1 Regression Models & Loss Reserve Variability Prakash Narayan Ph.D., ACAS 2001 Casualty Loss Reserve Seminar

2 Regression Models and Loss Reserve Variability Range for Loss Reserve: Ultimate loss will be different from any estimate A measure of variability or range of loss reserve is needed to monitor reserve levels We have methods that can be implemented in EXCEL to estimate reserve variability

3 Regression Models and Loss Reserve Variability Ad Hoc Methods of Reserve Ranges Ad Hoc Methods of Reserve Ranges: Use a % of ultimate reserve based on line of business and professional judgment.Use a % of ultimate reserve based on line of business and professional judgment. Use variety of methods and select a range.Use variety of methods and select a range. Use high and low development factors (various alternatives).Use high and low development factors (various alternatives).

4 Regression Models and Loss Reserve Variability Statistical Methods: Development Factor Variability Models (Mack/Murphy) Least Square Method (does not provide parameter uncertainty) Log Regression Method Variety of other methods (not discussed here)

5 Regression Models and Loss Reserve Variability Notation: x i, j = Losses Paid (reported) for the accident year i in development year j. (incremental losses) i,j = 1,... n. j y i,j = Σ x i k k=1 We observe x i, j for i = 1,... n; j=1,... n + 1 – i and we are interested in estimating y i, k and variability of these estimates for k = n + j – i,... n.

6 Regression Models and Loss Reserve Variability  Assumptions:   j independent of accident year  All future development dependent on most current evaluation   i, j are error and are independent of accident year and delay  Expected value of  i, j = o and variance may be a function of y i, j and the development year  Note:  We assume that losses are fully developed by period n and do not consider tail factors in this study.  These assumptions are helpful in deriving variance estimates of ultimate losses.  Regression Frame Work for Loss Development

7 Regression Models and Loss Reserve Variability  Alternate Loss Development - Method 1 Under the assumption simple average development factors (SAD) are best linear unbiased estimates (BLUE) for  j and are unbiased estimates of  j 2. are unbiased estimates of ultimate factors and ultimate losses respectively. And

8 Regression Models and Loss Reserve Variability  Alternate Loss Development - Method 2 Under the assumption weighted average development (WAD) factors are best linear unbiased estimates (BLUE) for  j and are unbiased estimates of  j 2. Define are unbiased estimates of ultimate losses, and then

9 Regression Models and Loss Reserve Variability  Variance Estimation-Loss Development The assumption of an underlying linear model allows estimation of variance of the estimated ultimate losses. These may be useful in deriving confidence intervals for ultimate losses. Variance of the prediction is The first term of this equation is parameter risk, the second term is process risk, and expected values of the cross product terms are all zero under our assumptions. Note: We have assumed k = n+1- i and used

10 Regression Models and Loss Reserve Variability  Variance Estimation (continued) Parameter Risk Hence Parameter Risk may be estimated by which is estimated by Note: We have used estimates as proposed by Thomas Mack (1994). Daniel Murphy (1994) uses slightly different estimators. Process Risk

11 Regression Models and Loss Reserve Variability  Variance Estimation (continued) The Process Risk is assumed independent of accident year. However, the age to ultimate factors are correlated. Therefore, to compute parameter risk for the total ultimate losses one must account for the covariance of the ultimate losses among accident year. The algebra is little messy, after simplification, the covariance estimate for accident years r and s (r<s) is given by and the estimate of total covariance for accident years 2 through n is

12 Regression Models and Loss Reserve Variability Least Square Method X i, j = r i c j + δ i j estimate r i,and c j by minimizing Σ Σ (X i j – r i c j ) 2 i j * It has been shown the least square equations do have a solution and that can be obtained by iterative numerical method. * Parameter uncertainty can not be estimated. * There are only 2n-1 parameters. Model unchanged if we multiply each r i by a constant and divide c j by the same.

13 Regression Models and Loss Reserve Variability Assume X i j = r i c j δ i j Taking logarithm and redefining notation, we can write Z i j = α i + β j + e i j or Z i j = μ + α i + β j + e i j with α 1 = β 1 = O In matrix notation Z = A θ + ε ~ Ê ε = O, V ( ε ) = σ 2 I ~ Least square estimates are ^ θ = (A´ A) -1 A ´ Z ^ ^ σ 2 = (Z´ Z - θ ´ A´ Z)/ r r = (n-1) (n-2)/2  Log Regression Model

14 Regression Models and Loss Reserve Variability  Log Regression Model The unknown elements of the loss process can be written as _ E Z = B θ _ ^ The vector Z can be estimated by B θ. However our aim is to estimate X i j and not Z i j. The formulae are a bit complex but unbiased estimate corresponding estimates can all be computed in EXCEL. Details are given in Verrall 1994 CAS Forum.

15 Regression Models and Loss Reserve Variability  Method of Log Regression Model (continued) * Model fitted has too many parameters * Many parameters may not be significant * Tail factor may be needed * Calendar year inflation may be distorting observed observation One can choose alternate models, for example Z i j = μ + α i + β j α i = (i - 1) α β j = γ log (j) Model parameters can be tested for statistical significance. If variance changes by payment year, weighted least square may be used.