advertisement

On Optimal Reinsurance Arrangement Yisheng Bu Liberty Mutual Group 1 Agenda 1. Related Literature and Introduction 2. A Simple Model 3. Numerical Simulations 4. Discrete Loss Distribution 5. The Value of and Contingent Capital Calls 6. Concluding Remarks R 2 1. Related Literature and Introduction Related Literature Optimality in Reinsurance Arrangement • Optimal portfolio sharing through quota-share contracts among insurers (Borch, 1961) • Optimal proportional reinsurance (Lampaert and Walhin, 2005) • Optimal stop-loss reinsurance contracts for minimizing the probability of ruin (Gajek and Zagrodny, 2004) 3 1. Related Literature and Introduction (cont.) Related Literature Value of Reinsurance (Venter, 2001) • Value of reinsurance comes from stability provided Aggregate Profile of Reinsurance Purchases (Froot, 2001) • Reinsurance contracts had been more often used to cover lower catastrophic risk layers rather than more severe but lower-probability layers. • Reinsurance contracts had been priced in such a way that higher reinsurance layers had higher ratios of premium to expected losses. 4 1. Related Literature and Introduction (cont.) Introduction: This Paper Standpoints: • From the ceding company’s perspective • Focus on the aggregate reinsurance portfolio of the insurer instead of individual reinsurance contracts Objectives: • Optimal Excess-of-Loss Reinsurance Arrangement for profit and stability maximization • Provide justifications for the profile of reinsurance purchases that had been observed for industry 5 2. A Simple Model Assumptions: The reinsurance market consists of one insurer and one reinsurer The insurer has no control over the pricing of its (aggregate) reinsurance portfolio The insurer knows about the reinsurance pricing rule and chooses the reinsurance layer for full coverage The insurer and reinsurer have access to the same information on the underlying loss distribution 6 2. A Simple Model (cont.) Reinsurance Pricing The reinsurance pricing rule of the aggregate reinsurance portfolio for insurer i: Z ( x ; a , b ) E[ L ( x ; a , b )] Var[ L ( x ; a , b )], i i i i i R i i i i R R i i i where can be considered as the market price of risk determined by the industry’s existing reinsurance portfolio, or ( Z E[ L ]) / Var[ L ] , (i) referring to all risks excluding contract i. R R (i) (i) (i) R R R 7 2. A Simple Model (cont.) Discussions: Addition of stochastically independent risks and additive property of reinsurance contracts No parameter uncertainty is considered The “down-side” variance vs. total variance Skewness and higher moments of the claim payments distribution under reinsurance contracts Supported by many empirical findings on reinsurance pricing (Kreps and Major, 2001; Lane, 2004) 8 The Insurer MIN : Z E[ L ( x; a, b)] Var[ L ( x; a, b)] a ,b S S S s.t.Z B -Choose a and b to minimize the sum of reinsurance costs and expected claim payments net of reinsurance recovery -Include a penalty term for the variation of net claim payments -Z: Reinsurance costs -B: Budget constraint for reinsurance purchase 9 The Insurer (cont.) For a given volume of premium from the underlying insurance contracts, this formulation virtually maximizes the expected value of the net income and its stability. How much the insurer values stability ( ) depends on the degree of its risk aversion. S 10 Optimality Conditions [1 dF ( x)] { (b a E[ L ]) ( x b a E[ L ])}dF ( x) a b R R S S [1 dF ( x)] { ( x a E[ L ]) (a E[ L ])}dF ( x) b a a R R S S dF ( x) { ( E[ L ]) ( x E[ L ])}dF ( x) (Z / a ) a a 0 R R S S [1 dF ( x)] { (b a E[ L ]) ( x b a E[ L ])}dF ( x) b b R R S S dF ( x) { ( x a E[ L ]) (a E[ L ])}dF ( x) b b a R R S S dF ( x) {( ( E[ L ]) ( x E[ L ])}dF ( x) (Z / b) a b 0 R R S S 11 Optimality Conditions (cont.) In essence, by choosing the optimal reinsurance coverage, the insurer attempts to achieve the optimal balance between the reduction in the cost of claim payment variation via reinsurance coverage and the price for shifting such variation to the reinsurer. 12 3. Numerical Simulations Methodology b Eq. (5) a0 Eq. (6) a0 a 13 Figure 1. Claim Payment Distribution (Gamma Distribution) xe f ( x) ! ( x/ ) ( 1 ) 14 Figure 2. Reinsurance Premium as a Function of a and b 15 Figure 3. The Value of the Insurer’s Objective Function as a Function of a and b 16 Table 1. Numerical Simulation Results with R 2 , S 2 Parameter Values, Expected Value and Variance of Underlying Loss Distribution (1) alpha 0 0 1 2 (2) beta 1 2 1 1 (3) E(x) 1 2 2 3 (4) Var(x) 1 4 2 3 (7) limit 1.594 3.187 2.008 2.351 (8) Z 0.806 2.648 1.531 2.218 case 1 case 2 benchmark case 3 Simulation Results (5) a 1.018 2.035 1.805 2.631 (6) b 2.611 5.222 3.813 4.982 (9) E(Lr) 0.288 0.576 0.497 0.670 (10) Obj 2.180 6.719 4.367 6.556 (11) ROL 0.506 0.831 0.762 0.943 (12) Z/E(Lr) 2.798 4.597 3.078 3.311 case 1 case 2 benchmark case 3 17 Results 1. 