If you pay someone $90 today and they promise to give you $100 in 12 months, you are making a loan. This is the same idea as buying a bond. To back out the simple interest rate denoted (ie assuming no compounding) we solve for r:
90 * (1+r) = 100
r = 100/90 – 1
r = 11.11%
If the loan was only for 6 months, then we’d annualize the interest rate by multiplying by 2 (12 months / 6 months) for a rate of 22.22%
Let’s return to the 12-month loan and say that the rate is compounded semi-annually. Then the computation is:
90 * (1+r/2)² = 100
r /2 = (100/90).5 – 1
r = 10.82%
If you compound more frequently than annually, it makes sense that the implied interest rate is lower. Consider the path of the principal + accrued interest:
Compounding semi-annually means interest gets credited at the 6-month mark. So the rate for the next 6 months is being applied to the higher accrued value amount which means the implied rate to end up at $100 (the same way the simple interest case ends up $100) must be lower than the simple interest case.
We can compound interest more frequently. Quarterly, monthly, daily. Since the number we are backing out, namely the implied rate, is being applied to a growing basket of principal + accrued interest at each checkpoint (I think of the compounding interval as a checkpoint where the accrued interest is rolled into the remaining loan balance), the implied rate to end up at $100 must be smaller. If we take this logic to the extreme and keep cutting the time interval into smaller increments we eventually hit the limit of Δt → 0. The derivatives world models everything in continuous time finance so interest rates get the same treatment.
Mechanically, the math is no harder.
To compute the continuously compounded interest rate we still just solve for r:
90 * ert = 100
t is a fraction of a year. So for the 12-month case:
90 * er*1 = 100
er = 100/90
r ln(e) = ln(100/90)
r = 10.54%
As expected, this is a lower implied rate than the 11.11% simple rate and the 10.82% semi-annual rate. Again, because we are compounding continuously.
Annualizing remains easy. If $90 grows to $100 in just 6 months, we compute the continuously compounded rate as follow:
90 * er*1/2 = 100
er*1/2 = 100/90
r *1/2 = ln(100/90)
r = 21.07%
This can be contrasted with the 22.22% 6-month loan using simple interest we computed earlier.
Application To Real Life
Note in all these cases, $90 is growing to $100. We are just seeing that the implied rate depends on the compounding assumption. In real life, when you see “compounded daily” or “compounded monthly” and so on, you are now equipped with the tools to compare rates on an apples-to-apples basis. If a rate is lower but compounds more frequently than another rate the relative value between both loans is ambiguous.
APYs disclosed on financial products make yields comparable. But now you understand how APYs convert different rate schedules into a single measure.