M,
Unless I'm missing something, I think we said essentially the same thing. My point was that the law of 72 is an ad-hoc correction to the continuous approximation to discretely compounded interest. Granted, it may come from using higher order terms in an approximation (or approximating about r=.1) but performing that exercise would be silly when one could just experimentally try 71, 72, 73, etc. until you found the best approximation for interest rates near 10....
P.S. ln(1+r)->r iff r->0, so I'm not sure what you mean by your remark about convergence. Of course, I'm not sure why you are even taking a limit at all.
It may not be precise mathematical jargon, but I read "ln(1+r)->r iff r->0" as equivalent to "In the limit as r approaches zero the natural logarithm of 1+r approaches r." "Converges" was probably a poor choice of words on my part. The Taylor series expansion of ln(1+r) for small |r| converges for any -1 < r <= 1,. but is approximated sufficiently by fewer and fewer terms the smaller |r| gets. It is perhaps slang to say that for small enough r it "converges" to r; approaches r is better.
I thought you were the one taking the limit by suggesting in your original post that
The rule itself comes from the formula for computing the future value of an investment that is compounded continuously. Really, you're dividing ln(2) by the rate as a decimal.
I read that as a statement that there is an exact mathematical formula for the limiting case of continuously compounded interest, and what followed that as a claim that the rule of 72 was created as you now say as an ad-hoc correction to that specific limiting case result. I was only pointing out that there is an exact formula for any other interest rate between zero and one that would serve as a valid starting point for an approximation, and that formula is little more complex than the limiting case of r->0. Maybe you have some historical anecdote to support your contention that the law of 72 is an ad-hoc correction to the continuous compounding result, but absent that evidence I find it just as silly to be making your claim as it is to claim the rule is based on higher order corrections to ln(1+r) or on any specific interest rate. What is far more likely is that over time people projecting returns on investments, etc, doing laborious calculations by hand for discrete return rates, possibly using the binomial theorem rather than logarithms came to the realization that for numerous cases of single digit and teen interest rates the time*rate product was around 72. It is most likely the empirical result you suggest, with the focus on 72 coming from the numerous whole number factors of that number I mentioned in my earlier reply, and not a whimsical adjustment to any logarithmic function calculated for one special case..
As a side note I'd be interested to know which is better for monthly compounding, a subject which I'm sure would interest short-term CC writers.
As it happens, monthly returns on the order of 8% to 15% come up all the time in the CC arena. It is exactly this that prompted me to post a reply at all. The flavor of your original comments, as I read them anyway, suggested that the rule of 72 was less accurate than the precise continuously compounding result, or a rule based on 70. In fact 8% is at the very heart of where the rule of 72 is a good approximation of the exact result for any rate-time combination, and for all rates above 8% the rule of 72 predicts the "time to double" at least 3% more accurately than the continuous compounding formula (at least 2% better than "rule of 70"). Yes, I did the simple spread sheet :)
Dan |
