On the Impossibility of Fair Risk Allocation

1 Introduction

If a firm (financial enterprise, bank, insurance company, investment fund, portfolio, etc.) consists of divisions (individuals, products, subportfolios, risk factors, etc.), not only is it important to measure properly the risk of the firm, but also to allocate the risk capital of the firm to the divisions. From now on we refer to this problem as risk allocation problem and use the terms firm and divisions while keeping all the possible applications in mind.

In terms of measuring risk, we would like to model Expected Shortfall (Acerbi and Tasche 2002), which is promoted by the Basel Committee on Banking Supervision (BCoBS 2014) as the future industry standard, replacing Value-at-Risk. In this setting, a measure of risk assigns a real number to the future profit and loss random variable from the perspective of present. In order to keep the paper as simple as possible and to reach readers who are not familiar with the advanced mathematics of finance, we work with a finite probability space, where the portfolios of divisions are represented by realization vectors. We emphasize that assuming a finite probability space is not an infringement of generality, all of our results hold when the portfolios are described by bounded random variables on a measure space. A coherent measure of risk (Artzner et al. 1999) satisfies four natural properties (see Definition 2.1). A prominent example is the k-Expected Shortfall (Acerbi and Tasche 2002), which is the average of the worst $100k$ percent of the losses for $0\le k\le 1$. On the other hand, Value-at-Risk is not a coherent measure of risk (Artzner et al. 1999).

When using a coherent measure of risk, the risk of the portfolio of the firm is at most as much as the sum of the risks of the portfolios of the divisions. Thus there is usually a diversification benefit, which should be allocated somehow. Risk (capital) allocation games (Denault 2001) are transferable utility (TU) cooperative games defined to model such risk allocation. In a TU game using the values (the negative of the risk) of the coalitions (subsets) of the players (divisions) a solution concept (a risk allocation rule) determines how to share the value of the grand coalition (the firm). An allocation is in the core if the total value of the grand coalition is allocated (Efficiency) in such a way that no coalition of the players fairs better by acting alone. A totally balanced game has a non-empty core in each of its subgames, where a subgame is obtained by considering only a subset of the players.

Csóka, Herings, and Kóczy (2009) showed that the class of risk allocation games (using coherent measures of risk) coincides with the class of totally balanced games. That is, on the one hand, for any risk allocation game there is a core allocation, a stable way to allocate risk using an allocation rule satisfying Core Compatibility (CC). On the other hand, any totally balanced game can be given by a properly chosen risk allocation game. Csóka and Herings (2014) defined risk allocation games when divisions have illiquid portfolios and show that the class of risk allocation games with liquidity also coincides with the class of totally balanced games. Csóka, Herings, and Kóczy (2009) also proved that the class of risk allocation games with no aggregate uncertainty equals the class of exact games (Schmeidler 1972), where for each coalition there is a core allocation allocating the stand-alone value of the coalition.

In addition to CC, in this paper we consider two further fairness properties of risk capital allocation rules: Equal Treatment Property (ETP) and Strong Monotonicity (SM). The motivation comes from Young’s axiomatization (Young 1985) of the Shapley value (Shapley 1953), where he showed that on the class of all games the Shapley value is the only solution concept satisfying Efficiency, ETP and SM. It is well-known that the Shapley value does not satisfy CC in general, hence the properties of CC, ETP and SM cannot be satisfied simultaneously on the class of all games.

However, the validity of an axiomatization of a solution can vary from subclass to subclass, e.g. Shapley’s axiomatization of the Shapley value (see Section 4) is valid on the class of monotone games but not valid on the class of strictly monotone games. In the case of risk allocation games (in particular totally balanced games or exact games) we generalize Young’s result, and show that his axiomatization remains valid on the classes of totally balanced and exact games. We will prove (Theorem 3.4) that on the class of risk allocation games there is no risk allocation rule satisfying CC, ETP and SM at the same time.

