Sam's Blog: Learn the math behind doubling your money: How to Use the Rule of 72

Learn the math behind doubling your money: How to Use the Rule of 72: "

The rule of 72 is useful for approximating doubling time in exponential growth.

The rule of 72 is a handy rule used in finance to estimate quickly the time it takes to double a sum of capital given an interest rate, or to estimate the interest rate it takes to double a sum of money within a given amount of time. The rule states that interest percentage times the number of years it takes to double a principal amount of money is approximately equal to 72.

The Rule of 72 is applicable in exponential growth (as in compound interest) or in exponential decay.

Steps

Exponential growth

Estimating doubling time

Let R * T = 72, where R = the rate of growth (for example, interest rate), T = doubling time (for example, time it takes to double an amount of money).
Plug in the value for R = rate of growth. For example, how long does it take to double $100 to $200 at an interest rate of 5% per annum? Substituting R = 5, we get 5 * T = 72.
Solve for the unknown variable. In the example given, divide both sides by R = 5, to get T = 72/5 = 14.4. So it takes 14.4 years to double $100 to $200 at an interest rate of 5% per annum.
Study these additional examples:
- How long does it take to double a given amount of money at a rate of 10% per annum? Let 10 * T = 72, so T = 7.2 years.
- How long does it take to turn $100 to $1600 at a rate of 7.2% per annum? Recognize that it takes 4 doubling to get from $100 to $1600 (double of $100 is $200, double of $200 is $400, double of $400 is $800, and double of $800 is $1600). For each doubling, 7.2 * T = 72, so T = 10. Multiply that by 4 yields 40 years.

Estimating growth rate

Let R * T = 72, where R = the rate of growth (for example, interest rate), T = doubling time (for example, the time it takes to double an amount of money).
Plug in value for T = doubling time. For example, if you want to double your money in ten years, what interest rate do you need? Substituting T = 10, we get R * 10 = 72.
Solve for the unknown variable. In the example given, divide both sides by T = 10, to get R = 72/10 = 7.2. So you will need 7.2% annual interest rate to double your money in ten years.

Estimating exponential decay

Estimate the time to lose half of your capital: as in the case of inflation. Solve T = 72/R, after plugging in value for R, analogous to estimating doubling time for exponential growth (it's the same as the doubling formula, but you think of the result as inflation rather than growth), for example:
- How long will it take for $100 to depreciate to $50 at an inflation rate of 5%?
  - Let 5 * T = 72, so 72/5 = T, so that T = 14.4 years for buying power to halve at an inflation rate of 5%.
Estimate the rate of decay for a certain time span: Solve R = 72/T, after plugging in value for T, analogous to estimating growth rate for exponential growth, for example:
- If the buying power of $100 becomes worth only $50 in ten years, what is the inflation rate per annum?
  - Let R * 10 = 72, where T = 10 so that we may find R = 72/10 = 7.2% for that one example.
Beware! Different things decay in value at various rates and over different times; a study of many products over time would be required to find a general trend (or average) of inflation – and 'out of bounds,' outliers, or odd examples are simply ignored, and dropped out of consideration.

Video

Tips

The value 72 is chosen as a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%). The approximations are less exact at higher interest rates.
To estimate doubling time for higher rates, adjust 72 by adding 1 for every 3 percentages greater than 8%. That is, T = [72 + (R - 8%)/3] / R. For example, if the interest rate is 32%, the time it takes to double a given amount of money is T = [72 + (32 - 8)/3] / 32 = 2.5 years. Note that 80 is used here instead of 72, which would have given 2.25 years for the doubling time.
For continuous compounding, 69.3 (or approximately 69) gives more accurate results, since ln(2) is approximately 69.3%, and R * T = ln(2), where R = growth (or decay) rate, T = the doubling (or halving) time, and ln(2) is the natural log of 2. 70 may also be used as an approximation for continuous or daily (which is close to continuous) compounding, for ease of calculation. These variations are known as rule of 69.3, rule of 69, or rule of 70.
- A similar accuracy adjustment for the rule of 69.3 is used for high rates with daily compounding: T = (69.3 + R/3) / R.
The Eckart-McHale second order rule, or E-M rule, gives a multiplicative correction to the Rule of 69.3 or 70 (but not 72), for better accuracy for higher interest rate ranges. To compute the E-M approximation, multiply the Rule of 69.3 (or 70) result by 200/(200-R), i.e., T = (69.3/R) * (200/(200-R)). For example, if the interest rate is 18%, the Rule of 69.3 says t = 3.85 years. The E-M Rule multiplies this by 200/(200-18), giving a doubling time of 4.23 years, which better approximates the actual doubling time 4.19 years at this rate.
- The third-order Padé approximant gives even better approximation, using the correction factor (600 + 4R) / (600 + R), i.e., T = (69.3/R) * ((600 + 4R) / (600 + R)). If the interest rate is 18%, the third-order Padé approximant gives T = 4.19 years.
Here is a table giving the number of years it takes to double any given amount of money at various interest rates, and comparing the approximation with various rules:

