Two deposits are made into a fund: 300 at time 0 and X at time 4. The force of interest for the fund is [math]\delta_t = t/50, \, t \geq 0 [/math]. The amount of interest earned from time 0 to time 10 is 4X.

Calculate X .

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

Solution: D

First, find [math]a(t)[/math]

[[math]] a(t)=\exp \left[\int_0^t \delta_r d r\right]=\exp \left[\int_0^t \frac{r}{50} d r\right]=\exp \left[t^2 / 100\right] [[/math]]

The balance in the account at time 10 is: [math]300 a(10)+X \frac{a(10)}{a(4)}=815.48+2.31637 X[/math] The total interest earned from [math]t=0[/math] to [math]t=10[/math] is:

[[math]]815.48+2.31637 X-(300+X)=4 X \Rightarrow X=192.08[[/math]]

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