# BM4.8 – Calculating the Term of the Investment

Chapter 4, Lesson 8

In this lesson students will:

• determine the number of compounds or the term of the investment, given information about the future value, principle amount and interest rate
• use the Rule of 72 to determine the length of an investment quickly

## Length of Investment

#### Number of Compounding Periods

Recall that there are two main ways to discus the length of a monetary investment: knowing how many compounding periods $n$ occur, or knowing how many years $t$ the money will be invested. We saw in Lesson 4.5 that these variable are connected through the formula $n=t \times f$, where $f$ is the frequency number.

Suppose we are interested in solving for the number of compounding periods, given some information about the investment or debt. We could do this by substituting $n$ into the compound interest formula and isolating for $n$. We first divide both sides by $P$, but we run into a problem.

\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{t \times f}\\ FV &= P \left( 1+ \frac{r}{f} \right)^{n}\\ \frac{FV}{P} &= \left( 1+ \frac{r}{f} \right)^{n} \end{aligned}

How do we remove the power of $n$? In Lesson 4.7 we used a root to remove the power, but this won’t help in this situation since we would like to isolate $n$. In this case, we must use one of the helpful properties of logarithms we saw in Lesson 4.3. Applying the natural logarithm to both sides of the equation allows us to “bring down” $n$ and isolate.

\begin{aligned} \frac{FV}{P} &= \left( 1+ \frac{r}{f} \right)^{n}\\ \ln \left( \frac{FV}{P} \right) &= \ln \left( 1+ \frac{r}{f} \right)^{n}\\ \ln \left( \frac{FV}{P} \right) &= n \times \ln \left( 1+ \frac{r}{f} \right)\\ n \times \ln \left( 1+ \frac{r}{f} \right) &= \ln \left( \frac{FV}{P} \right)\\ n &= \frac{\ln \left( \frac{FV}{P} \right)}{\ln \left( 1+ \frac{r}{f} \right)} \end{aligned}

#### Term in Years

Recall that the number of compounds and the term in years are connected by the formula $\displaystyle n = t \times f$. Dividing both sides by the frequency number, we obtain

\begin{aligned} t &= \frac{n}{f}. \end{aligned}

This means if we would like to determine the term of the investment in years, we can use the formula given in the last section, then divide the number of compounds $n$ by the frequency number $f$. Let’s work through a few examples using the our formulas.

## The Rule of 72

Suppose that you would like to determine how long in years it will take to double your money. The formulas above can be quite lengthy if all we need to do is perform a quick estimation. In this section we will develop a formula to quickly determine the doubling-time of an investment.

Let’s begin with our formula to calculate the number of compounding periods. If we would like to double our money, we note that this means we want $P$ to grow to a future value of $2P$ (twice as much as we started with). Substituting in $FV = 2P$ and simplifying we obtain

\begin{aligned} n &= \frac{\ln \left( \frac{FV}{P} \right)}{\ln \left( 1+ \frac{r}{f} \right)}\\ n &= \frac{\ln \left( \frac{2P}{P} \right)}{\ln \left( 1+ \frac{r}{f} \right)}\\ n &= \frac{\ln(2) }{\ln \left( 1+ \frac{r}{f} \right)}. \end{aligned}

Next, we know that $\displaystyle i = \frac{r}{f}$, so we can make this substitution. this allows us to use a property we saw in Lesson 4.3 to approximate the denominator of our fraction. Since $i$ is a small positive number, we know $\ln(1+i) \approx i$ and we obtain

\begin{aligned} n &= \frac{\ln(2) }{\ln \left( 1+ \frac{r}{f} \right)}\\ n &= \frac{\ln(2) }{\ln \left( 1+ i \right)}\\ n &\approx \frac{\ln(2)}{i}. \end{aligned}

Finally, while it is helpful in some instances to calculate the number of compounds $n$, we might as well go one extra step and try to solve for time in years $t$. Substituting $n = t \times f$, dividing both sides by $f$, and rewriting $\displaystyle i = \frac{r}{f}$, we obtain

\begin{aligned} n &\approx \frac{\ln(2)}{i}\\ t \times f &\approx \frac{\ln(2)}{i}\\ t &\approx \frac{\ln(2)}{i \times f}\\ t &\approx \frac{\ln(2)}{\frac{r}{f} \times f}\\ t &\approx \frac{\ln(2)}{r}. \end{aligned}

Now here is where things get a little murky. If we use our calculator to evaluate $\ln(2)$ we should get a decimal number approximately equal to $0.6931$. We also know that the nominal rate $r$ is typically given as a decimal number that represents a percent. If we were to multiply both the numerator and denominator by $100$ we would obtain

\begin{aligned} t &\approx \frac{0.6931}{r}\\ t &\approx \frac{0.6931}{r} \times \frac{100}{100}\\ t &\approx \frac{69.31}{r \times 100}. \end{aligned}

You might ask where does the $72$ come from in the Rule of 72? This is an excellent question! It seems as though someone decided to round $69.31$ up to $72$ (perhaps because the Rule of 69 didn’t have the same ring to it) somewhere along the way. This gives us our final formula for estimating the doubling-time of an investment. This formula is known as the Rule of 72. Divide the number $72$ by the nominal rate multiplied by $100$ and that’s it!

\begin{aligned} t &\approx \frac{72}{r \times 100} \end{aligned}

Let’s look at a few examples of how to use this in practice, and see how accurate it is compared with the typical formula.