Actuarial Outpost SOA #158
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#1
05-10-2009, 05:35 PM
 its_me Member Join Date: Jan 2003 Location: AO Posts: 1,772
SOA #158

Can someone please walk me through the solution to this problem? I can't why we subtract what we subtract. I reason ( i drew a graph for this problem) we subtract (10) v^10 A_{x+10:1}..the term insurance symbol here.
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#2
05-10-2009, 05:46 PM
 Jo.M. Member Join Date: Oct 2007 Posts: 288

The only thing missing from your reasoning is that you have to take into account the probability of surviving to age 50. Thus, you need to ''discount'' A_{x+10:1} using $_{10}p_{40}$
#3
05-10-2009, 05:47 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 26,466

(This is in response to the original post, not adjusted for Jo.M.'s post) You need to formulate your question better, especially why you use x+10 in your post instead of putting a value like 50 or 51.

The single premium at 40 is the actuarial present value at 40 of some death benefits. The single premium at 41 is the actuarial present value at 41 of some other benefits. Year by year, how much different are those death benefits? E.g., how much more death benefits at age 40 are being valued in the single premium at 40 than at 41? How much more at 41? At 42, ... at 50? (where the answer to "at 50" will be negative)?

So you need to adjust to get the same benefits being valued to match the age 40 coverage, then you need to adjust the discounting to be at age 40.
#4
05-10-2009, 06:09 PM
 its_me Member Join Date: Jan 2003 Location: AO Posts: 1,772

i used x+2 bcos i drew a graph starting 0..and named them generally.

so, doesnt the term become 10.(v^10)(10_p_40) ( v q_50)?
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#5
05-10-2009, 06:16 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 26,466

Quote:
 Originally Posted by its_me i used x+2 bcos i drew a graph starting 0..and named them generally. so, doesnt the term become 10.(v^10)(10_p_40) ( v q_50)?
That could, but does not necessarily, help. What are you subtracting it from?
#6
05-10-2009, 06:18 PM
 its_me Member Join Date: Jan 2003 Location: AO Posts: 1,772

i subract it from A_x:10 (term insur..the bottom rectangle in the graph) + (IA)_x+1:10 ( term incg insur)
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#7
05-10-2009, 06:21 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 26,466

Quote:
 Originally Posted by its_me i subract it from A_x:10 (term insur..the bottom rectangle in the graph) + (IA)_x+1:10 ( term incg insur)
OK, then maybe your only problem is that (IA)_x_1:10 is an actuarial present value at 41 and you are looking for an actuarial present value at 40.
#8
05-10-2009, 06:36 PM
 Jo.M. Member Join Date: Oct 2007 Posts: 288

Quote:
 Originally Posted by its_me so, doesnt the term become 10.(v^10)(10_p_40) ( v q_50)?
Yes.

The question asks for the APV of a 10-year increasing insurance starting at age 40. You are given the value of such an insurance starting at age 41.

You say you already drew a graph, so you see that you are missing a piece at the bottom of the graph; a 10-year term-insurance of 1 starting at age 40. However, you also need to remove the surplus; a 1-year term insurance of 10 paying upon death at age 50.

Whenever you play around with these pieces, you need to discount everything to age 40, hence the $vp_{40}$ factor on the IA_41:10 term and the $v^{10}*_{10}p_{40}$ on the 1-year term insurance $vq_{50}$.

I hope this helps.
#9
05-10-2009, 07:06 PM
 its_me Member Join Date: Jan 2003 Location: AO Posts: 1,772

got it..thanks a lot!
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