How many times have you heard of an investment advisor who promises to double your money? Is this really an amazing feat? That depends on how long it will take for your money to double. With enough patience, your funds eventually will double even if they earn only a very modest interest rate. Suppose your advisor promises to double your money in eight years? What interest rate is implicitly being promised?

$$ The advisor is promising a future value of $$2$ for every $$1$ invested today. Therefore, we find the interest rate by solving for r as follows:

$$ Future value$=$PV$\backslash$times $(1+$r$)$t$$2=$$$1\backslash$times $(1+$r$)8(1+$r$)=21/8=1\mathrm{.}0905=9\mathrm{.}05\%$

$$ By the way, there is a convenient rule of thumb that one can use to approximate the answer to this problem. The Rule of $72$ states that the time it will take for an investment to double in value equals approximately $72/$r$,$ where r is expressed as a percentage. Therefore, if the doubling period is eight years, the rule implies an $($approximate$)$ interest rate of $9$ percent $($since

$72/9=8$ years$).$ This is quite close to the exact solution of $9\mathrm{.}05$ percent.

The Rule of $72$ works best with relatively low interest rates. Suppose the time it will take for an investment to double in value is $12$ years. Find the interest rate. What is the approximate rate implied by the rule? Now suppose that the doubling period is only two years. Is the approximation better or worse in this case?