X5 Assignment, Question X5.5, part i

[Ayesha Khalil]

Hi,

For the question I've mentioned in the title, it tells us to show that Var(S)=288,952*n, however I don't understand how to do this since using the formula for aggregate claims, we have that:

Var(S) = E[N]*Var[X] + Var[N]*(E[X])^2

If N is the number of claims, then in this case, N=n*P, where n=number of independent policies and P=total annual claims from a single policy, then:

Var[N] = Var[n*P] = n^2 * Var[P]

Since n is a constant.

Since P has a compound Poisson distribution with parameter 0.25, then:

Var[N] = 0.25*n^2

I can get the figures for the rest of the components for Var(S) just fine, but I've realised that I think my Var[N] figure could be wrong since there is no n^2 term in the Var(S) figure we are being asked to show.

Please could you let me know where I'm going wrong using this method as the solution doesn't cover how to do it this way.

Thanks

 

[Andrew Martin]

Hello

If we say:

S = X1 + X2 + ... + XN

where Xi = Ci + Ei
Ci is the claim amount of the ith claim
Ei is the expense amount of the ith claim
N is the number of claims across all n policies.

Then as S is a compound Poisson random variable, we have:

Var(S) = E[N]*Var[X] + Var[N]*(E[X])^2

N here follows the Poisson(0.25 * n) distribution. This is because for each individual policy we have the Poi(0.25) distribution for the number of claims and there are n independent policies. Ie let Nj denote the number of claims on the jth policy, then:
N = N1 + N2 + ... + Nn
Nj ~ poi(0.25)
N ~ poi(0.25 * n)

It seems you've then defined P to be the total annual claims from a single policy. This would mean that:

\( P_j = C_{j,1} + C_{j,2} + ... + C_{j, Nj} \)

where Cj,k is the claim amount for the kth claim on the jth policy.

I think where you may be going wrong is the line N = n * P. If N is the number of claims across the entire portfolio, n is the number of policies and P, as you've defined it, is the amount of claims from a single policy then this does not hold.

I'm not sure which method you're attempting here - is it working out the relevant quantities per policy the summing over all policies? If so I've set out how to do this below:

Firstly let's redefine P to represent the total annual payments from a single policy (claims and expenses). Then we now have:

\( P_j = (C_{j,1} + E_{j,1}) + (C_{j,2} + E_{j,2}) + ... + (C_{j,Nj} + E_{j,Nj}) \)

Let:
\( R_{j, k} = C_{j,k} + E_{j,k} \)

So, now we have:

\( P_j = R_{j,1} + R_{j,2} + ... + R_{j,Nj} \)

with Nj ~ Poi(0.25)

As P_j is a compound Poisson random variable, we have:

E[Pj] = E[R] * E[Nj] = (E[C] + E[E]) * 0.25 = (600 + 60) * 0.25 = 165.

(dropping the j notation on R / C / E but keeping on Nj as N is sum(Nj) and R / C / E are just standing for any R_{j, k} / C_{j, k} / E_{j,k}.)

Var(Pj) = E[Nj] * var(R) + var(Nj) * E[R]^2
E[Nj] = 0.25
Var(Nj) = 0.25
E[R]^2 = 660^2
var(R) = var(C + E) = Var(C) + Var(E) by independence
= 720,000 + 208 1/3 = 720208 1/3

Var(Pj) = 0.25 * 720208 1/3 + 0.25 * 660^2 = 288952 1/12

S = P1 + P2 + ... + Pn
E = n * E[Pj] = 165 * n
var(S) = n * var(Pj) = 288952 1/12 n

Hope this helps!

Andy