Bank A lends a certain sum of money at an annual effective interest rate of 7%. The loan is to be repaid by 16 annual payments of 1000, with the first payment due after one year. Immediately after receiving the 8 th payment, Bank A sells the right to receive the remaining 8 payments to Bank B. Bank A’s yield on the entire transaction is an annual effective interest rate of 6%.

Calculate the amount that Bank B paid to assume the loan.

• 5159
• 5541
• 5655
• 5971
• 6210

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

Let K be the amount that Bank B paid. Equating the amount borrowed (the 16 payments discounted at 7%) to the actual payments received (using the 6% yield rate) gives the equation

[[math]] 1000 a_{\overline{1677 \%}}=1000 a_{\overline{8} | 6 \%}+\frac{K}{1.06^8} . [[/math]]

Then,

[[math]] \begin{aligned} K & =1000\left(a_{\overline{16} |7 \%}-a_{\overline{8} \mid 6 \%}\right) 1.06^8 \\ & =1000(9.44665-6.20979) 1.06^8 \\ & =3236.86(1.59385)=5159.06 . \end{aligned} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.