Sunday Function

"I'd like to open a savings account," you tell your banker.
"Excellent!, the banker says, "our savings accounts have an interest rate of 5% and an annual percentage yield of 5.127%. It's a great deal and I think you'll like it."

5% interest is a great deal on a savings account these days. But what's all this about the difference between interest rates and annual percentage yield? You often hear about the magic of compound interest, and this particular distinction between the various types of rates are a consequence of the way this magic works.

Say you've got $100, and your banker agrees to pay you 5% at the end of the year as your interest for saving with their bank. At the end of the year you'll have $105.

Or you might tell your banker that you'd rather have interest payments every six months, so at the end of each 6-month period you'd like half the yearly interest - 2.5%. So at the end of the first six months you'll have $102.50. But at the end of the second six months you'll have $105.06. Which is more than you'd have had if you took the entire interest payment at once at the end of the year.

The reason is that over the second six months, you're not being paid that five percent of a hundred bucks, you're being paid five percent on slightly over a hundred bucks.

Being a canny capitalist, you wonder if you could improve matters. Why not take one fourth of the interest four times per year? If you do, you'll end up with $105.10. Not much of an improvement, but an improvement nonetheless.

You stroke your chin and contemplate. What if you were willing to split that five percent into 365 payments that occurred once per day? Surely the magic of compounding will push your principal to lofty heights. And it will - slightly loftier, anyway. You'll have $105.13 at the end of the year, if your banker rounds up to the nearest cent. Which looks suspiciously like the APY the banker in our hypothetical conversation quoted.

The reason is that as the interval between compoundings grows shorter and shorter, the total you make keeps growing and growing. But it doesn't do so without limit. Instead, it converges in on a particular value. That value is what you get by continuously compounding, and it or something very close is what most banks pay. The interest rate is the interest rate, and the APY is how much you actually make due to that continuous compounding procedure.

But this is Sunday Function, not Sunday Exposition on Banking. Let's crank this into a function. If you start with an initial amount P and leave it in the bank for time t, with interest rate r per period (period in the same units as t), compounded n times per period, you'll end up with money equal to:

Let's play around with this a bit. Set r, t, and P equal to 1 (which is equivalent to putting $1 in for 1 year at 100% interest), just so we can see how the fractional increase in M changes as n increases.

n = 1
M = 2

n = 2
M = 2.25

n = 10
M = 2.59374

n = 100
M = 2.70481

n = 10000
M = 2.71814

Starting to look a little familiar? M seems to be closing in on the number e = 2.71828...

As in fact it is. It turns out that as n approaches infinity, we're going to end up with an exponential formula for continuously compounded interest:

The proof is not difficult, but longer than can easily fit into one post so we'll save it for another time. Meanwhile if you find a bank that's offering 5% interest on savings accounts, let me know.