The Rule of 72: A Quick Estimation for Doubling or Halving Values

The Rule of 72 is a useful tool to estimate how long it takes for an investment to double or halve in value, given a fixed annual interest rate. This rule provides a quick way to approximate the relationship between the rate of return and the number of years required for compounding (growth) or discounting (decline).

Formula

$n = \frac{72}{\text{rate}}$ where:

• n = Number of years required to double (compounding) or halve (discounting) the present value (PV).
• rate = Annual interest or discount rate (expressed as a percentage).

Examples for Understanding

Example 1: Fixed Deposit Growth

Suppose you invest in a bank fixed deposit that offers 3% per annum. How long will it take for your deposit to double?

Using the Rule of 72: $n = \frac{72}{3} = 24 ext{ years}$

Using the actual formula in Excel: $\text{NPER}(0.03, 0, -1, 2, 0) = 23.45 ext{ years}$ Thus, the Rule of 72 gives a close approximation.

Example 2: Inflation and Purchasing Power

If the current inflation rate is 6%, how long will it take for the price of your favorite coffee to double?

Using the Rule of 72: $n = \frac{72}{6} = 12 ext{ years}$

Using the actual formula in Excel: $\text{NPER}(0.06, 0, -1, 2, 0) = 11.89 ext{ years}$ Again, the approximation is quite close.

Theory Behind the Rule of 72

The Rule of 72 is derived from the Time Value of Money concept, particularly the compound interest formula: $FV = PV(1 + r)^t$ where:

• FV = Future Value
• PV = Present Value
• r = Annual interest rate (decimal form)
• t = Number of years

To estimate how long it takes to double an investment, set: $2 = (1 + r)^t$ Taking the natural logarithm (ln) on both sides: $\ln(2) = t \times \ln(1 + r)$ Approximating $\ln(1 + r) \approx r$ for small values of r: $0.693 = t \times r$ Rearranging: $t = \frac{0.693}{r}$ Since 0.693 is close to 72/100, multiplying by 100 results in: $t \approx \frac{72}{r}$ Thus, 72 is chosen because it has many factors (2, 3, 4, 6, 8, 9, 12, 18, 24, 36), making mental calculations easier.

Application in Daily Life

Example 3: Credit Card Debt

Consider credit card interest rates in different countries:

• Malaysia: 18% per annum
• Singapore: 24% per annum

Using the Rule of 72:

• Malaysia: $72 / 18 = 4$ years for debt to double.
• Singapore: $72 / 24 = 3$ years for debt to double.

Now, compare this with the local fixed deposit (FD) rate of 2.5% per annum: $72 / 2.5 = 28.8 ext{ years}$

Clearly, borrowing at high interest rates is detrimental, whereas lending or saving at low-risk interest rates is beneficial.

Conclusion

The Rule of 72 is a fast and effective tool for estimating the impact of interest rates on investments and debts. While not perfectly accurate, it provides a reliable approximation for financial decision-making. Understanding the Time Value of Money is crucial to making smart financial choices—whether saving, investing, or managing debt.

Final Advice: Be a lender, not a borrower—especially when it comes to high-interest credit cards!