Rate	Actual Years	Rule of 72	Rule of 70	Rule of 69.3	E-M rule
0.25%	277.605	288.000	280.000	277.200	277.547
0.5%	138.976	144.000	140.000	138.600	138.947
1%	69.661	72.000	70.000	69.300	69.648
2%	35.003	36.000	35.000	34.650	35.000
3%	23.450	24.000	23.333	23.100	23.452
4%	17.673	18.000	17.500	17.325	17.679
5%	14.207	14.400	14.000	13.860	14.215
6%	11.896	12.000	11.667	11.550	11.907
7%	10.245	10.286	10.000	9.900	10.259
8%	9.006	9.000	8.750	8.663	9.023
9%	8.043	8.000	7.778	7.700	8.062
10%	7.273	7.200	7.000	6.930	7.295
11%	6.642	6.545	6.364	6.300	6.667
12%	6.116	6.000	5.833	5.775	6.144
15%	4.959	4.800	4.667	4.620	4.995
18%	4.188	4.000	3.889	3.850	4.231
20%	3.802	3.600	3.500	3.465	3.850
25%	3.106	2.880	2.800	2.772	3.168
30%	2.642	2.400	2.333	2.310	2.718
40%	2.060	1.800	1.750	1.733	2.166
50%	1.710	1.440	1.400	1.386	1.848
60%	1.475	1.200	1.167	1.155	1.650
70%	1.306	1.029	1.000	0.990	1.523

Felix's Corollary to the Rule of 72 is used to approximate the future value of an annuity (a series of regular payments). It states that the future value of an annuity whose percentage interest rate and number of payments multiply to be 72 can be approximated by multiplying the sum of the payments times 1.5. For example, 12 periodic payments of $1000 growing at 6% per period will be worth approximately $18,000 after the last period. This is an application of Felix's Corollary to the Rule of 72 since 6 (the percentage interest rate) times 12 (the number of payments) equals 72, so the value of the annuity approximates 1.5 times 12 times $1000.
Let the rule of 72 work for you, by starting saving now. At a growth rate of 8% per annum (the approximate rate of return in the stock market), you would double your money in 9 years (8 * 9 = 72), quadruple your money in 18 years, and have 16 times your money in 36 years.

Derivation

Periodic compounding

For periodic compounding, FV = PV (1 + r)^T, where FV = future value, PV = present value, r = growth rate, T = time.
If money has doubled, FV = 2*PV, so 2PV = PV (1 + r)^T, or 2 = (1 + r)^T, assuming the present value is not zero.
Solve for T by taking the natural logs on both sides, and rearranging, to get T = ln(2) / ln(1 + r).
The Taylor series for ln(1 + r) around 0 is r - r²/2 + r³/3 - ... For low values of r, the contributions from the higher power terms are small, and the expression approximates r, so that t = ln(2) / r.
Note that ln(2) ~ 0.693, so that T ~ 0.693 / r (or T = 69.3 / R, expressing the interest rate as a percentage R from 0-100%), which is the rule of 69.3. Other numbers such as 69, 70, and 72 are used for easier calculations.

Continuous compounding

For periodic compounding with multiple compounding per year, the future value is given by FV = PV (1 + r/n)^nT, where FV = future value, PV = present value, r = growth rate, T = time, and n = number of compounding periods per year. For continuous compounding, n approaches infinity. Using the definition of e = lim (1 + 1/n)^n as n approaches infinity, the expression becomes FV = PV e^(rT).
If money has doubled, FV = 2*PV, so 2PV = PV e^(rT), or 2 = e^(rT), assuming the present value is not zero.
Solve for T by taking natural logs on both sides, and rearranging, to get T = ln(2)/r = 69.3/R (where R = 100r to express the growth rate as a percentage). This is the rule of 69.3.

Warnings

Don't let the rule of 72 work against you, when you take on high interest debt. Avoid credit card debt! At an average interest rate of 18%, the credit card debt doubles in just 4 years (18 * 4 = 72), and quadruples in only 8 years, and keeps escalating with time. Avoid credit cards like the plague.