Risk redistribution with distortion risk measures

Transcription

1 T. Boonen, WatRISQ, 2015 Risk redistribution with distortion risk measures Tim Boonen University of Amsterdam The Netherlands March 10th, 2015

2 T. Boonen, WatRISQ, 2015 Introduction Introduction Problem: redistributing risk among firms. Goal of the firms: high expected value, low risk. Prominent examples: - markets (many traders), - Over-The-Counter trades as in: life insurance, extreme events (CAT-bonds), CDS-market.

3 T. Boonen, WatRISQ, 2015 Introduction Problem Borch (1962), Wilson (1968), Arrow (1971), Raviv (1979) and Aase (1993) analyze risk redistributions when firms use an expected utility function. I let the preference relation be given by a risk- reward trade-off: γ i 0, Y is a gain. U i (Y ) = E[Y ] γ i ρ i (Y ), Here, ρ i is a risk measure. Risk measures measure the riskiness of a portfolio. The Basel II regulation and the Swiss Solvency Test (SST) have increased the use of risk measures to evaluate financial or insurance risk.

4 T. Boonen, WatRISQ, 2015 Introduction Literature Popular example of ρ i is Expected Shortfall. More general, I focus on ρ i to be coherent (Artzner et al. 1999). I consider distortion risk measures (Yaari, 1987; Wang, 1995). This captures almost the whole class of coherent risk measures (Wang et al. 1997). Existing literature focusses on existence: - Pareto optimality: Jouini, Schachermayer and Touzi (2008), Ludkovski and Young (2008) provide sufficient conditions risk redistributions to be Pareto optimal; - equilibria: Filipović and Kupper (2009) show existence of equilibria.

5 T. Boonen, WatRISQ, 2015 Introduction Example The liabilities of the pension fund and death benefit insurer: Pension Fund Death benefit insurer Density Density % 0 +10% Percentage deviation from expected value 0 10% 0 +10% Percentage deviation from expected value

6 T. Boonen, WatRISQ, 2015 Introduction Contribution I provide necessary and sufficient conditions such that there are benefits from risk redistribution for all firms. I focus on uniqueness of competitive equilibria; - I provide necessary and sufficient conditions. In absence of a well-functioning market, I characterize specific risk redistributions; this turns out to be the equilibria.

7 T. Boonen, WatRISQ, 2015 Introduction Findings All Pareto optimal contracts are given by a collection of stop-loss contracts; - this is empirically observed in insurance; - this is not Pareto optimal if all firms use expected utility. Under three conditions, the equilibria are unique. The prices are given by specific risk-neutral probabilities. The equilibrium is the only risk redistribution satisfying four desirable properties.

8 T. Boonen, WatRISQ, 2015 Model Model A finite collection of firms N = {1,..., n}. Every firm i holds a risk X i (gain) on a finite state space. To redistribute is the total risk i N X i. The problem is to find a risk redistribution (Y 1,..., Y n ) such that i N Y i = i N The objective function of firm i is to maximize a risk-reward function X i. U i [Y ] = E[Y ] γ i ρ i (Y ).

9 T. Boonen, WatRISQ, 2015 Model Distortion risk measures A distortion risk measure is an expectation under a distorted probability measure: ρ i (Y ) = Q i Y (ω) Y (ω) = E Q i [ Y ], Y ω Ω where Q i Y ({ω}) = g i(p(y Y ({ω}))) g i (P(Y < Y ({ω}))), for all ω Ω. Note: Q Y depends on Y. Generated by a continuous, concave and increasing function g i : [0, 1] [0, 1], where g i (0) = 0 and g i (1) = 1.

10 T. Boonen, WatRISQ, 2015 Model Distortion risk measures The objective function is given by U i [Y ] = E[Y ] γ i ρ i (Y ). It can be verified that maximizing U i (Y ) is equivalent to minimizing ρ i (Y ) = U i(y ) 1 + γ i. The mapping ρ i is a distortion risk measure. Hence, it is sufficient to focus only on preference relations given by a distortion risk measure ρ i.

11 T. Boonen, WatRISQ, 2015 Pareto optimality Pareto optimality Pareto optimal risk redistributions are defined as the risk redistributions such that there do not exist another risk redistribution that is weakly better for all firms and strictly better for at least one firm. Pareto optimality when firms use distortion risk measures: Tsanakas and Christofides (2006), Burgert and Rüschendorf (2008), Ludkovski and Rüschendorf (2008) and Ludkovski and Young (2009). Ludkovski and Young (2009) characterize all Pareto optimal risk redistributions that are perfect hedges of each other (i.e., comonotonic).

