Chapter 18: Credibility Theory

[Purple]

In Section 2.4 of Chapter 18, the Pure Premium formula, (PP) is given as (X1+X2+X3+.....+XN)/n, where n=exposures, while aggregate losses, L=X1+X2+......+XN.

Why are the formulae for the variance and Expected value of PP|N as follows?

Var(PP|N) = Nσ(s )^2
E(PP|N)=Nμ(s )

Shouldn't the above be the variance and expected value of L|N?

Shouldn't the variance and expected value of PP|N be as follows:

Var(PP|N) = (Nσ(s )^2)/(n^2)
E(PP|N)=(Nμ(s ))/n


Please advise. Thank you.

 

[Katherine Young]

Hi Purple,

The top of page 20 says that we have a given number of claims N (in other words we know with certainty that the number of claims is N). Since claims are independent, we can sum the individual variances (and we've been told there are N of them) to get the total variance.

The same argument holds for the expectation.

Kind regards,

Katherine.

 

[iActuary]

Standards for Full Credibility - Severity

The "final" formula for standards for full credibility (severity) is:
N = (n0) x (CVs)^2

My question is whether this standard will change according to the distribution of the claims frequency?

The original derivation shows that N = (y/k)^2 x (CVs)^2. Obviously, this does not depend on the distribution of claims frequency.

BUT, if we use the "final" formula, distribution of claims frequency does affect the standards for full credibility (severity).

Isn't it more appropriate to show N = (y/k)^2 x (CVs)^2? Any idea?

The same arithmetic confusion arises for Premiums (Poisson frequency) vs. Premiums (general).

 

[Katherine Young]

Dear iActuary,

You're quite right, the formula you speak of uses the simplifying assumption of Poisson claim frequency.

If you prefer, you could use the general formula for n0, given on page 15.

Remember as well, that the standard for full credibility (severity) uses a normal approximation for observed severity (see pages 16 and 17).

In my opinion, the important point is that an actuary should be aware of the assumptions underlying the model he / she is using, and the sensitivity of the model output to those assumptions.

So well done for questioning it!

Kind regards,

Katherine.