Risk capital allocation and cooperative pricing of insurance liabilities
Introduction
We discuss the problem of allocating aggregate capital requirements for a portfolio of pooled liabilities to the instruments that the portfolio consists of. Cooperative game theory (Shapley, 1953, Aumann and Shapley, 1974, Aubin, 1981) provides a suitable framework for cost allocation problems (e.g. Lemaire, 1984). Typically, provisions are made so that the allocation of total costs does not produce disincentives for cooperation to any player (instrument) in the game or coalition of players (sub-portfolio of instruments). A closely related approach has been to determine an allocation scheme by imposing economically motivated axioms on the system of prices that it would produce (Billera and Heath, 1982, Mirman and Tauman, 1982). The solution concept that emerges from the latter approaches is the celebrated Aumann and Shapley (1974) value. In the context of capital allocation, cooperation can be understood as the pooling of risky portfolios, and the cost as the ‘risk capital’ that a regulator decides that the holder of the portfolios should carry. This case was discussed in depth by Denault (2001), who used a cost functional based on a coherent risk measure (Artzner et al., 1999), and the Aumann–Shapley value emerged again as an appropriate solution concept. Explicit calculations of the Aumann–Shapley value were provided when the risk measure used is Expected Shortfall (this problem was solved by Tasche (2000a) in the context of performance measurement) and for the case of a risk measure used by the Securities and Exchange Commission.
In this paper we calculate an analytic formula for the Aumann–Shapley value using quantile derivatives (Tasche, 2000b), for the case that the risk measure belongs to the class of distortion principles (Denneberg, 1990, Wang et al., 1997). We obtain a representation of the Aumann–Shapley value, i.e. the capital allocated to each portfolio, as the expected value of the portfolio under a change of probability measure. This representation creates a formal link between problems of allocating capital and pricing risks. We discuss this relationship through the example of a pool, which covers specific liabilities carried by a number of insurers, who in turn make cash contributions to the pool, that can be interpreted as risk premia.
It was shown by Mirman and Tauman (1982) that the Aumann–Shapley value yields equilibrium prices in a monopolistic production economy. Motivated by this work, we generalise the example to a case where the different insurers choose the extent of coverage received from the pool, by expected utility maximisation. This set-up is quite different from the equilibrium models usually found in the literature on risk sharing, for example, Borch (1962), Bühlmann (1980), Taylor (1995), Aase (2002). In these papers, market prices are obtained via a clearing condition, which is not applicable to the problem that we discuss. Finally, we provide a simple numerical example, where the pool offers stop-loss protection to the participating insurers.
Section snippets
Coherent risk measures and distortion principles
A coherent risk measure is defined by Artzner et al. (1999) as a functional ρ(X) on a collection of random cashflows (in our case X will be a non-negative random variable representing liabilities) that satisfies the following properties:
- •
Monotonicity: If X≤Y a.s. then ρ(X)≤ρ(Y),
- •
Subadditivity: ρ(X+Y)≤ρ(X)+ρ(Y),
- •
Positive homogeneity: If then ρ(aX)=aρ(X),
- •
Translation invariance: If then ρ(X+a)=ρ(X)+a.
ρ(X) is interpreted as “the minimum extra cash that the agent has to add to the risky
The cost functional
In our application, costs corresponding to the capital that the holder of a risky portfolio (e.g. of insurance liabilities) is obliged to hold will be allocated to the different instruments (or sub-portfolios) it consists of. The cost functional used will be derived from the distortion principle defined in the previous section.
Consider a portfolio Zu composed of n liabilities Xj, with portfolio weights u∈[0,1]n:Then, as in Denault (2001), we define the cost functional :
The Aumann–Shapley value for distortion principles
Explicit calculations of the Aumann–Shapley value have been given in the bibliography for the cases that the cost functional is derived from the risk measure Expected Shortfall (Tasche, 2000a) and a risk measure used by the Securities and Exchange Commission (Denault, 2001). Here we provide an explicit formula for the Aumann–Shapley value in the case that the cost functional corresponds to a distortion principle, as defined in Section 2.