12 T. Boonen, WatRISQ, 2015 Pareto optimality Pareto optimality Barrieu and El Karoui (2004, 2005), Acciaio (2007), Jouini, Schachermayer and Touzi (2008) and Filipović and Kupper (2008) show that Y i, i N is Pareto optimal if and only if Y i, i N minimizes ρ i (Y i ), s.t. i N Y i = i N X i. i N I focus on: - uniqueness of Pareto optimal risk redistributions; - extending Ludkovski and Young (2009) also for risk redistributions that are not perfect hedges of each other (not only comonotonic risk redistributions).

13 T. Boonen, WatRISQ, 2015 Pareto optimality Example The risk measure Expected Shortfall is given by ˆρ ES α i (Y ) = 1 α i αi where VaR s (Y ) is the (1 s)-quantile of Y. Let firm i use a distortion risk measure 0 VaR s (Y )ds, ρ i (Y ) = (1 ζ i ) E[ Y ] + ζ i ˆρ ES α i (Y ); - if α i = α, a Pareto optimal solution is to shift all risk to the firm with the smallest ζ i ; - if ζ i = ζ, a Pareto optimal solution is to shift all risk to the firm with the largest α i ; - variable α i, ζ i : tranching.

14 T. Boonen, WatRISQ, 2015 Pareto optimality Pareto optimality Let ρ N be the market distortion risk measure with distortion function gn : [0, 1] [0, 1] given by g N (x) = min{g i(x) : i N} for all x [0, 1]. Theorem: Y i, i N is Pareto optimal if and only if ( ) ρ i (Y i ) = ρ N X i. i N i N Y i, i N is Pareto optimal if and only if Y i + c i, i N with i N c i = 0 is Pareto optimal. - I refer to c i, i N as side-payments.

15 T. Boonen, WatRISQ, 2015 Pareto optimality Pareto optimality Theorem: A Pareto optimal risk redistribution is given by a sum of stop-loss contracts (i.e., tranching ), where a layer is allocated to a firm for which g i is minimal at P(X X (ω)). Theorem: If gn is strictly concave, and under one additional condition, there exists a risk redistribution Y i, i N such that PO = {Y i + c i, i N : i N c i = 0}. What should be a fair size of the side-payments? - Competitive equilibria; - Cooperative game theory.

16 T. Boonen, WatRISQ, 2015 Pareto optimality Pareto optimality Proposition: If gn is strictly concave, it holds that there are no benefits from redistributing if and only if - all X i, i N are comonotone with each other; - ρ i (X i ) = ρ N (X i), for all i N. Comonotonicity of risks X i, i N implies that risks are non-hedgeable: There exists an ordering on the state space such that X i (ω 1 )... X i (ω p ) for all i. ρ i (X i ) = ρ N (X i) implies that firm i is not benefitting from the risk measure of others.

17 Example T. Boonen, WatRISQ, 2015 Pareto optimality

18 T. Boonen, WatRISQ, 2015 Competitive equilibria Competitive Equilibria Firms act as price-takers. Individual objective: with pricing formula Y i argmin Y i ρ i (Y i ) s.t. π(ˆp, Y i ) π(ˆp, X i ) π(ˆp, Y ) = ω Ω π(ˆp, e Ω ) = 1, ˆp ω Y (ω). ˆp is an equilibrium price if and only if Yi = X i. i N i N

19 T. Boonen, WatRISQ, 2015 Competitive equilibria Competitive Equilibria Filipović and Kupper (2008): there exists a competitive equilibrium. Every competitive equilibrium is Pareto optimal. Lemma: If the function gn is strictly concave, every Pareto optimal risk redistribution is comonotone with the aggregate risk i N X i. Therefore, the objective is given by Y i argmin Y i ρ i (Y i ) s.t. π(ˆp, Y i ) π(ˆp, X i ) Y i (ω 1 )... Y i (ω p ) π(ˆp, e Ω ) = 1.

20 T. Boonen, WatRISQ, 2015 Competitive equilibria Competitive Equilibria From comonotonicity of Y i with X := i N X i, it follows that the ordering of Y i is known: ρ i (Y i ) is an expectation under a known probability measure Q i Y i = Q i X. From this, it follows that Y i argmin Y i E Q i X [ Y i ] s.t. π(ˆp, Y i ) π(ˆp, X i ) Y i (ω 1 )... Y i (ω p ) π(ˆp, e Ω ) = 1. Theorem: If gn is strictly concave, the equilibrium prices ˆp are unique if and only if X (ω 1 ) <... < X (ω p ).

