Reserving Risk and Solvency II

Transcription

1 Reserving Risk and Solvency II Peter England, PhD Partner, EMB Consultancy LLP Applied Probability & Financial Mathematics Seminar King s College London November EMB. All rights reserved. Slide 1

2 Agenda Part 1 Solvency II Background Part 2 Reserving and Reserving Risk The traditional actuarial view (looking over the lifetime of the liabilities) Solvency II and the 1 year view Note: Everything in this presentation is an oversimplification of methods that are used in practice 21 EMB. All rights reserved. Slide 2

3 Theoretical requirements for estimating capital Risk Profile Risk Measure A distribution of profit/loss? A distribution of (some definition of) net assets? Standard Deviation? Value-at-Risk? Tail Value-at-Risk? Etc Risk Tolerance 3 x SD 99.5% VaR 95% TVaR Time Horizon 21 EMB. All rights reserved. Slide 3 One year? Ultimate?

4 DIRECTIVE OF THE EUROPEAN PARLIAMENT Article 11 The Solvency Capital Requirement shall be calibrated so as to ensure that all quantifiable risks to which an insurance or reinsurance undertaking is exposed are taken into account. With respect to existing business, it shall cover unexpected losses. It shall correspond to the Value-at-Risk of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a oneyear period. So it seems straightforward to estimate the SCR using a simulation-based model: simply create a simulated distribution of the basic own funds over 1 year, then calculate the 99.5%. 21 EMB. All rights reserved. Slide 4

5 DIRECTIVE OF THE EUROPEAN PARLIAMENT Articles 88 and 75 Article 88 Basic own funds shall consist of the following items: (1) the excess of assets over liabilities, valued in accordance with Article 75 and Section 2 ; (2) subordinated liabilities. Article 75 Member States shall ensure that, unless otherwise stated, insurance and reinsurance undertakings value assets and liabilities as follows: (a) assets shall be valued at the amount for which they could be exchanged between knowledgeable willing parties in an arm's length transaction; (b) liabilities shall be valued at the amount for which they could be transferred, or settled, between knowledgeable willing parties in an arm's length transaction. 21 EMB. All rights reserved. Slide 5

6 A Projected Balance Sheet View When projecting Balance Sheets for solvency, we have an opening balance sheet with expected outstanding liabilities Opening Balance Sheet The bulk of those liabilities are the reserves (provisions) set aside to pay unsettled claims that have arisen on policies sold in the past Year 1 Balance Sheet We then project one year forwards, simulating the payments that emerge in the year, and require a closing balance sheet, with (simulated) expected outstanding liabilities conditional on the payments in the year 21 EMB. All rights reserved. Slide 6

7 Solvency Capital Requirements Non-Life Companies Asset Risk: Movement in market value of assets Default Risk on assets, reinsurance and debtors Operational risk SCR for Overall Company Reserve risk on existing obligations 21 EMB. All rights reserved. Slide 7 Underwriting risk on new business Catastrophe risk on existing obligations and new business

8 Building simulation-based internal models For each component of the capital model, a statistical model needs to be proposed to describe the observed phenomena These are representations of real life By necessity they are simplifications Each component model can be criticised All models are wrong, but some are useful The components need to be connected together in a meaningful way Recognising the dependencies between risk types and risk drivers (Using copulas or causal relationships described by formulae) 21 EMB. All rights reserved. Slide 8

9 Reserving & Reserving Risk 21 EMB. All rights reserved. Slide 9

10 Example Data for One Line of Business Observed incremental values , ,94 61, ,94 527, , , ,95 227,229 67, , ,21 933,894 1,183, ,745 32, ,84 266, , ,57 1,1, ,219 1,16,654 75, , ,992 28, ,68 1,18,25 776,189 1,562,4 272, ,53 26, ,16 693,19 991, ,488 54,851 47, , ,85 847,498 85,37 75, , ,631 1,131,398 1,63, ,48 1,61,648 1,443, , , ,14 Dev Factors EMB. All rights reserved. Slide

11 Taylor & Ashe Data Fitted incremental values (chain ladder model) Reserve 1 27,61 672,617 74, , ,35 292, , ,35 272,66 67, , , ,176 1,49, ,26 47, , , ,669 94,634 94, , , ,264 1,38, ,388 43, ,957 25, ,833 93, , , , ,652 1,23, , ,29 364, ,19 37,179 92,268 79, , , , ,2 519, , , , ,456 84, , , , , ,53 546, , , , ,132 89,16 1,419, , ,923 1,22,175 1,93,189 65, ,51 389, , ,534 98,588 2,177, ,648 1,169,77 1,225,143 1,31, ,788 58, ,66 316, ,73 118,164 3,92, , ,733 1,18,834 1,89,616 63, , ,76 263, ,241 98,266 4,278, ,14 856,84 897,41 959, , , , , ,255 86,555 4,625,811 Total 21 EMB. All rights reserved. Slide 11 18,68,856

