Question

Determine the time necessary for p dollars to double when it is invested at interest rate r compounded annually, monthly, daily, and continuously. (round your answers to two decimal places.) r = 11%

Answer 1

Annual = Years = 6.64; Actually 7 years Monthly = Years = 6.33; 6 Years, 4 months Daily = Years = 6.30; 6 Years, 111 days Continuously = 6.30; 6 Years, 110 days The formula for compound interest is FV = P*(1 + R/n)^(nt) where FV = Future Value P = Principle R = Annual interest rate n = number of periods per year t = number of years For this problem, we can ignore p and concentrate on the (1+R/n)^(nt) term, looking for where it becomes 2. So let's use this simplified formula: 2 = (1 + R/n)^(nt) With R, n, and t having the same meaning as in the original formula. For for the case of compounding annually 2 = (1 + R/n)^(nt) 2 = (1 + 0.11/1)^(1t) 2 = (1.11)^t The above equation is effectively asking for the logarithm of 2 using a base of 1.11. To do this take the log of 2 and divide by the log of 1.11. So log(2) / log(1.11) = 0.301029996 / 0.045322979 = 6.641884618 This explanation of creating logarithms to arbitrary bases will not be repeated for the other problems. The value of 6.641884618 indicates that many periods is needed. 6 is too low giving an increase of 1.11^6 =1.870414552 and 7 is too high, giving an increase of 1.11^7 = 2.076160153 But for the purpose of this problem, I'll say you double your money after 7 years. For compounding monthly: 2 = (1 + R/n)^(nt) 2 = (1 + 0.11/12)^(12t) 2 = (1 + 0.009166667)^(12t) 2 = 1.009166667^(12t) log(2)/log(1.009166667) = 0.301029996 / 0.003962897 = 75.96210258 And since the logarithm is actually 12*t, divide by 12 75.96210258 / 12 = 6.330175215 Which is 6 years and 4 months. For compounding daily: 2 = (1 + 0.11/365)^(365t) 2 = (1 + 0.00030137)^(365t) 2 = 1.00030137^(365t) log(2)/log(1.00030137) = 0.301029996 / 0.000130864 = 2300.334928 2300.334928 / 365 = 6.302287474 Continuously: For continuous compounding, there's a bit of calculus required and the final formula is FV = Pe^(rt) where FV = Future value P = Principle e = mathematical constant e. Approximately 2.718281828 r = Interest rate t = time in years Just as before, we'll simplify the formula and use 2 = e^(rt) Since we have the function ln(x) which is the natural log of x, I won't bother doing log conversions. rt = ln(2) 0.11 * t = 0.693147181 t = 0.693147181 / 0.11 t = 6.301338005