Mildenhall Aggregate Loss Tools

Home Contents Back Next Help

Example 15: Claim Count Development Model

Notes

• This page displays a general method for computing the distribution of claim count factors-to-ultimate (FTUs).
• The method can be applied to open or closed claims, and can be adapted to consider only claims closed with amount, closed without amount, incurred claims or reported claims.
• At a given development period n, each claim can be in one of four states, as shown below:

  	Closed	Outstanding	Not yet opened
  With Amount	A	B	C
  Without Amount	D	E	F

• Notice that at all developments A+B+C+D+E+F = ultimate reported counts. A, D, B+E are easily obtained from standard triangles. B+C, E+F, B+E and C+F can be derived from a standard triangle analysis. However, determining B and E, C and F separately requires making some assumption about how outstanding and not-yet-incurred claims will CWOP.
• To model closed with amount claims p₀₁= 1- p₁₁ and p₁₁= A/(A+B+C), the other two parameters are zero.
• To model reported claims p₀₁= 1- p₁₁ and p₁₁ = (A+B+D+E)/(A+B+...+F), the other two parameters are zero.
• To model incurred claims is more difficult, because of the possibility of factors-to-ultimate less than 1.0. Let U= A+B+C+D+E+F be the number of ultimate claims. Then the four probabilities are:
  1. The claim will ultimately close with indemnity, but at development n it has not opened. The probability of being in this state is p₀₁ = C/U.
  2. The claim is currently open with amount and it will ultimately close with amount, or has already closed with amount. The probability of being in this state is p₁₁= (A+C)/U.
  3. The claim will ultimately close without payment, and at development n it is not open, or it has already CWOPed. The probability of being in this state is p₀₀= (D+F)/U.
  4. The claim is currently open with amount, but will ultimately close without amount. The probability of being in this state is p₁₀= E/U.
  Diagramatically we have, for closed with amount, reported and incurred:

Ulimate CWA Counts	p01=1-p11		p11=A/(A+B+C)
Frequency: A+B+C	p00=0		p10=0
			CWA to Date

Ultimate Reported Counts	p01=(C+F)/U		p11=1-p00
Frequency: U	p00=0		p10=0
			Reported to Date

Ultimate Incurred Counts	p01=C/U		p11=(A+C)/U
Frequency: U	p00=(D+F)/U		p10=E/U
			Incurred to Date

• Aggregating the p_ij over the number of expected claims, or more generally, convolving with a suitable frequency distribution to allow a variable number of claims, produces the bivariate distribution of observed (CWA, reported or incurred) claims vs. ultimate claims.
• The bivariate distribution (shown in the three dimensional plot) can then be converted into a univariate distribution of factors-to-ultimate. This is shown in the lower two dimenensional plot. Vertical slices correspond to Bayesian posterior distributions.
• Usage:
  • Num_L2 is the log_2 of the number of points in the XY plane. Seven corresponds to a 128 x 128 grid. This number must be greater than the maximum likely claim count to avoid aliasing. Unless you have a very fast PC, avoid setting Num_L2 greater than eight.
  • Max FTU is the largest FTU used to bucket the empirical FTU distribution.
  • FTU Num_L2 is the log_2 of the number of buckets used to discretize the claim count distribution. It should be selected slightly lower than Num_L2.

References

• Mildenhall, Working Paper on Loss Development, 2001. I am also developing a version of this method to work with paid or incurred losses.