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How many years will it take to double your money if you invest in a fund with each of the given interest rates below, compounded continuously? a. 5% b. 6% c. 8% d. 10% e. Financial managers use a quick calculation that does not involve logarithms to approximate how long it will take to double your money given a specific interest rate. They call it the "rule of 72". Determine the relationship between the interest rate, the doubling time and the number 72. Write a function r(x) to approximate the number of years it would take for your money to double given an interest rate of x. f. A savvy financial investor might want to convince their customers that they know how to make their money double faster, by using a more accurate calculation. Determine the best number, rounded to the nearest tenth, to use for a doubling rule. Explain how this number relates to natural logarithms.
(a) If we want to double an investment, the future value will be equal to , where is the present value. To calculate the time we need to double an investment we will apply formula for future value, which is
where is time expressed in years, is per annum interest rate compounded times per year.We know that and because we compound interest monthly, so substituting this into the previous equation we will get
Dividing the previous equation by we will get
Using the definition of the logarithm we have now that
We used the Change-of-Base formula for logarithms in the last step. So, the time needed to double the investment if it is invested at compounded monthly is approximately 8.69 years.
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