What is the lowest interest rate that gives you a doubling time less than 9 years?

Use a spreadsheet to find the number of years it will take for an investment to double at 3% annual interest. Find the product of the interest rate and the doubling time. Find a formula for doubling time as a function of interest rate. Use the compound interest formula.|||Steps


Exponential growth


Estimating doubling time


1Let R * T = 72, where R = the rate of growth (for example, interest rate), T = doubling time (for example, time it takes to double an amount of money).


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2Plug in the value for R = rate of growth. For example, how long does it take to double $100 to $200 at an interest rate of 5% per annum? Substituting R = 5, we get 5 * T = 72.3Solve for the unknown variable. In the example given, divide both sides by R = 5, to get T = 72/5 = 14.4. So it takes 14.4 years to double $100 to $200 at an interest rate of 5% per annum.4Study these additional examples:


How long does it take to double a given amount of money at a rate of 10% per annum? Let 10 * T = 72, so T = 7.2 years.How long does it take to turn $100 to $1600 at a rate of 7.2% per annum? Recognize that it takes 4 doubling to get from $100 to $1600 (double of $100 is $200, double of $200 is $400, double of $400 is $800, and double of $800 is $1600). For each doubling, 7.2 * T = 72, so T = 10. Multiply that by 4 yields 40 years.


Estimating growth rate


1Let R * T = 72, where R = the rate of growth (for example, interest rate), T = doubling time (for example, the time it takes to double an amount of money).2Plug in value for T = doubling time. For example, if you want to double your money in ten years, what interest rate do you need? Substituting T = 10, we get R * 10 = 72.3Solve for the unknown variable. In the example given, divide both sides by T = 10, to get R = 72/10 = 7.2. So you will need 7.2% annual interest rate to double your money in ten years.


Estimating exponential decay1Estimate the time to lose half of your capital: as in the case of inflation. Solve T = 72/R, after plugging in value for R, analogous to estimating doubling time for exponential growth (it's the same as the doubling formula, but you think of the result as inflation rather than growth), for example:


How long will it take for $100 to depreciate to $50 at an inflation rate of 5%?


Let 5 * T = 72, so 72/5 = T, so that T = 14.4 years for buying power to halve at an inflation rate of 5%.2Estimate the rate of decay for a certain time span: Solve R = 72/T, after plugging in value for T, analogous to estimating growth rate for exponential growth, for example:


If the buying power of $100 becomes worth only $50 in ten years, what is the inflation rate per annum?


Let R * 10 = 72, where T = 10 so that we may find R = 72/10 = 7.2% for that one example.3Beware! Different things decay in value at various rates and over different times; a study of many products over time would be required to find a general trend (or average) of inflation 鈥?and "out of bounds," outliers, or odd examples are simply ignored, and dropped out of consideration.|||FV = PV*(1 + 0.03)^n





FV/PV = 2





2 = (1 + 0.03)^n





ln(2) = n*ln(1.03)





n = ln(2)/(ln(1.03))





Find the product of the interest rate and the doubling time = 2*0.03 = 0.06





Find a formula for doubling time as a function of interest rate: 2 = (1.03)^(ln(2)/(ln(1.03)))