Risk Allocation Problem Research Articles

If Past is Prologue, then the Future is Bleak: Contracts, Covid–19, and the Changed Circumstances Doctrines

At the heart of most of the systemic problems currently confronting individuals and businesses as a result of the COVID–19 pandemic is quite literally a contract. Housing. Insurance. Food. Health care. Child care. Employment. Manufacturing. Construction. Supply chains. You name it. Contracts are implicated everywhere. So make no mistake: How contract law addresses these ostensibly private contracts will have profound social consequences. If the past really is prologue, then the future is indeed bleak. The empirical study conducted for this Article establishes what the conventional wisdom has claimed for the last 70 years. More specifically, the empirical study here shows that the common law’s changed circumstances doctrines (“CCDs”)—namely, impossibility, impracticability of performance, and frustration of purpose—will generally not excuse a party from performing his obligations under a contract, regardless of the changed circumstance he alleges. Contrary to all the CCD literature that addresses this issue, this Article makes the unconventional argument that the CCDs should be more broadly available, meaning they should be more successful in excusing contract performance when triggered by catastrophic circumstances. And unlike the rest of the field, which focuses on the CCDs themselves, this Article argues that to effectively address the allocation of unforeseen risks in general and catastrophic risks like a pandemic in particular, we must reframe the legal approach to contract formation. From there, given that the solution to the changed circumstances problem preferred by courts and commentators is an explicit risk-allocation term in the parties’ contract, the solution proposed in this Article to the risk-allocation problem literally suggests itself. A risk-and-loss-allocation clause should be mandated in most contracts as a part of contract formation. The type of risk-and-loss-allocation clause and how the clause would work would depend on whether the contract is co-drafted or adhesive. Generally, the inclusion of a risk-and-loss-allocation clause would facilitate transactions and encourage contracting by ensuring that contracts remain efficient and predictable. The main difference between the risk-and-loss-allocation clause proposed here and existing contract law, of course, is who ends up bearing all the risk and loss occasioned by the catastrophic changed circumstance. To be clear, if nothing changes and our approach to contract formation remains the same as it is right now, then all of the risk and all of the attendant loss will generally be left to lie where it falls—namely, on the party trying to get out of the contract because of the changed circumstances—and this will be the result regardless of the legal theory used to justify (or demonize) the CCDs or any changes made to the doctrines themselves. But if we finally acknowledge the public aspects of contracts and contract law, namely, that they do in fact produce social consequences that extend beyond the individual contract and contracting parties, then contracts and contract law may well be part of the solutions to some of the most pressing problems currently confronting American society now and into the future.

Covariance Steering With Optimal Risk Allocation

This article extends the optimal covariance steering (CS) problem for linear stochastic systems subject to chance constraints so as to account for an optimal allocation of the risk. Previous works have assumed a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uniform</i> risk allocation in order to cast the optimal control problem as a semidefinite program, which can be solved efficiently using standard SDP solvers. An iterative risk allocation (IRA) formalism is used to solve the optimal risk allocation problem for CS using a two-stage approach. The upper-stage of IRA optimizes the risk, which is a convex problem, while the lower-stage optimizes the controller with the new constraints. The process is applied iteratively until the optimal risk allocation that achieves the lowest total cost is obtained. The proposed framework results in solutions that tend to maximize the terminal covariance, while still satisfying the chance constraints, thus leading to less conservative solutions than previous methodologies. In this article, we consider both polyhedral and cone state chance constraints. Finally, we demonstrate the approach to a spacecraft rendezvous problem and compare the results with other competing approaches.

If Past is Prologue, Then The Future is Bleak: Contracts, COVID-19 and the Changed Circumstances Doctrines

