Question

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)?

Answer 1

Answer and Explanation:

The computation is shown below:

Required Beginning account balance of 21st year:

= $100,000 ÷ 10%

= $1,000,000

a) The annual contribution is

But before that we need to do the following calculations

Future value = Present value × (1 + interest rate)^number of years

= $50,000 × (1 + 10%)^20

= $336,375

Now

Extra future value needed

= $1,000,000-$336,375

= $663,625

Now

Future value of annuity = P×[(1+interest rate)^number of years - 1 ] ÷ interest rate

$663,625 = P × [(1+10%)^20-1] ÷ 10%

Annual contribution P = $11,586.64

b)

Future value of $30,000:

Future value = Present value × (1 + interest rate)^number of years

= $30,000 × (1+10%)^20

= $201,825

Extra future value needed

= $1,000,000 - $201,825

= $798,175

Now

Future value of annuity = Present value × [(1 + interest rate)^number of years -1]÷interest rate

$798,175 = P×[(1+10%)^20-1]÷10%

P = $13,935.84