12 Reserving Risk The traditional actuarial view Looking over the lifetime of the liabilities Solvency II and the 1 year view 21 EMB. All rights reserved. Slide 12

13 Reserving Risk The Traditional Actuarial View Reserving is concerned with forecasting outstanding liabilities There is uncertainty associated with any forecast Reserving risk attempts to capture that uncertainty We are interested in the predictive distributions of ultimate cost of claims AND the associated cash flows We need methods that can provide those distributions The methods are still evolving 21 EMB. All rights reserved. Slide 13

14 Stochastic claims reserving This has become a new academic discipline It has spawned several PhDs Numerous papers appearing in academic journals Presentations at every actuarial conference A book has appeared There is a Wikipedia page 21 EMB. All rights reserved. Slide 14

15 Sources of Uncertainty There is uncertainty in the number of claims Claims reporting delays - the IBNR problem There is uncertainty in the size and timing of payments Some claims take many years to settle The case reserves are just estimates Settlements may be set by courts, and may be subjective Legislation may change Inflation is uncertain And so on Actuarial and statistical modelling just uses the uncertainty observed in the past data to help measure the uncertainty in the forecast This uncertainty may be over or under represented in the data Actuaries propose models, which contain parameters, to assist with the forecasting problem The parameters are estimated from the observed data sample, and are subject to uncertainty There are several models that could be proposed, but which one is correct? Actually, they will all be wrong Some will be useful 21 EMB. All rights reserved. Slide 15

16 Sources of Uncertainty Process Error Variability due to the underlying claims generating process Parameter Error Taken into account when modelling Variability due to uncertainty in the parameters Model Error Variability due to mis-specification of the model 21 EMB. All rights reserved. Slide 16 Not usually quantified, but not ignored

17 Conceptual Framework Reserve Estimate (Measure of Location) Variability (Prediction Error) Traditional deterministic methods Statistical assumptions required Prediction Error = SD of Forecast Can be estimated analytically Predictive Distribution 21 EMB. All rights reserved. Slide 17 Usually cannot be obtained analytically Simulation methods required

18 We can do this the easy way, or we can do it the hard way 21 EMB. All rights reserved. Slide 18

19 Stochastic Reserving Over-dispersed Poisson Model Doing it the HARD way 21 EMB. All rights reserved. Slide 19

20 Over-Dispersed Poisson Model Cij Incremental claims in origin year i and development year j Cij ~ ODP( ij, j ) Var C Cij E Cij ij ij j ij With constant scale parameters, j j Writing log( ij ) ij with ηij c ai b j gives the same forecasts as the chain ladder model 21 EMB. All rights reserved. Slide 2 Variance proportional to expected value

21 MCMC Results ODP - Chain Ladder Model Only Maximum Likelihood Parameter Number 21 EMB. All rights reserved. Slide 21 Name GIBBS Value Standard Error Expected Value Standard Error 1 Mean Row (1) Row (2) Row (3) Row (4) Row (5) Row (6) Row (7) Row (8) Row (9) Row (1) Col (1) Col (2) Col (3) Col (4) Col (5) Col (6) Col (7) Col (8) Col (9) Col (1)

22 Variability in Claims Reserves Variability of a forecast Includes estimation variance and process variance prediction error (process variance estimation variance) Calculated analytically, the problem reduces to estimating the two components. See, for example, E & V (1999, 22) This is doing it the HARD way Note: prediction error is also known as root mean square error of prediction (RMSEP), and is just the SD of the forecast 21 EMB. All rights reserved. Slide

23 Prediction Variance Individual cell MSE 2 j ij ij Var ( ij ) Row/Overall total MSE 21 EMB. All rights reserved. Slide 23 2 Cov j ij ij 2 Var ( ij ) ( ij ik ) ij ik

24 Prediction Errors (%) ODP Chain ladder model (constant scale) Year Total 21 EMB. All rights reserved. Slide 24 ODP Expected Reserves ODP Prediction Error 94, % 469,511 46% 79,638 37% 984,889 31% 1,419,459 26% 2,177,641 23% 3,92,31 2% 4,278,972 24% 4,625,811 43% 18,68,856 16%

