# The Rule of 72

If money is invested in an account paying compound interest, the rule of 72 is a quick way to estimate the time it will take for the investment to double in value.

The rule of 72 states that if a lump sum is invested at a periodic discount rate (i), then the the number of periods (n) it takes to double the value of the investment is given by the rule of 72 formula:

`Number of periods to double investment n = 72 / i`

It is important to note that the discount rate (i) is per period, and is entered into the formula as a number not as a percentage, i.e. 10% is entered in the rule of 72 formula as 10.

## Rule of 72 Examples

The idea of the rule of 72 is best understood by looking at a simple example.

If a lump sum of money is invested today at an interest rate of 10% per period, then the number of periods it takes to double the investment is given by the rule of 72 formula as follows:

```n = 72 / i
n = 72 / 10
n = 7.2 periods
```

It will take approximately 7.2 periods to double the investment.

The periods can be any value, providing the discount rate is quoted per period. For example, if the amount was invested in an account paying 7% per year, then the rule of 72 would estimate the number of years to double the investment as:

```n = 72 / i
n = 72 / 7
n = 10.29 years
```

In this example, it will take approximately 10.29 years to double the investment.

The rule of 72 is an approximation, the actual amount of time it would take to double an investment is given by the lump sum number of periods formula.

`n = LN(FV / PV) / LN(1 + i)`
Variables used in the formula
PV = Present Value
FV = Future Value
i = Discount rate
n = Number of periods

In this special case, the lump sum invested at a discount rate (i) doubles in value after a number of periods (n), so the future value (FV) must be double the present value (PV), and FV / PV = 2, so the above formula can be simplified as follows:

```n = LN(FV / PV) / LN(1 + i)
n = LN(2) / LN(1 + i)
```

If the discount rate is 7% (as in the example above), then the lump sum number of periods formula gives the number of years is takes to double the investment as follows:

```n = LN(FV / PV) / LN(1 + i)
n = LN(2) / LN(1 + i)
n = LN(2) / LN(1 + 7%)
n = 10.24 years
```

In this case the number of years taken to double the investment is 10.24 years compared to the approximation given by the rule of 72 of 10.29 years.

Generally, the rule of 72 becomes less accurate at very low discount rates, provides good approximations between 6% and 12%, and gets progressively less accurate again as the discount rates increases.

For example is the discount rate is 0.25%, the rule of 72 gives the value of n as 288 periods, whereas the correct answer using the lump sum number of periods formula is 277.61 periods. Again, if the discount rate is 75% the the rule of 72 gives n as 0.96 periods, wheres the correct answer is 1.24 periods.

## Rearranging the Rule of 72 Formula

The rule of 72 is used to calculate the number of periods it takes to double an investment, however, by rearranging the formula, it is possible to estimate the discount rate needed to double an investment in a known number of periods.

As a example, suppose a venture capitalist wanted to double the value of their investment in 4 years, then the discount rate (return on investment) needed to do this would be estimated as follows:

```n = 72 / i
i = 72 / n
i = 72 / 4
i = 18%
```

An annual return of approximately 18% would double the value of the investment in 4 years, this compares to the actual value of 18.92% given by using the lump sum discount rate formula.

The rule of 72 is useful in calculating approximations for the number of periods or the discount rate needed to double the value of an investment. It should be used with care at very low or high discount rates, but provides a useful way to check time value of money calculations.

The Rule of 72 November 6th, 2016

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