When I was a child, I heard an advertisement for a bank that compounded the interest on your savings account with every heartbeat. I thought that was an odd thing to say and wondered what it meant. If you have a rapid heart rate, does your money compound more frequently?
I figured there was probably some fine print, such as saying interest was compounded once a second or something like that. Beyond some frequency it doesn’t matter that much how often interest is compounded, and that’s essentially what continuously compounded interest is: interest compounded so often that it doesn’t matter how often it is compounded [1].
So how often do you need to compound interest before the difference between discretely compounded interest and continuously compounded interest doesn’t matter? Well, that depends on what you think matters. The more demanding you are about what matters, the finer the discrete compounding needs to be. It also matters what the interest rate is. The following Python function gives the difference between continuous compounding and compounding n times per year, at a percentage rate r and with principle P.
def f(P, n, r) : return P*(exp(r) - (1 + r/n)**n)
Let’s first say that the frequency of compounding matters if it makes a difference of more than $1 on a loan of $1,000,000 over a year. The difference between continuous interest and compounding daily at 6% is $5.24. If we increase the frequency of compounding to hourly, the difference is $0.22, which we are saying does not matter.
When the interest rate goes up, the difference between continuous and discrete compounding also goes up. If we triple the interest rate to 18%, now the difference is $2.21, but if we go to compounding every minute, the difference is $0.04.
Now if we’re more demanding, and we want the difference in interest to be less than a cent on a principle of one million dollars, we need to compound even more often. In that case compounding once a second is enough, given an interest rate of 18%, which means that’s frequent enough for any lower interest rate.
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[1] You could make this statement rigorous by saying for every definition of what matters, i.e. for every tolerance ε, there exists an N such that for all n > N the difference between continuous compounding and compounding with n periods is less than ε.
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