The Rule of 70
The Rule of 70 is commonly used in accounting and finance as a way of estimating the number of years (t) it will take for the principal investment (P) to double in value given a particular interest rate (r) and an annual compounding period. Although it is not difficult to obtain a formula for the exact doubling time, the rule of 70 remains useful when finding quick estimates and doing mental math. The Rule of 70 says that the doubling time is close to 70 %r. For example, if r = 5 %, then t ≈ 70 %5 % = 0.700.05 = 14 years.
Why Does It Work?
The future value (FV) of an investment using compound interest is given by the equation FV=PV⁢1+rt, where PV is the present value of the investment, r is the interest rate, and t is the number of compounding periods.
When applying the rule of 70, we want to find out how many years it will take for the investment to grow to twice its size, so we can let PV = P, the amount of the initial or principal investment, and FV=2⁢P. Also, we are assuming a compounding period of 1 year, so the number of compounding periods, t, will equal the number of years.
To find the exact doubling time, we can solve the equation 2⁢P=P⁢1+rt for t to get t=ln⁡2ln⁡1+r, then substitute the current annual interest rate for r. Noting that ln2 ≈ 0.70 and that for small values of r (as is common for annual interest rates and other growth rates), ln1 + r ≈ r, this equation becomes t ≈ 0.70r.
This rule applies to any process which experiences exponential growth if the annual growth rate is sufficiently small. It is often also used in the fields of economics and geography to estimate the doubling time of a nation's real GDP, price level or population.
Use the slider below to change the current interest rate and see how this affects the doubling time. Notice how the difference between the actual number of years and the estimated number of years grows larger as the interest rate increases.
Annual Interest Rate, [r]
Actual Number of Years Until Principal Doubles:
Estimated Number of Years Until Principal Doubles:
MathApps/Finance and Economics
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