At the heart of most of the systemic problems currently confronting individuals and businesses as a result of the COVID-19 pandemic is quite literally a contract. Housing. Insurance. Food. Health Care. Child Care. Employment. Manufacturing. Construction. Supply Chains. You name it. Contracts are implicated everywhere. So, make no mistake, how contract law addresses these ostensibly private contracts will have profound social consequences. If the past really is prologue, then the future is indeed bleak. The empirical study conducted for this Article establishes what the conventional wisdom has claimed for the last 70 years. More specifically, the empirical study here shows that the common law’s changed circumstances doctrines [“CCDs”] (i.e., impossibility, impracticability of performance and frustration of purpose) will generally not excuse a party from performing his contract, regardless of the changed circumstance being alleged. Contrary to all of the CCD literature that addresses this issue, this Article makes the unconventional argument that the CCDs should be more broadly available, meaning they should be more successful in excusing contract performance when triggered by catastrophic circumstances. And, unlike the rest of the field which focuses on the CCDs themselves, this Article argues that to effectively address the allocation of unforeseen risks, in general, and catastrophic risks like a pandemic, in particular, we must reframe the legal approach to contract formation. If contract formation is reframed to more accurately reflect contracting in the real world, and given that the solution to the changed circumstances problem preferred by courts and commentators is an explicit risk allocation term in the parties’ contract, then the solution proposed in this Article to the risk allocation problem literally suggests itself. A risk allocation clause should be mandated in most contracts as part of contract formation. The type of risk allocation and loss clause and the way the clause would work would depend on whether the contract is co-drafted or adhesive. The inclusion of a risk and loss allocation clause would facilitate transactions and encourage contracting in general by ensuring that contracts remain efficient and predictable. The main difference between the risk and loss allocation clause proposed here and existing contract law, of course, is who ends up bearing all of the risk and loss occasioned by the catastrophic changed circumstance. To be clear, if nothing changes and our approach to contract formation remains the same as it is right now, then all of the risk and all of the attendant loss will generally be left to lie where it falls, namely, on the party trying to get out of the contract because of the changed circumstances; and this will be result regardless of the legal theory used to justify (or demonize) the CCDs or any changes made to the doctrines themselves. But, If we finally acknowledge the public aspects of contracts and contract law, namely, that they do in fact produce social consequences that extend beyond the individual contract and contracting parties, then contracts and contract law may well be part of the solutions to some of the most pressing problems currently confronting American society now and into the future.

Generalized Systematic Risk

We generalize the concept of “systematic risk” to a broad class of risk measures potentially accounting for high distribution moments, downside risk, rare disasters, as well as other risk attributes. We offer two different approaches. First is an equilibrium framework generalizing the Capital Asset Pricing Model, two-fund separation, and the security market line. Second is an axiomatic approach resulting in a systematic risk measure as the unique solution to a risk allocation problem. Both approaches lead to similar results extending the traditional beta to capture multiple dimensions of risk. The results lend themselves naturally to empirical investigation. (JEL D81, G11, G12)

Optimal Risk Allocation in a Market with Non-Convex Preferences

The aims of this study are twofold. First, we consider an optimal risk allocation problem with non-convex preferences. By establishing an infimal representation for distortion risk measures, we give some necessary and sufficient conditions for the existence of optimal and asymptotic optimal allocations. We will show that, similar to a market with convex preferences, in a non-convex framework with distortion risk measures the boundedness of the optimal risk allocation problem depends only on the preferences. Second, we consider the same optimal allocation problem by adding a further assumption that allocations are co-monotone. We characterize the co-monotone optimal risk allocations within which we prove the "marginal risk allocations" take only the values zero or one. Remarkably, we can separate the role of the market preferences and the total risk in our representation.

Optimal risk allocation for convex risk functionals in general risk domains

Abstract In this paper, we formulate the classical optimal risk allocation problem for convex risk functionals defined on products of real Banach spaces as risk domains. This generality includes in particular the classical case of Lp risks but also allows to describe the influence of dependence in the risk allocation problem. We characterize optimal allocations and complete known existence and uniqueness results from the literature. We discuss in detail an application to expected risk functionals. This case can be dealt with by the Banach space approach applied to Orlicz hearts associated to the risk functionals. We give a detailed discussion of the necessary continuity and differentiability properties. Based on ordering results for Orlicz hearts we obtain extensions of the optimal allocation results to different Orlicz hearts as domain of risk functionals and establish a general form of the classical Borch theorem. In some numerical examples, optimal redistributions are determined for the expected risk case and the precision of the numerical calculation is checked.

ON SOME PROPERTIES OF A CLASS OF MULTIVARIATE ERLANG MIXTURES WITH INSURANCE APPLICATIONS

AbstractWe discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.

