3.Mr. D. plans to retire exactly twenty years from now (t=0), and he would like
ID: 2731459 • Letter: 3
Question
3.Mr. D. plans to retire exactly twenty years from now (t=0), and he would like to have accumulated, by retirement, enough money to enjoy a $100,000 per year retirement income beginning in year 21 and continuing in perpetuity thereafter. So far he has saved up $50,000, all in stocks (that is, at t=0 his pension account contains $50,000).
a) What must his annual contributions be if he is to achieve his goal (assume he makes 20 payments)? On average he expects to earn 10% on his money.
b) The stock market collapses. By the end of the day (it is still t=0)his accumulated wealth has fallen to $30,000. Assuming he still expects on average to earn 10%, how much must he now contribute (assume 20 equal payments)?
Explanation / Answer
Required account balance at the beginning of 21st year:
= $100,000÷10%
= $1,000,000
a)
Future value of $50,000:
Future value = P×(1+r)^n
P is payment
r is interest rate per period
n is number of periods
= $50,000×(1+10%)^20
= $336,375
Additional future value required:
= $1,000,000-$336,375
= $663,625
Future value of annuity = P×[(1+r)^n-1]÷r
r is interest rate per period
P is payment per period
n is number of payments
$663,625 = P×[(1+10%)^20-1]÷10%
Annual contribution, P = $11,586.64
b)
Future value of $30,000:
Future value = P×(1+r)^n
P is payment
r is interest rate per period
n is number of periods
= $30,000×(1+10%)^20
= $201,825
Additional future value required:
= $1,000,000-$201,825
= $798,175
Future value of annuity = P×[(1+r)^n-1]÷r
r is interest rate per period
P is payment per period
n is number of payments
$798,175 = P×[(1+10%)^20-1]÷10%
Annual contribution, P = $13,935.84
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