ACTUARIAL METHODS AND REINSURANCE COMPANY RESULTS

Stephen Mildenhall

Summary

Reinsurance and Reinsurance Company Results

Reserving

Credibility Theory

Credibility and Reserving using the Chain Ladder Method

Ultimate = (Expected Unpaid Losses) + Losses-to-Date.

Ultimate = (Expected Ultimate Losses) x (1 - 1/f) + Losses-to-Date.

Pricing

Applying Bornhuetter-Ferguson

Chicken and Egg

Adjustment for Changes in Rate Adequacy

Other Methods of Loss Development

Comparison of Methods

Summary and Conclusions

Appendix: Bühlman Credibility Estimate of Ultimate Losses

Let U be ultimate losses and L be losses observed at some evaluation (e.g. after 12 months, after 24 months, etc.) Bühmlann credibility is defined as the best linear approximation to the Bayesian estimate of U. A linear estimator means one of the form a+bL. If a = 0 and b = FTU this reduces to the usual Chain-Ladder method.

The Bühlmann credibility estimate is dervied by minimizing the expected squared error loss:

Q = E(U-a-bL)2.
To do this, differentiate with respect to a and b and set equal to zero giving
Q
a
= -2E(U-a-bL) = 0
and
Q
b
= -2EL(U-a-bL) = 0.

These give two equations

E(U) = a + bE(L)
and
E(LU) = aE(L) + bE(L2).
Multiplying the first equation by E(L) and subtracting from the second gives
E(LU)-E(L)E(U) = b(E(L2)-E(L)2)
or
Cov(L,U) = bVar(L).
Now, U = L+B where B is the (unknown) reserve, and we can assume L and B are independent. Thus
Cov(L,U) = Cov(L,L+B) = Var(L)
and so b = 1. Substituting into the first equation above we get a = E(U)-E(L) = E(B), expected unpaid or unreported losses. Thus we have shown that the Bühlmann credibility estimate of Ultimate losses is given by
E(U)-E(L) + L.
Note that L is a known quantity. This equation is exactly the Bournuetter-Ferguson estimate of ultimate losses, derived using a different technique in the 1970's.

We can estimate E(U)-E(L) using loss development factors, as follows. From the definition of link ratios we know

E(U) = fE(L)
where f is the factor-to-ultimate. Thus we can estimate E(U)-E(L) as E(U)-E(U)/f = E(U)(1-1/f). The estimate of E(U) is generally derived as premium times expected loss ratio. It is a prior estimate of ultimate losses.