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
Input triangle specification similar to Example 9.
Input number of years in the triangle and the first year.
Number of resamples should be = 10,000 or more. Smoothing can
be used to remove high frequency noise from the simulated data.
Select one year to analyze with 3D graphics using the
radio buttons. The default is no graphics. If your machine is struggling with
the graphics, change NumL2 to 7.
Prior Ultimates: The model will analyze the four most recent years. Select
whether to input mean, CV and skewness (from your own analysis, or determined
from an analysis in Example 2, Example
7 or Example 8) or shifted lognormal parameters
(also available in the same examples).
Input parameters, as indicated. You cannot mix
inputting mean, CV, and skewness with (mu, sigma, T)'s.
Select correlation between ultimate losses and
Factor-To-Ultimate FTU.
Select number of points to use in the discretization.
8 correponds to 256 points. Do not select more than 8 unless you have a very
hefty computer!
Left click in the 3D graph to pan, right click to zoom.
Press w to get a wireframe, s to go back to a surface. Press t to go to
trackball mode, j to return to joystick. Don't press e.
Compute: recomputes with new inputs. The plot type must change for
the graph to update. To recompute with the same plot type, click "Blank" first.
Change Plot Type: Select from plot type drop down and click this button. No other settings
are updated in the plot. Use Compute if you want to change the settings
Blank: Clears the 3D plot
Snapshot: Produces a JPG of the graph, stored in c:\temp as test1.jpg, test2.jpg etc.
Print: Prints the graphic to a random printer connected to your machine. Still under development!
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
¶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(L^{2}).
Multiplying the first equation by E(L) and subtracting from the second gives
E(LU)-E(L)E(U) = b(E(L^{2})-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
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.
Mildenhall, S., "Bayesian-Bootstrap Loss
Development" CAS DFA Seminar Presentation, July 1999.
Ostaszewski K. and G. Rempala "Applications of
Resampling Methods in Dynamic Financial Analysis"
1998 CAS DFA Call Papers, CAS (1998). Available on the CAS Website.
Taylor, G. "Development of an incurred loss
distribution over time" COTOR Working Paper, 1998
Efron, B. and R. Tribshirani, "An Introduction to the Bootstrap" Chapman & Hall (1993)