Question

You are chairperson of the investment fund for the Continental Soccer League. You are asked to set up a fund of semiannual payments to be compounded semiannually to accumulate a sum of $220,000 after nine years at a 12 percent annual rate (18 payments). The first payment into the fund is to take place six months from today, and the last payment is to take place at the end of the ninth year. Use Appendix A and Appendix C for an approximate answer, but calculate your final answer using the formula and financial calculator methods.

a. Determine how much the semiannual payment should be. (Do not round intermediate calculations. Round your final answer to 2 decimal places.)

On the day after the sixth payment is made (the beginning of the fourth year), the interest rate goes up to an annual rate of 14 percent. This new rate applies to the funds that have been accumulated as well as all future payments into the fund. Interest is to be compounded semiannually on all funds.

     
b. Determine how much the revised semiannual payments should be after this rate change (there are 12 payments and compounding dates). The next payment will be in the middle of the fourth year. (Do not round intermediate calculations. Round your final answer to 2 decimal places.)
  

Answer 1

Compute the semi-annual interest rate, using the equation as shown below:

Semi-annual rate = Annual rate/ 2

                            = 12%/ 2

                            = 6%

Hence, the semi-annual interest rate is 6%.

Compute the present value of future payments, using the equation as shown below:

Present value = Future value/ (1 + Rate)^Time

                      = $220,000/ (1 + 0.06)^18

                      = $220,000/ 2.85433915282

                      = $77,075.6340509

Hence, the present value of future payments is $77,075.6340509.

Compute the present value annuity factor (PVIFA), using the equation as shown below:

Hence, the present value annuity factor is 10.8276034802.

(a)

Compute the semi-annual payment, using the equation as shown below:

Hence, the semi-annual payment is $7,118.44.

(b)

Compute the value of deposits after 6 payments, using the equation as shown below:

Hence, the value of deposits after 6 payments is $49,653.3789731.

Compute the semi-annual interest rate after 6 payments, using the equation as shown below:

Semi-annual rate = Annual rate/ 2

                            = 14%/ 2

                            = 7%

Hence, the semi-annual interest rate after 6 payments is 7%.

Compute the value of accumulated sum at the end of 6^th payment, using the equation as shown below:

Present value = Future value/ (1 + Rate)^Time

                      = $220,000/ (1 + 0.07)^12

                      = $220,000/ 2.2521915889

                      = $97,682.6310355

Hence, the value of accumulated sum at the end of 6^th payment is $97,682.6310355.

Compute the present value annuity factor (PVIFA), using the equation as shown below:

Hence, the present value annuity factor is 7.94268629626.

Compute the semi-annual payment after 6 payments, using the equation as shown below:

Hence, the semi-annual payment after 6 payments is $6,046.98.