Question

Sam is just starting his career and wants to begin saving for retirement. Sam estimates he will need to withdrawal $8000/month to live comfortably while retired and based on life expectancy this would last for 25 years. If he begins saving in 2020 and makes annual contributions growing at 4%, how much will he need to make his first contribution to reach his retirement goal by 2060? Sam’s retirement investments are expected to earn a 9% return (both before AND after he retires)

A) Find the present value of $8000/month for 25 years (it is NOT = 8000x12x25)

B) Treat your answer to Part A as a future value in 2060, then find its PV

C) Use the answer to Part B as a present value in Sam’s retirement saving annuity equation and solve for its first CF.

Answer 1

a).

Present value of annuity is calculated as C*(1-(1+r)^-n)/r; where C is the periodic cashflow, r is the discount rate per period and n is the number period.

Given 9% per year. So, discount rate per month is calculated as ((1+9%)^(1/12))-1= 0.7207%.

So, Present value= 8000*(1-(1+0.7207%)^-(25*12))/0.7207%= 981259.9

b).

PV in 2020 of Future value of 981259.9 in 2060 is calculated as 981259.9/(1+9%)^40= $31240.94

c).

Present value of growing annuity is calculated using the formula: (P/(r-g))*(1-((1+g)/(1+r))^n), where P is first payment, r is interest rate, g is growth rate and n is number of years.

So, 31240.94= (P/(9%-4%))*(1-((1.04/1.09)^40))

31240.94= (P/5%)*(0.847147)

P= (31240.94/0.847147)*5%

P= 1843.89.

So, the first contribution should be $1843.89