We also interpret the requirements in the risk allocation setting as follows. CC is satisfied if all the risk of the firm is allocated in such a way that no group of the divisions can improve by allocating only the risk of the group, the risk allocation can rightly be seen stable. The blocking interpretations of the core can be questioned by saying that divisions cannot really leave the firm, see also Homburg and Scherpereel (2008). However, CC can also be viewed in an other way. The allocated risk to each coalition should be at least as much as the risk increment the coalition causes by joining the complementary coalition. ETP ensures that if two divisions have the same stand-alone risk and also they contribute the same risk to all the subsets of divisions not containing them, then they are treated equally, that is, the same risk capital is allocated to them. In other words, ETP states that if two divisions are not distinguishable from the point of view of risk, then they must be evaluated equally. SM requires that if a division weakly reduces its stand-alone risk and also its risk contribution to all the subsets of the other divisions (hence weakly increases its relative performance), then its allocated capital should not increase. Therefore, SM is closely related to Incentive Compatibility.

We will also prove (Proposition 3.5) that on the class of totally balanced (exact) games if a risk allocation rule meets CC and SM together, then there does exist a risk allocation rule which satisfies CC, SM and ETP together. However, we know from Theorem 3.4 that these three properties cannot hold at the same time, hence we find two (independent) requirements to blame for the impossibility result: CC and SM, one has to give up at least one of them.

Finally, in Example 3.8 we will illustrate that our impossibility result (Theorem 3.7) can be made stronger, because in all practical applications SM can be replaced by the following requirement: if a division’s stand-alone performance increases, then its allocated risk should not increase. That is, in practice there does not exist a risk allocation rule which satisfies CC and Incentive Compatibility at the same time.

The paper is organized as follows. In the following section we introduce the notations and notions for transferable utility cooperative games in general and risk allocations games in particular. In Section 3 we present our impossibility result. In Section 4 we show how our impossibility result is related to other results in the literature. The last section concludes.

2 Risk Allocation Games

We will use the following notations and notions: $|N|$ is for the cardinality of a finite set N and $2^N$ is the power set of N.

Let N denote the finite set ofplayers. A cooperative game with transferable utility (game, for short) is a function $v:2^N\to ℝ$ such that $v(\mathrm{\varnothing })=0$. The class of games with player set N is denoted by ${{G}}^N$. For a game $v\in {{G}}^N$ and a coalition $C\in 2^N,$ a subgame v_c is obtained by restricting v to the subsets of C.

An allocation is a vector $x\in {ℝ}^N$, where $x_i$ is the payoff of player $i\in N$. An allocation $x$ yields payoff $x(C)={\displaystyle {\sum }_{i\in C}}{x}_i$ to a coalition $C\in 2^N.$ An allocation $x\in {ℝ}^N$ is called Efficient, if $x(N)=v(N)$ and Coalitionally Rational if $x(C)\ge v(C)$ for all $C\in 2^N$. The core (Gillies 1959) is the set of Efficient and Coalitionally Rational allocations. The core of game v is denoted by $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(v)$.

Let $v\in {{G}}^N$ and $i\in N$ be a game and a player, and for all $C\subseteq N$ let $v_i^{\prime }(C)=v(C\cup \{i\})-v(C)$ denote player i’s marginal contribution to coalition C in game v. Then $v_i^{\prime }$ is called player i’s marginal contribution function in game v. Moreover, players i and j are equivalent in game v, $i{\sim }^{v}j$, if for every $C\subseteq N$ such that $i,j\notin C$ we have that $v_i^{\prime }(C)=v_j^{\prime }(C)$.

A game is totally balanced, if each of its subgame has a non-empty core. Let ${{G}}_{\mathrm{t}\mathrm{b}}^N$ denote the class of totally balanced games with player set N.

An interesting subclass of totally balanced games is the class of exact games (Schmeidler 1972). A game v is exact, if for each coalition C there exists an allocation $x\in \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(v)$ such that $x(C)=v(C)$. Let ${{G}}_{\mathrm{e}}^N$ denote the class of exact games with player set N.

Throughout the paper we consider single-valued solutions. The function $\mathrm{\psi }:A\to {ℝ}^N$, defined on $A\subseteq {{G}}^N$, is called solution on the class of games A. In the context of risk allocation, we refer to solutions as risk allocation rules.

To define risk allocation games we use the setup of Csóka, Herings, and Kóczy (2009). Let S denote the finite number of states of nature and consider the set ${ℝ}^S$ of realization vectors. State of nature s occurs with probability $p_s>0$, where ${\sum }_{s=1}^S{p}_s=1$. The vector $X\in {ℝ}^S$ represents a division’s possible profit and loss realizations. The amount $X_s$ is the division’s payoff in state of nature s. Negative values of $X_s$ correspond to losses.

