September 11, 2006 How Valid Are Your Assumptions? A Basic Introduction to Testing the Assumptions of Loss Reserve Variability Models Casualty Loss Reserve Seminar Renaissance Waverly Hotel Atlanta, Georgia presented by F. Douglas Ryan, FCAS, MAAA Philip E. Heckman, ACAS, MAAA, PhD

Class for Regional Affiliates Schedule 8:30 am Optional Class A – The mathematics of regression Optional Class B - The reserving problem in casualty insurance 10:00Full session 1 11:45Break for lunch and Regional Affiliate business 1:00Full session 2 1:30Break out into work groups of four to six persons 3:00Full session 3 5:00End of class Participants who do not wish to take an optional early session are expected to travel to the meeting in the morning and return home the same day.

Class for Regional Affiliates Who Is It For? The class is designed to be offered by Regional Affiliates to all people interested in the technical side of the estimation of liabilities using data in “loss development triangles”. Regional Affiliates are encouraged to invite students and faculty in actuarial programs, actuaries at all levels of experience with the setting of loss reserves, and senior management.

Class for Regional Affiliates The Scope from 10:00 am to 5:00 pm Introduce triangles of cumulative loss costs and note their obvious features Algorithms: The “actuarial methods” and Excel instructions viewed as algorithms BLUE: “Best” Linear Unbiased Estimates Multiple Models and “The Reserving Problem” Testing the BLUE Assumptions Testing a BLUE Model: Validation Model Design

Class for Regional Affiliates The Participant Should Be Able To: 1.1 Participate in a discussion of how two triangles differ 1.2 Contrast the insights gained from two, three or four methods of viewing triangles 1.3 Use Chart Wizard to create “XY Scatter” charts of possible functional relations

Class for Regional Affiliates The Participant Should Be Able To: 2.1 Contrast the insights gained from normalizing using exposure rather than premium 2.2 Contrast the uses of price deflators with the uses of time series 2.3 Appreciate the complications introduced by introducing a triangle such as claim counts

Class for Regional Affiliates The Participant Should Be Able To: 3.1 Explain to a non-actuary what an algorithm is, why the “actuarial methods” are algorithms, and why Mack and Venter have asked us to think of the chain-ladder method as an algorithm.

Class for Regional Affiliates The Participant Should Be Able To: 4.1 Answer questions about how Excel’s functions call on algorithms and, if appropriate, draw an analogy with Russian dolls or some other system of nesting subroutines 4.2 Define in the Excel-user’s style (not as a statistician) the meaning of each function: FORECAST, RSQD, INTERCEPT, STEYX, TREND and LINEST 4.3 Give examples of what the Excel Function Wizard means by “known_y’s” and “known_x’s” 4.4 Use each of these Excel functions for a specific, limited purpose (although not be able yet to assemble them into a more sophisticated algorithm)

Class for Regional Affiliates The Participant Should Be Able To: 5.1 Using the acronym BLUE, recall the four aspects of a regression algorithm. 5.2 Examine a triangle of data and note its qualities as a set of “actual” observations. 5.3 Discuss the argument that regression is inappropriate for extrapolating into the future. 5.4 Discuss the argument that big differences between actual and expected imply a poor tool for forecasting.

Class for Regional Affiliates The Participant Should Be Able To: 6.1 Adjust a triangle of data (e.g., by taking logs or using incremental data) and discuss how the adjusted data is more or less appropriate as a set of “actual” data. 6.2 Give examples of real-world problems for which the linear assumption of regression is clearly inappropriate. [This may be difficult, but if it is, it simply shows the usefulness of linear models.] 6.3 Contrast BLUE models of paid data with BLUE models of incurred data (paid+case reserves).

Class for Regional Affiliates The Participant Should Be Able To: 7.1 Explain the differences between accounting summaries of loss costs and the “sufficient statistics” of a statistical approach such as GLM. 7.2 Explain how the “actuarial methods” rely on a subset of the accounting data rather than the entire triangles of paid and incurred. 7.3 Discuss the value of having a number of estimators that reflect a diversity of possible world-views and are independent of one another.

Class for Regional Affiliates The Participant Should Be Able To: 8.1.List five ways that patterns in the residuals might indicate that one or more assumptions is not appropriate. 8.2.Create a chart in Excel of a set of residuals and comment about what the chart says about the appropriateness of the BLUE assumptions. 9.1.Give reasons why the fitted values should look like the observations 9.2.Give examples of using subsets of the larger data set to test the reasonableness of a regression estimate

Minimum Limits Auto Dataset

Data Limitations To fit on overhead – only 5 years More stable than “real” data –Only one state –Only closed claims –Only minimum limits

Chain Ladder Projections

Data Adjustment

Exploratory Graphical Analysis

Exploratory Graphical Analysis: Residual A residual = Actual value for Y i – Fitted value for Y i Used to assess fit of model There should be no pattern, just random points

