On January 1, 2010, Toni deposits $140 into an account. On June 1, 2010, when the amount in Toni’s account is equal to $X, a withdrawal W is made. No further deposits or withdrawals are made to Toni’s account for the remainder of the year. On December 31, 2010, the amount in Toni’s account is $100. The dollar-weighted return over the period is 15%. The time-weighted return over the 1-year period is 11%.

Calculate X.

• 123.81
• 107.91
• 98.15
• 126.73
• 172.02

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

Solution: A

Initial deposit = 140, Withdrawal on June 1: W

Final Amount= 100

Dollar-Weighted= 15%

Time-Weighted= 11%

Dollar-Weighted:

[[math]] \begin{aligned} 140(1.15) – W[1+7/12*.15]=100 \\ W[1+7/12*.15]=61 \\ W= 56.09 \end{aligned} [[/math]]

Time-Weighted:

[[math]] \begin{aligned} (X/140)(85/(X-56.09))-1=.11 \\ X= 123.81 \end{aligned} [[/math]]

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