About NSP of UL Option 2

[Sherwin]

Has anyone read Page 42 of the book LIFE INSURANCE AND MODIFIED ENDOWMENT? I feel puzzled about Formula (2.8), which is NSP for UL Option 2 as follows.

NSP=Ax=Sum[ v^(t+1)*q(x+t)+v^(100-x) ].

I really cannot understand it. Could anyone help me with it please? Many thanks.

[Gandalf]

Has anyone read Page 42 of the book LIFE INSURANCE AND MODIFIED ENDOWMENT? I feel puzzled about Formula (2.8), which is NSP for UL Option 2 as follows.

NSP=Ax=Sum[ v^(t+1)*q(x+t)+v^(100-x) ].

I really cannot understand it. Could anyone help me with it please? Many thanks.
I don't see how it could possibly be right as you have written it. Consider as an extreme case a UL issued at x=98 with q(98) = 0 and q(99)=1. Your formula, assuming it is a sum from t=0 to t=1 in this case, gives (v^1 * 0 + v^2) + (v^2 * 1 + v^2) = 3v^2, whereas the correct result in this extreme case would be roughly 2v^2. [The expressions I put in parentheses were the t=0 term and the t=1 term.]

Or same policy with q(98) = .9999 and q(99) = 1. Your formula gives (v^1 * .9999 + v^2) + (v^2 * 1 + v^2) = .9999 v + 3v^2 when the correct result is something close to 2v.*

*In your post you haven't described the coverage very well (maybe it's clear in the book). Is this a single premium UL type 2 or an annual premium UL type 2 or a monthly premium UL type 2? There would be an NSP for each, and they would be different. In my second example, the 2v is the approximate NSP for an annual premium UL type 2. For a single premium, it would be close to 3v, still quite different from what your formula has. In my first example, the NSP would be 2v^2 for either annual premium or single premium.

[slystarnes]

I spoke (over email) with the writers of the text (Adney and Hertz), and they feel the formula is correct as it was posted originally in this thread. Luckily for us, we just have to know what it is and when to use it, not how it was developed.

[Gandalf]

I spoke with the writers of the text (Adney and Hertz), and they feel the formula is correct as it was posted originally in this thread. Luckily for us, we just have to know what it is and when to use it, not how it was developed.
What do you mean "you spoke with them"? And how could it be correct as posted, since it doesn't explicitly list what the limits of the sum are? Do you think the intent is \sum_{t=0}^{100-x-1} [v^{t+1}*q(x+t)+v^{100-x}], where x is the issue age (and where the policy matures at age 100 for 1)?

If so, consider the situation where x = 98, q_98 = 0, q_99 = 1 and i = 0 (or some tiny value), and compare it to the situation where x = 99, and i = 0 (or the same tiny value).

The two NSPs should be nearly identical, shouldn't they? No one dies between age 98 and age 99, so the only difference should be 1 year's interest at a near 0 interest rate.

But the formula at x=98 is (v^1)*0 + v^2 + v^2*1+v^2 = 3v^2 (= 3 for all practical purposes for very small i).

The formula at x=99 is (v^1)*1 + v^1 = 2v^1 (= 2 for all practical purposes).

If the v^(100-x) is not inside the summation, the formula might possibly be correct, but in post 1 it definitely is inside the summation, and you said it is correct as posted here.

Edited to add: you might get some circularity in solving for the NSP if q_99 =1, but if q_99 = .9999 or some similar very high value the calculation is essentially unchanged from NSP at 99 almost exactly 2 and NSP at 98 almost exactly 3.

[cat20]

If it helps, there's an example in the book showing the formula being used. See page 46. GLP for DBO2.

[wat?]

What do you mean "you spoke with them"? And how could it be correct as posted, since it doesn't explicitly list what the limits of the sum are? Do you think the intent is \sum_{t=0}^{100-x-1} [v^{t+1}*q(x+t)+v^{100-x}], where x is the issue age (and where the policy matures at age 100 for 1)?

If so, consider the situation where x = 98, q_98 = 0, q_99 = 1 and i = 0 (or some tiny value), and compare it to the situation where x = 99, and i = 0 (or the same tiny value).

The two NSPs should be nearly identical, shouldn't they? No one dies between age 98 and age 99, so the only difference should be 1 year's interest at a near 0 interest rate.

But the formula at x=98 is (v^1)*0 + v^2 + v^2*1+v^2 = 3v^2 (= 3 for all practical purposes for very small i).

The formula at x=99 is (v^1)*1 + v^1 = 2v^1 (= 2 for all practical purposes).

If the v^(100-x) is not inside the summation, the formula might possibly be correct, but in post 1 it definitely is inside the summation, and you said it is correct as posted here.

Edited to add: you might get some circularity in solving for the NSP if q_99 =1, but if q_99 = .9999 or some similar very high value the calculation is essentially unchanged from NSP at 99 almost exactly 2 and NSP at 98 almost exactly 3.

It seems as though the v^(100-x) factor is located within the parentheses (something I didn't get either).

The exact writing of the formula is:

NSP = Ax = Sum SA x ( v^(t+1) x q_x+t x v^(100-x) ),

where "Sum" stands for the summation sign (no limits of summation) and the superscripts/subscripts belong where they look like they belong.

