Mr. Moore is 35 years old today and is beginning to plan for his retirement. He wants to set aside an equal amount at the end of each of the next 25 years so that he can retire at age 60. He expects to live to the maximum age of 80 and wants to be able to withdraw $25,000 per year from the account on his 61st through 80th birthdays. The account is expected to earn 10 percent per annum for the entire period of time. Determine the size of the annual deposits that must be made by Mr. Moore.
A) $212,850
B) $23,449
C) $2,164
D) $8,514

Question

Quizplus · Accepted Answer

To solve this, we use the concept of the time value of money, specifically the future value of an annuity (for the savings phase) and the present value of an annuity (for the withdrawal phase). First, calculate the present value of the withdrawals he plans to make in retirement, then determine the annual deposit required to reach this amount.1. Calculate the present value (PV) of the retirement withdrawals at age 60:$$ PV = \frac{PMT}{i} \left(1 - \frac{1}{(1+i)^n}\right) $$Where $PMT = \$25,000$, $i = 0.10$ (10% annual interest), and $n = 20$ years (from age 60 to 80).$$ PV = \frac{\$25,000}{0.10} \left(1 - \frac{1}{(1+0.10)^{20}}\right) = \$25,000 \times 8.5136 = \$212,840 $$(Note: The exact number might slightly vary due to rounding in the factor used.)2. Calculate the annual deposit (AD) required to accumulate this PV by his 60th birthday:$$ AD = \frac{PV \times i}{(1+i)^n - 1} $$Where $PV = \$212,840$ (rounded from the first calculation), $i = 0.10$, and $n = 25$ years (from now until age 60).$$ AD = \frac{\$212,840 \times 0.10}{(1+0.10)^{25} - 1} = \$2,164 $$This calculation shows that Mr. Moore needs to deposit approximately \$2,164 each year to meet his retirement goals.

Mr Moore Is 35 Years Old Today and Is Beginning