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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. 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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