# what interest rate would you have to earn if you wanted to double an investment in 3 years?

### James

Guys, does anyone know the answer?

get what interest rate would you have to earn if you wanted to double an investment in 3 years? from EN Bilgi.

## Rule of 72 Calculator

The Rule of 72: Divide 72 by the interest rate to get the number of years to double your investment. A good estimate for how long it takes to double your money.

Calculators > Financial > Saving and Investing > Rule of 72 Calculator

## Rule of 72 Calculator

Get a Widget for this Calculator

© Calculator Soup

Share this Calculator & Page

## 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.

Interest Rate

The annual nominal interest rate of your investment in percent.

Time Period in Years

The number of years the sum of money will remain invested. You can also input months or any period of time as long as the interest rate you input is compounded at the same frequency.

Compounding

This calculator assumes the frequency of compounding is once per period. It also assumes that accrued interest is compounded over time.

## 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×ln(1+r)=ln(2) t×ln(1+r)=ln(2)

and isolating t on the left:

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

We can rewrite this to an equivalent form:

t= ln(2) r × r ln(1+r) t=ln(2)r×rln(1+r)

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

t= 0.69 r × 0.08 ln(1.08) = 0.69 r (1.0395)

t=0.69r×0.08ln(1.08)=0.69r(1.0395)

Solving this equation for r times t:

rt=0.69×1.0395≈0.72 rt=0.69×1.0395≈0.72

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.

**Cite this content, page or calculator as:**

Furey, Edward "Rule of 72 Calculator" at https://www.calculatorsoup.com/calculators/financial/rule-of-72-calculator.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators

Source : **www.calculatorsoup.com**

## The Rule of 72 (with calculator)

## Rule of 72

Have you always wanted to be able to do compound interest problems in your head? Perhaps not... but it's a very useful skill to have because it gives you a lightning fast benchmark to determine how good (or not so good) a potential investment is likely to be.

The rule says that to find the number of years required to double your money at a given interest rate, you just divide the interest rate into 72. For example, if you want to know how long it will take to double your money at eight percent interest, divide 8 into 72 and get 9 years.

**Y = 72 / r**and

**r = 72 / Y**

where Y and r are the years and interest rate, respectively.

## Compound Interest Curve

Suppose you invest $100 at a compound interest rate of 10%. The rule of 72 tells you that your money will double every seven years, approximately:

**Years**

**Balance**

Now $100

7 $200 (doubles every

14 $400 seven years)

21 $800

If you graph these points, you start to see the familiar compound interest curve:

## Practice using the Rule of 72

It's good to practice with the rule of 72 to get an intuitive feeling for the way compound interest works. So...

## Why Stop at a Double?

There's nothing sacred about doubling your money. You can also get a simple estimate for other growth factors, as this calculator shows:

## Why Does the Rule of 72 Work?

If you want to know more, see this explanation of why the rule of 72 works. (Brace yourself, because it's slightly geeked out.)

## Rule of 72 Definition, Formula, & Calculation

The Rule of 72 is a shortcut or rule of thumb used to estimate the number of years required to double your money at a given annual rate of return and vice versa.

INVESTING INVESTING ESSENTIALS

## Rule of 72

By WILL KENTON Updated August 25, 2021

Reviewed by JEFREDA R. BROWN

## What Is the Rule of 72?

The Rule of 72 is a quick, useful formula that is popularly used to estimate the number of years required to double the invested money at a given annual rate of return.

While calculators and spreadsheet programs like Microsoft's Excel have inbuilt functions to accurately calculate the precise time required to double the invested money, the Rule of 72 comes in handy for mental calculations to quickly gauge an approximate value. Alternatively, it can compute the annual rate of compounded return from an investment given how many years it will take to double the investment.

### KEY TAKEAWAYS

The Rule of 72 is a simplified formula that calculates how long it'll take for an investment to double in value, based on its rate of return.

