Question

You are planning to retire after working for another 30 years. You believe that you can save 4% of your salary annually. Your last salary, which has just been paid, was $45,000. You anticipate that your salary will increase at the rate of 5% for the remainder of your working life. The savings will be placed in investments earning an average return of 9% per year. After retiring you expect to live on your savings for 20 years.

a. If you plan to withdraw an equal amount of money every year in retirement, how much will you be able to withdraw from your account every year?

b. Suppose that, to keep up with the rate of inflation, you want your retirement withdrawals to increase by 3% every year, how much will you withdraw in your last year of retirement?

c. Suppose that you want to withdraw an equal amount every year but you want to withdraw 5% more than your answer to (a.)

d. You plan to finance this increase by increasing your savings by $Y a year for the last 4 years of your working life. What is $Y?

(Manually Calculate)

Answer 1

a. FV at end of 30 yrs.of the 5%growing annuity of (45000*4%=)1800 will be

Using FV of growing annuity formula,

FV(ga)=Annuity amt.*((1+r)^n-(1+g)^n)/(r-g)

1800*(1.09^30-1.05^30)/4%

402558

so, $ 402558 is the PV of yearly withdrawals for the next 20 yrs. At 9% reqd. return

So, using PV of annuity formula

PVOA=Annuity amt.*(1-(1+r)^-n)/r

402558=Annuity amt.*(1-(1+0.09)^-20)/0.09

44099

Amt.that can be withdrawn every year will be = $ 44099

b.

Now this $ 402558 is the PV of a growing annuity at 3% (inflation rate)

Using PV of growing annuity formula,

PV(ga)=Annuity amt./(r-g)*(1-((1+g)/(1+r))^n)

ie. 402558=Annuity amt./(0.09-0.03)*(1-((1+0.03)/(1+0.09))^20)

Annuity amt.= 35639

So, the last,ie. 20 th year withdrawal be

35639*(1+0.03)^20=

64368

c.

Answer in a.=

Amt.that can be withdrawn every year will be = $ 44099

5% more than the above=44099*1.05=

46304

So, PV at end of 30 yrs. at 9% reqd. return,of the withdrawal- annuity of 46304,over next 20 yrs.

using PV of annuity formula

PVOA=46304*(1-(1+0.09)^-20)/0.09

422688

d. Difference in PVs (at end of 30 yrs.)=

422688-402558=

20130

Increase in savings for the last 4 years of working life=20130/4=

5032.5