Question

This is a challenging question. Please allocate time wisely.) Cipro Milda is planning a college fund for her child. When the child joins college in 15 years, the tuition payment for the first semester is expected to be $21,000. Thereafter, tuition payments will be due every six months for the following semester. These payments are expected to increase at an annual inflation rate of 6 percent. The child will graduate in eight semesters, so she will have to make a total of eight tuition payments. If she can earn 10 percent annual rate compounded monthly, how much will she have to deposit monthly in the college fund so that the balance in the fund is zero when the last tuition payment is made? She plans to continue making these deposits until the last tuition bill is paid.

Answer 1

Compute the effective annual rate (EAR), using the equation as shown below:

EAR = (1 + Rate/ Compounding period) ^{Compounding period} – 1

         = (1 + 0.10/12)¹² – 1

         = 10.471307%

Hence, the EAR is 10.471307%.

The EAR is 10.471307% after dividing with the semi-annual rate comes out to be 5.2356535%.

Compute the value of payments after 15 years, using the equation as shown below:

Value of annuity = {Payment/ (Required rate – Growth rate)}*{1 – [(1 + Growth rate)/ (1 + Required rate)]^{Number of periods}}

                           = {$21,000/ (5.2356535% – 3%)}*{1 – [(1 + 0.03)/ (1 + 0.052356535)]⁸}

                            = ($21,000/2.2356535%)*(1 – 0.84215994018)

                            = $939,322.663373*0.15784005982

                            = $148,262.745377

Hence, the value of annuity after 15 years is $148,262.745377.

The annual rate of interest is 10% after dividing the annual rate with 12 months, the monthly rate comes out to be 0.8333333%

Compute the PVIF at 0.8333333% and 15 years, using the equation as shown below:

PVIF = 1/ (1 + Rate)^{Number of periods}

              = 1/ (1 + 0.008333333)¹⁸⁰

         = 0.2245

Hence, the PVIF at 0.8333333% and 15 years is 0.2245.

Compute the PVIFA at 0.8333333% and 15 years, using the equation as shown below:

PVIFA = {1 – (1 + Rate)^{-Number of periods}}/ Rate

                   = {1 – (1 + 0.008333333)^-180}/ 0.8333333%

            = 93.0574

Hence, the PVIFA at 0.8333333% and 15 years is 93.0574.

Compute the monthly deposit, using the equation as shown below:

Monthly deposit = Annuity value after 15 years*PVIF_{0.8333333%, 180}/ PVIFA_{0.8333333%, 180}

                           = $148,262.745377*0.2245/93.0574

                           = $357.682315829

Hence, the monthly deposit amount in college fund is $357.682315829