Course 3 - Spring 95 exam question from CSM

[Agtuary]

The following is an email message sent to the course 3 study group. I would appreciate any help in answering it.


Hi

Could somebody please solve the following exam question for me? (1995Spring)
from CSM Manual, Section 7: Question H18:

For a special fuly discrete 20-year endowment insurance on (68), you are
given:

i = 0.05
1000P68 (i.e. premium 68) = 55.62
annuity due:68:20 = 9.351599
annuity due:85:3 = 2.521820
annuity due: 68 = 9.6816
annuity due: 70 = 9.07586

A68:20 = 554.69

Calculate reserve at end of year 2: i.e. 2V.

Answers: A 63 B 71 C84 D95 E105

__________________________
In the solutions: they've done simply: Reserve = 1 - (annuity due:70/annuity
due:68) = 63
But isn't this the formula for reserve for life insurance?

[Elisha]

I'll try and get back to you within a couple of days on this one.

[Toonces]

Saucywench, I looked up #22 on the Chapter 8 Batten handout. What you did not include in the post at that this is a "special" endowment insurance, where the premium is 1000P68 for all but the 5th year and the death benefit is 1000 for all but the 16th year. Given that, think of the reserve in the retrospective form; i.e. the accumulated value of premiums (1000P68*s(due)2) minus the accumulated value of "satisfaction" (1000*2k68). In this way, the reserve is exactly equal to what the reserve would be for a whole life policy of 1000. This is only true in this case because for the 1st 4 years, the premium is based on the corresponding whole life policy and benefits are exactly the same as for a whole life policy, so the first 4 reserves will be identical. Therefore, you can use the "Ch. 7 annuity formula" for whole life reserves.

[Elisha]

Thanks Toonces. I hadn't looked at that closely yet either, but that helps.

[Agtuary]

How do you get a(due)68, which isn't given in the original problem?

[Mopus]

See Bowers et. al. Page 218.

For an end ins, the formula kV = 1 - a_(x+k)/a_x is valid.

In the given problem, the answer should be 1000(1 - 9.07586/9.68616), or 63.0074

[Agtuary]

See Bowers et. al. Page 218.

For an end ins, the formula kV = 1 - a_(x+k)/a_x is valid.


You and I have different interpretations of what that page says. Here is how I read the bottom half of the page:

Fully discrete whole life benefit reserves are analogous to fully continuous whole life benefit reserves where equations (7.4.12) - (7.4.14) correspond to (7.3.8) - (7.3.10).

Endowment insurance benefit reserves also have an anologous relationship between fully discrete and fully continous with "similar special formulas". The similar special formulas substitute whole life with temporary annuities or endowment insurance or premiums, but does not imply that whole life = endowment.

I think Toonces explanation of the first 4 reserves being exactly equal because of special circumstances specific to this problem is good. But I do not believe it holds in the general case as samiam implies.

I still am unable to calculate a(due)68 from the information given in the actual problem, not the one posted above. The actual problem provides the following info:

1000P68=55.62, a(due)68:20=9.351599, 1000A68:20=554.69,a(due)85:3=2.521820, a(due)70=9.07586, a(due)77=6.97306, 5p68=.84356, 15p68=.42325, q73=.04330, i=.05.

Benefit premiums are 1000P68 for all premiums except the 6th and death benefits are 1000 except for the 16th year.

[Toonces]

Samiam, the formulas on 218 are for whole life. It says that similar formulas hold for endowment insurance, but not exactly the same, i.e. (1 - a_x+k:n-k/a_x:n). The only reason you can use the whole life formula here is because this is a "special" case of endowment insurance where the premiums (for the first 4 years) are defined to be equal to the corresponding whole life premium.

Saucywench, I was confused about a(due)68 myself, but you can calculate it from P68 and the interest rate, i.e. P68 = 1/a(due)68 - d or a(due)68 = 1/(P68 + d) = 9.686

[Mopus]

You're right. Sorry guys, I was being sloppy with my notation when I worked it out( omitting the :n's for convenience). I'm still not awake yet, so take this with a grain, but deriving things I get

kV = 1 - a_(x+k:n-k)/a_(x:n) with the .. on the a's.