Samantha age 40, buys a 20-year certain-and-life annuity due with a single premium. The annuity makes annual payments of $2,000$. If $i = 6\%$ and you are given the following table, find the reserve 5 years after issue. (hint: the benefit is an annuity)
$l_{40} = 39,143.64$
$l_{45} = 23,582.45$
$l_{60} = 400.49$
$\ddot{a}_{40} = 5.90503$
$\ddot{a}_{45} = 4.69803$
$\ddot{a}_{60} = 2.12522$
So far I have: (sorry, some of the notation is difficult to create)
Premium (per 1 dollar) = $\dfrac{A_{40}}{\ddot{a}_{40:\overline{10}|}} = \dfrac{1 - d\ddot{a}_{40}}{\ddot{a}_{40:\overline{10}|}} = \dfrac{1 - (0.06/1.06)(5.90503)}{ ? }$
** I can't seem to figure out what $\ddot{a}_{40:\overline{10}|}$ is?
Furthermore, I know how to finally finish off the question as:
$2000 {}_5 ^{20}$$V_{40}= (2000)(1 - d\ddot{a}_{45}) - (2000)(\text{Premium})(\ddot{a}_{45:\overline{5}|})$
Any help would be greatly appreciated!!
The actuarial present value of the benefits is $$2000\ddot{a}_{\overline{40:\overline{20}|}} = 2000(\ddot{a}_{\overline{20}|}+{}_{20|}\ddot{a}_{40}) = 2000(\ddot{a}_{\overline{20}|}+\ddot{a}_{60}\cdot {}_{20}E_{40})$$ (do you see why?) and assuming the equivalence principle, note that you have a single premium, so letting $P$ be the single premium, $$P = 2000(\ddot{a}_{\overline{20}|}+\ddot{a}_{60}\cdot {}_{20}E_{40})\text{.}$$ I'm assuming you know how to compute these. You should know $\ddot{a}_{\overline{20}|}$ from FM.
The formula you have for the premium is false, since 1) we're not working with a whole-life insurance as the benefit, but rather, an annuity; 2) we only have ONE premium (so $\ddot{a}_{40:\overline{10}|}$ is unnecessary).
The reserve (I like using the simple notation ${}_{5}V$ for this) is $${}_{5}V = 2000\ddot{a}_{\overline{15}|}+{}_{15|}\ddot{a}_{45}=2000\ddot{a}_{\overline{15}|}+\ddot{a}_{60}\cdot {}_{15}E_{45}\text{.}$$ Note here that NO PREMIUM is included. (Why?)