An annuity immediate pays an initial benefit of 1 per year, increasing by 10.25% every four years. The annuity is payable for 40 years. Using an annual effective rate of 5%, find the present value of this annuity.

• 21.5
• 22.3
• 23.8
• 24.1
• 24.6

References

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

Solution: C

Find accumulated amount of groups of 4 payments at years [math]4,8,12,16, \ldots, 40[/math]. Call them [math]K_1, K_2, \ldots, K_{10}[/math]. Then find PV. [math]\left\{K_1, \ldots, K_{10}\right\}=\left\{s_{\overline{4} \mid .05}, 1.1025 s_{\overline{4} \mid .05}, \ldots, 1.1025^9 s_{\overline{4} \mid .05}\right\}[/math]. Note that [math]1.05^2=1.1025[/math]. Thus

[[math]] \begin{aligned} & P V=K_1 1.05^{-4}+K_2 1.05^{-8}+\cdots=K_1 1.1025^{-2}+K_2 1.1025^{-4}+\ldots \\ & =s_{\overline{4} \mid .05}\left(1.1025^{-2}+1.1025^{-3}+\cdots+1.1025^{-11}\right)=s_{\overline{4} \mid .05}1.1025^{-1}\left(1.1025^{-1}+\cdots+1.1025^{-10}\right) \\ & =s_{\overline{4} \mid .05} 1.1025^{-1} a_{\overline{10} \mid .1025}=23.76580 \end{aligned} [[/math]]

References

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