Actuarial Outpost About a kind of Pareto model
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#1
12-16-2015, 05:46 AM
 Hiu Member Join Date: Jan 2008 Studying for Last FCAS exam-old 7U Posts: 193 Blog Entries: 1
About a kind of Pareto model

I am reading an old book "Foundations of Casualty Actuarial
Science (Fourth Edition)" by the CAS published in 2001, in order to learn some reinsruance pricing approaches.

In Chapter 7 "Reinsurance", there is a model called Censored Pareto model, as shown in the picture attached.

However, it can be seen that F(1) is not equal to 1 as there is no definition for x>1. So it is not a reasonable probability model, is it?

In addition, when it computed E[x;1], it utilized the limited expected value equation. But the latter only applies to a Pareto distribution which is not censored. So the result seems to have something wrong.

Anyone holds interest in that question? Thanks in advance.
Attached Images

#2
12-16-2015, 09:42 AM
 Colymbosathon ecplecticos Member Join Date: Dec 2003 Posts: 6,167

F(1) does equal 1 since you are given that 1-F(1) = 0.

There is a point mass at 1 which corresponds to the probability that the uncensored distribution is bigger than 1.
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#3
12-16-2015, 09:54 AM
 whoanonstop Member Non-Actuary Join Date: Aug 2013 Location: Los Angeles, CA Studying for Spark / Scala College: College of William and Mary Favorite beer: Orange Juice Posts: 5,946 Blog Entries: 1

Okay, so I agree that it is presented funny, but I don't have the full context, so I may be missing something. However, I'm pretty sure the misunderstanding is associated with this line:

Quote:
 where the claim size X is expressed as a percent of MPL
So MPL being 10,000,000, x is 1 when the claim size is 10,000,000, .5 when claim size is 5,000,000 and so on. Is it possible that the claim size could be 20,000,000? From the perspective of the reinsurer, Absolutely. So if you run the integrand over the distribution for all of x it will evaluate to 1 as expected.

However, from the perspective of the insurer, reinsurance kicks in once x exceeds 1. There will never be a claim size of 120% of MPL, for example. Since severity is capped at 100%, the "normal" pareto formula for E[X;1] will correctly calculate the expected fraction of MPL for the severity.

-Riley
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#4
12-16-2015, 12:50 PM
 Colonel Smoothie Member CAS Join Date: Sep 2010 College: Jamba Juice University Favorite beer: AO Amber Ale Posts: 49,956

You know, I don't regret missing that game. I told the boys I had to go see about a Pareto model.
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#5
12-18-2015, 07:35 AM
 Sherwin Member CAS SOA Join Date: Jun 2006 Location: China Reinsurer Posts: 220 Blog Entries: 10

The integral x*d(F(x)) from 0 to 1 is not equal to 1. But there is a stacked probability at x=1, Pr(x=1)=(0.1/1.1)^2=1/121. So the integral from 0 to 1 plus Pr(x=1) is equal to 1.
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#6
04-20-2016, 02:40 PM
 AbedNadir Member CAS SOA Join Date: Mar 2014 Studying for FCAS Posts: 2,774

Quote:
 Originally Posted by Colonel Smoothie You know, I don't regret missing that game. I told the boys I had to go see about a Pareto model.
that bastard stole my line
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#7
04-21-2016, 02:45 PM
 DeepPurple Member Join Date: Jun 2004 Posts: 4,333

Quote:
 Originally Posted by Colonel Smoothie You know, I don't regret missing that game. I told the boys I had to go see about a Pareto model.
Quote:
 Originally Posted by AbedNadir that bastard stole my line
RsN

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