25 Stochastic Reserving Over-dispersed Poisson Model Doing it the EASY way 21 EMB. All rights reserved. Slide 25

26 Stochastic Reserving: Bootstrapping Bootstrapping assumes the data are independent and identically distributed With regression type problems, the data are often assumed to be independent but are not identically distributed (the means are different for each observation) However, the residuals are usually i.i.d, or can be made so Therefore, with regression problems, it is common to bootstrap the residuals instead 21 EMB. All rights reserved. Slide 26

27 Reserving and Bootstrapping Define and fit statistical model Obtain residuals and pseudo data Re-fit statistical model to pseudo data Obtain forecast, including process error Any model that can be clearly defined can be bootstrapped 21 EMB. All rights reserved. Slide 27

28 Bootstrapping the Chain Ladder Over-dispersed Poisson model 1. Fit chain ladder model and obtain fitted incremental values 2. Obtain (scaled) Pearson residuals 3. Resample residuals1 4. Obtain pseudo data, given rij*, ij, j rij Cij ij j ij Cij* rij* j ij ij 5. 1 At Use chain ladder to re-fit model, and estimate future incremental payments this stage, we use the adjusted scaled residuals 21 EMB. All rights reserved. Slide 28

29 Bootstrapping the Chain Ladder 6. Simulate observation from process distribution assuming mean is incremental value obtained at Step 5 7. Repeat many times, storing the reserve estimates (this gives the predictive distribution) 8. Prediction error is then standard deviation of results Note: Where curve fitting has been used for smoothing and extrapolation (for tail estimation), replace the chain ladder model in steps 1 and 5 by the actual model used 21 EMB. All rights reserved. Slide 29

30 Example 21 EMB. All rights reserved. Slide 3

31 Taylor & Ashe Data Observed incremental values , ,94 61, ,94 527, , , ,95 227,229 67, , ,21 933,894 1,183, ,745 32, ,84 266, , ,57 1,1, ,219 1,16,654 75, , ,992 28, ,68 1,18,25 776,189 1,562,4 272, ,53 26, ,16 693,19 991, ,488 54,851 47, , ,85 847,498 85,37 75, , ,631 1,131,398 1,63, ,48 1,61,648 1,443, , , ,14 Dev Factors EMB. All rights reserved. Slide

32 Taylor & Ashe Data Fitted incremental values (chain ladder model) Reserve 1 27,61 672,617 74, , ,35 292, , ,35 272,66 67, , , ,176 1,49, ,26 47, , , ,669 94,634 94, , , ,264 1,38, ,388 43, ,957 25, ,833 93, , , , ,652 1,23, , ,29 364, ,19 37,179 92,268 79, , , , ,2 519, , , , ,456 84, , , , , ,53 546, , , , ,132 89,16 1,419, , ,923 1,22,175 1,93,189 65, ,51 389, , ,534 98,588 2,177, ,648 1,169,77 1,225,143 1,31, ,788 58, ,66 316, ,73 118,164 3,92, , ,733 1,18,834 1,89,616 63, , ,76 263, ,241 98,266 4,278, ,14 856,84 897,41 959, , , , , ,255 86,555 4,625,811 Total 21 EMB. All rights reserved. Slide 32 18,68,856

33 Taylor & Ashe Data Scaled residuals : ODP with constant scale parameter Scale^ EMB. All rights reserved. Slide

34 Prediction Errors (%) ODP Chain ladder model (constant scale) Year Total 21 EMB. All rights reserved. Slide 34 ODP Analytic Method ODP Bootstrap Method

35 Taylor & Ashe Data Scaled residuals : ODP with constant scale parameter Development Residuals (Scaled, Bias-Adjusted, Zero-Average) With Scale Values Residuals R e si d u a l ScaleValues(Initial) D e ve l o p m e n t Ye a r Note that the volatility is lower at the earlier and later development periods 21 EMB. All rights reserved. Slide 35 ScaleValues (Residuals) ScaleValues (Forecasting)

36 Taylor & Ashe Data Scaled residuals : ODP with non-constant scale parameter Scale^ EMB. All rights reserved. Slide

37 Taylor & Ashe Data Scaled residuals: ODP with non-constant scale parameter Development Residuals (Scaled) With Scale Values Residuals ScaleValues(Initial) 1. 3 R e si d u a l ScaleValues (Residuals).5 25 ScaleValues (Forecasting) D e ve l o p m e n t Ye a r Note that the residuals are standardised better when using non-constant scale parameters 21 EMB. All rights reserved. Slide 37

