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 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 …0 d 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 model data = trends + randomness No one model

log(p w,d ) = y w,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,  2 d ) 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

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 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

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

Correlation in residuals about 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 dist n of aggregate Gross 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) inflation higher in All1M, none in higher layer. Need to look 1 2 X

(other model diagnostics good) residual corr n very high about trends (0.96+) 1 2 X Residuals against calendar years

Forecasting LayerCV Mean($M) All1M 12%495 1MXS1M 12%237 All2M12% 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 CV All2M 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)

peak in development comes later for higher layers

Weighted Residual Correlations Between Datasets All 0 to to to to to to All Correlations higher for nearby layers

Forecasting Aggregate outstanding Layers CV % % %

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

Arial Bold 35 Arial 32

Arial Bold 35 Arial 32