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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!!

Clarinetist
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queence
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1 Answers1

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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?)

Clarinetist
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  • is it because there is only one single premium? Also, I don't have what a¨15 is in the last equation – queence Mar 07 '16 at 17:31
  • @queence There is no premium after time 5. Note that under the prospective method for reserves, we're looking at FUTURE premiums which will be subtracted. The $\ddot{a}_{15}$ is a typo. (See edit.) – Clarinetist Mar 07 '16 at 17:36
  • yes right, that makes sense. alright thanks a lot! – queence Mar 07 '16 at 18:08