Rule 72 – It is a name of a simple way for finding the time period required to double your money at certain interest rate per period (e.g., per year). It is mostly use for mental calculations or when only simple calculator is available for calculations. Basically, It is used for estimating an approximation of how much time do need to for investment to double itself.

**Rules similar to Rule 72**

The rule 72 In finance, the rule of 70 and the rule of 69.3 are methods for estimating the time period required to double an investment. This rule are used for a fixed range of compound interest rates for example rule 72 is used for range between 6 to 10. And the rule 73 may be used between range of 10 to 13.

The rule number (e.g., 72) is divided by the interest percentage per period (usually years) to obtain the approximate number of periods required for doubling.

Time period = | Rule Number |

Interest Rate |

And many more that depends on situations.

**Comparison between Rule 72 and Logarithmic equations for compound interest rates**.

For accuracy, we can use logarithmic equations to find time period. But it needs a scientific calculator to do logarithmic calculations or A logarithmic table.

For example if Rs. 100 dollar is invested at 9% per annum compound interest rate. Then to find how much time will it take to double the money i.e., Rs. 200 invested.

We can calculate it as = `72/r=8` years using Rule 72. But for an exact calculation, we can do, t = `(In (2))/(In (1+r%))`=8.0432 years.t = time required to double the money (annum).

r = compound rate of interest per annum.

So, from above example we can see that rule 72 is only an approximation, It doesn’t tell accurate precise answers, that logarithmic equations can tell.

For example if $ 100 invested at 23% compound interest rate. Then to find how much time will it take to double the money i.e., $200 invested.

We can calculate it as = `72/23`=3.1304 years. But for an exact calculation, we can do, t = `(In (2))/(In (1+ ( 23/100) ))`=3.34 years.**Limitations of Rule 72**

If one wants to know the time required to triple the investment amount, then we he can use logarithmic calculations but not Rule 72 because it only time required to double the money.

But we can use logarithmic equations for that purpose.

For example, we can simply replace the constant 2 in the numerator with 3. As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5.

**Advantages of Rule 72**

The rule of 72 is a useful shortcut, since the equations related to compound interest are too complicated for most people to do without a calculator. So they can use rule 72 instead of compound interest calculations.

T = `log (2) / log (1.08)` = 9.006 (using logarithmic equations).

**Most of the peoples cannot do logarithmic functions in their heads,**

But they can do `72/ 8` and get almost the same result or approximation.

If it takes 9 years to double a Rs.1,000 investment, then the investment will grow to Rs.2,000 in Year 9, Rs.4,000 in Year 18, Rs.8,000 in Year 27, and so on. Conveniently, 72 is divisible by 2, 3, 4, 6, 8, 9, and 12, making the calculation even more simpler for peoples.

**Usage of Rule 72**

It is not necessary that the units should be in terms of money for using rule 72. It can be anything that grows over time at certain rates. For example, we can use rule 72 for populations’ related calculations of any economy or for Gross Domestic product Calculations or National Income etc.

Let’s say Per capita income of India according to 2017 report is Rs. 20000. And it is assumed that it would increase 10 % per annum on compound basis.

We can use rule 72 to find time required to double the per capita income to Rs. 20000 = `72 / 10` = 7.2 years.

Someone can use Rule 72 to find anything which grows over time at compound rate like.

- Returns on mutual funds.
- Interest on Bank Balance etc.

**Rule 72 For Higher compound Interest Rates**

The rule of 72 is reasonably accurate for interest rates between 6% to 10%.

But When dealing with rates outside this range, We can see that rule 72 doesn’t gives accurate answers as before so we can be adjusted it by adding or subtracting 1 from 72 for every 3 numbers change in the interest rate from 8%.

So for 11% annual compounding interest, the rule of 73 would be more appropriate, it will provide more accurate answer than rule 72 itself; for 14%, it would be the rule of 74; for 5%, the rule of 71.

For example, say you have a 22% rate of return (Very Good you are investing at right place). The rule of 72 says the initial investment will double in 3.27 years.

Since 22 – 8 = 14, and we consider 3 number change so, 14 ÷ 3 is 4.67 ≈ 5. Then the adjusted rule would be something like => 72 + 5 = 77 for the numerator.

This gives a time period of 3.5, while rule 72 gave 3.27 years and period given by the logarithmic equation is 3.49, so the adjusted rule is more accurate for finding time period.

**Continuous Compounding with rule 72**

For daily or continuous compounding interest rates, 69.3 in the numerator gives a more accurate answer than 72. One may also use 69 or 70 for sake of simplicity.

For example, lets assume an invest giving 10% interest rate on continuous basis. And we invested Rs.1000. And now we want to know, how much time will it take to become Rs.2000.

So, time period required = `69/10`=6.9 yearsOr For accuracy we can use A = Pe^{rt}

Where A = Amount P = Principal

e = exponential number r = continuous rate of interest

t = time

`([log (2000) – log (1000)])/(0.4343 x(10/100))`= 6.93138374.