In the first part of this article series on the mathematics of finance, we discussed the compound interest formula and how it is used to calculate the accumulated value of money deposited over time. In this next part, we examine the different methods of compounding and the impact this has on the growth of your money. The interesting thing of all this compounding stuff is that you will see that regardless of how frequently we compound, at some point we reach an upper limit. In other words, frequent compounding helps but at some point you have to do more than just leave your money with “Frequent Compounding Bank USA” to get more of a return. Keep reading.
You may have noticed at your last visit to your local bank a billboard or sign promoting the bank’s aggressive savings and interest programs. You may have seen something like 5.5% rate on your money market with a net effective yield of 5.61%. Sometimes this net effective rate is referred to as the APR or annual percentage rate. Why would there be two different interest rates attached to the same savings plan? Simply put: this is where compounding comes in.
You see the bank might very well be compounding your money quarterly, monthly, or even daily. What does this mean? Quarterly compounding means the bank gives you interest every three months based on the given interest rate. Monthly and daily work similarly except that the compounding is more frequent. Let’s take a specific example to make this all clear.
Suppose you decide to deposit $1,000 with “Frequent Compounding Bank USA.” The bank advertises heavily its generous interest rate of 5.5%. As you learned in Part I of this series, if the bank compounds this money yearly, then at the end of the year you will receive $55 in interest. However, most banks don’t do this and actually compound more frequently, such as quarterly or even monthly. If the bank compounds quarterly, then it takes the rate of 5.5% and divides it by four to get a nominal rate of 0.01375 or 1.375%. The bank then plugs this into the compound interest formula and uses 4 as the number of compounding periods. The accumulated value at the end of the first year will be A = $1,000*(1.01375)^4 or $1,056.14.
Thus at the end of the year you will receive not $55 in interest but $56.14 in interest or $1.14 more. Consequently, by compounding four times instead of once, you will have an extra $1.14 in interest. Notice that this will be the same as if the bank had compounded only once and used an interest rate of 5.61% instead of 5.5%, hence the APR or net effective interest rate of 5.61% when compounding quarterly.
If the bank compounds monthly, then it takes the 5.5% rate and divides by 12 to get a nominal rate of 0.004583 or 0.4583%. The bank plugs this into the formula to get the accumulated amount at the end of the year of A = $1,000*(1.004583)^12 or $1,056.41. Thus by compounding monthly instead of yearly, you have an extra $1.41 in interest for a net effective interest rate of 5.64%. Notice that by going from quarterly to monthly, the amount of interest has gone up. So if we compound ever more rapidly will we get richer and richer?
Alas, no. As much as our local banker wants to be our friend, even he is limited by the compound interest formula. You see, as we compound more and more frequently, we eventually reach an upper limit. The limit is established by a formula which uses the transcendental number e. You can see how the limit starts to be reached in the example of daily compounding. Suppose we find the accumulated value using daily compounding. We take 5.5% and divide by 365 to get 0.0001507 or 0.01507%. Thus A = $1,000*(1.0001507)^365 or $1,056.54, only $0.13 more than the monthly example.
Too bad. If we could get more and more money by more frequent compounding, all we would have to do is cut a deal with our banker over at “Frequent Compounding Bank USA” and watch our money grow. Unfortunately, even mathematics has its limits. No pun intended. See you next time…
