Reinsurance Optimization GIE- AXA 06/07/2010

Transcription

1 Reinsurance Optimization GIE- AXA 06/07/2010 1

2 Agenda Introduction Theoretical Results Practical Reinsurance Optimization 2

3 Introduction As all optimization problem, solution strongly depend on criteria chosen and assumption made. Reinsurance optimization can be roughly separated into 2 different problematic: Pure theoretical : optimal contract design which try to find optimal reinsurance contract type (quota share versus XS) Economical Point of view : Knowing the type of contract, find the optimal parameters (retention or cession rate, capacity) under simple or sophisticated insurance wealth model Only second problematic has an practical interest 3

4 4 Some theoretical results

5 Statistical Approach: Optimal Structure Results based on Lehmann-Scheffe and Rao-Blackwell works and here, optimal reinsurance mean minimize insurer and reinsurer variance Let X = Xi + Xr aggregate loss amount (insurer part and reinsurer part) Let P = Pi + Pr Premium split which is defined by a crieteria (risk aversion for example) Hypothesis: X, Xi and Xr are square integrable Hypothesis : no loading, on insurance and reinsurance side. Results : optimal reinsurance contract is define by : Xi = min(x,c) with c define by E(Xi) = Pi 5

6 Statistical Approach: Optimal Structure Hypothesis : reinsurance loading is based on variance Results : optimal reinsurance contract is define by : with θ define by Xi = θx Var ( X i ) Var ( X ) 6

7 Economical Approach : Ruin Probability (1/2) Ruin Probability minimization Let B the insurer Benefit : Premiums Aggregated Losses B ( X ) X (1 ) E( X ) X Under a loading based on premium for insurer Remark E( B) E( X ) ( B) ( X ) Ruin event is defined by Losses Re serve Premiums X R ( X ) Ruin Probability is defined by Proba X R ( X ) 1 ProbaX R (1 ) E( X ) 7

8 Economical Approach : Ruin Probability (2/2) Ruin Probability With F Distribution of the VA: T the security coefficient : Proba X E( X ) ( X ) R E( B) ( B) Ruin 1 Proba 1 F( T) X E( X ) ( X ) R E( B) ( B) Minimize Ruin Probability is equivalent to maximize T Maximize T can be reach minimizing σ(b) using reinsurance Under simplify assumptions optimal treaty parameters could be computed in order to minimize Ruin Probability 8

9 Economical Approach : Value principle In 1957, Bruno Finetti proposed to change the point of view of actuaries, who focused on the risk of ruin. He insisted on the dividend paid to the shareholders. He introduced the model of value as the actualisation of expected cash flows. Initialy the model was quite simple: Value t E[ dividend 1 r t t K t ] Dividend depend on company result K is the needed capital 9

10 Economical Approach : Value principle Initial Finetti model weakness is the implicit risk neutral assumption. Dividend (what you win) have the same actualization rate as needed Capital (what you spend). Then, thanks to Modigliani-Miller theorem we know that risk management and consequently reinsurance is useless. To correct this weakness of the model, you should add more weight (in order to add fictional costs) on inflows as follow: Value t E[ dividend t (1 CoC) K ] 1 r t t Dividend depend on company result K is the needed capital CoC is the cost of holding capital The goal is to maximize Value 10

11 Economical Approach : Value principle Numerous articles generalize Finetti model to multi-period model. The common starting point of such programs is a cash flow equation (actually a stochastic differential equation) for the insurance company linking its inflows and outflows, which take the general form (budget equation): dw t ( Wt, ut ) dt dx ( Wt, ut ) dc( Ut ) dz( ut ) Where: W t represents the wealth of the firm at time t U t represents a vector of management controls (for example reinsurance) μ represents the expected rate of change of wealth (which is a function of current wealth and the controls X is the stochastic process representing the cumulative risks C is the cumulative external Capital supplied by shareholders Z is the cumulative dividends paid back to shareholders 11

12 Economical Approach : Value principle The objective is to maximize the market value of the firm over the set of all adapted increasing processes Z (dividend strategies) which are admissible (in the sense that it does not cause the bankruptcy of the firm) and over all management controls (Ut) The market Value of the firm is (Actualized Dividends less actualized injected capital): rt _ rt M ( W ) E e dzt (1 K) e dct W 0 W 0 0 M(W), called M-Curve, is the continuous version of Finetti model Such problem are optimal control problem and can be solved using Dynamic Programming techniques and leads to solves associated Hamilton-Jacoby-Bellman partial differential equation JeanBlanc & Shiryaev (1995) solved this model and showed that there exists a threshold u such that every excess of the capital above u is distributed as dividend instantaneously (barrier strategy for dividend). Frangos & al (2007) compute (using numerical simulations) the optimal dynamic quota share. Proportion retained by insurer is then linear with W, the company Wealth with U(0)=0. Y Krvavych & al (2004) solve the single period problem without frictional cost (in this case the optimal choice is no reinsurance) and with taxes (in this case there is demand for reinsurance) 12