A measure of risk is a function $\mathrm{\rho }:{ℝ}^S\to ℝ$ measuring the risk of a realization vector.

Let the matrix of realization vectors corresponding to the divisions be given by $X\in {ℝ}^{S\times N}$, and let $X_{\cdot i}$ denote column i of X, the realization vector of division i. For a coalition of divisions $C\in 2^N,$ let $X_C={\sum }_{i\in C}{X}_{\cdot i}$.

A risk environment is a tuple $(N,S,p,X,\mathrm{\rho })$, where N is the set of divisions, S indicates the number of states of nature, $p=(p_1,\dots ,p_S)$ is the vector of realization probabilities of the various states, X is the matrix of realization vectors, and $\mathrm{\rho }$ is a coherent measure of risk.

A risk allocation game assigns to each coalition of divisions the negative of the risk the coalition runs in its aggregate portfolio.

Let ${{G}}_{\mathrm{r}}^N$ denote the family of risk allocation games with player set N. In such games, according to eq. [1], the larger the risk of any subset of divisions, the lower its value. We illustrate the definition of the risk allocation game by the following example.

Note that for all $C\in 2^N\mathrm{\setminus }\{\mathrm{\varnothing }\}$ the value function is given by $v(C)=-\mathrm{\rho }(X_C)={min}_{s\in S}{X}_{C,s}$.

If the rows of a matrix of realization vectors sum up to the same number, then there is no aggregate uncertainty. Formally: a matrix of realization vectors $X\in {ℝ}^{S\times N}$ has no aggregate uncertainty, if there exists a number $\mathrm{\alpha }\in ℝ$ such that $X_N=\mathrm{\alpha }{1}^S$. Let ${{G}}_{\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{u}}^N$ denote the family of risk allocation games with no aggregate uncertainty and with player set N.

Note that by Theorem 2.4 all risk allocation games with or without liquidity are totally balanced (also the one in Example 2.3), and if there is no aggregate uncertainty, then all of them are exact. Moreover, for any totally balanced (exact) game there is a risk environment including a coherent measure of risk (with no aggregate uncertainty) that generates the game using eq. [1].

A risk allocation rule shows how to share the risk of the firm among the divisions. Since the situation can be converted into a TU game, a risk allocation rule can also be a solution in cooperative game theory, such as e.g. the Shapley solution. We illustrate the marginal contribution function and the fact that the Shapley solution is not always in the core by continuing Example 2.3 (Table 2).

Note that player 2 has a higher marginal contribution than the others, which is also expressed by the Shapley value, since $\mathrm{\varphi }(v)=(1/6,2/3,1/6)$. However, coalition $\{1,2\}$ is a blocking coalition for the Shapley solution in game v, since $v(\{1,2\})=1>5/6=1/6+2/3$, that is, $\mathrm{\varphi }(v)\notin \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(v)$.

3 The Impossibility Result

Next, we introduce the four basic properties (axioms) that we focus on.

Efficiency is implied by CC, since it requires that all the risk of the firm should be allocated to the divisions.

ETP makes sure that if two divisions have the same stand-alone risk and also they contribute the same risk to all the subsets of divisions not containing them, then they are treated equally, that is, the same risk capital is allocated to them. In other words, if two divisions are the same from the viewpoint of risk, then by ETP the same risk is assigned to both.

SM requires that if a division weakly reduces its stand-alone risk and also its risk contribution to all the subsets of the other divisions, hence weakly increases its relative performance, then its allocated capital should not increase. Thus as a kind of Incentive Compatibility notion, weakly better relative performance is weakly rewarded. Note that it follows from SM that for any $v,w\in A$, $i\in N$ such that $v_i^{\prime }=w_i^{\prime }$, we have ${\mathrm{\psi }}_i(v)={\mathrm{\psi }}_i(w)$ (called Marginality).

Note that risk allocation games form a proper subset of all games, since they are always totally balanced. In the following, we prove a Theorem 3.2 type result on the classes of totally balanced and exact games.

Only if: Let $u_T$ denote the unanimity game on coalition T, that is, for all $C\subseteq N$: $u_T(C)=\{\begin{array}{cc}1,& \mathrm{i}\mathrm{f}C\supseteq T\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$.