[Gandalf]

It seems as though the v^(100-x) factor is located within the parentheses (something I didn't get either).

The exact writing of the formula is:

NSP = Ax = Sum SA x ( v^(t+1) x q_x+t x v^(100-x) ),

where "Sum" stands for the summation sign (no limits of summation) and the superscripts/subscripts belong where they look like they belong.
wat?, I think you must have a typo, or your formula is substantially different from what was posted earlier in the thread (and even stranger). Isn't that x supposed to be a +?

I don't have access to the book, so I'm at a huge disadvantage here. I don't even know for sure what the policy design is (e.g., death benefits paid monthly or annually; might matter if the death benefit would be dropping during the year as reserves are released).

However, I'm guessing it's a policy that pays annually, that the death benefit is the Sum Assured plus the reserve (or account value) at the end of the year. Furthermore, that it endows for the Sum Assured at age 100. If that policy description is correct, then here's an example that should easily convince you that the book's formula is ridiculous. Suppose the q's are 0 at all ages. Then the present value of death benefits is 0 for all issue ages (no one collects a death benefit). The present value of the endowment benefit must be Sum Assured * v^(100-issue age), not the sum of 100-x terms.

[wat?]

wat?, I think you must have a typo, or your formula is substantially different from what was posted earlier in the thread (and even stranger). Isn't that x supposed to be a +? You are correct, sir. I'm starting to learn this "reading" thing. It's kinda important, huh?

I don't have access to the book, so I'm at a huge disadvantage here. I don't even know for sure what the policy design is (e.g., death benefits paid monthly or annually; might matter if the death benefit would be dropping during the year as reserves are released).

However, I'm guessing it's a policy that pays annually, that the death benefit is the Sum Assured plus the reserve (or account value) at the end of the year. Furthermore, that it endows for the Sum Assured at age 100. If that policy description is correct, then here's an example that should easily convince you that the book's formula is ridiculous. Suppose the q's are 0 at all ages. Then the present value of death benefits is 0 for all issue ages (no one collects a death benefit). The present value of the endowment benefit must be Sum Assured * v^(100-issue age), not the sum of 100-x terms.

BTW, SA = "specified amount". I know you don't have the book, so I'm just tossing that in to help you, in case that clarifies anything.

[Gandalf]

"Specified Amount", "Sum Assured", doesn't matter. I would call it the Face Amount, but I was pretty sure Face doesn't start with S. :-P

[Gandalf]

Definitely the latter, since I've been saying all along that the first way can't possibly be right and others have been saying that they can't understand why that term should be inside the sum.

If slystarnes is still reading this thread, he should point out to the authors that their book contains two different versions, and ask them if they're still going to maintain that the page 42 version is correct. They should be embarrassed. A few typos are understandable and excusable, but when someone asks if a formula is correct, and they reiterate that it is when it isn't, that's bad. (I'm going by what slystarnes said about his question and their response.)

[Gandalf]

Testing that formula directly would seem like minutiae to me, though I don't know how prominent single premium type 2 UL is in the study material. If that type of policy is considered important, maybe they should care if you know how to calculate the NSP.

[slystarnes]

In the response to my question, one of the men referred me to the book from which the formula is drawn. That book is Life Contingencies by Chester Wallace Jordan, Jr., and the material is in section 12 of chapter 5, problems 34-36. This is appparently a book that an older actuary at your company may have as it was the predecessor to Actuarial Mathematics. Someone could look there and verify which is correct. I would suspect (as others before me have) that the one on pg. 46 is right now that I have looked at both.

[wat?]

In the response to my question, one of the men referred me to the book from which the formula is drawn. That book is Life Contingencies by Chester Wallace Jordan, Jr., and the material is in section 12 of chapter 5, problems 34-36. This is appparently a book that an older actuary at your company may have as it was the predecessor to Actuarial Mathematics. Someone could look there and verify which is correct. I would suspect (as others before me have) that the one on pg. 46 is right now that I have looked at both.

Lol, thanks for the effort, but I cracked the Jordan book open, and I see too many words for my liking. :)

I'll take your word for it that the p. 46 one is right. :tup:

[tom_47]

Just two additional comments:

1. As for the endowment part (i.e., v^100 - x), on p. 42 of the text it's definitely inside the summation. On p. 46 of the text it is (arguably) outside the summation.

And let's face it, inside the summation makes no sense at all.

2. The text referenced above, from whence this DB Option 2 is alledged to have come, does not expressly give the formula.

In addition, the text itself pre-dates the advent of UL and so cannot be directly applicable to Option 2 policies.

And finally, the original text refers to a policy where "the reserve" is part of the death benefit. I guess we're to assume that reserve and AV are the same thing?

Let's face it, the authors screwed the pooch on this one, whether they care to admit it or not.

But if it shows up on the exam I'll sure try and parrot it back for the powers that be....but which form do I recite?

[RG]

Your timing is perfect, I was staring at this last night. I don't have pg 46 in front of me but the missing paranthesis was only one problem: why aren't we discounting for survivorship? They even say in the text not to discount for survivorship.

[CDesRochers]

The formula was discussed in the thread above. The endowment is properly outside the summation. We'll fix it in the next edition.