13 13 Reinsurance Optimization in Practice

14 Optimal Reinsurance in Practice : Main Steps Step 1: Strategic decisions Optimal Reinsurance structure choice Proportional Reinsurance is suitable for new business - Quota Share for equivalent share between reinsurer and cedante - Surplus treaties in order to transfer a most important part of large risks to reinsurer Non proportional Reinsurance is adapted to extreme risk - Per Event treaties for Natural Catastrophe - Per Risk treaties for industrial risk - Stop-Loss is suitable for frequent small losses Step 2: Tactic decisions Value Optimization Optimal Reinsurance structure parameters computation is based on Value Creation under some Risk Management Constraints and define Risk Appetite Framework: Retention constraints could be simply based on determined acceptable Combined Ratio Loss, return Period, Communication constraints, etc Limit constraints could be based on Solvency constraints (for example 200Y return period), rating agency constraints (250Y return period for AM-Best), etc 14

15 Optimal Reinsurance in Practice : Value Optimization Economic Value Added A simplify version of Finetti model (one year horizon) and based on value created in addition to capital required K. Economic Value Added is defined by EVA E( result net) CoC K Result net : Underlying Result Net of Reinsurance and Net of taxes K is the Capital CoC is the cost of capital to meet shareholders required return on capital Our goal is to maximize EVA under some Risk Management Constraints 15

16 Optimal Reinsurance in Practice : Value Optimization Capital K Needed Capital correspond to a risk measure. Numerous risk measure could be used but Solvency II propose a Capital based on 200y Return Period Value at Risk K E ( 99.5% X ) Var 1 ( X ) Cost of Capital CoC Cost of capital depend on return expected by shareholders. The return that investors demand for the use of their Capital is composed of the base cost of capital (the return that investors could have obtained by investing those funds in financial market directly) plus frictional capital cost. Frictional costs are: - Cost of double taxation (Insurance companies are taxed on their investment return before it can be distributed to shareholder) - Costs of financial distress - Agency costs (shareholders expects management to act in their best interests, but this is difficult to control due to an intrinsic lack of transparency) CoC depends on Company and LoB (second order). Its value is typically into the range [5%- 10%] 16

17 Optimal Reinsurance in Practice : Value Optimization EVA Practical use as decision criteria In order to optimize reinsurance, we compare EVA and choose reinsurance structure which leads to maximum EVA. EVA1 is the EVA computed with Reinsurance structure 1 EVA2 is the EVA computed with Reinsurance structure 2 We choose Structure 1 if EVA1 > EVA2 and if Structure 1 17

18 Optimal Reinsurance in Practice : IsoValue Consider a reference reinsurance structure Consider a reinsurance option (for example reference structure plus one layer) Let EVA ref (resp EVA opt ) the Economic Value Add corresponding to the reference program (resp to the optional structure). We define IsoValue, the option price in order such: EVAref EVA opt As a consequence if the option price is smaller than IsoValue, the option create value, conversely the option destroy value IsoValue is a price tacking into account risk price and Capital saved by reinsurance IsoValue should also take into account model uncertainty 18

19 Optimal Reinsurance in Practice : IsoValue Consider a reinsurance option based on reference structure plus one XL layer XL layer IsoValue price is defined by EVAopt EVA ref XL layer IsoValue computation in a simplified case: E( R ref P Recov eries ) CoC K E( R ) CoC K ISO XL opt ref ref P ISO E(Re cov eries XL ) CoC ( K ref K opt ) P ISO PP CoC K 19

20 Optimal Reinsurance in Practice : IsoValue Suppose an Insurance company, expose to Natural Catastrophe damage. The insurer faces risk of hurricanes and earthquakes. This hypothetical firm has an opportunity to buy an excess of loss catastrophe reinsurance program. The pure premium of the program is suppose to be 20 m Capital saved by the program is suppose to be 250 m Company CoC is 5% IsoValue Price is : %250 = 32,5 m If price asked by Reinsurer is greater than 32,5 m, the program destroy value Nevertheless the program could be bought by the insurer if some Risk Management constraints are taking into account 20

21 Risk Appetite Why Risk Appetite in Reinsurance? Questions that arise at each Renewal: Retention Levels: Do the retention levels efficiently protect the Group Earnings? Shortfalls: Do shortfalls overly expose the Group to catastrophe events? Is it better to keep shortfalls or to place layers no matter what the prices are? Capacities: Is the Capacity adequate? Rationalize these Answers: Risk Appetite enables to rationalize answers by inferring ALERT and LIMIT levels on functional indicators from prior management decisions. This will simplify the decision process during the reinsurance renewal period. In particular it will answer the question about the use of Iso-values during the reinsurance placement. 21

22 Risk Appetite Methodology Net Retention is the proposed Functional Indicator The proposed methodology consists in assessing the Net Retention corresponding to Reference Scenarios. By looking at previous RISC decisions related to reinsurance protections, Alert and Limit levels corresponding to the Group Risk Appetite will be inferred. Reference Scenarios used Frequency Scenarios: They correspond to scenarios in order to test reinsurance retentions. Extreme Scenarios: They enable to test capacities. Additional Scenarios: Additional Scenarios are used in order to test independently shortfalls located at different levels (bottom, middle, and top) of the reinsurance program. 22

23 Risk Appetite Reinsurance Placement Decision Tree for a Layer Placement Compare Layer Premium to Iso-Value Placement no matter the Group Risk Level Placement only if the Limit Level is breached. Sequential Placement To ensure the optimal reinsurance placement using the Iso-Value indicators: Layers whose Premium < Iso-Value Min shall be secured first since reinsurance creates Value; Layers whose Premium > Iso-Value Max: Finally place the remaining layers up to the point where the Limit Level is no longer breached. 23