Chain Ladder Method Residual

Chain Ladder Method Residual -1

Chain Ladder Method Residual -2

Chain Ladder Method Residual -3

Chain Ladder Method Residual -4

Chain Ladder Method Residual -5

Exploratory Graphical Analysis: Scatterplots Plot of points of dependent variable vs independent variable Should suggest the shape of the relationship between the variables –Does it look like a linear relationship? –Does it look non-linear? –Is any relationship suggested: If not – mean of Y is its best estimate

Incremental vs. Cumulative - 1

Incremental vs. Cumulative - 2

Incremental vs. Cumulative - 3

Assumes a Linear Function

Linear Regression A to B incremental against age A cumulative Use Excel function to model y = mx + b LINEST(known y’s, known x’s,const,stat) – Result is a 5 x 2 array – Arguments known y’s – column of incremental known x’s – column of cumulative const – “false” sets b = 0 stat – “true” provides full array

Input Data for 1 to 2 Incremental vs. 1 Cumulative

Create Array of Regression Statistics Select 5 x 2 matrix of cells Type LINEST function in upper left cell Keystroke

Statistics Based on Residual Compute variance of regression as sum of squared residuals divided by the degrees of freedom (the mean square error, MSE) –Its square root, s, is standard error of regression The R 2 or percent of explained variance: 1-R 2 = divide MSE by total variance Standard deviation of constant –Use to test significance of constant Standard deviation of coefficient –Use to test significance of coefficient

Selecting Array Components INDEX Function in Excel will return array elements Form - INDEX(ARRAY, ROW #, COLUMN #) –INDEX(LINEST, 1, 1) = M –INDEX(LINEST, 1, 2) = B –INDEX(LINEST, 2, 1) =SE m –INDEX(LINEST, 2, 2) =SE b –INDEX(LINEST, 4, 2) =Degrees of Freedom

Test Significance Of Intercept Calculate Student t statistic, B / SE b Select Significance Level –Judgment (selected 0.05 for this example) Excel Function for probability value from Student t distribution -TDIST(ABS(B/ SE b ), Degrees Freedom, Tails) -Set Tails = 2 For 2-tail Distribution Compare result to selected significance level

Results Significance of Intercept

The “One Factor Model” Age-toAge d Factor= incremental d loss/cumulative d-1 Incremental loss d = f(d) * cumulative d-1 This is equivalent to weighted regression where –Y is incremental at d –X is cumulative at d-1 –B, the coefficient, is the (LDF – 1.0) –The regression has no constant term –Weights are x i /  x j instead of x i 2 /  x j 2

Linear Regression Through the Origin-Statistic Array

Results Linear Regression Through the Origin

Linear Regression Projections

Residual Results

Graphs Based on Residual Plot residual versus Y Plot residual versus predicted Y Plot residual versus X Plot residual versus variables not in regression (i.e., age, calendar year)

Chain Ladder Model Residual -1

Chain Ladder Model Residual -2

Chain Ladder Model Residual -3

Chain Ladder Model Residual -4

Chain Ladder Model Residual -5

Additional Models What to do if tests of assumptions underlying chain ladder fail? Alternative Models –Linear with Constant –Additive Model –Bornhuetter-Ferguson (BF) –Cape Cod-Special Case of BF (CC)

Approach to CC and BF Estimate Ultimate Claim Counts Parameterize incremental severity by functional form f(d)*h(w), where f(d) varies by development age and h(w) varies by accident year Fitting accomplished by an iterative approach with constraints (Solver add-in to Excel) Cape Cod assumes h(w) constant over all accident years (reduces parameters)

Claim Count

Chain Ladder Method-Claim Count

Claim Count Incremental vs Cumulative-1

Claim Count Incremental vs Cumulative-2

Claim Count Incremental vs Cumulative-3

Claim Count Residual - 1

Claim Count Residual - 2

Significance of Intercept-Claim Count

Linear Regression Results-Claim Count

Linear Regression Projections

Observed Severity

Initial Seed for f(d) and h(w) Cape Cod F(d) BASED ON THE PAYMENT PATTERN FROM THE CHAIN LADDER MODEL H(w) BASED ON THE ULTIMATES FROM THE CHAIN LADDER MODELS (LOSS DIVIDED BY CLAIM COUNTS) 5,943 FOR ALL YEARS

Iterative Process Results-CC

Fitted Ultimates and Projected Development-CC

Initial Seed for f(d) and h(w) Bornhuetter-Ferguson F(d) BASED ON FINAL PATTERN FROM CAPE COD MODEL H(w) BASED ON THE ULTIMATES FROM THE CHAIN LADDER MODELS (LOSS DIVIDED BY CLAIM COUNTS)

Iterative Process Results-BF

Fitted Ultimates and Projected Development-BF

Summary of Projected Ultimates

Comparison of Prediction of Incremental Loss