2. The results justify the aggregate profile of reinsurance purchases observed in Froot (2001): • To stabilize its book of business and maximize net income, it is optimal to use reinsurance protection against risks of moderate size, but leave the most severe loss scenarios uncovered or self-insured. The results suggest that the retention be set to be comparable to the expected ground-up claim payments. 18 Results (cont.) 3. In situations where • underlying claim payments are more dispersed (higher ), • events of higher severity occur with larger probabilities (higher ), the insurer should purchase more protection against more severe events, or higher limit and higher retention. As a result, the optimal reinsurance layer in the above situations also has higher ROL and higher ratio of premium to expected losses. 19 Results (cont.) 4. 5. The optimal choices of the reinsurance layer can be very sensitive to the chosen values of the model parameters, which implies that parameter uncertainty is an important consideration in reinsurance purchase. To the extent that higher reinsurance layers are more vulnerable to prediction errors from engineering models, parameter uncertainty may well explain the observed high prices for lowprobability layers. 20 Varying the value of R . Figure 4. Retention and Limit As Reinsurance Load Changes (assuming S 2 ) Retention/Limit 4.500 4.000 3.500 Retention + Limit 3.000 2.500 Retention 2.000 1.500 1.000 0.500 0.000 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 Reinsurance Load 21 4. Discrete Loss Distribution Loss Scenarios: Scenario 1 Scenario 2 Scenario 3 Total Cat Loss s1 s2 s3 Probability f1 f2 f3 0s s s 1 2 3 f f f 0 1 2 3 f f f 1 s 0 1 2 3 1 Scenario 1: little or no loss occurrence Scenario 2: moderate losses Scenario 3: most severe losses 22 4. Discrete Loss Distribution (cont.) It can be shown that: At the optimum, 0a s b s * * 2 3 which implies that it is advisable for the insurer to purchase some reinsurance protection against both moderate and most severe cat loss scenarios rather than against any one particular scenario only. Specifically, the optimal reinsurance layer boundaries are given by s a * 2 R R s s s b a * S * 3 R S 2 S R R 3 S S 23 4. Discrete Loss Distribution (cont.) The layer limit is independent of the probability with which each event occurs (not intuitive) and satisfies that s s l b a * * * 2 R R 3 S S The minimum (optimal) value of the insurer’s value function is equal to the rate on line of the reinsurance contract, or Obj Z /(b a ) * * * min 24 5. The Value of and Contingent Capital Calls R The capital consumption approach to reinsurance pricing uses the value of potential capital usages as the risk load (Mango, 2004) The reinsurer attempts to maximize the firm’s expected net income after adjusting for the capital costs in the unprofitable states. 25 5. The Value of and Contingent Capital Calls (cont.) R The objective function of the reinsurer is formulated as MAX : Var[ L ] g ( x (a Z ))dF ( x) b R R R a z g (b (a Z )) dF ( x) b where the function g(*) is the capital call charge function and satisfies g ' () 0 and g" () 0 . 26 5. The Value of and Contingent Capital Calls (cont.) R For the purpose of simulation, g(*) is specified as g ( ) c 2 where ( 0 ) is the amount of capital calls and c ( c 0 ) is the rate at which the marginal cost of capital calls increases. With higher values of c , it is more costly for the reinsurer to underwrite more severe cat events. 27 Figure 5. The Choice of and the Value of the Reinsurer’s Objective Function R 0.20 c=4 0.10 c=5 0.00 0 .0 10 50 9. 00 9. 50 8. 00 8. 50 7. 00 7. 50 6. 00 6. 50 5. 00 5. 50 4. 00 4. 50 3. 00 3. -0.10 50 2. 00 2. Value of Reinsurer's Objective Function 0.30 c=8 -0.20 -0.30 -0.40 Gam m aR 28 5. The Value of and Contingent Capital Calls (cont.) R Results from Figure 5: When the marginal cost of capital calls increases relatively faster for the reinsurer (higher c), the reinsurer sets higher and the insurer tends to purchase reinsurance protection for moderate losses only and leave higher layers uncovered. R 29 6. Concluding Remarks Summary Optimal excess-of-loss reinsurance purchase when the insurer maximizes net income and stability for both the discrete and continuous loss distribution Optimal Reinsurance Purchase Other Considerations Can it be optimized? The cost of reinsurance capital Empirical measurement of . R 30