Multi-Dimensional CredibilityMulti-Dimensional CredibilityExcess Work Comp ApplicationExcess Work Comp Application

Simplified Version of Least Simplified Version of Least Squares Credibility in Squares Credibility in GeneralGeneral

3

Loss Models Setup

An individual insured (policyholder) has n iid observations X1,…,Xn whose distribution is from a parameter

is an instance of a random variable with density

Define = E(Xj|) and v() = Var(Xj|)

is called the hypothetical mean and v() is the process variance– In classical statistics, is called the population mean, but Charles

Hewitt, a Bayesian, considered that to be a model construct, not a truly existing entity, and so called it hypothetical, and the terminology has persisted

Let = E v = Ev, a = Var[ v is the expected process variance and a is the variance of hypothetical

means

Bühlmann: estimate linearly by a0+ajXj minimizing expected squared error

Answer is a0 = (1 – z), ai = z/n i>0 where z = n/(n+k), k = v/a

Estimates by zX* + (1 – z) = + z(X* – ) = EX* + z(X* – EX*)

We will generalize the left side, but derive the right side

4

Simplified Version

Let X* be the mean of the Xj’s

Bühlmann’s result is to estimate by zX* + (1 – z).

Derivation of z is much simpler if you start with that instead of a0+ajXj.

Not giving up much by this simplification because best linear estimate of the mean is the sample mean.

Assumptions imply = X* + v()½ = + a½, where and are independent mean 0, variance 1 deviations.

Generalize this to having two estimators X and Y of C with expected squared errors of s2 and t2, respectively, where s and t might even be random variables themselves.

Find z that minimizes E{[C – zX + (z–1)Y]2}

5

Finding z

Find z that minimizes E{[C – zX + (z–1)Y]2} X = C + s, Y = C + t

Set derivative to zero– 0 = E{[C – zX + (z–1)Y][Y–X]} = E{[–zs + (z–1)t][t–

s]} = E[zs22 + (z–1)t22] = zE[s2] + (z–1)E[t2]– Thus z = E(t2) / [E(s2) + E(t2)]

In the credibility model E(s2) is the expected process variance and t2 is already a constant – the variance of the hypothetical means

Also z = [1/E(s2)] / [1/E(s2) + 1/E(t2)] so the weight on X is proportional to the reciprocal of its variance, and similarly for Y

This is a standard statistical result

Excess Pricing for Work Excess Pricing for Work Comp ClassesComp Classes

7

Workers Compensation Excess Pricing Model

Bureau excess prices traditionally based on hazard groups

Excess potential - very different across hazard groups– but also within hazard groups

Bureau methodology weights injury-type severity distributions by hazard group injury-type frequency splits

Can do that by class– Requires credibility procedure to get class distribution of

losses by injury type

8

Severity by Injury Type, Massachusetts:Large Loss Potential Is Driven by Fatal, PT

Fatal PT Major Minor TT

Mean $ 411,287

$ 896,725 $ 137,163

$ 15,826

$ 12,367

95th Percentile

$ 1,285,878

$ 2,566,482 $ 307,876

$ 42,187

$ 49,050

Ratio to TT Mean

33.3

72.5

11.1

1.3

1.0

95th Percentile

26.2

52.3

6.3

0.9

1.0

9

Differences in Injury-Type Frequencies Across and Within Hazard Groups: Ratios to Temporary Total

Means

HGFatal:TT PT:TT

Major:TT

1 0.21%0.33

% 6.10%

2 0.28%0.44

% 7.06%

3 0.69%0.74

% 11.61%

4 1.83%1.44

% 27.27%

95th Percentile Class*

HG Fatal:TT PT:TT

1 0.86% 0.74%

2 0.97% 1.47%

3 2.82% 2.66%

4 4.79% 2.77%

*95th percentile of larger classes

Hazard group means are very different but significant variation exists within each hazard group

10

Correlation of Ratios to TT Across Classes Hazard Group III

PT Major Minor

Fatal 39% 45% 20%

PT 52% 31%

Major 28%

Use correlations to better estimate class frequencies.

