John deposits money into an account that has a payment of $25,000 at the end of 5 years. Sally deposits money into 2 accounts. One has a payment of 4,000 at the end of year t and one has a payment of $17, 000 at the end of year 2t. The sum of Sally’s present value is equal to John’s present value and is equal to a deposit with payment of $7,000 at time 0.

Find the value of the payment $14,000 at the end of year t+4 if all interest rates are equal for all deposits.

• $2,704
• $3,894
• $58,956
• $26,737
• $3,498,106

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

Solution: A

John's deposit: pv [math]=25,000 \mathrm{v}^5[/math] Sally's deposit: [math]\mathrm{pv}=4,000 \mathrm{v}^{\mathrm{t}}[/math]

[[math]] \mathrm{pv}=17,000 \mathrm{v}^{2 \mathrm{t}} [[/math]]

setting them equal to 7.000 :

[[math]] \begin{gathered} 7,000=25,000 \mathrm{v}^5=4,000 \mathrm{v}^{\mathrm{t}}+17,000 \mathrm{v}^{2 \mathrm{t}} \\ \mathrm{v}^5=.28 \end{gathered} [[/math]]

Since we want to find the pv at time equals [math]t+4[/math] of a payment of [math]14,000: p v=14,000 v^{2 t}[/math]

We must then solve the quadratic with [math]x=v^t: 17,000 x^2+4,000 x-7,000=0[/math]

[[math]] \mathrm{X}=.53474=\mathrm{v}^{\mathrm{t}} [[/math]]

Thus,

[[math]] \begin{aligned} \mathrm{pv} & =14,000 \mathrm{v}^{\mathrm{t}+4} \\ & =14,000 \mathrm{v}^{\mathrm{t}} \mathrm{v}^4 \\ & =14,000 \mathrm{v}^{\mathrm{t}} \mathrm{v}^{5(4 / 5)} \\ & =14,000^* .53474 * .28^{4 / 5} \\ \mathrm{pv} & =2,703.94 \end{aligned} [[/math]]

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