At an annual effective interest rate of i, i > 0, both of the following annuities have a present value of X:

1. a 20-year annuity-immediate with annual payments of 55
2. a 30-year annuity-immediate with annual payments that pays 30 per year for the first 10 years, 60 per year for the second 10 years, and 90 per year for the final 10 years.

Calculate X.

• 575
• 585
• 595
• 605
• 615

References

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

Solution: A

[math]\begin{aligned} & 55 a_{\overline{20} \mid}=30 a_{\overline{10} \mid}+60 a_{\overline{10} \mid} v^{10}+90 a_{\overline{10} \mid} v^{20} \text { so } \\ & 55\left(1-v^{20}\right)=55\left(1-v^{10}\right)\left(1+v^{10}\right)=30\left(1-v^{10}\right)+60\left(1-v^{10}\right) v^{10}+90\left(1-v^{10}\right) v^{20} \text {. But } v^{10} \neq 1 \text { so } \\ & 55\left(1+v^{10}\right)=30+60 v^{10}+90 v^{20} \text { so } 11(1+x)=6+12 x+18 x^2 \text { where } x=v^{10} \text {. Thus } 18 x^2+x-5=0 \text {. So } \\ & v^{10}=x=\frac{-1 \pm \sqrt{1+4(18)(5)}}{36}=18 / 36=.5 \text { so } i=2^{.1}-1=.072 \text {. } \\ & \text { Thus } P V=55 a_{\overline{20} \mid}=55 \frac{1-v^{20}}{i}=55 \frac{1-.5^2}{.072}=574.725 \\ & \end{aligned}[/math]

References

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