Question

Your client is 26 years old. She wants to begin saving for retirement, with the first payment to come one year from now. She can save $7,000 per year, and you advise her to invest it in the stock market, which you expect to provide an average return of 10% in the future.

If she follows your advice, how much money will she have at 65? Do not round intermediate calculations. Round your answer to the nearest cent.

$

How much will she have at 70? Do not round intermediate calculations. Round your answer to the nearest cent.

$

She expects to live for 20 years if she retires at 65 and for 15 years if she retires at 70. If her investments continue to earn the same rate, how much will she be able to withdraw at the end of each year after retirement at each retirement age? Do not round intermediate calculations. Round your answers to the nearest cent.

Annual withdrawals if she retires at 65: $

Annual withdrawals if she retires at 70: $

Answer 1

(a) Number of years to turn 65 = n = 65 - 26 = 39
Amount Invested each Year = P = $7000
Interest Rate = r = 10%
Hence, Future Value = FV = P(1+r)^n-1 +....+ P(1+r)² + P(1+r) + P = P[(1+r)ⁿ -1]/r = 7000[(1+0.10)³⁹ -1]/0.10 = $2810134.45

(b) Number of years to turn 70 = n = 70 - 26 = 44
Amount Invested each Year = P = $7000
Interest Rate = r = 10%
Hence, Future Value = FV = P(1+r)^n-1 +....+ P(1+r)² + P(1+r) + P = P[(1+r)ⁿ -1]/r = 7000[(1+0.10)⁴⁴ -1]/0.10 = $4568485.33

(c) Let the annual withdrawals be X
Value of investments = PV = $2810134.45
Number of years = n = 20
Present Value of withdrawals = X/(1+r) + X/(1+r)² +....+ X/(1+r)ⁿ = X[1- (1+r)^-n]/r = X[1- (1+0.10)^-20]/0.10
This should be equal to the value of investments
=> X[1- (1+0.10)^-20]/0.10 = 2810134.45
=> X = 2810134.45*0.10 / [1- (1+0.10)^-20]
=> X = $330077.33

(d) Let the annual withdrawals be X
Value of investments = PV = $4568485.33
Number of years = n = 15
Present Value of withdrawals = X/(1+r) + X/(1+r)² +....+ X/(1+r)ⁿ = X[1- (1+r)^-n]/r = X[1- (1+0.10)^-15]/0.10
This should be equal to the value of investments
=> X[1- (1+0.10)^-15]/0.10 = 4568485.33
=> X = 4568485.33*0.10 / [1- (1+0.10)^-15]
=> X = $600636.02