At the end of each year for 60 years, Marilyn makes a deposit to a bank account that credits interest at an annual effective interest rate of i. She deposits 2 at the end of the first year, and each subsequent year her deposit increases by 2. At the end of 60 years, Marilyn uses the accumulated amount to purchase a 5-year annuity-immediate paying annually at the same annual interest rate of i, with a first payment of X and each subsequent payment increasing by 5%. Which of the following expressions represents a correct equation of value?

• [[math]]\quad \frac{2\left(s_{60}-60\right)}{i}=\frac{X\left(1-(1.05 v)^5\right)}{i-0.05}[[/math]]
• [[math]]\quad 2(I a)_{\overline{60}}=X v^{60}\left(1+1.05 v+\ldots+(1.05 v)^4\right)[[/math]]
• [[math]]\quad 2 \sum_{t=0}^{59} s_{\overline{60-t}}=X\left(v+1.05 v^2+\ldots+1.05^4 v^5\right)[[/math]]

• I only
• II only
• III only
• I, II, and III
• The correct answer is not given by (A), (B), (C), or (D).

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

Solution: C

Statement I should have an [math]\ddot{s}_{\overline{60}|}[/math] on the left.

Statement II has an annuity-due rather than an annuity-immediate on the right.

Statement III is correct.

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