Reinsurance of Long Tail Liabilities

Dr Glen Barnett and Professor Ben Zehnwirth

Where this started

• Were looking at modelling related ’s◤segments, LoBs

• started looking at a variety of indiv. XoLdata sets

Non proportional reinsurance

• Typical covers include individual excess of loss and ADC (retrospective and prospective)

• Major aim is to alter the cedant’s risk . profile (e.g. reduce risk based capital%)

(spreading risk → proportional)

In this talk -

• Develop multivariate model for related triangles

• discover sometimes coefficient of variation of aggregate losses net of some non-proportional reinsurance is not smaller than for gross.

Trends occur in three directions

Payment year trends

• project past the _ end of the data

• very important to _ model changes

11

22

1 …00d

t = w+d

Development year

Calendar (Payment) year

Accident year

w

Projection of trends

Inflation

• payment year trend

• acts in percentage terms (multiplicative)

• acts on incremental payments

• additive on log scale

• constant % trends are linear in logs

• trends often fairly stable for some years

Simple model• Model changing trends in log-incrementals _ (“percentage” changes)

• directions not independent _ ⇒ can’t have linear trends in all 3

• trends most needed in payment and _ development directions

⇒ model accident years as (changing) levels

Probabilistic modeldata = trends + randomness

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9

-2

-1.5

-1

-0.5

0

0.3365+-0.1096

-0.4761+-0.0357

-0.2770+-0.0284

Wtd Std Res vs Dev. Yr

0 1 2 3 4 5 6 7 8 9

-1.5

-1

-0.5

0

0.5

1

1.5

2

No one model

log(pw,d) = yw,d = w+ i + j + w,d

d

i=1

w+d

j=1

levels for acci. years Payment year trends

adjust for economic inflation, exposure (where sensible)

Development trends

randomness

N(0,2d)

Framework – designing a model

• The normal error term on the log scale (i.e. w,d ~ N(0,2

d) ) - integral part of model.

• The volatility in the past is projected into the future.

• Would never use all those parameters at the same time (no predictive ability)

• parsimony as important as flexibility (even more so when forecasting).

• Model “too closely” and out of sample predictive error becomes huge

• Beware hidden parameters (no free lunch)

• Just model the main features. Then

• Check the assumptions!

• Be sure you can at least predict recent past

Wtd Std Res vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 13

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Wtd Res Normality Plot

N = 85, P-value is greater than 0.5, R^2 = 0.9895-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1-0.8-0.6-0.4-0.2

00.20.4

0.60.8

Prediction

• Project distributions (in this case logN)

• Predictive distributions are correlated

• Simulate distribution of aggregates

Related triangles (layers, segments, …)

• multivariate model

• each triangle has a model capturing _ trends and randomness about trend

• correlated errors (⇒ 2 kinds of corr.)

• possibly shared percentage trends

• find trends often change together

• often, correlated residuals

-2.5-2

-1.5-1

-0.50

0.51

1.52

2.5

-3 -2 -1 0 1 2 3

LOB1 vs LOB3 Residuals

Correlation in logs generally good – check!

good framework ⇒

understand what’s happening in data

Find out things we didn’t know before

Net/Gross data (non-proportional reins)

• find a reasonable combined model Wtd Std Res vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Wtd Std Res vs Acc. Yr

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

2 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Wtd Std Res vs Cal. Yr

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Wtd Std Res vs Fitted

-6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Wtd Std Res vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-2-1.5

-1

-0.5

00.5

1

1.5

22.5

Wtd Std Res vs Acc. Yr

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

2 2

-2-1.5

-1

-0.5

00.5

1

1.5

22.5

Wtd Std Res vs Cal. Yr

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

-2-1.5

-1

-0.5

00.5

1

1.5

22.5

Wtd Std Res vs Fitted

-6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

• trend changes in the same place (but generally different percentage changes).

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0.4926+-0.2006

-0.3613+-0.0238

Acc. Yr Trends

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

0.4199+-0.1629

-0.4199+-0.1629 -0.4605

+-0.1246

Cal. Yr Trends

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0.0682+-0.0131

MLE Variance vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.50.55

0.9471

0.3562

2.8073

1.0559

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0.5

1

1.5

2

2.5

3

3.5

0.4926+-0.2006

-0.3110+-0.0250

Acc. Yr Trends

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

0.2781+-0.1933

-0.2781+-0.1933 -0.1354

+-0.1194

Cal. Yr Trends

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.0000+-0.0000

MLE Variance vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9471

0.3562

2.8073

1.0559

• Correlation in residuals about 0.84.

• Gross has superimposed inflation running at about 7%, Net has 0 inflation (or very slightly –ve; “ceded the inflation”)

• Bad for the reinsurer? Not if priced in.

