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#1
04-18-2011, 12:53 PM
 Elsaball Member CAS Join Date: Jun 2010 Studying for Exam 8 Posts: 999
Multiple Life DML

Does anyone know of any shortcut to calculate multiple life insurance or annuities using DML (without integrating)?

Thanks
#2
04-18-2011, 01:40 PM
 Bballry1234 Member SOA Join Date: Sep 2010 Location: Your mom's bed Studying for NOTHING! Favorite beer: Does Redbull count? Posts: 761

Quote:
 Originally Posted by Elsaball Does anyone know of any shortcut to calculate multiple life insurance or annuities using DML (without integrating)? Thanks
...cheat and bring a TI 83 to the exam... haha j/k, but seriously, if an exam question came up, they would probably give you a ux=uy situation where you just add up the alphas to get to a nice Pxy.
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#3
04-18-2011, 02:20 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College 1962 (!) Posts: 1,756

With joint DML statuses, an insurance APV is simpler–––one integration by parts rather than two–––to evaluate as an integral than is an annuity APV. Both are easy should the interest rate i = 0.

Jim Daniel
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#4
04-18-2011, 04:41 PM
 Bama Gambler James Washer / Notes Contributor SOA Join Date: Jan 2002 Location: B'ham, AL Posts: 16,178

Do you know this formula under DML?

$\bar{A}_{x} = \frac{\bar{a}_{\overline{\omega - x}|}}{\omega - x}$

If we add a contingency on (x) dying before (y) then:

$\bar{A}_{\overset{1}{x}y} = \frac{\bar{a}_{y:\overline{\omega - x}|}}{\omega - x}$

This make sense right? Because if the death benefit is contingent on (x) dying before (y) then make my annuity contingent on y being alive. The proof of the formula is not hard (you can do it by setting up the integral and then simplifying).

Now we can write the joint status in terms of two contingent insurances:

$\bar{A}_{xy} = \bar{A}_{\overset{1}{x}y} + \bar{A}_{x\overset{1}{y}}$

Does that help?
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#5
04-18-2011, 04:43 PM
 Bama Gambler James Washer / Notes Contributor SOA Join Date: Jan 2002 Location: B'ham, AL Posts: 16,178

Oh and then once you have the insurance factor you can easily find the annuity factor:

$\bar{a}_{xy} = \frac{1 - \bar{A}_{xy}}{\delta}$
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#6
04-18-2011, 04:49 PM
 Bama Gambler James Washer / Notes Contributor SOA Join Date: Jan 2002 Location: B'ham, AL Posts: 16,178

Sorry for the triple post. Here is the proof:

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#7
04-19-2011, 02:09 PM
 Bballry1234 Member SOA Join Date: Sep 2010 Location: Your mom's bed Studying for NOTHING! Favorite beer: Does Redbull count? Posts: 761

Quote:
 Originally Posted by Bama Gambler Do you know this formula under DML? $\bar{A}_{x} = \frac{\bar{a}_{\overline{\omega - x}|}}{\omega - x}$ If we add a contingency on (x) dying before (y) then: $\bar{A}_{\overset{1}{x}y} = \frac{\bar{a}_{y:\overline{\omega - x}|}}{\omega - x}$ This make sense right? Because if the death benefit is contingent on (x) dying before (y) then make my annuity contingent on y being alive. The proof of the formula is not hard (you can do it by setting up the integral and then simplifying). Now we can write the joint status in terms of two contingent insurances: $\bar{A}_{xy} = \bar{A}_{\overset{1}{x}y} + \bar{A}_{x\overset{1}{y}}$ Does that help?
If w-x = 50 with alpha1 = 2 and w-y =26 and alpha2 = 1/3, that formula is not going to help you much on an exam... it would still be too much for an exam question IMHO.
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