The Rule of 72 applies to compounded interest rates and is reasonably accurate for interest rates that fall in the range of 6% and 10%.

The Rule of 72 can be applied to anything that increases exponentially, such as GDP or inflation; it can also indicate the long-term effect of annual fees on an investment's growth.

## The Formula for the Rule of 72

\begin{aligned} &\text{Years to Double} = \frac{ 72 }{ \text{Interest Rate} } \\ &\textbf{where:}\\ &\text{Interest Rate} = \text{Rate of return on an investment} \\ \end{aligned}

Years to Double= Interest Rate 72 where:

Interest Rate=Rate of return on an investment

0 seconds of 0 secondsVolume 75%

1:10

### Rule Of 72

## How to Use the Rule of 72

The Rule of 72 could apply to anything that grows at a compounded rate, such as population, macroeconomic numbers, charges, or loans. If the gross domestic product (GDP) grows at 4% annually, the economy will be expected to double in 72 / 4 = 18 years.

With regards to the fee that eats into investment gains, the Rule of 72 can be used to demonstrate the long-term effects of these costs. A mutual fund that charges 3% in annual expense fees will reduce the investment principal to half in around 24 years. A borrower who pays 12% interest on their credit card (or any other form of loan that is charging compound interest) will double the amount they owe in six years.

The rule can also be used to find the amount of time it takes for money's value to halve due to inflation. If inflation is 6%, then a given purchasing power of the money will be worth half in around 12 years (72 / 6 = 12). If inflation decreases from 6% to 4%, an investment will be expected to lose half its value in 18 years, instead of 12 years.

Additionally, the Rule of 72 can be applied across all kinds of durations provided the rate of return is compounded annually. If the interest per quarter is 4% (but interest is only compounded annually), then it will take (72 / 4) = 18 quarters or 4.5 years to double the principal. If the population of a nation increases at the rate of 1% per month, it will double in 72 months, or six years.

## Rule of 72 FAQs

### Who Came Up With the Rule of 72?

People love money, and they love to see it grow even more. Getting a rough estimate of how much time it will take to double your money also helps the average Joe or Jane to compare different investment options. However, mathematical calculations that project an investment's appreciation can be complex for common individuals to do without the help of log tables or a calculator, especially those involving compound interest.

The Rule of 72 offers a useful shortcut. It's a simplified version of a logarithmic calculation that involves complex functions like taking the natural log of numbers. The rule applies to the exponential growth of an investment based on a compounded rate of return.

### How Do You Calculate the Rule of 72?

Here's how the Rule of 72 works. You take the number 72 and divide it by the investment's projected annual return. The result is the number of years, approximately, it'll take for your money to double.

For example, if an investment scheme promises an 8% annual compounded rate of return, it will take approximately nine years (72 / 8 = 9) to double the invested money. Note that a compound annual return of 8% is plugged into this equation as 8, and not 0.08, giving a result of nine years (and not 900).

If it takes nine years to double a $1,000 investment, then the investment will grow to $2,000 in year 9, $4,000 in year 18, $8,000 in year 27, and so on.

### How Accurate Is the Rule of 72?

The Rule of 72 formula provides a reasonably accurate, but approximate, timeline—reflecting the fact that it's a simplification of a more complex logarithmic equation. To get the exact doubling time, you'd need to do the entire calculation.

The precise formula for calculating the exact doubling time for an investment earning a compounded interest rate of r% per period is:

\begin{aligned} &T = \frac{ \ln( 2 ) }{ \ln \left ( 1 + \frac{ r } { 100 } \right ) } \simeq \frac{ 72 }{ r } \\ &\textbf{where:}\\ &T = \text{Time to double} \\ &\ln = \text{Natural log function} \\ &r = \text{Compounded interest rate per period} \\ &\simeq = \text{Approximately equal to} \\ \end{aligned}

T= ln(1+ 100 r ) ln(2) ≃

Guys, does anyone know the answer?