Rule of 72

Do you use the rule of 72? Is it helpful to you? Can you connect it to some higher order mathematics?

If you are earning 9% for 9 years do you expect to more or less double your money? (Choose more or less)

At 9% how many years to quadruple your investment.

Quote: pacomartin

Do you use the rule of 72? Is it helpful to you? Can you connect it to some higher order mathematics?

If you are earning 9% for 9 years do you expect to more or less double your money? (Choose more or less)

At 9% how many years to quadruple your investment.

18 years

edit: damn, I misread the question. He threw me off with the 9 years thing

Quote: pacomartin

Do you use the rule of 72? Is it helpful to you? Can you connect it to some higher order mathematics?

If you are earning 9% for 9 years do you expect to more or less double your money? (Choose more or less)

At 9% how many years to quadruple your investment.

More.

Double in 8 years. Thus quadruple in 16 years.

I don't use it really, since I essentially always have a calculator available. Besides, the "72" is an approximation of a number that varies with the interest rate or number of periods. For 5 years, it is 74.349; for 9 years, 72.053; and for 15 years, 70.941. Not at all a fixed "72", just an approximation.

At the current interest rates I'm being paid -- just a fraction of a percent -- it's more like lifetimes to double the money.

In answer to your question, 9 years at 9% more than doubles.

Quote: pacomartin

I use it as a general guidepost in financial planning. It should be taught more in schools. I had to learn it on my own before it was briefly covered in a college finance course. Prof in that course told about 4 of us he wasn't calling on us for the rest of the semester about 10 weeks in.

I use the rule of 100 ln 2 (which is about 69.3147).

If it pays X percent a year, then after 1 year, $1 is now (1 + X/100); in 2 years, it is (1 + X/100)², and so on.

It doubles when (1 + X/100)^N = 2
ln 2 = N ln (1 + X/100)
N = ln 2 / ln (1 + X/100)
NX = X ln 2 / ln (1 + X/100) = ln 2 * X / ln (1 + X/100)

There was a reason I did this, but it escapes me at the moment...

I teach an introductory calculus class, and I always have my students derive this rule.

Start from A = P*e^(rt), which calculates the value, A, of an investment with principle, P, annual interest rate, r (compounded continuously), and time, t years. To double your money, you want to reach an A-value equal to 2P.

2P = P*e^(rt)
2 = e^(rt)
ln(2) = rt
t = ln(2)/r

Note that ln(2) is about 0.69. Since 72 is pretty close to 69 (and has lots of factors), it is commonly used for a quick estimate. For r = 9%, t = ln(2)/0.09 = 7.7 years, so the 8 years given by the Rule of 72 is pretty close.

By the way, if you want to quadruple your money, use ln(4) instead. ln(4) is about 1.39, so you could use a "Rule of 140" or "Rule of 144" there. Of course, ln(4) = 2*ln(2), which means quadrupling your money will take twice as long as doubling it.

Also, A = P*e^(rt) can be derived by solving the separable differential equation dA/dt = rA.

LN(2) ~= 9/13
1 sigma ~= 13/19

These two were approximations used in H.S. in the mid 70's.

Quote: drebbin37

Start from A = P*e^(rt), which calculates the value, A, of an investment with principle, P, annual interest rate, r (compounded continuously), and time, t years. To double your money, you want to reach an A-value equal to 2P.

2P = P*e^(rt)
2 = e^(rt)
ln(2) = rt

Thanks - I knew that something approaching infinity was in there somewhere (in this case, the number of times per year that interest is compounded), but couldn't remember where.

The "rule of 72" is a relic from the days before everyone had instant access to a good calculator.

Take out your phone, express the interest rate as a decimal, add 1, take the reciprocal of the natural log, and multiply by .693. There was a time that would have been appreciably harder than the rule of 72, but that time is past.

Though for what it's worth, I'm getting that it works best around 8%. More that that, it'll give you too short a time, less than that, too long of one, although I'm a bit surprised how slowly it actually moves. I can see how it became as popular as it did.