Percentile Premiums

[777888]

1) When a question refers to "percentile premiums", it often will be saying things like "find the smallest annual premium P such that the insurer's probability of a positive financial loss is at most 0.25".
Is there any conceptual difference in saying that and saying "find the exact annual premium P such that the insurer's probability of a positive financial loss is exactly 0.25"? Are they the same thing?

2) For a fully discrete whole life insurance of 1 on (x), you're given:
A_x=0.19, 2^A_x = 0.064 (i.e. at doubled force of interest), d=0.057
The level annual premium payable is 0.019.
The insurer has N such insurances and the losses are independent.
N is the smallest number of such insurances for which the probability, using the normal approximation, of a positive aggregate loss is less than 0.05. Calculate N.
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I believe the aggregate loss-at-issue S here would be a DISCRETE random variable because it is a function of curtate future lifetimes K, so I think we need the continuity correction. But how can we use the "continuity correction" in this case? The random variable S is discrete but the possible values are irregularly spaced and not equally spaced. (because we have terms like v^(K+1) There is no way we can figure out all the distances/gaps between the possible values. Then how can we figure out what the adjustment factor is going to be?


I hope someone can kindly explain these. Any help is much appreciated!

[777888]

In the normal table provided with the SOA exam, we are actually explicitly asked to use the continuity correction when approximating a DISCRETE r.v. with normal distribution. In question 2 above, aggregate loss-at-issue S here would be a discrete random variable, but how can we figure out what adjustment factor to use for correction? Does anyone have any idea?

[alekhine4149]

It's a good question. You are right that the distribution of S is discrete, but it does not always take on integral values. The way the continuity correction method is taught, it's evident that it is intended to be used when the approximated distribution can only take on integral values. If K were being approximated, then we might be in business. I would not use it.

[alekhine4149]

1) When a question refers to "percentile premiums", it often will be saying things like "find the smallest annual premium P such that the insurer's probability of a positive financial loss is at most 0.25".
Is there any conceptual difference in saying that and saying "find the exact annual premium P such that the insurer's probability of a positive financial loss is exactly 0.25"? Are they the same thing?

Regarding this first question of yours, they are the same if there exists a premium such that probability of loss is exactly 0.25. However, there may not be such a premium depending on how the problem is set up (if they make you round your premium, or something of that nature). In this case, you have to view the problem as an optimization problem given a constraint. Basically, your idea is right though, yes.