A loan of 10,000 is taken out on March 1, 1995 at an effective rate of interest of 8% per year. Interest is paid annually, and a sining fund is established to repay the principal on March 1, 2002. Payments are made annually into the fund beginning on March 11,1996 and the fund earns interest at 9% per year.
(a) Find the amount of each payment made to the sinking fund
(b) Find total amount the borrower must pay each year.
(c) From the view of the borrower, what rate of interest is she really paying each year?
How does each payment of sinking found works? So each payment doesn't go to the loan and its pay to the loan all at once in 2002? So the total amount needed to pay to the loan is 10,000*(1.08)^7?
With the sinking fund method, the borrower
Let be $L=10,000$, $n=7$ years, $i=8\%$, $j=9\%$.
The interest paid each year will be $$I=iL=800$$.
To accumulate $L$ at time $n$, each sinking fund installment is $$ {L}=S\,{s_{\overline{n}|j}}={S}\,{\frac{(1+j)^n-1}{j}}\quad\Longrightarrow\quad S=\frac{L}{s_{\overline{n}|j}}\approx 1086.91 $$
Thus the total amount the borrower must pay each year is $$ P=I+S=iL+\frac{L}{s_{\overline{n}|j}}=L\left(i+\frac{1}{s_{\overline{n}|j}}\right)\approx 1886.91 $$
In a traditional amortization schedule for a loan $L=10,000$ and a payment $P\approx 1886.91$ for $n=7$ years we would have an interest rate $r$ such that $$ L=P\,a_{\overline{n}|r}=P\,\frac{1-(1+r)^{-n}}{r}\quad\Longrightarrow r\approx 7.48\%\;\text{(solved numerically)} $$ Thus from the view of the borrower, the rate of interest she really is paying each year is $r\approx 7.48\%<i=8\%$.