To derive Aumann–Shapley prices, we will need to calculate
Simple model of a pool
It was shown that the Aumann–Shapley value produces a cost allocation mechanism that takes the form of an expectation under a change of probability measure. This representation provides a formal link between cost allocation and pricing problems. Here we make this link explicit by an example. Consider n (re)insurers, exposed to some specific very high risks. In order to protect themselves against these risks, they form a pool. The pool takes on the individual insurers’ liabilities, that is, it
Summary and future research
We discuss the problem of allocating capital requirements for a portfolio of stochastic liabilities to the different instruments (and sub-portfolios) that the portfolio consists of. The allocation functional used is the Aumann–Shapley value from cooperative game theory, which has been proposed as a cost allocation mechanism by several authors (Mirman and Tauman, 1982, Billera and Heath, 1982, Denault, 2001). In our application, we used as a risk measure the distortion principle, which is an
References (29)
- et al.
Axiomatic characterization of insurance prices
Insurance: Mathematics and Economics
(1997) Families of update rules for non-additive measures: applications in pricing risks
Insurance: Mathematics and Economics
(1998)Perspectives of risk sharing
Scandinavian Actuarial Journal
(2002)- et al.
Coherent measures of risk
Mathematical Finance
(1999) Cooperative fuzzy games
Mathematics of Operations Research
(1981)- Aumann, R.J., Shapley, L.S., 1974. Values of Non-atomic Games. Princeton University Press,...
- et al.
Allocation of shared costs: a set of axioms yielding a unique procedure
Mathematics of Operations Research
(1982) Equilibrium in a reinsurance market
Econometrica
(1962)An economic premium principle
ASTIN Bulletin
(1980)- et al.
Choquet pricing in financial markets with frictions
Mathematical Finance
(1996)
Theory of capacities
Annales de l’Institut Fourier
Coherent allocation of risk capital
Journal of Risk
Distorted probabilities and insurance premiums
Methods of Operations Research
Cited by (52)
Can a regulatory risk measure induce profit-maximizing risk capital allocations? The case of conditional tail expectation
2021, Insurance: Mathematics and EconomicsCitation Excerpt :Also, for loss RVs with continuous cumulative distribution functions (CDFs), allocation rule (1) coincides with the RC allocation rule induced by the Expected Shortfall risk measure (Kalkbrener, 2005; Wang and Zitikis, 2021). Last but not least, allocation rule (1) belongs to the class of distorted (Tsanakas and Barnett, 2003) and weighted (Furman and Zitikis, 2008b) RC allocation rules and is optimal in the sense of Laeven and Goovaerts (2004) and Dhaene et al. (2012). The number of works that evaluate allocation rule (1) for random losses having distinct joint CDFs is really overwhelming.
τ-value for risk capital allocation problems
2020, Operations Research LettersA generalization of the Aumann–Shapley value for risk capital allocation problems
2020, European Journal of Operational ResearchCitation Excerpt :The allocation rule then determines how the benefits of this hedge potential are allocated to the divisions. There is a large literature on capital allocation rules, with approaches based on finance (e.g., Major, 2018; Myers & Read, 2001; Tasche, 1999) optimization (e.g., Dhaene, Goovaerts, & Kaas, 2003) and game theory (e.g., Boonen, Tsanakas, & Wüthrich, 2017; Csóka, Herings, & Kóczy, 2009; Denault, 2001; Powers, 2007; Tsanakas, 2004; Tsanakas & Barnett, 2003). Our focus in this paper is on game-theoretic approaches to allocating risk capital.
Tail risk measures and risk allocation for the class of multivariate normal mean–variance mixture distributions
2019, Insurance: Mathematics and EconomicsCapital allocation à la Aumann–Shapley for non-differentiable risk measures
2018, European Journal of Operational ResearchCitation Excerpt :Another work that deals with non-differentiability, but is limited to the coherent case, is Boonen, De Waegenaere, and Norde (2012). The reader is also advised to refer to Denault (2001) and Tsanakas and Barnett (2003) for further considerations on the topic. The purpose of the present work is to try to plug these gaps, though not in full generality.
Distortion measures and homogeneous financial derivatives
2018, Insurance: Mathematics and EconomicsCitation Excerpt :The conclusion is that, if the risk measure is sufficiently differentiable, the Aumann–Shapley value “is the only linear coherent allocation principle”. Equivalent axioms and conclusions appear in Aubin (1981), Billera et al. (1981), Tasche (1999), and Tsanakas and Barnett (2003). It should be noted that the Aumann–Shapleyallocation principle is also compatible with the optimization approaches to capital allocation in Laeven and Goovaerts (2004).