Rates of interest and Euler's number

joão pestana

In a previous post I mentioned interest rates, but I didn't told you what they are. I simply assumed that you knew and, to a certain extent, I believe that's a reasonable assumption. This time I want to bring back the subject and focus just on what an interest rate is and not on what it's used for — which is probably what you know intrinsically.

I'm a firm believer that if you can put a story behind your explanations you should do it. Single case examples go first and we end with a general statement that's universal. Imagine our dear friend Sarah has 1 € and she has the opportunity to double it after a year by investing it in a savings account. If that seems like it's not much, think again. It's a 100% rate of return on investment! Mathematically, we can write it as:

1 € + 1 € xx 100% = 2 €

This is what is expected to happen if the back credits the account once every end of year. It doesn't have to be like that and some banks do credit the accounts more often. Let's say that Sarah's bank credits her account every semester, everything else remaining the same. Will we get the same result? Let's see...

1 € + 1 € xx (100%)/2 = 1.50 €

1.5 € + 1.5 € xx (100%)/2 = 2.25 €

Because we're being credited twice a year, the rate of return for each time is halved. On the first semester we get 0.50 € from our investment of 1 €, but on the second semester we receive 0.75€. This sums up to a total of 2.25 € and is bigger than the first example where the credit happens only once. What happens is that on the second semester, the amount already paid is considered reinvested and contributes positively to the final result. What if the bank paid every quarter year, would we get an even better deal?

1 € + 1 € xx (100%)/4 = 1.25 €

1.25 € + 1.25 € xx (100%)/4 = 1.56 €

1.56 € + 1.56 € xx (100%)/4 = 1.95 €

1.95 € + 1.95 € xx (100%)/4 = 2.44 €

These numbers are rounded, but clearly the answer is yes! The more frequently the bank credits your account, the better is your return at the end of the year due to new credit being considered a reinvestment. Let's take a big leap here and ask ourselves what if the back paid us every second? Would we get an astronomically high rate of return? I'm going to give out the answer right way and prove it to you afterwards.

The answer is a bit ambiguous, so let me clarify it. It always happens that, if you increase the frequency of credit, your final return is higher, but it has an upper limit. There is a number in which the return grows ever so closely with each increase in the credit frequency, but it cannot surpass it.

We can see, from our previous results, that even if we always get a bigger value than before, the ratio between each successive increase diminishes. From 2 € to 2.25 €, we have a 12.5% increase, but from 2.25 € to 2.44 € that increase falls off to just about 8.51%. You can check for yourself that this pattern goes on to eventually reach a value very close to 0%. This is the scenario where the bank is crediting your account with micro-payments every second.

It's time for the revelation of what that upper limit I told you about just before is. If you proceed endlessly the above calculation, each time increasing the frequency of credit from a quarter a year to monthly, weekly, daily, hourly ou per second, you'll reach an incredible number — Euler's number or e. The general form for the accumulation factor of a compound interest rate on an investment is well-known to many people as:

a(t)=(1+i/n)^(nt)

To find our answer — and the actual value of Euler's number — we just need to use it generally as we did before. We don't actually have to do it indefinitely, because in mathematics we can take the limit of a function as the variable tends towards infinity.

lim_(n->oo) (1+1/n)^n=e

Indeed, even if we do just some iterations of the previous method we do get a very good approximation. After 1000 steps — I used python, but it can easily be done in Excel or any other spreadsheet — you get the value 2.7169239322355936... which is really close to the actual value of e=2.718281828459045... In fact, if you take the ratio of the absolute difference between the two over e, you'll find that the error is merely 0.04995% which makes it a pretty good approximation.

This was just a complicated way to say that no matter how frequently your bank credits your account, you cannot surpass the value of Euler's number — a both irrational and transcendental number.