You might
have heard of the rule where you divide 72 by your interest rate to determine
the time it takes for your investment to double. That seems too simple to be
true, but it also seems to work. Why?

To figure out the doubling time, you need to solve the following:

That in no way looks
like time is 72 divided by interest rate. Can we get it close? If you restrict
yourself to interest rates much smaller than 100% (which is reasonable), you can
use the Taylor expansion for ln(1 + x) when x is small, which is:

Trying some values with
r (x in the equation) equal to 10%, n = 1 yields 10%, n = 3 yields 9.53%, and
if you continue, you see that the values are pretty close to 10%. Trying it
with different numbers, you end up with the first term dominating, and the
first term in this case is simply x. Thus, you can approximate ln(1 + r) as
just r when r is small. That leaves us with:

That’s still not right.
Noting that the rule divides by 100 times the interest rate rather than the
actual interest rate (i.e., if your rate is 8%, divide by 8 instead of 0.08). Calling
R = 100*r we get:

ln(2) is a constant, so
we can go ahead and calculate its value:

That’s still not 72/R,
but it’s close. Why would people replace 69.3 with 72? It’s all about what’s
easy to do in your head. No integers divide evenly into 69.3. We could round it
to 69. Then, 3 and 23 are factors. Can we do better? Try 70…that gives us 2, 5,
7, 10, and 14. That’s a lot better. 71 is prime so it doesn’t help us at all.
72 gives us 2, 3, 4, 6, 8, 9, and 12. That’s the best we can do for any of the
numbers near 69, so it works out.

One interesting note is
that replacing ln(1 + r) with r will make the denominator larger than it
actually is which will make our estimates for time smaller than the actual
values while replacing 69.3 with 72 will make our estimates for time larger
than the actual values. Thus, the errors introduced by our two approximations
shift the result in opposite directions so two approximations might be better
than one in this case.

All that’s left is to
check out how good these approximations are.

#### What About A Rule For Tripling Time?

All you have to do is go back to the original math, and replace the '2' with a '3'. This makes the numerator 100*ln(3), or 109.9. Going through the same factor evaluation, 108 gives us 2, 3, 4, 6, and 9, 109 is prime, 110 gives us 2, 5, and 10, 111 gives us 3, and 112 gives us 2, 4, 7, and 8. 108 and 112 look to be the best of those, and based on the error analysis earlier it's better to err slightly high in the numerator, so if you find yourself needing to estimate how long it will take your money to triple in value, just divide 112 by the interest rate.

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