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.

EFFECT OF COMPOUNDING
A 'story' for High Schools Math teachers to teach Compounding and another concept.
So, in this country banks in their efforts to attract Deposits started to incerase the interest rates they give on deposits.
Bank A gave 10%. Bank B gave 15% to steal the deposits.
Bank A increased the rate to 20% to get back the deposits. Bank B went to 20%.
And so on to 30%, 40%, 50%, 70%, 80%
Untill the Central Bank of the country intervened and set as Maximum Nominal Interest rate of 100%.
So all banks were giving 100% with no further possibility of increase to beat the competition.
Until the CEO of Bank A thought. Why not give them the Nominal of 100% interest at 6 months intervals. ie 50% per 6 months.
This way $100 will become $150 at 6 months but will become $225 in the year with the Compounding, better than the $200 with Interest at the end of the year.
So Bank A offered compounding at 6 months with 100% nominal rate and stole all the deposits.
Bank B went further with Quartely compounding and 100% nominal rate for even better return.
And the competition went like that, with Monthly Compounding, Weekly, Daily, Hourly, By Minute, By second, by 100th of a second.

Untill the CEO of a bank came and said that we offer Infinite compounding.
The Boards of Directors of the bank got confused. You can do that. Infinite Compounding means we will pay infinite Interest and we cannot pay that?
But the CEO was not worried.

So how much wil the bank pay out on a $100 for 100% Interest with Infinite Compounding.

(For the Math guys, I appreciate this is a trivial question.)

Quote: AceTwo

So how much wil the bank pay out on a $100 for 100% Interest with Infinite Compounding.

Shouldn't you say "Instantaneous" compounding instead of "Infinite"? It does not actually pay anything infinite.

Quote: 24Bingo

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

I don't think basic comprehension is a relic. A calculator tells me that if I want to double my money in 8 years, I need an interest rate of 9.0508%
If I get 7% for 10 years then I will have 196.715% of what I began with.

I still think it is valuable to be able to instantly comprehend that the answer to the first question is roughly 9%, and the answer to the second question is about double.

I had a friend tell me she was in a financial planning seminar for professionals. The teacher said that with income tax rates at 36% that if a client wanted to take home $100K after taxes they needed to make $136K. She interrupted and said wouldn't the client have to bring home more than $150K before taxes. The teacher started screaming at her and saying that the tax rate was not 50%, and wasn't she paying attention. The teacher had no clue, nor did the other half a dozen professionals in the class. Calculators are for refining numbers, but you should still be able to estimate.

It's a meaningless rule in practice because the true return will always be much less percentage-wise due to inflation and taxes. You have to much more than double the dollar amount to actually double your money.

With most investments these days paying 3% or less, I suggest the "Rule of Eternity"--you will never actually make a dime when the interest rate you earn is barely equal to the inflation rate. Take THAT one to the bank.