Rule Of 72 Questions

Hello,

Figure that this must be the right and best group for this question. Have read the Google links on it, but still unsure.

Dividing 72 by the interest rate gives the number of years to double an investments value (approx.).. Fine.

Questions:

I guess the assumption is that when they say, e.g., 6%, they mean at the end of every year, 6% is added in, and left in for the subsequent years. This is certainly compounding, but not very real world.

Or, do they mean a "certain amount" is added daily, but at the end of the year it amounts to the Principal + 6% ?

I guess I am asking what the compunding "Period" implied in the formula actually is ?

**And also, how to adjust the formula for different periods ? e.g., compunded every 3 months, monthly, or even daily perhaps ? **

Thanks, Bob

Reply to
Robert11
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See

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for a very good answer.-- Doug

Reply to
Douglas Johnson

Robert11 wrote: ...

As you said, the rule of 72 is an approximation. Is it worth agonizing over the exact compounding schedule of something that is only an approximation in the first place? The approximation remains approximately correct whether compounding is done yearly, quarterly, or daily.

Over the course of the year, you have 365 different principals. Which one of those are you referring to? If the principal at the start of the year, then ending the year at that principal plus 6% is functionally equivalent to annual compounding at 6%.

Xho

Reply to
Xho Jingleheimerschmidt

The Rule of 72 is only an approximation. It isn't exact for any given compounding period (or interest rate, for that matter). That being said, the approximation is closest to annual compounding. However, it works reasonably well for shorter compounding periods.

If only there were a mathematician in the house...

Honestly, at this point, I would just throw away the approximation and solve for the doubling time exactly. Let r be the interest rate, n be the number of compounding periods per year and t be time (in years). The equation you need to solve is:

(1 + r / n)^(nt) = 2

Solving for t, we get:

t = ln(2) / (n * ln(1 + r / n))

If you really wanted a rule like the Rule of 72, you could use a Maclaurin series to approximate the natural log in the denominator:

t ~= ln(2) / (n * (r / n - (r / n)^2 / 2)) ~= ln(2) / (r * (1 - r / (2 * n)))

t ~= ln(2) / (r * (1 - .07 / (2 * 1))) ~= ln(2) / (.965r) ~= .72 / r

Ta da! If you wanted a rule that's more appropriate for, say, monthly compounding and interest rates around 6.3%, you'd get:

t ~= ln(2) / (r * (1 - .063 / (2 * 12))) ~= ln(2) / (.997r) ~= .695 / r

Sadly, the Rule of 69.5 just doesn't have the same ring to it. That alludes to another reason why the Rule of 72 is so popular. 72 is a highly composite number. It's evenly divisible by 2, 3, 4, 6, 8, 9,

12, 18, 24 and 36. So if you use the Rule of 72 to approximate the doubling time for 6%, it comes out an even 12 years. For 8%, you get 9 years. It's handy that way.

I hope none of my students read this newsgroup. This would make an excellent exam question for next semester...

--Bill

Reply to
Bill Woessner

I hope my high school algebra teacher doesn't take away my diploma if I get this wrong.

R = rate (simple or same as if you compound yearly) C = number of times you compound a year Y= number of years to double money ^ = exponential 2^3 is 2 to the 3 or 8

  • = multiply 2*3 is 6 log = log base 10, log100=2

doubling formula is: (1+(R/C))^(Y*C) =2

crunch around

Y = log(2)/(C*log(1+R/C))

At 6%, monthly compounding, it takes 11.5813 years to double your money. At 6%, daily compounding, it takes 11.5534 years to double your money. Thats a difference of 10 days interest over 11 years. Don't even think about asking about leap year :)

Reply to
camgere

Its not that precise of a rule to matter. But I find it useful for quick mental calculations.

For example I smile when a guy brags about selling his house for four times the purchase prices after 30 years. Thats a doubling period of 15 years (two doublings). By the 72-rule thats just under 5% or about the same return as long term treasuries.

Useful rules are doubling 5 years = 14% (a decent goal),

7 years = 10%, 10 years = 7 percent.
Reply to
rick++

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