Completely confused about a Sinking Fund Question in Kellison

[CalloftheBlade]

Kellison 6.4 #25

The question reads: "A borrows $12,000 for 10 years and agrees to make semiannual payments of $1000. The lender receives 12% convertible semiannually on the investment each year for the first 5 years and 10% convertible semiannually for the second 5 years. The balance of each payment is invested in a sinking fund earning 8% convertible semiannually. Find the amount by which the sinking fund is short of repaying the loan at the end of the 10 years. Answer to the nearest dollar." The answer is.. $2221. I do not understand this problem at all, why does the question say that the borrower is making semiannual payments if he is investing in a sinking fund... doesn't that defeat the whole purpose of making a sinking fund in the first place? I am also having trouble distinguishing between a lot of the terminology... payment, principle, balance, etc. Any help is appreciated.

[Ken]

I think your FV for both the loan and (sinking fund + extra payment) are supposed to be equal. The loan's accumulated at some rate for 5 years and another for 5 years. Your sinking fund is your $1000 payments accumulated for 10 years.

[krzysio]

Think of the sinking fund as a separate account, where money accumulates at a different rate than the rate applicable outside of that account. Just out there, and it has its own interest rate, and that should always help.
Yours,
Krzys'

[CalloftheBlade]

Ok I think I understand now, so there are 20 payment periods, between periods 1 and 10 the interest of the loan is 720, and between periods 11 and 20 the interest payment is 600. So the principle for the sinking fund from periods 1 to 10 is 280, and from periods 11 to 20 is 400. So I found the future value of the sinking fund by the equation 280 S(10) (1.04)^10 + 400 S(10), which equals 9778.594856, and since the loan value remains the same, I take the future value of the sinking fund and subtract it from 12000 (the loan amount) to arrive at the value the sinking fund is short, which is 2221.