If annual payments of 4000 are made into an ordinary annuity

If annual payments of $4,000 are made into an ordinary annuity that 5% interest compounded annually, how long will it take until the annuity is worth $100,000? (Round your answer to the nearest whole number.)

Solution

The formula for the Futurevalue, F of an annuity is F = A [ {(1+r)ⁿ -1 } / r ] .where A is the periodic payment, r is the rate of interest in decimals, n is the number of periods over which payments are made. Here, F = $ 100000, A = $ 4000 and r = 5 % = 5/100 = 0.05. We have to find n.

Now, we have, by substitution in the formula, 100000 = 4000 [ { ( 1+ 0.05)ⁿ - 1} / 0.05] or, 100000/4000 = [ {(1.05)ⁿ - 1}/0.05] or, 25 * 0.05 = (1.05)ⁿ - 1 or,   1.25 + 1 = ( 1.05)ⁿ or, ( 1.05)ⁿ = 2.25. On taking logarithms of both the sides, we have n log 1.05 = log 2.25 or, n = log 2.25/ log 1.05 = (0.352182518) / (0.021189299) = 16.6207725 years = 16 years, 7.45 months approximately. On rounding off to a whole number, the answer is 16 years, 7 months or, 17 years.