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
I Standard Optimal Reinsurance Problem (Arrow (1963)) n o max E [U (W − π − X + I (X ))] : (Arrow) I ,π 0 6 I (X ) 6 X , E[I (X )] 6 K . I Optimal contract: stop loss I ∗ (X ) = (X − d)+ , 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 I Reinsurer (insurance seller) can default. Let Θ be the percentage of the indemnity paid back to the insurance company (insurance buyer). I Optimal Reinsurance under counterparty risk I with fair premium n o max E [U (W − π − X + ΘI (X ))] : I ,π 0 6 I (x) 6 x, E[ΘI (X )] 6 K .
(A)
I with asymmetry of information (reinsurer does not take into account its own default.) n o max E [U (W − π − X + ΘI (X ))] : (AS) I ,π 0 6 I (x) 6 x, E[I (X )] 6 K . Carole Bernard
Cost-Efficiency in Portfolio Management
3
Setting
Dependence Modelling
Optimal Reinsurance
Illustration
Information Asymmetry
Conclusions
Credit Risk Modeling I Structural Approach: The reinsurer can default because its assets are not enough to pay I (X ). Biffis and Milossovitch (2010). I Intensity Approach: The default of the reinsurer is not entirely driven by the level of X and W . I A reinsurer has many policyholders and it is hard to imagine that default is triggered by one claim. I The default risk is “higher” when the claim X is big... I Because insurance indemnities are in general non-decreasing functions of X . I Because if one company has a big loss, likely other insurance companies also suffer from big losses and might therefore also claim reinsurance.
I 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
I No need to specify a copula. I Credit risk of the reinsurer measured by the recovery rate Θ. I Θ = 1, no default. I Θ ∈ (0, 1), partial default. I Θ = 0, total default. See Cummins and Mahul (2003).
I 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 I We denote by Θ ↓st X when Θ is stochastically decreasing with X . I Formal definition A random variable Θ is stochastically decreasing in another random variable X if x 7→ 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 I Model an “increase in risk” and a negative (or positive) dependence between 2 risks. I 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. I 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 X1 is the insured risk related to the fire damage of the house, and X2 is the uninsured risk related to the fire damage of the car, the assumption X2 ↑ X1 is quite reasonable. I 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: n o ∀x ∈ [0, x¯], Pr {f (X ) 6 x} = Pr f˜(X ) 6 x . I Variant of Hardy-Littlewood inequality. Theorem If L : [0, x¯]2 → R is C 1 such that for all t ∈ [0, x¯], the application x → ∂L ∂t (x, t) is increasing (supermodularity condition), then h i E L X , f˜(X ) > E [L(X , 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
I In the standard case (Arrow) L : (x, t) 7→ U (W − π − x + t) satisfies the previous constraint. Therefore the optimum is non-decreasing! (optimal stop-loss). I In the general case (with counterparty risk) L : (x, t) 7→ E θ [U (W − π − x + Θt) |X = x] does not satisfy the supermodularity condition. I 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 ) 7→ 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. Iλ∗ (x) = max (min(yλ,x , x), 0) . ∂yλ,x ∂x
Moreover
> 1, where yλ,x is the unique solution to
E θ ΘU 0 (W − π − x + Θyλ,x ) − λE θ [Θ] = 0. If Θ = 1 a.s. (no default) then yλ,x = x − W + π + [U 0 ]−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 θ0 ∈ [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 θ0 = 0 (no recovery), the optimum is a stop-loss contract −1 Iλ∗ (x) = (x − W + π + U 0 (λ))+ . • When 0 < θ0 < 1, 0 if ∗ Iλ (x) = min(yλ,x , x) if
x 6 W − π − [U 0 ]−1 (λ) x > W − π − [U 0 ]−1 (λ)
∂y
where ∂xλ,x at the deductible d = W − π − [U 0 ]−1 (λ) is strictly greater than 1. Locally the optimal indemnity exceeds the stop loss 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 when θ0 = 0 or θ0 = 1, the optimal indemnity is a stop-loss; when 0 < θ0 < 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
I an exponential loss X ∼ Exp(m), I probability of recovery p(x) =
c c+x
I m = 0.3, c = 0.9, ρ = 0.2, W = 5. I CARA utility U(x) = −e −γx with γ = 2.75. I θ0 = 0.6. Figure 1: Effect of counterparty risk on optimal premium π. I π = 1. Figure 2: Effect of recovery rate θ0 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
Setting
Dependence Modelling
Optimal Reinsurance
Illustration
Information Asymmetry
Conclusions
Presence of Asymmetric information Assume Θ can take only two possible values 1 and θ0 ∈ [0, 1), and for x > 0, p(x) , E[Θ = 1|X = x]. is non-increasing, differentiable and takes values in (0, 1). π = E [I (X )] (instead of π = E[ΘI (X )].) Theorem (Optimal Indemnity) • When θ0 = 0 (no recovery), the optimum is Iλ∗ (x)
+ 0 −1 λ = x −W +π+ U . p(x)
• When 0 < θ0 < 1, Iλ∗ (x) = max(min(zλ,x , x), 0) and where zλ,x is defined implicitly by E ΘU 0 (W − π − x + Θzλ,x ) |X = x − λ = 0, Carole Bernard
Cost-Efficiency in Portfolio Management
17
Setting
Dependence Modelling
Optimal Reinsurance
Illustration
Information Asymmetry
Conclusions
Numerical Illustration
Optimal shape can change a lot! I probability of recovery p(x) =
c c+x
I Effect of the recovery rate θ0 on the shape of the optimal indemnity. I m = 0.3, c = 0.2, W = 3, λ = 0.01. I CARA utility U(x) = −e −γx with γ = 2.75.