38 Taylor & Ashe Data ODP: Constant vs Non-Constant Scale Parameters Simulated Simulated Constant Scale Non-Constant Scale Accident Prediction Prediction Prediction Prediction Year Error Error % Error Error % Total 21 EMB. All rights reserved. Slide , , ,934 36, ,745 5, ,481 1,6,473 2,25,898 2,992,296.% 119.% 46.2% 36.9% 31.% 26.4% 22.9% 2.1% 24.7% 43.3% 15.9% 43,882 19, ,59 256,31 398, , ,245 89,457 1,285,56 2,228,677.% 45.3% 23.% 19.8% 25.7% 27.8% 24.2% 18.7% 18.9% 27.6% 11.9%

39 Distribution of Ultimates by Origin Period Ultim ates by - Accident Year 13,, 12,, 9% Percentile 11,, 75% Percentile 1,, C l a i m Amo u n ts 9,, 25% Percentile 8,, 1% Percentile 7,, Mean 6,, 5,, 4,, 3,, 2,, 1,, Acci d e n t Ye a r 21 EMB. All rights reserved. Slide

40 Claims Development Percentile Fan Chart Single Origin Year Unscaled Paid Claims Development ,, Paid Claims C l a i m Amo u n ts 7,, 9% Percentile 6,, 75% Percentile 5,, 25% Percentile 1% Percentile 4,, Mean 3,, 2,, 1,, D e ve l o p m e n t Ye a r 21 EMB. All rights reserved. Slide

41 MCMC Methods 21 EMB. All rights reserved. Slide 41

42 Reserving and Bayesian Methods Define statistical model Obtain distribution of parameters using MCMC methods Obtain forecast, including process error 21 EMB. All rights reserved. Slide 42

43 MCMC Results ODP - Chain Ladder Model Only Uniform Priors Maximum Likelihood Parameter Number 21 EMB. All rights reserved. Slide 43 Name GIBBS Value Standard Error Expected Value Standard Error 1 Mean Row (1) Row (2) Row (3) Row (4) Row (5) Row (6) Row (7) Row (8) Row (9) Row (1) Col (1) Col (2) Col (3) Col (4) Col (5) Col (6) Col (7) Col (8) Col (9) Col (1)

44 Prediction Errors (%) ODP Chain ladder model (constant scale) Year Total 21 EMB. All rights reserved. Slide 44 ODP Analytic Method ODP ODP Bootstrap Bayesian Method Method

45 Total Outstanding Liabilities Density Chart Bootstrap vs MCMC Results Total Outstanding Liabilities 2E-7 1.8E-7 MCMC 1.6E-7 1.4E-7 Density 1.2E-7 Bootstrap 1E-7 8E-8 6E-8 4E-8 2E-8 1,, 12,, 14,, 16,, 18,, 2,, 22,, Range 21 EMB. All rights reserved. Slide 45 24,, 26,, 28,, 3,, 32,,

46 Solvency II For Solvency II, a 1 year perspective is taken, requiring a distribution of the expected value of the liabilities after 1 year, for the 1 year ahead balance sheet in internal capital models Under Solvency II, reserve risk is based on the distribution of profit/loss on reserves after 1 year and is measured using the standard deviation of that distribution Note: This is different from the traditional actuarial view of reserve risk 21 EMB. All rights reserved. Slide 46 Opening Balance Sheet Year 1 Balance Sheet

47 The one-year run-off result (undiscounted) (the view of profit or loss on reserves after one year ) For a particular origin year, let: The opening reserve estimate be R The reserve estimate after one year be R1 The payments in the year be The run-off result (claims development result) be C1 CDR1 Then CDR1 R C1 R1 U U1 Where the opening estimate of ultimate claims and the estimate of the ultimate after one year are U,U1 21 EMB. All rights reserved. Slide 47

48 The one-year run-off result in a simulation model The EASY way For a particular origin year, let: The opening reserve estimate be R The expected reserve estimate after one year be R1(i ) The payments in the year be The run-off result (claims development result) be C1(i ) CDR1(i ) Then CDR1(i ) R C1(i ) R1(i ) U U1(i ) Where the opening estimate of ultimate claims and the expected ultimate after one year are U, U1( i ) for each simulation i 21 EMB. All rights reserved. Slide 48

49 The one-year run-off result in a simulation model The EASY way 1. Given the opening reserve triangle, simulate all future claim payments to ultimate using bootstrap (or Bayesian MCMC) techniques. 2. Now forget that we have already simulated what the future holds. 3. Move one year ahead. Augment the opening reserve triangle by one diagonal, that is, by the simulated payments from step 1 in the next calendar year only. An actuary only sees what emerges in the year. 4. For each simulation, estimate the outstanding liabilities, conditional only on what has emerged to date. (The future is still unknown ). 5. A reserving methodology is required for each simulation an actuary-in-thebox is required*. We call this re-reserving. 6. For a one-year model, this will underestimate the true volatility at the end of that year (even if the mean across all simulations is correct). * The term actuary-in-the-box was coined by Esbjörn Ohlsson 21 EMB. All rights reserved. Slide 49

