NOTE & BLOG

Compound Interest Formula

Compound interest refers to the process of calculating interest not only on the principal amount but also on the interest that has already been added to it. The formula for compound interest is:

$$A = P \left(1 + \frac{r}{n}\right)^{nt}$$

where:

• $A$ is the final amount of the investment or loan (principal + interest).
• $P$ is the initial principal.
• $r$ is the annual interest rate (expressed as a decimal, e.g., 5% is 0.05).
• $n$ is the number of times interest is compounded per year (if compounded annually, $n = 1$; if compounded quarterly, $n = 4$).
• $t$ is the time the money is invested or borrowed for, in years.

Example: If you invest $1,000 at an annual interest rate of 5%, compounded annually, after 5 years, your investment will grow to:

$$A = 1000 \times \left(1 + \frac{0.05}{1}\right)^{1 \times 5} \\ = 1000 \times (1.05)^5 \\ = 1276.28 \text{ dollars}$$

The Rule of 72

The Rule of 72 is a simple heuristic used to estimate the number of years required to double the investment under compound interest. The formula is:

$$t \approx \frac{72}{r}$$

where:

• $t$ is the time required to double the investment, in years.
• $r$ is the annual interest rate (as a percentage).

Example: If the annual interest rate is 6%, then according to the Rule of 72, the time required to double the investment is approximately:

$$t \approx \frac{72}{6} = 12 \text{ years}$$

The Rule of 72 is particularly convenient for quickly estimating the doubling time, especially when the interest rate is not very high (it is generally recommended to use the Rule of 72 for annual interest rates between 6% and 10% for minimal error).

Proof of the Rule of 72

The mathematical derivation of the Rule of 72 is based on an approximation of the compound interest formula. The following is the mathematical proof of the Rule of 72:

1. Compound Interest Formula

We start with the compound interest formula:

$$A = P \left(1 + \frac{r}{n}\right)^{nt}$$

Here, we assume interest is compounded annually (i.e., $n = 1$), and we are interested in the time $t$ when the investment doubles, so the formula simplifies to:

$$2P = P \left(1 + r\right)^t$$

2. Solving for Time $t$

Dividing both sides by $P$, we get:

$$2 = \left(1 + r\right)^t$$

Next, we take the natural logarithm (ln) of both sides:

$$\ln(2) = t \cdot \ln(1 + r)$$

Solving for $t$:

$$t = \frac{\ln(2)}{\ln(1 + r)}$$

3. Approximation for Small $r$

For small $r$, we can use the approximation for the natural logarithm:

$$\ln(1 + r) \approx r$$

Therefore, the formula can be further simplified to:

$$t \approx \frac{\ln(2)}{r}$$

4. Calculation and Approximation

We know $\ln(2) \approx 0.693$, so the formula becomes:

$$t \approx \frac{0.693}{r}$$

For ease of estimation, 0.693 is often approximated as 0.72 in finance, leading to the Rule of 72:

$$t \approx \frac{72}{r}$$

5. Conclusion

The Rule of 72, $t \approx \frac{72}{r}$, is an approximate result derived from the above calculations and is suitable for situations with small annual interest rates (typically between 6% and 10%). While the Rule of 72 is not an exact formula, it provides a very practical tool for estimating doubling times in real-world applications.