An enhanced multi-objective optimization approach for risk allocation in public–private partnership projects: a case study of Malaysia

The decision making for risk allocation problems in public–private partnership (PPP) projects is a vital process that directly affects the timeliness, cost, and quality of the project. Fair risk allocation is a vital factor to achieve success in the implementation of these projects. It is essential for private and public sectors to apply efficient risk allocation approaches to experience a more effective process of agreement arbitration and to reduce the appearance of dispute during the concession period. The aim of this study is to develop an optimization approach to enhance risk allocation process in PPP projects. The shared risks in projects are identified through comprehensive literature review and questionnaire survey obtained from Malaysian professionals involved in PPP projects. Objective functions are then developed to minimize the total time and cost of the project and maximize the quality while satisfying risk threshold constraints. The combinatorial nature of the risk allocation problem describes a multi-objective situation that can be simulated as a knapsack problem (KP). The formulation of the KP is described and solved applying genetic algorithm (GA). Due to the flexibility of GA, the results are Pareto Optimal solutions that describe the combinations of risk percentages for shared risks in PPP projects.

Generalized Systematic Risk

We generalize the concept of to a broad class of risk measures potentially accounting for high distribution moments, downside risk, rare disasters, as well as other risk attributes. We offer two different approaches. First is an equilibrium framework generalizing the Capital Asset Pricing Model, two-fund separation, and the security market line. Second is an axiomatic approach resulting in a systematic risk measure as the unique solution to a risk allocation problem. Both approaches lead to similar results extending the traditional beta to capture multiple dimensions of risk. The results lend themselves naturally to empirical investigation.

Short note on inf-convolution preserving the Fatou property

We model agents’ preferences by cash-invariant concave functionals defined on L∞, and formulate the optimal risk allocation problem as their infimal-convolution. We study the case of agents whose choice functionals are law-invariant with respect to different probability measures and show how, in this case, the value function preserves a desirable dual representation (equivalent to the Fatou property).

On convex risk measures on L p -spaces

Much of the recent literature on risk measures is concerned with essentially bounded risks in L∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on Lp spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on Lp we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for Lp risks.

Optimal capital and risk allocations for law- and cash-invariant convex functions

In this paper we provide a complete solution to the existence and characterization problem of optimal capital and risk allocations for not necessarily monotone, law-invariant convex risk measures on the model space L p for any p∈[1,∞]. Our main result says that the capital and risk allocation problem always admits a solution via contracts whose payoffs are defined as increasing Lipschitz-continuous functions of the aggregate risk.

Allocation of risks and equilibrium in markets with finitely many traders

The optimal risk allocation problem, equivalently the optimal risk sharing problem, in a market with n traders endowed with risk measures ϱ 1 , … , ϱ n is a classical problem in insurance and mathematical finance. This problem however only makes sense under a condition motivated from game theory which is called Pareto equilibrium. There are many situations of practical interest, where this condition does not hold. This is the case if the risk measures are based on essential different views towards risk. In this paper we introduce and analyze a meaningful extension of the optimal risk allocation (risk sharing) problem without assuming the equilibrium condition. The main point of this is to introduce a suitable and well motivated restriction on the class of admissible allocations which prevents effects of artificial ‘risk arbitrage’. As a result we obtain a new coherent risk measure which describes the inherent risk which remains after using admissible risk exchange in an optimal way.

Correction note: On the optimal risk allocation problem

Correction note: On the optimal risk allocation problem

On the optimal risk allocation problem

SUMMARY The optimal risk allocation problem or equivalently the problem of risk sharing is the problem to allocate a risk in an optimal way to n traders endowed with risk measures ϱ1, …, ϱ n . This problem has a long history in mathematical economics and insurance. In the first part of the paper we review some mathematical tools and discuss their applications to various problems on risk measures related to the allocation problem like to monotonicity properties of optimal allocations, to optimal investment problems or to an appropriate definition of the conditional value at risk. We then consider the risk allocation problem for convex risk measures ϱ i . In general the optimal risk allocation problem is well defined only under an equilibrium condition. This condition can be characterized by the existence of a common scenario measure. We formulate ameaningful modification of the optimal risk allocation problem also formarkets without assuming the equilibrium condition and characterize optimal solutions. The basic idea is to restrict the class of admissible allocations in a proper way.