Assume that you just had a child, and you are now planning for her college education.You would like to make 43 equal payments over the next 21 years (the first payment to be made immediately, all other payments to be made at 6-month intervals, with the final payment to be made at her 21st birthday) so that you will be able to cover her expected expenses while in school.You expect to pay expenses on her 18th, 19th, 20th, and 21st birthdays.Assume that the current (time period 0) annual cost of college is R6,000, that you expect annual inflation to be 8 percent for the next 5 years, and then 5 percent thereafter.If you expect to be able to earn a return of 4 percent every 6 months on your investments (a simple rate of 8 percent with semi-annual compounding), what will be the amount of each of the 43 payments?

Question

Quizplus · Accepted Answer

First, we need to calculate the future cost of college for each year. Using the given inflation rates, we can calculate the future costs as follows:

Year 1: R6,000 * (1 + 8%) = R6,480
Year 2: R6,480 * (1 + 8%) = R7,022.40
Year 3: R7,022.40 * (1 + 8%) = R7,618.17
Year 4: R7,618.17 * (1 + 8%) = R8,269.97
Year 5: R8,269.97 * (1 + 8%) = R8,980.97
Year 6: R8,980.97 * (1 + 5%) = R9,429.02
Year 7: R9,429.02 * (1 + 5%) = R9,900.47
Year 8: R9,900.47 * (1 + 5%) = R10,395.49
Year 9: R10,395.49 * (1 + 5%) = R10,914.28
Year 10: R10,914.28 * (1 + 5%) = R11,457.06
Year 11: R11,457.06 * (1 + 5%) = R12,024.07
Year 12: R12,024.07 * (1 + 5%) = R12,615.55
Year 13: R12,615.55 * (1 + 5%) = R13,231.74
Year 14: R13,231.74 * (1 + 5%) = R13,872.9
Year 15: R13,872.9 * (1 + 5%) = R14,539.28
Year 16: R14,539.28 * (1 + 5%) = R15,231.13
Year 17: R15,231.13 * (1 + 5%) = R15,948.72
Year 18: R15,948.72 * (1 + 5%) = R16,692.29
Year 19: R16,692.29 * (1 + 5%) = R17,462.1
Year 20: R17,462.1 * (1 + 5%) = R18,258.4
Year 21: R18,258.4 * (1 + 5%) = R19,081.45

Next, we need to calculate the present value of these future costs, using a semi-annual discount rate of 4%. We have 43 payments over 21 years, so each payment is made every 6 months for a total of 42 payments, plus a final payment made at the end of the 21st year. Using the present value formula for an annuity, we can calculate the amount of each payment:

PV = Payment * (1 - (1 + r)^-n) / r

where PV is the present value of the future college expenses, Payment is the amount of each equal payment we need to make to cover those expenses, r is the semi-annual discount rate (4%), and n is the number of payments (42).

Plugging in the numbers, we get:

PV = R6,000 + R6,480 / (1 + 4%)^1 + R7,022.40 / (1 + 4%)^2 + ... + R19,081.45 / (1 + 4%)^42

PV = R195,557.35

Now we can solve for Payment:

R195,557.35 = Payment * (1 - (1 + 4%)^-42) / 4%

Payment = R731.93

So the amount of each of the 43 payments will be R731.93. Therefore, the best choice is B.

Assume That You Just Had a Child, and You Are