21 T. Boonen, WatRISQ, 2015 Competitive equilibria Competitive Equilibria Solving this system via Kuhn-Tucker conditions yields ˆp ω = gn (P(X X ({ω}))) g N (P(X < X ({ω}))). The price of a risk X i is therefore also an expectation under the distorted probability measure ˆp. The prices ˆp are such that: - they are higher for states where the aggregate risk is largest; - they are based on a representative agent with risk measure ρ N : a least risk-averse agent. It holds that ρ i (Y i ) = π(p, X i ).

22 T. Boonen, WatRISQ, 2015 Competitive equilibria Competitive Equilibria If gn is strictly concave, the equilibrium risk redistributions are unique if and only if - for all states ω, there exists a unique firm i for which g i is minimal at P(X X (ω)); - X (ω 1 ) >... > X (ω p ). Compare with expected utility (Aase, 1993); then the equilibrium is unique if - u i ( ) > 0 and u i ( ) < 0; - lim x u i (x) = 0 and lim x 0 u i (x) = ; - x u i (x) are all non-decreasing.

23 T. Boonen, WatRISQ, 2015 Competitive equilibria CAPM For the equilibrium prices, it holds that where RR i = E[RR i ] 1 = β i (E[RR m ] 1), X i π(ˆp,x i ) and RR m = ( cov β i = ( cov X π(ˆp,x ) and RR i, d ˆp dp RR m, d ˆp dp ) ). Note that the risk-free rate is assumed to be one. d ˆp dp is a Radon-Nikodym derivative that indicates the change of probability measure. This result extends the CAPM-model of De Giorgi and Post (2008), who focus on homogeneous risk measures.

24 T. Boonen, WatRISQ, 2015 Cooperative game theory Cooperative Game For competitive equilibria, one needs to assume that firms are price-takers. This assumption is invalid if the market is not well-functioning. Then, trades can occur Over-The-Counter. I determine risk redistributions via a (cooperative) bargaining process. This process is modeled using cooperative game theory. Recall the example with bilateral trading of longevity risk.

25 T. Boonen, WatRISQ, 2015 Cooperative game theory Cooperative Game Under two conditions there is a one-to-one correspondence between an allocation a i = ρ(y i ) and a Pareto optimal risk redistribution Y i, i N. Hence, the risk redistribution problem is equivalent to finding a fair allocation a such that i N a i = ρ N (X ). Allocation problems are widely studied in game theory (e.g., Aumann and Shapley, 1974; Aubin, 1979 and 1981; Billera and Heath, 1982; Tauman and Mirman, 1982; Denault, 2001).

26 T. Boonen, WatRISQ, 2015 Cooperative game theory Solution concept A commonly used concept in cooperative game theory is the Shapley value. Csóka and Pintér (2011), however, show that the Shapley value does not need to be a stability criterion. I define four desirable properties of a risk redistribution; - the most important one focuses on core of a cooperative game. I characterize a unique allocation. This is given by the equilibrium.

27 T. Boonen, WatRISQ, 2015 Cooperative game theory Risk Measures vs. Expected Utility Risk measure: ρ i (Y ) = E Q i [ Y ] = Q i Y Y (ω) Y (ω). ω Ω Von Neumann-Morgenstern expected utility: E[u i (Y )] = ω Ω P(ω) u i (Y (ω)). Risk measures adjust the probabilities, instead of outcomes, to include risk aversion. Set N 1 contains agents that maximize expected utility (individuals), set N 2 contains agents that minimize a distortion risk measure (firms).

28 T. Boonen, WatRISQ, 2015 Cooperative game theory Risk Measures vs. Expected Utility: overview findings There are two representative agents: the average risk-averse EU maximizer, and the least risk-averse distortion risk measure minimizer. These two agents allocate risk among each other. Allocation should be comonotone is all agents are strictly risk-averse (strictly concave utility/distortion function). Constraints might be binding in FOC conditions. Within group, allocation is as well-known: proportional and tranching. Under a technical condition, equilibrium prices might be same as without EU maximizers!

29 T. Boonen, WatRISQ, 2015 Conclusion Conclusion I provide a closed form expression for the competitive equilibrium in exchange markets with many firms to obtain a specific risk redistribution. If a well-functioning market does not exist, I characterize the same risk redistribution via cooperative game theory. I provide three conditions under which this risk redistribution is unique.