A bank issues a loan to be repaid by level end-of-year payments for ten years at an annual effective interest rate of 10%. The bank invests these payments at an annual effective interest rate of 10% for the first four years and 7% for the next six years.

Calculate the bank’s annual effective yield rate on this loan over the ten-year period.

• 7.90%
• 8.20%
• 8.33%
• 8.67%
• 9.10%

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

Solution: D

Amount of loan = L

Initial expected yield rate = 10.00%

Annual payment = [math]L/a_{\overline{10}|10\%}[/math]

Accumulated value at time 10 = [math]L/a_{\overline{10}|10\%})(S_{\overline{4}|10\%}1.07^6 + s_{6|7\%})[/math]

[[math]] \begin{aligned} \textrm{Yield Rate} = \left(\frac{\textrm{Accum Value}}{L}\right)^{1/10}-1 \\ =\left(\frac{s_{\overline{4}|10\%}+ s_{\overline{6}|7\%}}{a_{\overline{10}|10\%}}\right)^{1/10} - 1 \\ =\left(\frac{4.6410(1.5007)+7.1533}{6.1446}\right)^{1/10}-1 \\ =8.67\% \\ \end{aligned} [[/math]]

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