A loan of $4000 is to be repaid over a period of 8 years. During the first years, exactly half of the loan principal is to be repaid (along with accumulated compound interest) by a uniform series of payments A1 dollar per year. The other half of the loan principal is to be repaid over four years with accumulated interest by a uniform series of A2 dollar per year. If interest rate=9% per year what are A1 and A2?
In the book answers are:
A1=797.47 A2=617.4
Let be $L_1=L_2=2000,\,L=L_1+L_2=4000, \, i=9\%,\,n_1=n_2=4,\, n=n_1+n_2=8$.
The loan $L$ during the first $n_1$ years will produce the amount $S_1=L(1+i)^{n_1}=5643.33$ and the interest over the first $n_1$ years is $I_1=S_1-L=1643.33$.
So at the end of the $n_1$-th year we have to repay the value $FV_1=L_1+I_1=3643.33$ with a uniform stream of payments $A_1$ such that $$ FV_1(i,n_1,A_1) = A_1 \times s_{\overline{n_1}\!|i} \quad \text{where}\quad s_{\overline{n_1}\!|i} =\frac{(1+i)^{n_1}-1}{i} $$ and so we find $A_1=797.34$.
Over the $n_2$ years we have to repay the loan $L_2=PV_2(i,n_2,A_2)$ as the present value of a stream of payments satisfying the annuity relation $$ PV_2(i,n_1,A_1) = A_2 \times a_{\overline{n_2}\!|i} \quad \text{where}\quad a_{\overline{n_2}\!|i} =\frac{1-(1+i)^{-n_2}}{i} $$ that is $A_2=617.34$.