• But maybe not so good for the cedant:

CV of predictive distn of aggregateGross 15%Net 17%

(process var. on log scale larger for Net)

 here ⇒ no gain in CV of outstanding

 

Don’t know exact reins arrangements,

But this reinsurance not doing the job

(in terms of, CV. RBC as a %)

(CV most appropriate when pred. distn of aggregate near logN)

Another data set

Three XoL layers

A: <$1M (All1M)

B: <$2M (All2M)

C: $1M-$2M (1MXS1M)

 

(C = B-A)

 Similar trend changes

(dev. peak shifts later)

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0

1

2

3

4

5

6

1.0919+-0.1094

0.0000+-0.0000 -0.3786

+-0.0482

Acc. Yr Trends

85 86 87 88 89 90 91 92 93 94 95 96 97 98

5

6

7

8

9

10

-0.4689+-0.1641

0.4689+-0.1641

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98

-2

-1

0

1

2

3

0.1115+-0.0075

MLE Variance vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.3127 2.5826

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0

1

2

3

4

5

6

74.8824

+-0.4335

1.2665+-0.1194

0.1259+-0.0154

-0.2747+-0.0529

Acc. Yr Trends

85 86 87 88 89 90 91 92 93 94 95 96 97 98

4

5

6

7

8

9

10

11

-0.3858+-0.1770

0.5210+-0.1764

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98

-3

-2

-1

0

1

2

3

0.0000+-0.0000

MLE Variance vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.3127 2.5826

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0

1

2

3

4

5

6

74.6244

+-0.4111

1.1438+-0.1125

0.0472+-0.0051 -0.3403

+-0.0495

Acc. Yr Trends

85 86 87 88 89 90 91 92 93 94 95 96 97 98

5

6

7

8

9

10

11

-0.0154+-0.0061

-0.4632+-0.1678

0.4786+-0.1677

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98

-3

-2

-1

0

1

2

3

0.0716+-0.0056

MLE Variance vs Dev. Yr

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.3127 2.5826

inflation higher in All1M, none in higher layer. Need to look

1

2X

(other model diagnostics good)

residual corrn very high about trends (0.96+)

Wtd Std Res vs Cal. Yr

89 90 91 92 93 94 95 96 97 98

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Wtd Std Res vs Cal. Yr

89 90 91 92 93 94 95 96 97 98

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Wtd Std Res vs Cal. Yr

89 90 91 92 93 94 95 96 97 98

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

1

2X

Residuals against calendar years

Forecasting

Layer CV Mean($M)All1M 12% 4951MXS1M 12% 237All2M 12%  731

ceding 1MXS1M from All2M doesn’t reduce CV 

consistent 

Scenario

Reinsure losses >$2M?

Not many losses. >$1M?

Not any better

Retrospective ADC

250M XS 750M on All2M

Layer CVAll2M 12%Retained 8%Ceded 179%

“Layers” (Q’ly data)

• decides to segment

• many XoL layers

• similar trends – e.g. calendar trend change 2nd qtr 97

some shared % trends

(e.g. low layers share with ground-up)

Cal. Qtr Trends

91 92 93 94 95 96 97 98 99 00 01 02 032 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2

-3

-2

-1

0

1

2

3

4

0.0516+-0.0026

0.0217+-0.0016

Cal. Qtr Trends

91 92 93 94 95 96 97 98 99 00 01 02 032 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2

-2

-1

0

1

2

3

4

0.0612+-0.0045

0.0312+-0.0019

• peak in development comes later for higher layers

Dev. Qtr Trends

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

1.2541+-0.0258

-0.6124+-0.0121

-0.4172+-0.0141

-0.3581+-0.0081

-0.2819+-0.0045

-0.1583+-0.0052

Dev. Qtr Trends

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48

-4

-3

-2

-1

0

1

2

1.2541+-0.0258

0.3860+-0.0139

0.0000+-0.0000

-0.2819+-0.0045

-0.1300+-0.0041

0-25 50-75

Weighted Residual Correlations Between Datasets 

  0-25 25-50 50-75 75-100 100-150 150-250 All

0 to 25 1 0.30 0.13 0.09 0.08 0.00 0.37

25 to 50 0.30 1 0.30 0.13 0.08 0.02 0.39

50 to 75 0.13 0.30 1 0.45 0.22 0.05 0.48

75 to 100 0.09 0.13 0.45 1 0.50 0.16 0.55

100 to 150 0.08 0.08 0.22 0.50 1 0.34 0.63

150 to 250 0.00 0.02 0.05 0.16 0.34 1 0.57

All 0.37 0.39 0.48 0.55 0.63 0.57 1

• Correlations higher for nearby layers

Forecasting

Aggregate outstanding 

Layers CV 0-25 4.2% 0-100 3.9% 0+ 3.9%

• Individual excess of loss not really helping here

• Retrospective ADC – 25M XS 400M

⇒ cedant’s CV drops from 3.9% to 3.4%

 

Summary

• CV should reduce as add risks

• non-proportional cover should reduce CV as we cede risk

Summary

• XoL often not reducing CV

• Suitable ADC/Stop-Loss type covers generally do reduce cedant CV

 

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