Next, we generalize Theorem 3.2 for such classes of games where if a game v is in the class, then for all $\mathrm{\alpha }>0$ and coalition T we have that $v+\mathrm{\alpha }{u}_T$ is also in the class. Notice that the class of totally balanced (exact) games is such class of games, since we get the required core allocations by distributing $\mathrm{\alpha }$ among the members of coalition T in a proper way.

Let v be a totally balanced (exact) game and let us decompose it to the unique sum of unanimity games such as $v={\sum }_{T\subseteq N}{\mathrm{\alpha }}_T{u}_T$. Moreover, let ${\mathrm{\alpha }}^m={max}_{T\subseteq N}{\mathrm{\alpha }}_T$, $v^{\ast }={\mathrm{\alpha }}^m{\sum }_{T\subseteq N}{u}_T$, and $v_d=v^{\ast }-v$.

Notice that $v^{\ast }$ is a totally balanced (exact) game and in $v_d={\sum }_{T\subseteq N}{\mathrm{\beta }}_T{u}_T$ we have that ${\mathrm{\beta }}_T\ge 0$ for all $T\subseteq N$. For each game w define the index $I(w)$ as follows: $I(w)=|\{{\mathrm{\gamma }}_T\ne 0:w={\sum }_{T\subseteq N}{\mathrm{\gamma }}_T{u}_T\}|$.

The proof goes by induction on $I(v_d)$.

If $I(v_d)=0$, then $v=v^{\ast }$, all players are equivalent in game v, so Efficiency and ETP imply that solution $\mathrm{\psi }$ is well-defined (unique) for game v.

Let k be an integer such that $0<k<2^{|N|}-1$. Assume that for each totally balanced (exact) game w such that $I(w_d)\le k$, $\mathrm{\psi }(w)$ is well-defined. Then, let v be a totally balanced (exact) game such that $I(v_d)=k+1$. Consider the decomposition $v_d={\sum }_{T\subseteq N}{\mathrm{\beta }}_T{u}_T$, that is, $v=v^{\ast }-{\sum }_{T\subseteq N}{\mathrm{\beta }}_T{u}_T$, where ${\mathrm{\beta }}_T\ge 0$ for all $T\subseteq N$.

First, if it exists, take any player i for which there exists $T\subseteq N$, ${\mathrm{\beta }}_T>0$ such that $i\notin T$. Let $v^k=v+{\mathrm{\beta }}_T{u}_T$. Notice that since ${\mathrm{\beta }}_T>0$ we have that is totally balanced (exact). Moreover, $I(v_d^k)=k$ and by the induction hypothesis $\mathrm{\psi }(v^k)$ is well-defined. Since $v_i^{\prime }=(v+{\mathrm{\beta }}_T{u}_T{)}_i^{\prime }$, SM (Marginality) implies that ${\mathrm{\psi }}_i(v)={\mathrm{\psi }}_i(v^k)$.

Second, if they exist, take the remaining players, that is, take all i such that $i\in T\subseteq N$ for all $T\subseteq N$ where ${\mathrm{\beta }}_T>0$. These players are equivalent in v (since they are equivalent in all games ${\mathrm{\beta }}_T{u}_T$, ${\mathrm{\beta }}_T>0$), so by ETP they get the same value.

Finally, by Efficiency, solution $\mathrm{\psi }$ is well-defined for game v.

Since the Shapley solution satisfies Efficiency, ETP and SM (Marginality) and $\mathrm{\psi }$ is well-defined (unique) using these properties, we have $\mathrm{\psi }=\mathrm{\varphi }$.□

Using Theorem 3.3 we can state the following.

So far we have established that the Shapley value is the only risk allocation rule satisfying Efficiency, ETP and SM (Marginality) on the class of risk allocation games with or without liquidity (with or without aggregate uncertainty), but it does not satisfy CC in general, hence there is no hope to satisfy CC, ETP and SM (Marginality) at the same time. Next, we show that Theorem 3.4 can be refined using the following proposition.

