Question

How much will you have accumulated, if you annually invest $1,500 into an IRA at 8% interest compounded monthly for:

a. 5 year

b. 20 years

c. 40 years

d. How long will it take to earn your first million dollars? Your answer should be exact rounded within 2 decimal places. Please use logarithms to solve. (Total of 15 points)

Answer 1

The formula for compounding of interest is:
A = P (1 + r/n) ^nt , where A is the future value of the investment, including interest, P is the principal/ initial amount invested, r is the annual interest rate in decimals, n is the number of times that interest is compounded in a year and t is the number of years the money is invested. Here, P = $ 1500, r = 0.08, and n = 12.

a. Here, t = 5 so that A = 1500(1+0.08/12)⁶⁰ = 1500*1.489845709 = 2234.77 ( on rounding off to the nearest cent).

b. Here, t = 20 so that A = 1500(1+0.08/12)²⁴⁰ = 1500*4.926802775 = $ 7390.20( on rounding off to the nearest cent).

c. Here, t = 40 so that A = 1500(1+0.08/12)⁴⁸⁰ = 1500*24.27338558 = $ 36410.08( on rounding off to the nearest cent).

d. Let the amount increase to $ 1000000 in t years. Then 1000000 = 1500(1+0.08/12)^12t or, (12.08/12)^12t = 1000000/1500 = 2000/3. Now, on taking log of both the sides, we get log (2000/3) = 12t(log 12.08/12) or, t=(log2000-log3)/12(log12.08-log12) = (3.301029996-0.477121254)/12*(1.082066934-1.079181246) = 2.823908742/12*0.00288568829 =2.823908742/0.034628259 = 81.54925554 = 81.55 years ( on rounding off to 2 decimal places).