Investment Math

Question
Recently, a mathematical question popped into my head: "What annually compounding interest rate is required to double your money every five years?" The answer is just under 14.87%. Good luck with that!

First some definitions:

• Annually Compounding Interest: Suppose you have $10 invested in something that returns 10% per year. After one year, you have $11 = $10 + $10 * 10%. After two years you have $12.10 = $11 + $11 * 10%. Each year, your current amount (not your starting amount) grows by 10%.
• Annually means yearly.
• Non-compounding interest: Taking the same example as above, First you have $10, then you have $11, then you have $12. Each year, you get 10% of your starting amount.

Math Time

Let's start by defining some variables:

• S = starting amount
• F = final amount
• y = number of years to go from starting amount to final amount
• r = annually compounding interest rate

1. What is the relationship between these numbers? We know that when y = 0, S = F regardless of what r is.
2. When y = 1, S + S*r = F, or by factoring out an S, S(1+r) = F.
3. For y = 2, S + S*r + (S + S*r) * r = F. The original amount S, plus the first year's interest S*r, plus the second year's interest which is the interest rate times the amount after the first year's interest is added, or (S + S*r) * r. After distributing the r we get S + S*r + S*r + S*r*r which after factoring out the S becomes S(1 + 2r + r^2). And of course 1 + 2r + r^2 = (1 + r)^2. So to reiterate, S(1+r)^2 = F.
4. Following this trend it becomes obvious that for number of years y, S(1+r)^y = F (which can be proven formally using mathematical induction).

So now we know how to find F given S, r, and y. What about each given the others?

Final Answers

Using mostly basic algebra we get:

• F = S(1+r)^y
• S = F/(1+r)^y
• r = (F/S)^1/y - 1
• y = log(F/S) / log(1+r)

If you're a calculus nerd, and you only know two of the four variables, then you can set the first derivative equal to zero to find local maxima or minima ... nerd.

So what annually compounding interest rate is required to double your money every five years?
r = (2S/S)^1/5 - 1 = 2^1/5 - 1. The fifth root of 2 is roughly 1.1487, so r = .1487 or 14.87%.