What rate of interest compounded annually is required to double an investment in 10 years?

Double Your Money: The Rule of 72

The Rule of 72 is a quick and simple technique for estimating one of two things:

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• Double Your Money: The Rule of 72
• 1 Expert Answer
• Still looking for help? Get the right answer, fast.
• Calculator Use
• Rule of 72 Formula
• Derivation of the Rule of 72 Formula
• Example Calculations in Years
• Example Calculation in Months
• At what rate of interest money will double in 10 years?
• At what interest rate compounded annually will money double itself in 10 years?
• What rate of interest compounded annually is required to triple an investment in 10 years?
• What annual rate of interest compounded annually is required to double an investment?

• The time it takes for a single amount of money to double with a known interest rate.
• The rate of interest you need to earn for an amount to double within a known time period.

The rule states that an investment or a cost will double when:

[Investment Rate per year as a percent] x [Number of Years] = 72.

When interest is compounded annually, a single amount will double in each of the following situations:

The Rule of 72 indicates than an investment earning 9% per year compounded annually will double in 8 years. The rule also means if you want your money to double in 4 years, you need to find an investment that earns 18% per year compounded annually.

You can confirm the rationality of the Rule of 72 as follows: Find factors on the FV of 1 Table that are close to 2.000. (The factor of 2.000 tells you that the present value of 1.000 had doubled to the future value of 2.000.) When you find a factor close to 2.000, look at the interest rate at the top of the column and look at the number of periods (n) in the far left column of the row containing the factor. Multiply that interest rate times the number of periods and you will get the product 72.

To use the Rule of 72 in order to determine the approximate length of time it will take for your money to double, simply divide 72 by the annual interest rate. For example, if the interest rate earned is 6%, it will take 12 years (72 divided by 6) for your money to double. If you want your money to double every 8 years, you will need to earn an interest rate of 9% (72 divided by 8).

Here's another way to demonstrate that the Rule of 72 works. Assume you make a single deposit of $1,000 to an account and wish for it to grow to a future value of $2,000 in nine years. What annual interest rate compounded annually will the account have to pay? The Rule of 72 indicates that the rate must be 8% (72 divided by 9 years). Let's verify the rate with the format we used with the FV Table:

To finish solving the equation, we search only the "n = 9" row of the FV of 1 Table for the FV factor that is closest to 2.000. The factor closest to 2.000 in the row where n = 9 is 1.999 and it is in the column where i = 8%. An investment at 8% per year compounded annually for 9 years will cause the investment to double (8 x 9 = 72).

1 Expert Answer

Mark M. answered • 12/03/20

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2 = (1 + r)12

12√2 = 1 + r

1.059463094 ≈ 1 + r

0.059463094 ≈ r

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Calculator Use

Use the Rule of 72 to estimate how long it will take to double an investment at a given interest rate. Divide 72 by the interest rate to see how long it will take to double your money on an investment.

Alternatively you can calculate what interest rate you need to double your investment within a certain time period. For example if you wanted to double an investment in 5 years, divide 72 by 5 to learn that you'll need to earn 14.4% interest annually on your investment for 5 years: 14.4 × 5 = 72.

The Rule of 72 is a simplified version of the more involved compound interest calculation. It is a useful rule of thumb for estimating the doubling of an investment. This calculator provides both the Rule of 72 estimate as well as the precise answer resulting from the formal compound interest calculation.

Rule of 72 Formula

The Rule of 72 is a simple way to estimate a compound interest calculation for doubling an investment. The formula is interest rate multiplied by the number of time periods = 72:

R * t = 72

where

• R = interest rate per period as a percentage
• t = number of periods

Commonly, periods are years so R is the interest rate per year and t is the number of years. You can calculate the number of years to double your investment at some known interest rate by solving for t: t = 72 ÷ R. You can also calculate the interest rate required to double your money within a known time frame by solving for R: R = 72 ÷ t.

Derivation of the Rule of 72 Formula

The basic compound interest formula is:

A = P(1 + r)t,

where A is the accrued amount, P is the principal investment, r is the interest rate per period in decimal form, and t is the number of periods. If we change this formula to show that the accrued amount is twice the principal investment, P, then we have A = 2P. Rewriting the formula:

2P = P(1 + r)t , and dividing by P on both sides gives us

(1 + r)t = 2

We can solve this equation for t by taking the natural log, ln(), of both sides,

\( t \times ln(1+r)=ln(2) \)

and isolating t on the left:

\( t = \dfrac{ln(2)}{ln(1+r)} \)

We can rewrite this to an equivalent form:

\( t = \dfrac{ln(2)}{r}\times\dfrac{r}{ln(1+r)} \)

Solving ln(2) = 0.69 rounded to 2 decimal places and solving the second term for 8% (r=0.08):*

\( t = \dfrac{0.69}{r}\times\dfrac{0.08}{ln(1.08)}=\dfrac{0.69}{r}(1.0395) \)

Solving this equation for r times t:

\( rt=0.69\times1.0395\approx0.72 \)

Finally, multiply both sides by 100 to put the decimal rate r into the percentage rate R:

R*t = 72

*8% is used as a common average and makes this formula most accurate for interest rates from 6% to 10%.

Example Calculations in Years

If you invest a sum of money at 6% interest per year, how long will it take you to double your investment?

t=72/R = 72/6 = 12 years

What interest rate do you need to double your money in 10 years?

R = 72/t = 72/10 = 7.2%

Example Calculation in Months

If you invest a sum of money at 0.5% interest per month, how long will it take you to double your investment?

t=72/R = 72/0.5 = 144 months (since R is a monthly rate the answer is in months rather than years)

144 months = 144 months / 12 months per years = 12 years

References

Vaaler, Leslie Jane Federer; Daniel, James W. Mathematical Interest Theory (Second Edition), Washington DC: The Mathematical Association of America, 2009, page 75.

Weisstein, Eric W. "Rule of 72." From MathWorld--A Wolfram Web Resource, Rule of 72.

At what rate of interest money will double in 10 years?

At what interest rate compounded annually will money double itself in 10 years?

What rate of interest compounded annually is required to triple an investment in 10 years?

What annual rate of interest compounded annually is required to double an investment?