Impact of Counterparty Risk on the Reinsurance Market Carole Bernard This talk is joint work with Mike Ludkovski (University of Santa Barbara California) IME 2011, Trieste, June 2011. Carole Bernard Cost-Efficiency in Portfolio Management 1
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Optimal Reinsurance Design ◮ Standard Optimal Reinsurance Problem (Arrow (1963)) � � max E [ U ( W − π − X + I ( X ))] : (Arrow) I ,π � 0 � I ( X ) � X , E [ I ( X )] � K . ◮ Optimal contract: ❼ stop loss I ∗ ( X ) = ( X − d ) + , ❼ it is non-decreasing in X . Carole Bernard Cost-Efficiency in Portfolio Management 2
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Reinsurer Credit Risk : Counterparty Risk for the Insurer ◮ Reinsurer (insurance seller) can default. Let Θ be the percentage of the indemnity paid back to the insurance company (insurance buyer). ◮ Optimal Reinsurance under counterparty risk ◮ with fair premium � � max E [ U ( W − π − X + Θ I ( X ))] : (A) I ,π � 0 � I ( x ) � x , E [Θ I ( X )] � K . ◮ with asymmetry of information (reinsurer does not take into account its own default.) � � max E [ U ( W − π − X + Θ I ( X ))] : (AS) I ,π � 0 � I ( x ) � x , E [ I ( X )] � K . Carole Bernard Cost-Efficiency in Portfolio Management 3
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Credit Risk Modeling ◮ Structural Approach: The reinsurer can default because its assets are not enough to pay I ( X ). Biffis and Milossovitch (2010). ◮ Intensity Approach: The default of the reinsurer is not entirely driven by the level of X and W . ◮ A reinsurer has many policyholders and it is hard to imagine that default is triggered by one claim. ◮ The default risk is “higher” when the claim X is big... ◮ Because insurance indemnities are in general non-decreasing functions of X . ◮ Because if one company has a big loss, likely other insurance companies also suffer from big losses and might therefore also claim reinsurance. ◮ To model these effects, we need to model the dependence between the amount of loss of the insurer X and the default risk of the reinsurer. Carole Bernard Cost-Efficiency in Portfolio Management 4
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Credit Risk & Dependence Modelling ◮ No need to specify a copula. ◮ Credit risk of the reinsurer measured by the recovery rate Θ. ◮ Θ = 1, no default. ◮ Θ ∈ (0 , 1), partial default. ◮ Θ = 0, total default. See Cummins and Mahul (2003). ◮ We assume that Θ is “stochastically decreasing” in X . Θ ↓ st X Carole Bernard Cost-Efficiency in Portfolio Management 5
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Credit Risk & Dependence Modelling ◮ We denote by Θ ↓ st X when Θ is stochastically decreasing with X . ◮ Formal definition A random variable Θ is stochastically decreasing in another random variable X if x �→ E [ f (Θ) | X = x ] is nonincreasing for every nondecreasing function f for which expectations exist. Carole Bernard Cost-Efficiency in Portfolio Management 6
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Other Applications in Actuarial Science ◮ Model an “increase in risk” and a negative (or positive) dependence between 2 risks. ◮ A household with two individuals: one of them has health insurance, the other does not. Two people living in the same environment, are prone to get sick at the same time, and therefore there is positive dependence between the risks related to the health of the two individuals. ◮ An individual may insure her house but not her car against fire. If the car is parked in the driveway of the house and for some reason it gets on fire there is a positive probability that the fire spreads to the house and damages it, or vice versa. Therefore, if X 1 is the insured risk related to the fire damage of the house, and X 2 is the uninsured risk related to the fire damage of the car, the assumption X 2 ↑ X 1 is quite reasonable. ◮ Many other situations in actuarial science... Carole Bernard Cost-Efficiency in Portfolio Management 7
