202104-6

[ykai]

Claims on a particular type of insurance policy follow a compound Poisson process
with an annual claim rate per policy of 0.4. Individual claim amounts are
Exponentially distributed with mean 120. In addition, for a given claim and
independent of its size, there is a probability of 20% that an extra claim handling
expense of 30 is incurred. The insurer charges an annual premium of 60 per policy.
Estimate, using a Normal approximation, the minimum number of policies to be sold
so that the insurer has at least a 99% probability of making a profit.


Let the individual total claim costs be denoted by X.
Then X = Y + Z where Y is the cost of the claim and Z is the claim handling expense.
Var(Y) = 120^2 = 14,400
Var(Z) = 0.2 * 30^2 - 6^2 = 144

Where did 6^2 comes from?

 

[Andrew Martin]

Hello

The formula here is:

Var(Z) = E[Z^2] - E[Z]^2

E[Z] = 0.2 * 30 = 6.

Hope this helps!

Andy

 

[ykai]

Thank you for your reponse! I have fully understand.

 

[Tom17]

Hi, I would like to additionally ask, from what is E(Z^2) = 0.2*30^2 derived? Do we assume that Z~Bin(30,0.2), in which case E(Z^2) = 30*0.2*0.8 + (30*0.2)^2?

 

[ykai]

Hi, I would like to additionally ask, from what is E(Z^2) = 0.2*30^2 derived? Do we assume that Z~Bin(30,0.2), in which case E(Z^2) = 30*0.2*0.8 + (30*0.2)^2?

S=sum(X)~compound Poisson
N~Poisson(lambda)
X~?(parameter)

Var(X)=E(X^2)-E^2(X)=>E(X^2)=Var(X)+E^2(X)
Becasue E(N) & VAR(N) of Poisson are lambda, we could combine original equation of Var(X)=E(N)*Var(X)+Var(N)*E^2(X) become lambda*Var(X)+lambda*E^2(X)=lambda*E(X^2).

 

[Andrew Martin]

Hi, I would like to additionally ask, from what is E(Z^2) = 0.2*30^2 derived? Do we assume that Z~Bin(30,0.2), in which case E(Z^2) = 30*0.2*0.8 + (30*0.2)^2?

Hello

Z is a discrete RV that can take the values 0 or 30 with probabilities 0.8 and 0.2, respectively.

For a discrete RV, we have:

E[h(Z)] = sum(z):[h(z) * P(Z = z)]

So:

E[Z^2] = 0^2 * 0.8 + 30^2 * 0.2 = 0.2 * 30^2.

Hope that helps!

Andy

 

[Tom17]

Hello

Z is a discrete RV that can take the values 0 or 30 with probabilities 0.8 and 0.2, respectively.

For a discrete RV, we have:

E[h(Z)] = sum(z):[h(z) * P(Z = z)]

So:

E[Z^2] = 0^2 * 0.8 + 30^2 * 0.2 = 0.2 * 30^2.

Hope that helps!

Andy

Thank you very much, that is clear now.