50 The standard actuarial perspective: forecasting outcomes over the lifetime of the liabilities, to their ultimate position A single accident year, 4 years developed Actual simulated future amounts 21 EMB. All rights reserved. Slide 5

51 One year ahead forecast 21 EMB. All rights reserved. Slide 51

52 Actual simulated future amounts 21 EMB. All rights reserved. Slide 52 Expected payments conditional on year 1 position

53 EMB ResQ Example 21 EMB. All rights reserved. Slide 53

54 Multiple 1 yr ahead CDRs An interesting result Creating cascading CDRs over all years gives the following results (SDs): Accident Year Total Number of years ahead Sqrt(Sum of ODP Squares) "Ultimo" 1 Yr 2 Yrs 3 Yrs 4 Yrs 5 Yrs 6 Yrs 7 Yrs 8 Yrs 9 Yrs 112, , ,617 23, ,44 316, , ,383 1,782,215 88,89 174, , , , , ,666 74,88 83, , , , , , ,819 75, ,28 149, ,99 218,26 277,58 77,428 17, , ,24 24,738 81, ,16 143, ,997 87, , ,235 78,74 153,65 72, , , , , , 262,356 36,61 38,68 378, , ,167 51,73 79,78 788,175 1,55,73 1,59,224 2,36,122 2,51,95 2,43,242 1,262, , ,53 453, , , ,21 72,515 2,999,369 3,7,867 The sum of the variances of the repeated 1 yr ahead CDRs (over all years) equals the variance over the lifetime of the liabilities The simulation based results are approximate This means that we expect the risk under the 1 year view to be lower than the standard ultimo perspective 21 EMB. All rights reserved. Slide 54

55 Summary Reserving Risk The Solvency II one-year view of reserve risk is different from the traditional actuarial view The two views can be reconciled in a simulation framework It is possible to use results from the ultimo perspective to obtain a one-year perspective In a simulation based framework, we can obtain both: A distribution of the expected outstanding liabilities after 1 year A standard deviation of the profit/loss on reserves after 1 year There are many complexities when modelling in practice 21 EMB. All rights reserved. Slide 55

56 Summary Solvency II Internal Models Solvency II Internal Models comprise a set of inter-connected statistical models for each risk type The models may be very complex, and make use of familiar statistical/financial techniques Monte Carlo simulation, extreme value theory, time series analysis, copulas, bootstrapping, MCMC methods, GLMs Economic scenario generators, coherent risk measures, capital allocation The models are not only used for estimating regulatory capital, but also for assisting with strategic business decisions Optimising reinsurance programmes, or business mix The power of computers is harnessed to assist with modelling, but the computational aspects are challenging Managing RAM and hard disk space, efficient algorithms, distributed computing 21 EMB. All rights reserved. Slide 56

57 Risk Theory and Simulation Modern computer simulation techniques open up a wide field of practical applications for risk theory concepts, without the restrictive assumptions, and sophisticated mathematics, of many traditional aspects of risk theory. - Daykin, Pentikainen and Pesonen, EMB. All rights reserved. Slide 57

58 References Mack, T (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin, 23, pp England, P and Verrall, R (1999). Analytic and bootstrap estimates of prediction errors in claims reserving, Insurance: Mathematics and Economics 25, pp England, P (22). Addendum to Analytic and bootstrap estimates of prediction errors in claims reserving, Insurance: Mathematics and Economics 31, pp England, PD & Verrall, RJ (22). Stochastic Claims Reserving in General Insurance, British Actuarial Journal 8, III, pp England, PD & Verrall, RJ (26). Predictive distributions of outstanding claims in general insurance, Annals of Actuarial Science 1, II, pp AISAM/ACME (27). AISAM/ACME study on non-life long tail liabilities. Merz, M & Wuthrich, MV (28). Modelling the Claims Development Result for Solvency Purposes. ASTIN Colloquium, Manchester. Ohlsson, E & Lauzeningks, J (28). The one-year non-life insurance risk. ASTIN Colloquium presentation, Manchester. Ohlsson, E & Lauzeningks, J (29). The one-year non-life insurance risk. Insurance: Mathematics and Economics 45, pp Diers, D (29). Stochastic re-reserving in multi-year internal models An approach based on simulations. ASTIN Colloquium, Helsinki. CP71 and CP75 (29) EMB. All rights reserved. Slide 58