Carole Bernard
Cost-Efficiency in Portfolio Management
18
Setting
Dependence Modelling
Optimal Reinsurance
Illustration
Information Asymmetry
Conclusions
Optimal Contract when reinsurance is overpriced
Carole Bernard
Cost-Efficiency in Portfolio Management
19
Setting
Dependence Modelling
Optimal Reinsurance
Illustration
Information Asymmetry
Conclusions
Numerical Comparison of Problems (A) and (AS)
I an exponential loss X ∼ Exp(m), I probability of recovery p(x) =
c c+x
I m = 0.3, c = 0.9, ρ = 0.2, W = 5, π = 1. I CARA utility U(x) = −e −γx with γ = 2.75. I Comparison I In red: Fair premium: optimal indemnity max(0, min(yλ,x , x)) with λ such that the premium level is optimal equal to π ∗ = 0.9. I In blue: Asymmetric information: optimal indemnity max(0, min(zλ,x , x)) with λ such that the premium level is optimal equal to π ∗ = 1.275.
Carole Bernard
Cost-Efficiency in Portfolio Management
20
Setting
Dependence Modelling
Optimal Reinsurance
Illustration
Information Asymmetry
Conclusions
Comparison of Optimal Indemnities
12 I(x)=max(0,min(y
,x))
λ,x
10
Corresponding Deductible I(x)=max(0,min(z ,x)) λ,x
Corresponding Deductible 8
6
4
2
0 0
Carole Bernard
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4
6
8
10
12
Cost-Efficiency in Portfolio Management
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Setting
Dependence Modelling
Optimal Reinsurance
Illustration
Information Asymmetry
Conclusions
Conclusion Qualitative investigation of counterparty default on optimal
reinsurance Extension of the standard theory of insurance design to
account for systemic credit risk (multiplicative loss-dependent background risk). Impact of asymmetric views of credit risk of the reinsurer from insurer’s & reinsurer’s perspective Moral Hazard issues : non-decreasing indemnities are not optimal, marginal overinsurance is possible. Important effects: when fairly priced the presence of counterparty risk increases the reinsurance demand in the tail but at the same time decreases the optimal premium level (amount spent by the buyer in the reinsurance market). Static one-period model. Most reinsurance contracts are signed for several years, need to further analyze dynamic multi-period models.
Carole Bernard
Cost-Efficiency in Portfolio Management
22
Setting
Dependence Modelling
Optimal Reinsurance
Illustration
Information Asymmetry
Conclusions
References I Arrow, K. J. (1963): “Uncertainty and the Welfare Economics of Medical Care,” American Economic Review, 53(5), 941–973. I Bernard, C., W. Tian (2009): “Optimal Insurance Policies When Insurers Implement Risk Management Metrics,” The Geneva Risk and Insurance Review, 34, 74–107. I Biffis, E., P. Millossovich (2010): “Optimal Insurance with Counterparty Default Risk,” Working Paper - Available at SSRN: http://ssrn.com/abstract=1634883. ´raa ` , P., J. Lefoll (1983): “Aversion pour le risque croissante avec la I Cape richesse initiale al´ eatoire,” Econometrica, 53(2), 473–475. I Carlier, G., R.-A. Dana (2003): “Pareto Efficient Insurance Contracts When The Insurer’s Cost Function Is Discontinuous,” Economic Theory, 21, 871–893. I Cummins, J., O. Mahul (2003): “Optimal Insurance with Divergent Beliefs about Insurer Total Default Risk,” The Journal of Risk and Uncertainty, 27, 121–138. I Dana, R.-A., M. Scarsini (2007): “Optimal Risk Sharing with Background Risk,” Journal of Economic Theory, 133(1), 152–176. I Eeckhoudt, L., C. Gollier, and H. Schlesinger (2005): Economic and Financial Decisions under Risk. Princeton University Press. I Mahul, O., B. Wright (2007): “Optimal Coverage for Incompletely Reliable Insurance,” Economic Letters, 95, 456–461. Carole Bernard
Cost-Efficiency in Portfolio Management
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