Then it is easy to see that for each $\mathrm{\pi }\in \mathrm{\Pi }(N)$ game $v\circ \mathrm{\pi }$ is a totally balanced (exact) game, and $v_i^{\prime }\ge w_i^{\prime }$ if and only if $(v\circ \mathrm{\pi }{)}_{\mathrm{\pi }(i)}^{\prime }\ge (w\circ \mathrm{\pi }{)}_{\mathrm{\pi }(i)}^{\prime }$. Therefore $\bar{\mathrm{\psi }}$ also meets CC and SM.

Take a game $v\in {{G}}_{tb}^N$ ($v\in {{G}}_e^N$) such that players i and j are equivalent in game v. Let ${\mathrm{\pi }}_{ij}\in \mathrm{\Pi }(N)$ be the permutation such that ${\mathrm{\pi }}_{ij}(k)=k$ if $k\in N\mathrm{\setminus }\{i,j\}$ and ${\mathrm{\pi }}_{ij}(i)=j$. Then for each permutation $\mathrm{\pi }\in \mathrm{\Pi }(N)$ we have that $v\circ \mathrm{\pi }\circ {\mathrm{\pi }}_{ij}=v\circ \mathrm{\pi }$. Therefore ${\sum }_{\mathrm{\pi }\in \mathrm{\Pi }(N)}\mathrm{\psi }(v\circ \mathrm{\pi }{)}_{\mathrm{\pi }(i)}={\sum }_{\mathrm{\pi }\in \mathrm{\Pi }(N)}\mathrm{\psi }(v\circ \mathrm{\pi }{)}_{\mathrm{\pi }(j)}$.□

Thus ETP is not independent from CC and SM on the class of totally balanced (exact) games.

Using Proposition 3.5 we have the following theorem.

The following example demonstrates that in practice SM can be replaced by the following requirement: if a divisions’ stand-alone (reported) performance increases, then its allocated risk should not increase. That is, in practice SM can be replaced by Incentive Compatibility.

Let the risk allocation games w and v be generated by risk environments $(N,S,p,X,\mathrm{\rho })$ and $(N,S,p,Y,\mathrm{\rho })$, respectively. It is easy to see that we get the games by Young (1985, Theorem 1, p. 69), where $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(w)=(3,0,0,6,3)$ and $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(v)=(0,1,2,7,1)$. Moreover, $Y_{\{2\}}<X_{\{2\}}$, thus in this case division 2 can outsmart any risk allocation method satisfying CC by performing worse or reporting less profit (like in Y), because then it gets 1 instead of 0. In other words there is no risk allocation rule satisfying CC and Incentive Compatibility.

4 Relations to Other Axiomatic Approaches

Other axiomatic approaches in the literature related to our work will be discussed below.

5 Conclusion

We have shown that by using coherent measures of risk it is impossible to allocate risk satisfying simultaneously the natural requirements of Core Compatibility and Strong Monotonicity (in practice Incentive Compatibility). To obtain the result we have characterized the Shapley value on the class of totally balanced games and also on the class of exact games as being the only risk allocation method satisfying Strong Monotonicity, Equal Treatment Property and Efficiency. Both classes are proper subsets of all TU games, hence it is not obvious that the characterization by Young (1985) holds on them. We have also interpreted the axioms and our results in the risk allocation setting and clarified their relation to the existing literature.

Since due to the results in this paper the Shapley value is the only risk allocation method satisfying Strong Monotonicity, Equal Treatment Property and Efficiency, the deepness of our impossibility result can be captured by checking the necessary and sufficient condition for the Shapley value to be in the core. Since no such condition is known, one has to resort to simulation. Balog et al. (2014) showed that our analytical result is not only a theoretical possibility. For randomly generated risk allocation games with 3 or 4 divisions the Shapley value is not in the core about 40-60% of the cases on the average. Csóka (2015) reported only a few percent lower numbers for risk allocation games with liquidity. Moreover, for a higher number of divisions in general we expect higher numbers.

Therefore our result raises the practical problem: one has to give up one of Core Compatibility or Incentive Compatibility to allocate risk in practice too. Depending on the application at hand, the literature on cooperative games can help in deciding requirement to drop in light of the trade-off between theoretical versus computational gains and losses. For instance (as we have seen) the Shapley solution meets Efficiency, Equal Treatment Property and Strong Monotonicity (in practice Incentive Compatibility). Moreover, it is well-known that the nucleolus (Schmeidler 1969) meets Core Compatibility and Equal Treatment Property, and there are many other well-analyzed solution concepts to choose from.