A perpetuity-immediate pays 100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8% greater than the preceding payment. Immediately after the 10th payment of the 25-year annuity, the annuity will be exchanged for a perpetuity-immediate paying Y per year. The annual effective rate of interest is 8%.

Calculate Y.

• 110
• 120
• 130
• 140
• 150

References

Hlynka, Myron. "University of Windsor Old Tests 62-392 Theory of Interest". web2.uwindsor.ca. Retrieved November 23, 2023.

Solution: C

At time 5, the value of the perpetuity and annuity must be equal so

[[math]] \begin{gathered} 100 a_{\overline{\infty} \mid}=X v+X(1.08) v^2+\cdots+X(1.08)^{24} v^{25} \text { or } \\ 100 / i=25 X v . \end{gathered} [[/math]]

At time 15, the remaining value of the annuity must equal the value of the perpetuity of [math]Y[/math]. Thus

[[math]] \begin{gathered} X(1.08)^{10} v+X(1.08)^{11} v^2+\cdots+X(1.08)^{24} v^{15}=Y a_{\overline{\infty}} \\ 15 X v(1.08)^{10}=Y / i . \end{gathered} [[/math]]

Thus [math]Y=\frac{15}{25}(1.08)^{10}=129.5355[/math].

References

Hlynka, Myron. "University of Windsor Old Tests 62-392 Theory of Interest". web2.uwindsor.ca. Retrieved November 23, 2023.