chapter 7

[Neetu Verma]

in chapter 7 ,topic parameter variability/uncertainty...
please tell me what is the main difference between variability in a heterogeneous and homogenous portfolio????
and why r change the formula of variance for the whole portfolio in the examples given in this topic?????

thanks
neetu

 

[nageshmcl]

Neetu,
I am able to understand the concepts like below:

Heterogeneous Portfolio:
Assume this portfolio is having 10 policies, each poicy covers a building in an area.
Het Portfolio = Policy1 + Policy2 + .....+ Policy10

Policy1 is issued to provide coverage for a building in Area1
Policy2 is issued to provide coverage for a building in Area2
......
......
Policy10 is issued to provide coverage for a building in Area10

Here the policies provides the coverage for different buildings located in different Areas.

Homogeneous Portfolio:
Assume this portfolio is having 10 policies, each poicy covers a building in an area.

Homo Portfolio = Policy1 + Policy2 + .....+ Policy10

Policy1 is issued to provide coverage for a building in Area1
Policy2 is issued to provide coverage for a building in Area1
......
......
Policy10 is issued to provide coverage for a building in Area1

Here the policies provides the coverage for different buildings located in only one Area.

Assume that if a catastrophe happens in Area1, extreme damage occurs and all the policies issued in that area can raise claims. By knowing the number of claims for policy1, we can easily guess the number of claims for other policy too since all the buildings are located in the same area and all are subjected to the same type loss.

Based on this we can say that, the number of claims for each policy under a homogeneous portfolio is similar.

Coming to heterogeneos portfolio, since the policies covers buildings situated at different areas, the number of claims raised by any two policies may not be the same. Thats why they are heterogeous. For more explanation on this go thru the egMotor insurance portfolio) given in the material.

********************
Coming to the calc of variance for the whole portfolio:

For this we need to understand the difference between the
'Two random variables are independent' and
'Two random variables are Conditionally Independent'

E(X1X2) = E(X1) E(X2) - Two Random variables (X1,X2) are independent.
E(X1X2/Y) = E(X1/Y)E(X2/Y) - Two Random variables (x1,x2) are conditionally independent.

To get the clear idea on this please have a look at the first eg: given in the credibility theory.

Just as discussed above in the homogeneous portfolio, the number of claims for each policy is almost similar, lets say it is 'lamda'.

Based on the credibility theory eg, we can say that aggregate claims 'Si'/lamda for i = 1 to n are independent & identically distributed since the claim distributions F(x) are also similar.

In the case of homo si s are just independent i.e, not conditionally independent. So overall, here the si s are iid.

The variance formula for 'Sum of aggregate cliams where the aggregate claims are independent' is different from the 'Sum of aggregate claims where the aggregate claims are conditionally independent'

Hetero:
Var[Sigma si] = n var(si) since si s are iid, var(s1) = var(s2) = ...= var(sn)

Homo:
Var(S) = E(Var(s/lamda)) + var(E(s/lamad)), since aggregate claim depends on lamda. In the summation we need to apply this for every si and known that si/lamda are iids.

 

[Neetu Verma]

In the case of homo si s are just independent i.e, not conditionally independent. So overall, here the si s are iid.

The variance formula for 'Sum of aggregate cliams where the aggregate claims are independent' is different from the 'Sum of aggregate claims where the aggregate claims are conditionally independent'

Hetero:
Var[Sigma si] = n var(si) since si s are iid, var(s1) = var(s2) = ...= var(sn)

Homo:
Var(S) = E(Var(s/lamda)) + var(E(s/lamad)), since aggregate claim depends on lamda. In the summation we need to apply this for every si and known that si/lamda are iids.

i am confuse here.....
u write.....
"In the case of homo si s are just independent i.e, not conditionally independent."
&
In last paragraph u use si as a conditionally independent in homo case.....
plz clarify this point..
thanks
neetu

 

[nageshmcl]

Neetu, yes you are right. Sorry, in hurry i have written wrongly. Further explanation is given below.

Homo:
As per our understanding Aggregate claim 'Si' for each policy should depend on the claim frequency. Since the portfolio is Homo, the claim frequency(λ) for each policy is same. So S1,S2,S3...Sn depends on λ. Then as per the credibility theory basic eg:If the value of λ is known, then S1,S2,S3..,Sn are not independent, but S1,S2,S3...Sn are conditionally independent on λ.i.e,

E(S1S2S3...Sn/λ) = E(S1/λ).E(S2/λ)...E(Sn/λ)

We need to find the variance of a RV which conditionally depends on another RV. In this case we have the following formula:

Var(S) = E(Var(s/λ))+var(E(s/λ))

Var(Sigma Si) = E[var(sigma si/λ)] + var[E(sigma si/λ)] -- Here sigma Si depends on λ.

Var(sigma si/λ) = var(s1/λ) + var(s2/λ) + var(s3/λ) + ....+var(sn/λ) based on the below formulas.

Var(X+Y) = Var(X) + var(Y) - This formula is valid if X & Y are independent,

otherwise Var(X+Y) = Var(X) + var(Y)- cov(X,Y) - If there exists a dependency.

Hetero:

Here s1,s2,s3....sn doens't depend on λ, instead each si depends on λi. That means claim frequency is not same for each si.

So si s are not depending on λ, then as per the credibility theory basic eg:,
si s are independent, since each si has a different λi.

Since si s are iid,then
var(sigma si) = var(s1) + var(s2) + ....+var(sn) = n var(si) - This is due to the fact that si s are not depending on λ, so sigma si doesn't depend on λ.

Then each si has the variance formula like below:
var(si) = E[var(si/λi)] + var[E(si/λi)], to get the answer this value is multiplied by n.

Thanks,
Nageswar.

 

[Neetu Verma]

now my all confusions r remove...
thank a lot nageswar for spending time for my query