Who never got confused about the differences between APR and APY ? I want to explain you in this small post why the difference matters, in a simple mathematical way, because I do believe that everybody should be able to understand it. I'm a physicist, and I always ejoy to explain stuff to others.
As any good mathematical explaination, let's start with some definitions:
Well, those words don't tell you what's the difference actually; they look to have both the same meaning. I believe there should be a more convenient way to call them, because in this way it is confusing. However, the only difference is that :
Ok, fine. What's compounding? It is adding the previous interests earned to the new basis for the next interest payout calculations.
Everything depend upon the frequency of the compounding.
If you have interests paid once per month, you will compound 12 times in a year; if you get them daily, compounding will be effective 365 times. A general property is that APY > APR , so often in ads on the web we find APY for marketing, because it is greater and looks better. They can be equal only if the compounding frequency is indeed 1 year, so that the difference appears only after multiple years.
Let's call X the amount put in a savings account, in dollars for example; If you invest, say, 100$, we will pose X= 100$ to state that, on day 1, the amount in the savings account is 100$, as it should be.
Now, call K the annual interest rate; at the end of the year, your savings account will be worth 100$ plus the earnings, which are worth K * 100$, right?
In formula, calling F the final amount:
F = X + K * X = X * ( 1 + K ) ( remember elementary math, we have just grouped for X)
K is a small number, usually not far from 0. Remember that if one says 5%, what is understood is 0.05, or 5/100.
So what we are doing to find F is simply multiply X by 1+K, and if K=0.05 and X=100 , then we get
F = X* 1.05 = 105 . Of course, when you multiply a number by another which is greater than 1, you obtain a bigger number.
Ps : aren't you happy to earn 5$ per year? :-) ahahah
Actually, of course the APR case is straightforward, and I put it there just to start with the math notations. In the case of an APY, we have to take care of compounding. This is not difficult. Actually, I think there is a big misleading about this. We will see why later.
Consider now that, in one year, there are a certain number N of interest payments, called periods; it could be N=12 (monthly), N=52 (weekly) or N=365 (daily) or whatever else. Suppose that after each payment period, you are paid with an interest rate of R. This is not the annual interest ( APY or APR), but the one effective for the period alone. Let's call X(1), X(2), X(3), . . . , X(N) the amounts worth of your interest account at each period. Just to be clear, X(1)= 100$ and X(N) will be the amount after 1 year, made of N periods.
For each period, we just have to use the same formula as before, because the new amount will be worth the old one plus the interest paid during one period, which is calculated with R :
X(2) = X(1) * ( 1 + R ) , X(3) = X(2) * ( 1 + R ) and so on.
But if in the expression for X(3) we substitute, for X(2), his own expression in terms of X(1), we find
X(3) = X(1) * ( 1 + R ) * ( 1 + R ) = X(1) * ( 1 + R )^2
Where ^2 means squared, that is elevated to the second power. Indeed, this is because X(3) can be calculated using X(1) , just you have to put 2 factors (1+R) . Now you can guess by yourself what will look like X(N), the final amount after 1 year:
X(N) = X(1) * ( 1 + R )^N
This is the power of compounding, you have a power law entering you gains! This is much better than a linear growth, as for the APR. However, for short times or for small amounts, the difference can be neglegible.
But, wait one second! The R we used above is the interest rate per period, be it daily, monthly or whatever. Nobody tell you this number, they all claim in big letters the APY, which is still on an annual basis.
So, nobody tell you R, but only the APY. If you've read carefully, X(N) is the final amount, so let's call it also F, as before with the APR calculations. Let's also replace the X(1) symbol in favour of simple X, because we don't need anymore this notation. We can conclude that
To come at a resolution about this misleading, we have to re-write the interest earned compounding in a simpler form.
Now, the expression ( 1 + R )^N will evaluate to a number, a bit larger than 1 usually. If interest is compounded monthly, say, and so N=12, and they pay 1% per month, so that R= 0.01 , this expression evaluate to:
( 1 + 0.01 )^12 = 1.1268... = 1 + 0.1268...
and so we end up with
F = X * 1.1268 = X * ( 1 + 0.1268... ) = X * ( 1 + S )
which is an expression identical to the one for the APR case, but of course S and K are not the same. This is to say that if 1% is paid monthly, and with compounding, is like having a 12.68% APR, simple, without compounding!!!! Remark that this is a bit larger than simply multiply 12 for 1%, and correspond to a 12.68% APY = S .
What we have just done is to evaluate the APY with our knowledg of R, the interest rate per period.
So, what if I want to know R, the interest per period, if we now the APY? This means: what is the interest rate per month, say, if I know the APY, which is for 1 year?
If you look back at the formulas, we have just done this trasformation
( 1 + R )^N = ( 1 + S )
which can be reverted to give
R = N_square_root_(1 + S) - 1
where "N_square_root" is the N-th square rooth of the number considered, and this is equal to exponentiate to the 1 over N power, 1/N.
So suppose I know that the APY is equal 15%, and so S=0.15, for example , I now can evaluate R, with compounding monthly, N=12
R = (1 + 0.15 )^(1/12) - 1 = (1.15)^(1/12) - 1 = 0.0117... = 1.17 %
So this mean that we will receive a 1.17 % interest on a monthly basis!!! Notice that 1.17*12=14.05 , a bit smaller than our 15% APY, because of compounding.
Ok, now I have concluded with the explainations, and I hope you are still reading and you didn't died for being too bored!
Thank you very much for your tips! Have a nice day
