claim amount increase impacts

[Molly]

Hi all,

I keep seeing it popping up for the pareto distribution that a claim amount increase of 5% for example would have impact only on lambda. I cant find any justification on this in the course notes, does anyone know of a rule for this?

also, im a bit worried now that there might be rules for the other distributions that im not aware of, regarding claim amount increases - is there some sort of general rule?

Could someone please help on this? (I understand its easter, so its a bit of a long-shot but any help would be really appreciated!)

 

[Andrew Martin]

Hello

One way to approach this is to introduce some random variables and consider CDFs. Let X be the original claim amount and let it follow the Pa(a,b) distribution.
Let Y =1.05X then:

P(Y <= y) = P(1.05X <= y) = P(X <= y/1.05)

= 1 - (b/(b+y/1.05))^a
= 1 - (1.05b/(1.05b+y))^a
Let b'=1.05b
FY(y) = 1 - (b'/(b'+y))^a
This is the CDF of the Pa(b', a) distribution.

Hope this helps!

Andy

 

[Molly]

Hi Andy,

thats a really interesting way of thinking about it, thank you so much thats really helpful!!!

Hello

One way to approach this is to introduce some random variables and consider CDFs. Let X be the original claim amount and let it follow the Pa(a,b) distribution.
Let Y =1.05X then:

P(Y <= y) = P(1.05X <= y) = P(X <= y/1.05)

= 1 - (b/(b+y/1.05))^a
= 1 - (1.05b/(1.05b+y))^a
Let b'=b
FY(y) = 1 - (b'/(b'+y))^a
This is the CDF of the Pa(b', a) distribution.

Hope this helps!

Andy

 

[Andrew Martin]

No problem! I noticed a typo in my post, I had b'=b instead of b'=1.05b, I have updated it now.