Major predictive of fatal and PT.

Credibility Including Credibility Including CorrelationCorrelation

12

Credibility with Correlation

Denote by V, W, X, Y - class ratios to TT for Fatal, PT, Major & Minor

Credibility Formula for Fatal for Class i:

– Evi + b(Vi – EVi) + c(Wi – EWi) + d(Xi – EXi) +e(Yi – EYi)

– Here Evi = EVi is the hazard group mean for Fatal:TT; b is usual z

Example credibilities for fatal for a class in HG III with 300 TT claims– b = 32.6%, c = 5.0%, d = 1.3%, e = 0.2%

Major frequency - over 15 times fatal – so factor of 1.3% is in ballpark of being like 20% for fatal

Minor frequency - over 50 times fatal – so factor of 0.2% has impact of a factor of 10% for fatal (assuming

differences from mean are of same magnitude as the mean)

How are these estimated?

13

Denote four injury types by V, W, X, and Y.

For the ith class, denote the population mean ratios (i.e., the true conditional, or “hypothetical” means) as vi , wi , xi , and yi .

Here these are mean ratios to TT.

Credibility with Correlation

14

We observe each class i for each time period t.

Denote by Wi the class sample mean ratio for all time periods weighted by exposures mit (TT claims), where there are N periods of observation.

Similarly for V, X, and Y.

Let mi denote the sum over the time periods t of the mit

m is the sum over classes i of the mi . Then within Var(Wit|wi ) = Wi

2/mi

N

t

N

tititit mWm

1 1

/

Notation

15

Assume a linear model and minimize expected squared error, where expectation is taken across all classes in the hazard group.

For PT this can be expressed as minimizing:E[(a + bVi + cWi + dXi + eYi – wi )

2]

The coefficients sought are a, b, c, d, and e. Differentiating wrt a gives:

a = – E( bVi + cWi + dXi + eYi – wi )

Plugging in that for a makes the estimate of wi =

Ewi + b(Vi – EVi ) + c(Wi – EWi ) + d(Xi – EXi ) + e(Yi – EYi )

estimate of wi

16

We have wi = Ewi + b(Vi – EVi ) + c(Wi – EWi ) + d(Xi – EXi ) + e(Yi – EYi )

Since in taking the mean across classes Ewi = EWi , c is the traditional credibility factor z.

The derivative of E[(a + bVi + cWi + dXi + eYi – wi )2] wrt b gives:

aEVi + E[Vi ( bVi + cWi + dXi + eYi – wi )] = 0

Plugging in for a then yields: 0 = E(bVi + cWi + dXi + eYi – wi )EVi + E[bVi

2 + cViWi + dViXi + eViYi – Viwi]

Using Cov(X,Y) = E[XY] – EXEY ,this can be rearranged to give: Cov(Vi ,wi) = b Var(Vi ) + c Cov(Vi ,Wi ) + d Cov(Vi ,Xi ) + e Cov(Vi ,Yi )

Doing the same for c, d, and e will yield three more equations that look like (3), but with the variance moving over one position each time.

Thus you will end up with four equations that can be written as a single matrix equation:

17

where C is the covariance matrix of the class by injury-type sample means Cov(Vi ,Yi ) etc.

e

d

c

b

C

,wY

,wX

,wW

,wV

ii

ii

ii

ii

)Cov(

)Cov(

)Cov(

)Cov(

You need estimates of all covariances - like estimating the EPV and VHM

But…with these you can solve this equation for b, c, d, and e to be used for PT. Repeat for the other injury types.

18

How’s That Working for How’s That Working for You?You?

