Juliana takes out a loan for $200,000 with 25 yearly payments at the end of each year. She makes payments which are twice the interest due for the first 24 months and pays off the remaining balance with the 25th payment. If the interest on the loan is 4%, what is the final payment equal to?

• $72,079.34
• $117,167.27
• $78,085.96
• $75,082.65
• Insufficient information to solve problem

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

Solution: C

[math]\begin{aligned} & \mathrm{OB}_{\mathrm{o}}=200,000 \\ & \mathrm{i}=.04 \\ & \mathrm{OB}_1=\mathrm{OB}_0(1+\mathrm{i})-2 \mathrm{OB}_{\mathrm{o}} \mathrm{i}=\mathrm{OB}_{\mathrm{o}}(1-\mathrm{i}) \\ & \mathrm{OB}_2=\mathrm{OB}_1(1+\mathrm{i})-2 \mathrm{OB}_1 \mathrm{i}=\mathrm{OB}_1(1-\mathrm{i})=\mathrm{OB}_{\mathrm{o}}(1-\mathrm{i})^2 \\ & \mathrm{OB}_3=\mathrm{OB}_2(1+\mathrm{i})-2 \mathrm{OB}_2 \mathrm{i}=\mathrm{OB}_2(1-\mathrm{i})=\mathrm{OB}_{\mathrm{o}}(1-\mathrm{i})^3 \\ & \mathrm{OB}_{\mathrm{t}} =\mathrm{OB}_0(1-\mathrm{i})=\mathrm{OB}_{\mathrm{o}}(1-\mathrm{i})^{\mathrm{t}} \end{aligned}[/math]

After the [math]24^{\text {th }}[/math] payment

[[math]] \begin{aligned} & \mathrm{OB}_{24}=200,000(.96)^{24} \\ & =75,082.65 \end{aligned} [[/math]]

Thus, she will owe

[[math]] \begin{aligned} & \mathrm{OB}_{25}=\mathrm{OB}_{24}(1+\mathrm{i})=75,082.65(1.04) \\ & =78,085.96 \end{aligned} [[/math]]

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