Actuarial Outpost Understanding temporary life annuities
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#1
03-21-2014, 04:05 PM
 pdk17 SOA Join Date: Mar 2014 College: Senior College Student Posts: 24
Understanding temporary life annuities

Hey everyone, this is my first post on this website. I'm sure you'll see more of me in the future, and hopefully I'll be able to return the help I get.

Anyway, I'm having a difficult time trying to conceptualize the formula for temporary annuities. The specific formula I'm referring to is slide #7 from this site: http://www.math.binghamton.edu/arcon...c/sect-5-3.pdf

I don't fully understand why there is a second term to this. I understand that the formula is important for mean and variance, and also that it can easily be shown that it's equal to the current payment formula. But where I get tripped up is that when I look at a timeline, it seems like all the possible payments are covered in the summation. So how does it intuitively make sense that there should be a term after the summation? Any help would be appreciated. Thanks.

-pdk17
#2
03-21-2014, 04:37 PM
 BruteForce Member SOA AAA Join Date: Apr 2013 Studying for More Money Favorite beer: Wurzel Bier Posts: 11,721

I don't really agree with the formula, especially after seeing the example on slide 11. I would have calculated $\ddot{a}_{x:3} = 1+vp_x + v^{2} \ast (_{2}p_x)$. Maybe it's the same as the way they're calculating it, it's just not the formula that I'm used to. I've done ASM and TIA and didn't see that formula in either place (not that I can remember at least). I don't know how to do the angle over the 3 in Latex
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Last edited by BruteForce; 03-21-2014 at 04:49 PM..
#3
03-21-2014, 04:40 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,727

Quote:
 Originally Posted by pdk17 Hey everyone, this is my first post on this website. I'm sure you'll see more of me in the future, and hopefully I'll be able to return the help I get. Anyway, I'm having a difficult time trying to conceptualize the formula for temporary annuities. The specific formula I'm referring to is slide #7 from this site: http://www.math.binghamton.edu/arcon...c/sect-5-3.pdf I don't fully understand why there is a second term to this. I understand that the formula is important for mean and variance, and also that it can easily be shown that it's equal to the current payment formula. But where I get tripped up is that when I look at a timeline, it seems like all the possible payments are covered in the summation. So how does it intuitively make sense that there should be a term after the summation? Any help would be appreciated. Thanks. -pdk17
If (x) survives n-1 years then (x) has gotten all n payments. To get just k < n payments, (x) must survive k-1 years and then die the following year.

Jim Daniel
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#4
03-21-2014, 04:58 PM
 pdk17 SOA Join Date: Mar 2014 College: Senior College Student Posts: 24

Quote:
 Originally Posted by BruteForce I don't really agree with the formula, especially after seeing the example on slide 11. I would have calculated $\ddot{a}_{x:3} = 1+vp_x + v^{2} \ast (_{2}p_x)$. Maybe it's the same as the way they're calculating it, it's just not the formula that I'm used to. I've done ASM and TIA and didn't see that formula in either place (not that I can remember at least). I don't know how to do the angle over the 3 in Latex
What you're doing is the current payment formula. It's the one that makes intuitive sense, and stops at n. I like it way better. However, you can't determine the mean and variance from it, which is why we have the other formula (the one I'm trying to make sense of).

Quote:
 Originally Posted by Jim Daniel If (x) survives n-1 years then (x) has gotten all n payments. To get just k < n payments, (x) must survive k-1 years and then die the following year. Jim Daniel
That makes sense. Thank you for your response. However, I think I should have worded my question in a different way.

If (x) survives n-1 years, then all of the payments are covered. Why is it that we need an extra term after that for those that survive n years? Why do they matter to us?

Last edited by pdk17; 03-21-2014 at 05:01 PM.. Reason: To respond to someone else as well
#5
03-21-2014, 05:47 PM
 Phileas Fogg Member SOA Join Date: Dec 2012 Posts: 1,458

Quote:
 Originally Posted by pdk17 That makes sense. Thank you for your response. However, I think I should have worded my question in a different way. If (x) survives n-1 years, then all of the payments are covered. Why is it that we need an extra term after that for those that survive n years? Why do they matter to us?
For this specific example, you don't. However, if you change the formula to that for an annuity-immediate instead of an annuity-due, you will need that "final" term which distinguishes between those who have survived for n-1 years and those who have survived to n.

In the longer formula with the "extra" term you can transition between those two formulae just by replacing a-double-dot with a, which makes it (arguably) more convenient as a general formula.
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#6
03-21-2014, 07:20 PM
 pdk17 SOA Join Date: Mar 2014 College: Senior College Student Posts: 24

Quote:
 Originally Posted by Phileas Fogg For this specific example, you don't. However, if you change the formula to that for an annuity-immediate instead of an annuity-due, you will need that "final" term which distinguishes between those who have survived for n-1 years and those who have survived to n. In the longer formula with the "extra" term you can transition between those two formulae just by replacing a-double-dot with a, which makes it (arguably) more convenient as a general formula.
Thank you!
#7
03-21-2014, 07:56 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,727

Quote:
 Originally Posted by pdk17 That makes sense. Thank you for your response. However, I think I should have worded my question in a different way. If (x) survives n-1 years, then all of the payments are covered. Why is it that we need an extra term after that for those that survive n years? Why do they matter to us?
Consider the first of the two formulas on Slide #7. The last term in the summation treats those who receive payment #n and then die that year, while the term outside the summation treats those who receive that payment and survive throughout that year.

The second formula on Slide #7 just comes from combining the final term in the summation with the term outside the summation.

Jim Daniel
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