Brian purchases a 7 year annuity with payments at the end of every quarter for $X. The first payment is $350 and each subsequent payment is $50 more.

How much did Brian pay for the annuity if the interest was 14% convertible quarterly?

• $75,990.43
• $16,155.86
• $50,816.33
• $16,721.00
• $1,982.40

Hardiek, Aaron (June 2010). "Study Questions for Actuarial Exam 2/FM". digitalcommons.calpoly.edu. Retrieved November 20, 2023.

Solution: B

The first thing we must do is recognize the arithmetic pattern which we must separate from the other payments.

Thus the annuity payments are: 300, 300, 300, . . .

And the increasing annuity pay is 50, 100, 150, . . .

Thus the present value would be:

[[math]] \begin{aligned} & \mathrm{pv}=300 \mathrm{a}_{\overline{28} |.035}+50\left(\mathrm{I}_{\mathrm{a}}\right)_{\overline{28} | .035} \\ & =300\left(1-\mathrm{v}^{28} / .035\right)+50\left(\ddot{\mathrm{a}}_{\overline{28} | .035}-28 \mathrm{v}^{28} / .035\right) \\ & =300(17.667)+50(217.1155) \\ & =16,155.55\end{aligned} [[/math]]

Hardiek, Aaron (June 2010). "Study Questions for Actuarial Exam 2/FM". digitalcommons.calpoly.edu. Retrieved November 20, 2023.