need help!!Pleaaaaaaaaaaasssssssssseeeeeeee

Hi guys!!
can you please help me with this question . I am really stcuk and not sure what to do?

An insurance company issues 4 different types of policy as follows:

Type ~ Probability of claim ~ Amount of Claim ~ No. of policies sold
1 ~ 0.1 ~ 5 ~ 1000
2 ~ 0.1 ~ 3 ~ 2000
3 ~ 0.2 ~ 5 ~ 1000
4 ~ 0.2 ~ 3 ~ 2000


You may assume the claims from all policies are independent.
(a) Calculate the mean and variance of the claims cost from a single
policy of each type.
(b) Hence calculate the mean and variance of the claims cost from the
portfolio S.
(c) The company has a profit loading of theta so that the total premium
income is equal to (1 + theta)E. Calculate theta if it is set such that
the probability that the total claims cost is less than the total
premium income is 95%. (The 95th percentile on the standard
normal distribution is 1.645).




Thanks

 

This may help

Hi,

Thanks for the question. Quite a challenging one. Here's what I think:

Let Ni denote the number of claims all policyholders that hold policy i for i in {1,2,3,4}.

So: N1 ~ Bin (1000;0.1),
Similarly:
N2 ~ Bin(2000;0.1),
N3 ~ Bin(1000;0.2) and
N4 ~ Bin(2000;0.2).

Let Xi denote that claim amount for all policyholders that hold policy i for i in {1,2,3,4}.

Now: X1 = 5 with prob=1. So E(X1) = 5, and Var(X1) = 0.
Similarly for X2, X3 and X4 (getting tired of typing now

Now define the compound distribution for policy j, j in [1,4] as:
Sj = Sum(Xji overall i from 1 to Nj).
So: S1 = Sum(X1i overall i from 1 to N1).
Similarly for for S2, S3 and S4.

Lastly define S = Sum(Si overall i from 1 to 4).

For Question (i), We need to find E(Sj) and Var(Sj) for j in [1,4].
For these the usual Binomial expectation and variance formulae can be used.

For Question (ii), We need to E(S).
Clearly E(S) = Sum(E(Si))...
and Var(S) = Sum(Var(Si))...as the 4 compound distributions are independant from the fact that the claims are independant, and hence also the claim numbers.

*To whomever's reading this: Please correct me if I'm wrong in Q (ii) above), but that's what I sincerely think is the answer/method*


For Question (iii), we need to find theta that satistifes:

Pr [ (1+theta)E(S) > S] = 0.95...

At this point we need to assume that S is assymptotically Normally distributed.

Using the above assumption then, we can normalize the above using E(S) and Var(S), and solve for theta.

And that's that!

Once again, that's how I would solve it. ActEd tutors, if I'm mistaken please correct me as well.

Hope this helped.

Good luck in your preparation,

Goku

 

Thank you

 

No worries man

 

Please help me to understand this

in the red bit,I dont undersatnd how u got prob of 1. so does that mean when we do E(X2) we gwt 3, E(X3)= 10, E(X4)= 6. and why did u get var(X1)=0. Isnt Var(x)= E(X^2)-{E(X)}^2 = 20??
Please help me to understand this.
THanks