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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. XoL data sets. Non proportional reinsurance.

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slide1

Reinsurance of Long Tail Liabilities

Dr Glen Barnett and Professor Ben Zehnwirth

slide2

Where this started

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

• started looking at a variety of indiv. XoL data sets

slide3

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)

slide4

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.

slide5

Development year

0

1 …

d

1

2

Calendar (Payment) year

t = w+d

w

Accident year

Trends occur in three directions

Projection of trends

Payment year trends

• project past the _ end of the data

• very important to _ model changes

slide6

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

slide7

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

slide8

Probabilistic model

data = trends + randomness

No one model

slide9

randomness

N(0,2d)

d

i=1

w+d

j=1

log(pw,d) = yw,d = w+ i + j + w,d

Payment year trends

levels for acci. years

adjust for economic inflation, exposure (where sensible)

Development trends

Framework – designing a model

slide10

• The normal error term on the log scale (i.e. w,d ~ N(0,2d) ) - integral part of model.

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

slide11

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

•parsimony as important as flexibility (even moreso when forecasting).

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

•Beware hidden parameters (no free lunch)

slide12

•Just model the main features. Then

•Check the assumptions!

•Be sure you can at least predict recent past

slide13

Prediction

•Project distributions (in this case logN)

•Predictive distributions are correlated

•Simulate distribution of aggregates

slide14

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

slide15

LOB1 vs LOB3 Residuals

2.5

2

1.5

1

0.5

0

-3

-2

-1

0

1

2

3

-0.5

-1

-1.5

-2

-2.5

• find trends often change together

• often, correlated residuals

Correlation in logs generally good – check!

slide16

good framework ⇒

understand what’s happening in data

Find out things we didn’t know before

slide17

Net/Gross data (non-proportional reins)

• find a reasonable combined model

slide19

• 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.

slide20

• But maybe not so good for the cedant:

CV of predictive distn of aggregate

Gross 15%

Net 17%

(process var. on log scale larger for Net)

 here ⇒ no gain in CV of outstanding

slide21

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)

slide22

Another data set

Three XoL layers

A: <$1M (All1M)

B: <$2M (All2M)

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

(C = B-A)

slide23

 Similar trend changes

(dev. peak shifts later)

1

X

2

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

slide24

Residuals against calendar years

1

residual corrn very high about trends (0.96+)

X

2

(other model diagnostics good)

slide25

Forecasting

Layer CV Mean($M)

All1M 12% 495

1MXS1M 12% 237

All2M 12%  731

ceding 1MXS1M from All2M doesn’t reduce CV 

consistent

slide26

Scenario

Reinsure losses >$2M?

Not many losses. >$1M?

Not any better

slide27

Retrospective ADC

250M XS 750M on All2M

Layer CV

All2M 12%

Retained 8%

Ceded 179%

slide28

“Layers” (Q’ly data)

• decides to segment

• many XoL layers

slide29

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

some shared % trends

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

slide31

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

slide32

Forecasting

Aggregate outstanding 

Layers CV

0-25 4.2%

0-100 3.9%

0+ 3.9%

slide33

• Individual excess of loss not really helping here

• Retrospective ADC – 25M XS 400M

⇒ cedant’s CV drops from 3.9% to 3.4%

slide34

Summary

• CV should reduce as add risks

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

slide35

Summary

• XoL often not reducing CV

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

slide36

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