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Main Tool Used Non-decreasing rearrangement ˜ f Given f : [0 , ¯ x ] → [0 , ¯ x ], a measurable function. There exists a unique non-decreasing function ˜ f such that: � � ˜ Pr { f ( X ) � x } = Pr f ( X ) � x ∀ x ∈ [0 , ¯ x ] , . ◮ Variant of Hardy-Littlewood inequality. Theorem x ] 2 → R is C 1 such that for all t ∈ [0 , ¯ If L : [0 , ¯ x ] , the application x → ∂ L ∂ t ( x , t ) is increasing (supermodularity condition), then � � �� X , ˜ � E [ L ( X , f ( X ))] . E L f ( X ) and the inequality is strict unless f ( · ) is non-decreasing Carole Bernard Cost-Efficiency in Portfolio Management 8
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions ◮ In the standard case (Arrow) L : ( x , t ) �→ U ( W − π − x + t ) satisfies the previous constraint. Therefore the optimum is non-decreasing! (optimal stop-loss). ◮ In the general case ( with counterparty risk ) L : ( x , t ) �→ E θ [ U ( W − π − x + Θ t ) | X = x ] does not satisfy the supermodularity condition. ◮ In general in the presence of counterparty risk, the optimum is not always non-decreasing (moral hazard issue). Carole Bernard Cost-Efficiency in Portfolio Management 9
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Related literature on background risk Credit risk can be seen as a multiplicative background risk . There is a major difference between the presence of an additive background risk and a multiplicative background risk. For an additive background risk Υ, Dana and Scarsini (2007) proved that L : ( x , y ) �→ E [ U ( W − π − x − Υ + y ) | X = x ] is supermodular when Υ ↓ st X so that the optimal indemnity is non-decreasing. Such supermodularity property does not hold with a multiplicative background risk. Carole Bernard Cost-Efficiency in Portfolio Management 10
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Θ and X are independent Theorem Under the assumption of independence between Θ and X and when K ∈ (0 , E [ X ]) , the optimal solution to Problem A (with fair premium) exists, is unique, and is non-decreasing with respect to X. I ∗ λ ( x ) = max (min( y λ, x , x ) , 0) . Moreover ∂ y λ, x � 1 , where y λ, x is the unique solution to ∂ x Θ U ′ ( W − π − x + Θ y λ, x ) − λE θ [Θ] = 0 . E θ � � If Θ = 1 a.s. (no default) then y λ, x = x − W + π + [ U ′ ] − 1 ( λ ) Carole Bernard Cost-Efficiency in Portfolio Management 11
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Bernoulli distribution for Θ Assume Θ can take only 2 values 1 and θ 0 ∈ [0 , 1), and for x � 0, p ( x ) � E [Θ = 1 | X = x ] . is non-increasing, differentiable and takes values in (0 , 1). Theorem (Optimal Indemnity) • When θ 0 = 0 (no recovery), the optimum is a stop-loss contract U ′ � − 1 ( λ )) + . � I ∗ λ ( x ) = ( x − W + π + • When 0 < θ 0 < 1 , � 0 x � W − π − [ U ′ ] − 1 ( λ ) if I ∗ λ ( x ) = x > W − π − [ U ′ ] − 1 ( λ ) min( y λ, x , x ) if where ∂ y λ, x at the deductible d = W − π − [ U ′ ] − 1 ( λ ) is strictly ∂ x greater than 1. Locally the optimal indemnity exceeds the stop loss contract at d. Carole Bernard Cost-Efficiency in Portfolio Management 12
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Bernoulli distribution for Θ Theorem (Suppose p ( x ) takes only two values,) then ❼ when θ 0 = 0 or θ 0 = 1 , the optimal indemnity is a stop-loss; ❼ when 0 < θ 0 < 1 , the optimal indemnity is non-decreasing and ∂ I ∗ ∂ x � 1 when I ∗ ( x ) > 0 . Carole Bernard Cost-Efficiency in Portfolio Management 13
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Numerical Illustration ◮ an exponential loss X ∼ Exp ( m ), c ◮ probability of recovery p ( x ) = c + x ◮ m = 0 . 3, c = 0 . 9, ρ = 0 . 2, W = 5. ◮ CARA utility U ( x ) = − e − γ x with γ = 2 . 75. ◮ θ 0 = 0 . 6. Figure 1: Effect of counterparty risk on optimal premium π . ◮ π = 1. Figure 2: Effect of recovery rate θ 0 on the shape of the optimal indemnity. Carole Bernard Cost-Efficiency in Portfolio Management 14
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Expected Utility w.r.t. Premium Carole Bernard Cost-Efficiency in Portfolio Management 15
Setting Dependence Modelling Optimal Reinsurance Illustration Information Asymmetry Conclusions Optimal Shape with fair premium π = 1 Carole Bernard Cost-Efficiency in Portfolio Management 16
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