19

Comparison to NCCI Hazard GroupsSum of Squared Errors for PT/TT RatiosThree Odd Years Predicted from Three Even Years

(1) (2) (3) (4)Prediction Prediction Prediction

Hazard Based on Based on Based onGroup HG Raw Even Cred. Proc.A 33.4 52.9 33.0 B 206.9 363.2 205.1 C 250.8 479.3 248.0 D 89.9 129.9 89.3 E 333.2 484.0 330.8 F 247.1 306.3 242.6 G 263.7 386.2 256.9

Total 1,425.0 2,201.7 1,405.6

20

Comparison to NCCI Hazard GroupsSum of Squared Errors for Injury Type Ratios to TTThree Odd Years Predicted from Three Even Years

(1) (2) (3) (4)Prediction Prediction Prediction

Injury Based on Based on Based onType HG Raw Even Cred. Proc.

Fatal 43.6 65.2 43.5 PT 34.7 83.6 34.6 Major PP 1,425.0 2,201.7 1,405.6 Minor PP 6,756.9 10,360.3 6,558.0 Med. Only 417,260.8 434,837.9 351,270.8

Conclusion: Slight improvement by this measure

21

Other Tests

Individual class ratios are highly variable

Grouping classes might show up the effects better

Quintiles test for a hazard group– Group the classes in the hazard group into 5 sets based on

ranking predicted ratio of injury count types to TT– Look at actual vs. predicted for those sets

22

Hazard Group D Quintiles Test for PT / TT Ratios

(1) (2) (3) (4) (5)Prediction Prediction Prediction

Odd Based on Based on Based onQuintile Relativity HG Raw Even Cred. Procedure

1 0.4951 1.0000 0.3065 0.5648 2 0.8634 1.0000 0.4260 0.8732 3 0.9861 1.0000 0.7513 1.0000 4 1.1269 1.0000 1.3473 1.1038 5 1.5215 1.0000 2.1547 1.4519

Mean 1.0000 1.0000 1.0000 1.0000 SSE 0.5618 0.7315 0.0105

23

Prediction Prediction PredictionInjury Based on Based on Based on

HG Type HG Raw Even Cred. Proc.A Fatal 0.13227 0.94431 0.18952 B Fatal 0.32630 1.79940 0.05637 C Fatal 0.83604 1.39413 0.03376 D Fatal 0.97498 0.87260 0.12111 E Fatal 0.49691 1.44023 0.05096 F Fatal 0.39060 1.35362 0.07280 G Fatal 0.55650 1.23015 0.06035

A PT 0.03941 1.94151 0.57993 B PT 0.38273 1.34145 0.11044 C PT 0.56175 0.55609 0.01180 D PT 0.56183 0.73151 0.01053 E PT 0.73195 0.82350 0.07050 F PT 0.56872 0.53817 0.01812 G PT 1.09139 0.52326 0.07946

A Major 0.58481 0.01988 0.05079 B Major 0.33888 0.03729 0.00870 C Major 0.38001 0.04108 0.00738 D Major 0.18900 0.03928 0.01850 E Major 0.28775 0.07476 0.01030 F Major 0.32418 0.04703 0.01781 G Major 0.58518 0.14046 0.00538

Sum of squared

prediction errors

Credibility better

except for HG A Fatal

and PT

24

Distribution of Credibility Indicated Class Means within Hazard GroupsRatio of PT / TT Counts

0

40

80A

0

40

80B

0

40

80C

0

40

80D

0

40

80E

0

40

80F

0 0. 00810585 0. 01621169 0. 02431754 0. 03242339 0. 04052923 0. 04863508 0. 05674093

0

40

80G

PT: TT

hg7

25

Distribution of Credibility Indicated Class Means within Hazard GroupsRatio of Major / TT Counts

0

15

30A

0

1530B

0

15

30C

0

15

30D

0

15

30E

0

15

30F

0. 014 0. 054 0. 094 0. 134 0. 174 0. 214 0. 254 0. 294 0. 334 0. 374 0. 414 0. 454 0. 494 0. 534

0

15

30G

Maj or : TT

hg7