Mildenhall Aggregate

Loss Tools

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Example 14: Bootstrap and Revision of Ultimates

Loss Development Triangle
Number of Years Resamples
FirstYear Smoothing
Prior Ultimate
Year Parameters Plot 3D    No 3D plot 
You must select parameters for the prior ultimate distribution that make sense! The example is only set up for the default triangle. Use Example 2 to determine suitable parameters for the other triangles.
Model Options
Correlation Num L2
Tau Type of Plot
Results of triangle analysis for a Simulated Triangle

Click compute for results...


Bornhuetter-Ferguson is Buhlmann Credibility!

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
= -2E(U-a-bL) = 0
= -2EL(U-a-bL) = 0.

These give two equations

E(U) = a + bE(L)
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)
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
 Bühlmann Ultimate = E(U)-E(L) + L.
Note that L is a known quantity. This equation is exactly the Bornhuetter-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.

References to Wang