04.15.07
Rule of 72
Logarithms are one of the most universally hated/feared topics in high school mathematics. The notation is arcane, their properties are confusing, and few teachers (let alone students) will even attempt to estimate them without a calculator. Unfortunately they are indispensable for asking one very useful investment question: “Assuming exponential growth of R% per year, how long will it take for $X to grow into $Y?”
Luckily, there is a simple rule of thumb that you can calculate using pencil and paper. At a growth rate of R% per year, it takes approximately 72/R years for an investment to double. Investing at a 4% rate of return? Your money will double in approximately 72/4 or 18 years. Wait 36 years and it will redouble (think Backgammon or Bridge) to *four* times the initial investment — thus the power of starting your retirement savings early.
Now double the rate of return to 8%. At this rate, your money will double every nine years. In 36 years it would double, double, double, and double again to a total of *16* times the initial investment. Due to compound growth, doubling your rate of return doesn’t merely double your final portfolio value — it effectively squares the overall multiplier.
Note also the insidious toll that investment fees can take on these results. Suppose for the sake of argument that an investment can be expected to return 10% annually over the long run, against a backdrop of 4% inflation. This is a 6% “real rate of return”, so by the “rule of 72″ its purchasing power should double every 12 years (and increase by a factor of 8 over 36 years). Now tack on a 2% “management fee”, as many annuities and retail funds seem to do, and that real return drops to 4%. Now if you invest your money for 36 years it increases by only a factor of 4, not the factor of 8 that it might have done without being weighed down by the fees. To reach your goal, you would need to either save twice as much or delay retirement until you are 83!
A final warning! All of the above examples were working with “whole” doubling periods. If you split a doubling period in half (e.g. 9 years at a 4% return), you’ll only end up with a little more than 40% increase — not the 50% that you might have expected. Still, this approximation should get you in the general ballpark. For anything more precise, find a financial calculator (or a financial adviser). Good luck!
For those who are curious, there is a mathematical basis for this rule. The calculation for “continuous compounding” is simply: A = Pe^(rt) where P is the principal invested, r is the rate of return (expressed as a decimal, e.g. 0.06 instead of 6%), and t is the number of years. Solving this equation for the “doubling period”:2P = Pe^(rt)
2 = e^(rt)
rt = 0.69
t = 0.69/r or 69/r (if r is expressed as a percent)So why 72? With continuous (or even daily) compounding, the annualized rate of return is slightly higher than the stated rate of return. This is the difference between “APR” and “APY” on a certificate of deposit. Since the “rule of 72″ assumes that we are already working with an annualized rate, we use the slightly higher target. It isn’t mathematically robust, but gives “pretty good” estimates for most common situations.