Compound Interest and 72 Formulation

    22 Aug, 2024

    Compound Interest Formula and The Rule of 72 are commonly used tools in finance for calculating investment growth and estimating the time it takes for an investment to double.

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

    where:

    • AA is the final amount of the investment or loan (principal + interest).
    • PP is the initial principal.
    • rr is the annual interest rate (expressed as a decimal, e.g., 5% is 0.05).
    • nn is the number of times interest is compounded per year (if compounded annually, n=1n = 1; if compounded quarterly, n=4n = 4).
    • tt 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×(1+0.051)1×5=1000×(1.05)5=1276.28 dollarsA = 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:

    t72rt \approx \frac{72}{r}

    where:

    • tt is the time required to double the investment, in years.
    • rr 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:

    t726=12 yearst \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(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}

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

    2P=P(1+r)t2P = P \left(1 + r\right)^t

    2. Solving for Time tt

    Dividing both sides by PP, we get:

    2=(1+r)t2 = \left(1 + r\right)^t

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

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

    Solving for tt:

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

    3. Approximation for Small rr

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

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

    Therefore, the formula can be further simplified to:

    tln(2)rt \approx \frac{\ln(2)}{r}

    4. Calculation and Approximation

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

    t0.693rt \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:

    t72rt \approx \frac{72}{r}

    5. Conclusion

    The Rule of 72, t72rt \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.