Question

Burt is saving up for his retirement. Today is his 36th birthday. Burt first started saving when he was 27 years old. On his 27th birthday, Burt made the first contribution to his retirement account when he deposited $2,000. Each year on his birthday, Burt has contributed another $2,000 to the account. The 10th (and last) of these contributions was made earlier today on his 36th birthday. The account has paid an effective annual rate of return of 5.4%. a) How much will Burt have in the account on his 36th birthday (after the contribution mentioned above)? Burt wants to close the account tomorrow and move the money to a stock fund which is expected to earn an effective return of 8.4% a year. Burt’s plan is to continue making contributions to this account each year on his birthday. His next contribution will be one year from today (age 37) and his final planned contribution will be on his 60th birthday (24 additional contributions in all). Burt plans on being able to withdraw $8,000 a month for 20 years after he retires (240 withdrawals starting one month after his 60th birthday). b) How much does Burt need in his account on his 60th birthday to fund the withdrawals? c) How much does each of the 24 annual contributions (from age 36 to 60) need to be to reach this goal?

Answer 1

a)

future value of annuity = P*[(1+r)^n - 1 / r ]

P = annual contributions

r = rate of interest

n = number of periods

future value (i.e., amount on his 36th birthday) = 2000*[(1+5.4%)^10 - 1 / 5.4% ]

= 2000 * 12.81523

= $25,630.46

b)

Present value of annuity = P*[1 - (1+r)^-n / r ]

EAR = (1 + x)^12 - 1

where x = monthly interest rate

8.4% = (1 + x)^12 - 1

so monthly rate = (1.084)^1/12 - 1

= 0.674%

Present value (amount required on his 60th birthday) = 8000*[1 - (1+0.674%)^-240 / 0.674% ]

= 8000 * 118.7318

= $949,854.08

c)

Burt needs 949,854.08 in 24 years

Balance at the age of 36 = 25,630.46

future value = Present value*(1+r)^n

future value = 25,630.46*(1+8.4%)^24

= 177,606.63

so balance required = 949,854.08 - 177,606.63 = 772,247.46

Using future value of annuity formula

772,247.46 = P*[(1+8.4%)^24 - 1 / 8.4% ]

P = 772,247.46 / 70.58945

so Amount of annual contributions (P) = $10,940

(in case of any further explanation please comment)