Financial Mathematics: Periodic factor, APR and APY

So in the first piece on the Time Value of Money, we looked at scenarios in which interest was compounded annually. But interest is not always compounded annually. Some assets compound semiannually, quarterly, monthly or even daily. So we need to develop some tools to deal with these.

To begin with we will introduce some terminology.

Annual Percentage Rate (APR) or Nominal Interest Rate

The annual percentage rate (APR) or nominal interest rate is interest that you would earn over year provided that the compounding occurred annually. This is even the case when interest is paid at more frequent intervals.

Many assets have their yields stated in terms of the APR or nominal interest rate. Banks often prefer quoting APR figures with regard to loans since this rate is lower (so it sounds better from a borrower's perspective.)

We'll denote this rate as $i$ as we have in the past.

Annual Percentage Yield (APY) or Effective Interest Rate

The annual percentage yield (APY) or effective interest rate takes into account the fact that compounding occurs at more frequent intervals. As a result, provided that the compounding intervals are smaller than a year (say, monthly), APY will be higher than APR. As a result, banks often prefer quoting APY figures when advertising for savings products (such as Certificates of Deposit) as the higher sounding figure would attract savers.

We'll denote this rate as $r$. 

Example 1


Suppose you have $ \$ 100 $ and you invest it in an account that has an APR of 12%. If this is paid annually, then at the end of the year you will have $ \$ 112  $ and your APY will be:
$$ \frac{\$ 112}{\$ 100} - 1 = 12 \%$$
But let's suppose that instead the interest is paid semiannually. In other words, you receive $ \$6 $ after 6 months (which you withdraw from your account) and another $ \$6 $ at the end of the year. What's your APY in this case?

Well, isn't that like earnings 6% every 6 months? And if we were to compound that (i.o.w. you didn't withdraw the funds), then you would compound it like this:
$$1.06 \times 1.06 = 1.1236$$
So if you have an APR of $i=12 \%$ and this is paid out semiannually then you end up with an APY of $r = 12.36 \% $.

Now let's be more methodical about what we did here (we're going to generalize this so that it applies to other situations such as quarterly, monthly or even daily payments). We started with the APR of 12%. We then divided this by 2 to get 12%/2 = 6%. We then added 1 to this and squared it and then subtracted 1 to get the APY or in other words:
$$r=\left(1 + \frac{12\%}{2}\right)^2 - 1=12.36\% $$

Example 2

Now suppose we do the same thing but we do this on a monthly basis. What do we have there? Now instead of 6% every 6 months we'll be paying out $\frac{12\%}{12} = 1\%$ per month. If we compound 1% per month and annualize it looks something like this:
$$ 1.01 \times 1.01 \times 1.01 \times 1.01 \times 1.01 \times 1.01 \times 1.01 \times 1.01 \times 1.01 \times 1.01 \times 1.01 \times 1.01$$
$$ = 1.01^{12} \approx  1.1268$$
So we started with an APR of $i=12\%$ and this was paid out monthly to obtain an APY of $r=12.68\%$.
And just in the last example, let's be more explicit about what was done here. We started with the 12% APR, we divided this by 12 months, we then added 1, we then raised that to the power of 12 and then subtracted 1 again to obtain APY or:
$$r=\left(1 + \frac{12\%}{12}\right)^{12} - 1=12.68\% $$
Generalization

Now we're ready to generalize the result. Suppose that instead of 2 periods (semiannually), 4 periods (quarterly) or 12 periods (monthly) there are $n$ periods. Then the APY will look like this (where $i$ is the APR):
$$r=\left(1 + \frac{i}{n}\right)^n - 1$$
We can add 1 to both sides to obtain:
 $$1+ r=\left(1 + \frac{i}{n}\right)^n$$
If we invert this and raise it to the power of $t$ for time in years, we get something like the Discount Factor discussed here:
$$D = \frac{1}{\left(1+r \right)^t} = \frac{1}{\left(1+\frac{i}{n}\right)^{nt}}$$
This result will be useful when we begin discounting things like bonds with coupon payments at intervals that are less than a year.

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