Assume that your father is now 50 years old, that he plans to retire in 10 years
ID: 2613637 • Letter: A
Question
Assume that your father is now 50 years old, that he plans to retire in 10 years, and that he expects to live for 25 years after he retires - that is, until he is 85. He wants his first retirement payment to have the same purchasing power at the time he retires as $60,000 has today. He wants all his subsequent retirement payments to be equal to his first retirement payment. (Do not let the retirement payments grow with inflation: Your father realizes that the real value of his retirement income will decline year by year after he retires). His retirement income will begin the day he retires, 10 years from today, and he will then get 24 additional annual payments. Inflation is expected to be 6% per year from today forward. He currently has $100,000 saved up; and he expects to earn a return on his savings of 10% per year with annual compounding. To the nearest dollar, how much must he save during each of the next 10 years (with equal deposits being made at the end of each year, beginning a year from today) to meet his retirement goal? (Note: Neither the amount he saves nor the amount he withdraws upon retirement is a growing annuity.)
Explanation / Answer
Answer:
The annuity that he expectes to have at retirement is the future value of $60,000, growing at the rate of inflation, so that he can have same purchasing power as today.
= $60,000(1+0.06)10 = $107,450.86 is the annuity.
So Present value of the annuity that he will receive at the end 10th year (Annuity due - received at from the date of retirement - beginning of the year)
= Annuity * [{1-(1+i)-n}/i]*(1+i) where i = interest rate, and n = no of periods
= $107,450.86 * [{1-(1+0.1)-25/0.1} * (1+0.1) = $107,450.86 * 9.077 * 1.1 = $1,072,869.33
He need to accumulate the above amount at the time of retirement to be able fund is annuity after retirement.
So in order to accumulate $1,072,869.33 at the time of retirement, he should deposit an annuity. Let that annuity is "X".
So Now the future value annuity regular (deposits amde at the end of each year)
=> $1,072,869.33 = X {(1+i)n-1}/i => $1,072,869.33 = X {(1+0.1)10 -1}/0.1
=> $1,072,869.33 = X * 15.9374 =>X = $1,072,869.33/15.9374 = $63.318(ans)