On the impossibility of fair risk allocation

Measuring and allocating risk properly are crucial for performance evaluation and internal capital allocation of portfolios held by banks, insurance companies, investment funds and other entities subject to financial risk. We show that by using a coherent measure of risk it is impossible to allocate risk satisfying the natural requirements of (Solution) Core Compatibility, Equal Treatment Property and Strong Monotonicity. To obtain the result we characterize the Shapley value on the class of totally balanced games and also on the class of exact games

Fair risk allocation in illiquid markets

Let us consider a financially constrained leveraged financial firm having some divisions which have invested into some risky assets. Using c oherent measures of risk the sum of the capital requirements of the divisions is larger tha n the capital requirement of the firm itself, there is some diversification benefit that should b e allocated somehow for proper performance evaluation of the divisions. In this pa per we use cooperative game theory and simulation to assess the possibility to jointly sat isfy three natural fairness requirements for allocating risk capital in illiquid markets: Core C ompatibility, Equal Treatment Property and Strong Monotonicity. Core Compatibility can be viewed as the allocated r isk to each coalition (subset) of divisions should be at least as much as the risk increment th e coalition causes by joining the rest of the divisions. Equal Treatment Property guarantees that if two divisions have the same stand� alone risk and also they contribute the same risk t o all the subsets of divisions not containing them, then the same risk capital should be allocate d to them. Strong Monotonicity requires that if a division weakly reduces its stand�alone r isk and also its risk contribution to all the subsets of the other divisions, then as an incentiv e its allocated risk capital should not increase. Analyzing the simulation results we concl ude that in most of the cases it is not possible to allocate risk in illiquid markets satis fying the three fairness notions at the same time, one has to give up at least one of them

Az arányos csődszabály karakterizációja körbetartozások esetén

Az arányos csődszabály első használata egészen Arisztotelészig vezethető vissza. A tanulmányban olyan pénzügyi hálózatokat vizsgálunk, ahol az ágenseknek van induló pénzkészlete, és mindenki tartozhat mindenkinek. Egy adott pénzügyi hálózatban a csődszabály meghatároz egy fizetési mátrixot, amelynek elemei megmondják, hogy ki mennyit fizessen a többi szereplőnek. Egy szereplő eszközei az induló pénzkészletéből és a többiektől kapott fizetésekből állnak. A rendszerkockázati irodalomban gyakran használt arányos csődszabály azt követeli meg, hogy az ágensek a tartozásaikkal arányosan fizessenek eszközeikből, maximum a tartozások erejéig. Ha érvényes az arányos csődszabály, akkor az eszközök értéke endogén módon határozódik meg, mivel a fizetések egymástól függhetnek. Cikkünkben részletesen bemutatjuk az arányos csődszabály egyik karakterizációját, olyan tulajdonságokat, amelyek közül mindegyiket csak ez a csődszabály teljesíti: a követelések felsőkorlát-jellegét, a korlátolt felelősséget, a hitelezők elsőbbségét, a pártatlanságot, az azonos ágensek általi manipulálhatatlanságot és a folytonosságot

Risk allocation under liquidity constraints

Abstract Risk allocation games are cooperative games that are used to attribute the risk of a financial entity to its divisions. In this paper, we extend the literature on risk allocation games by incorporating liquidity considerations. A liquidity policy specifies state-dependent liquidity requirements that a portfolio should obey. To comply with the liquidity policy, a financial entity may have to liquidate part of its assets, which is costly. The definition of a risk allocation game under liquidity constraints is not straightforward, since the presence of a liquidity policy leads to externalities. We argue that the standard worst case approach should not be used here and present an alternative definition. We show that the resulting class of transferable utility games coincides with the class of totally balanced games. It follows from our results that also when taking liquidity considerations into account there is always a stable way to allocate risk

On the impossibility of fair risk allocation

Measuring and allocating risk properly are crucial for performance evaluation and internal capital allocation of portfolios held by banks, insurance companies, investment funds and other entities subject to financial risk. We show that by using coherent measures of risk it is impossible to allocate risk satisfying simultaneously the natural requirements of Core Compatibility, Equal Treatment Property and Strong Monotonicity. To obtain the result we characterize the Shapley value on the class of totally balanced games and also on the class of exact games

Coherent Measures of Risk from a General Equilibrium Perspective

Coherent measures of risk defined by the axioms of monotonicity, subadditivity, positive homogeneity, and translation invariance are recent tools in risk management to assess the amount of risk agents are exposed to. If they also satisfy law invariance and comonotonic additivity, then we get a subclass of them: spectral measures of risk. Expected shortfall is a well-known spectral measure of risk is. We investigate the above mentioned six axioms using tools from general equi- librium (GE) theory. Coherent and spectral measures of risk are compared to the natural measure of risk derived from an exchange economy model, that we call GE measure of risk. We prove that GE measures of risk are coherent measures of risk. We also show that spectral measures of risk can be represented by GE measures of risk only under stringent conditions, since spectral measures of risk do not take the regulated entity's relation to the market portfolio into account. To give more insights, we characterize the set of GE measures of risk.Coherent Measures of Risk, General Equilibrium Theory, Exchange Economies, Asset Pricing

Az összekapcsoltság hatása a rendszerkockázatra homogén bankrendszerben

A pénzügyi rendszerkockázat legfontosabb formája a modern pénzügyi hálózatokban bekövetkező fertőzések veszélye. A cikkben egy olyan bankrendszert vizsgálunk, ahol homogének a bankok (mérlegfőösszegük és preferenciájuk azonos) és egymás eszközeit tulajdonolják. Ezen egyszerűsítő feltevéseket felhasználva egy analitikusan kiszámítható mérőszámot adunk a rendszerkockázatból adódó veszteségre, amely a bankok várható veszteségét adja meg egy rendszerbeli intézmény csődje esetén. E mérőszám tulajdonságait vizsgálva azt találjuk, hogy a banki eszközök volatilitásának növekedése, illetve a saját tőke arányának csökkenése emeli a lehetséges rendszerkockázati veszteséget, továbbá, hogy a bankrendszer felépítésének (a banki eszközök kereszttulajdonlásának) hatása kettős. Egyrészt az összekapcsoltság növelése erősíti a diverzifikációs hatást, mivel az adott bank más bankok eszközeivel fedezheti veszteségeit. Másrészt ha már eleve szorosan együttműködnek a bankok, akkor az összekapcsoltság további erősítése a fertőzés megnövekedett esélye következtében növeli a rendszerkockázatból fakadó potenciális veszteséget

An Axiomatization of the Pairwise Netting Proportional Rule in Financial Networks

We consider financial networks where agents are linked to each other via mutualliabilities. In case of bankruptcy, there are potentially many bankruptcy rules, ways to distribute the assets of a bankrupt agent over the other agents. One common approach is to first apply pairwise netting of agents that have mutual liabilities and next use the proportional rule to determine the payments on the basis of the net liabilities. We refer to this as the pairwise netting proportional rule. The pairwise netting proportional rule satisfies the basic requirements of claims boundedness, limited liability, priority of creditors, and continuity. It also satisfies the desirable properties of net impartiality, an agent that has two creditors with the same net claims pays the same amount to both creditors on top of pairwise netting, and invariance to mitosis, an agent that splits into a number of identical agents is not a_ecting the payments of the other agents. We demonstrate that if net impartiality and invariance to mitosis, together with the basic requirements, are regarded as imperative properties, then payments should be determined by the pairwise netting proportional rule