Question

You are planning to save for retirement over the next 35 years. To do this, you will invest £400 a month in a share account and £500 a month in a bond account. The annual return of the share account is expected to be 7 per cent, and the bond account will pay 4 per cent annually. When you retire, you will combine your money into an account with a 6 per cent annual return.

How much can you withdraw each month from your account, assuming a 25-year withdrawal period?

1. (a) £7,585.

2. (b) £8,650.

3. (c) £9,000.

4. (d) £9,985.

5. (e) I choose not to answer.

Answer 1

Amount invested per month in share = S = $400
Monthly Interest Rate earned in share = r_S = 0.07/12
Number of months for which investment is made = n_S = 35*12 = 420
Value of share investment after 35 years = = S(1+r_S)^n_S-1 +....+ S(1+r_S)² + S(1+r_S) + S = S[(1+r_S)^n_S -1]/r_S = 400[(1+0.07/12)⁴²⁰ -1]/(0.07/12) = $720421.84

Amount invested per month in bond = B = $500
Monthly Interest Rate earned in share = r_B = 0.04/12
Number of months for which investment is made = n_B = 35*12 = 420
Value of share investment after 35 years = = B(1+r_B)^n_B-1 +....+ B(1+r_B)² + B(1+r_B) + B = B[(1+r_B)^n_B -1]/r_B = 500[(1+0.04/12)⁴²⁰ -1]/(0.04/12) = $456865.47

Hence, total value in account after 35 years = FV = 720421.84 + 456865.47 = $1177287.31

Let the amount withdrawn each month be P
Interest Rate = r = 0.06/12
Number of months = n = 25*12 = 300 months
The present value of all the future deposits will be equal to the amount accrued at the end of 35 years
=> PV = P/(1+r) + P/(1+r)² +....+ P/(1+r)ⁿ = P[1- (1+r)^-n]/r
=> 1177287.31 = P[1- (1+0.06/12)^-300]/(0.06/12)
=> P = 1177287.31*(0.06/12)/[1- (1+0.06/12)^-300] = $7585.28 = $7,585