, which was evaluated at one year!