Joel just won the lottery. He has two options to take the money. He can take the lump sum of $3,000,000 or he can take the level payments of $500,000 over 6 years. If he takes the lump sum, Joel will deposit the money into an account earning i% annually. If Joel takes the payment plan, he will deposit the payments at the end of each year at a compounded interest of 14%.

After 16 years, the accounts will be equal. Calculate i.

• .1095
• .0563
• .1065
• .371
• .022

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

Solution: A

Lump sum: 3,000,000 interest i

Payments: 500,000 over 6 years i = .14

First find the accumulated value of the payment plan for the 16 years

[[math]] \begin{aligned} & 500,000 \mathrm{~s}_{\overline{6}|0.14}(1+.14)^{10} \\ & =500,000\left[\left((1+.14)^6-1\right) / .14\right](1.14)^{10} \\ & =15,821,528.50 \end{aligned} [[/math]]

Set this equal to the lump sum accumulated value:

[[math]] \begin{aligned} & 15,821,528.50=3,000,000(1+i)^{16} \\ & 1.1095=1+i \\ & i=.1095 \end{aligned} [[/math]]

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