How long will it take for an investment to double at 6% compounded monthly?
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Question 540573: How long does it take for an investment to double in value if it is investedat 6% compounded monthly? Compounded contiuously? Answer by stanbon(75887) You can put this solution on YOUR website! Show 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 RateThe annual nominal interest rate of your investment in percent.Time Period in YearsThe 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.CompoundingThis calculator assumes the frequency of compounding is once per period. It also assumes that accrued interest is compounded over time.Rule of 72 FormulaThe 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
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 FormulaThe 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 YearsIf 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 MonthsIf 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 ReferencesVaaler, Leslie Jane Federer; Daniel, James W. Mathematical Interest Theory (Second Edition), Washington DC: The Mathematical Association of America, 2009, page 75. 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. Interest Rate:%Years Required for Principal to Double Exact Answer: Rule of 72 Estimate:(We're assuming the interest is annually compounded, by the way.) As you can see, the "rule" is remarkably accurate, as long as the interest rate is less than about twenty percent; at higher rates the error starts to become significant. You can also run it backwards: if you want to double your money in six years, just divide 6 into 72 to find that it will require an interest rate of about 12 percent. Years to doubleyour investmentRequired Interest Rate Exact Answer:% Rule of 72 Estimate:% Y = 72 / r and r = 72 / Y Compound Interest CurveSuppose 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: YearsBalanceNow$1007$200(doubles every14$400 seven years)21$800If you graph these points, you start to see the familiar compound interest curve: Practice using the Rule of 72It'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: Find years needed for my money to ...Double Triple Grow by a factor of Estimate to UseDivide the interest rate into 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.) |
