Actuarial Outpost Term Insurance Question
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 Long-Term Actuarial Math Old Exam MLC Forum

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#1
01-26-2019, 11:20 PM
 SweepingRocks Member SOA Join Date: Jun 2017 College: Bentley University (Class of 2019ish) Posts: 174
Term Insurance Question

https://imgur.com/a/wtvpRHh

So that's a link to a question I'm stuck on. I tried using first principles instead of using the shortcut, since the shortcut isn't that intuitive for me. I can understand and calculate the pure endowment part, but for the term insurance this is what I did:

tP45=1-(t/58) and to find the force of mortality between 45 and 65, I took 20P45=.65517=e^(20ux). Then solved for ux as .02114. Then I integrated between 0 and 20 for (1-(t/58))*.02114*(e^.06t), or tP45*ux*v^t.

I ended up with an answer of .28037 for the 20 term insurance, while the solution has .200806. Can someone provide clarity on what I did wrong or why my approach may not work? Thank you in advance!
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#2
01-27-2019, 12:08 AM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,720

You're acting as though mu is a constant, but it's not a constant when you have a uniform distribution on (0, 58) for the future lifetime T_{45}. If you want to use basic principles, note that the density function for T_{45} is the constant 1/58.

Quote:
 Originally Posted by SweepingRocks https://imgur.com/a/wtvpRHh So that's a link to a question I'm stuck on. I tried using first principles instead of using the shortcut, since the shortcut isn't that intuitive for me. I can understand and calculate the pure endowment part, but for the term insurance this is what I did: tP45=1-(t/58) and to find the force of mortality between 45 and 65, I took 20P45=.65517=e^(20ux). Then solved for ux as .02114. Then I integrated between 0 and 20 for (1-(t/58))*.02114*(e^.06t), or tP45*ux*v^t. I ended up with an answer of .28037 for the 20 term insurance, while the solution has .200806. Can someone provide clarity on what I did wrong or why my approach may not work? Thank you in advance!
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#3
01-27-2019, 12:13 AM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 5,139

The shortcut:

tp45 = l(45+t)/l(45); mu(45+t) =-dl(45+t)/l(45+t)

Multiply and cancel l(45+t) which leaves -dl(45+t)/l(45) = -(-1)/(103-45) = 1/(103-45)
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#4
01-27-2019, 10:22 AM
 SweepingRocks Member SOA Join Date: Jun 2017 College: Bentley University (Class of 2019ish) Posts: 174

Quote:
 Originally Posted by Breadmaker The shortcut: tp45 = l(45+t)/l(45); mu(45+t) =-dl(45+t)/l(45+t) Multiply and cancel l(45+t) which leaves -dl(45+t)/l(45) = -(-1)/(103-45) = 1/(103-45)
Dumb question that I should know the answer to: does that mu equation apply all the time or just when deaths are uniform?
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#5
01-27-2019, 11:42 AM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 5,139

Applies all the time.
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