Al Corbin is 25 years old today and wishes to accumulate enough money over the next 35 years to provide for a 20-year retirement annuity of $100,000 at the beginning of each year, starting with his 60th birthday. He can save $2,000 at the end of each of the next 10 years and $3,000 each year for the following 10 years. How much must he save each year at the end of years 21 through 35 to obtain his goal? Assume that the average rate of return over the entire period will be 10%.

Question

Quizplus · Accepted Answer

First, we need to calculate the total amount that Al needs to accumulate in 35 years to provide for the retirement annuity. Using the present value of an annuity formula, we get:

PV = PMT x [(1 - (1 + r)^-n) / r]

where PV is the present value of the annuity, PMT is the annual payment, r is the annual interest rate, and n is the number of payments.

In this case, PV = $100,000 x [(1 - (1 + 0.1)^-20) / 0.1] = $1,371,038.16

Next, we need to calculate how much Al will accumulate over the first 20 years of saving. Using the future value of an annuity formula, we get:

FV = PMT x [((1 + r)^n - 1) / r]

For the first 10 years, PMT = $2,000, r = 0.1, and n = 10. Plugging these values into the formula, we get:

FV1 = $2,000 x [((1 + 0.1)^10 - 1) / 0.1] = $39,074.20

For the next 10 years, PMT = $3,000, r = 0.1, and n = 10. Plugging these values into the formula, we get:

FV2 = $3,000 x [((1 + 0.1)^10 - 1) / 0.1] x (1 + 0.1)^10 = $95,946.41

Therefore, the total amount that Al will accumulate over the first 20 years is:

FV1 + FV2 = $39,074.20 + $95,946.41 = $135,020.61

Now, we can use the future value of a single sum formula to calculate how much Al must save each year for the last 15 years to reach his goal:

FV = PV x (1 + r)^n

where FV is the future value of the savings, PV is the present value of the annuity, r is the annual interest rate, and n is the number of periods.

In this case, PV = $1,371,038.16, r = 0.1, and n = 15. Plugging these values into the formula, we get:

FV3 = $1,371,038.16 / (1 + 0.1)^15 = $251,649.22

To find the annual savings needed, we can use the present value of a single sum formula:

PV = FV / (1 + r)^n

where PV is the present value of the savings, FV is the future value of the savings, r is the annual interest rate, and n is the number of periods.

In this case, PV = 0, FV = $251,649.22, r = 0.1, and n = 15. Plugging these values into the formula, we get:

PV = $251,649.22 / (1 + 0.1)^15 = $74,163.29

Therefore, Al needs to save $74,163.29 at the end of each year for the last 15 years to reach his goal. However, since he is also saving $2,000 and $3,000 in the first 20 years, the actual amount he needs to save each year for the last 15 years is:

$74,163.29 - ($135,020.61 / 15) = $12,321.46

Rounded to the nearest dollar, the answer is $12,321, which is choice C.

Al Corbin Is 25 Years Old Today and Wishes to Accumulate