Risk-sharing Rules Research Articles

A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules

A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules

Optimal commissions and subscriptions in mutual aid platforms

AbstractThis paper investigates an operation mechanism for mutual aid platforms to develop more sustainably and profitably. A mutual aid platform is an online risk-sharing platform for risk-heterogeneous participants, and the platform extracts revenues by charging participants commission and subscription fees. A modeling framework is proposed to identify the optimal commissions and subscriptions for mutual aid platforms. Participants are divided into different types based on their loss probabilities and values derived from the platform. We present how these commissions and subscriptions should be set in a mutual aid plan to maximize the platform’s revenues. Our analysis emphasized the importance of accounting for risk heterogeneity in mutual aid platforms. Specifically, different types of participants should be charged different commissions/subscriptions depending on their loss probabilities and values on the platform. Participants’ shared costs should be determined based on their loss probabilities. Adverse selection occurs on the platform if participants with different risks pay the same shared costs. Our results also show that the platform’s maximum revenue will be lower if the platform charges the same fee to all participants. The numerical results of a practical example illustrate that the optimal commission/subscription scheme and risk-sharing rule result in considerable improvements in platform revenue over the current scheme implemented by the platform.

Conditional mean risk sharing of losses at occurrence time in the compound Poisson surplus model

This paper proposes a new risk-sharing procedure, framed into the classical insurance surplus process. Compared to the standard setting where total losses are shared at the end of the period, losses are allocated among participants at their occurrence time in the proposed model. The conditional mean risk-sharing rule proposed by Denuit and Dhaene (2012) is applied to this end. The analysis adopts two different points of views: a collective one for the pool and an individual one for sharing losses and adjusting the amounts of contributions among participants. These two views are compatible under the compound Poisson risk process. Guarantees can also be added by partnering with an insurer.

Why are risk-sharing rules uncertain? A sociological study of local financial governance

This article discusses the uncertainty of risk-sharing rules in local financial governance. That is, when the formal risk-sharing rules of financial transactions are agreed upon in advance, actual operations are uncertain. First, the institutional contradiction at the macro-level is an important structural source of the uncertain rules at the micro-level. Second, institutional contradictions endow actors with conflicting bases of legitimacy and driving forces of interest, which induces games of norms and interests among investors, local governments, and intermediaries with regard to risk-sharing rules and leads to the competitive pattern of varied risk-sharing rules. Last, the combination of multiple legitimacy claims and multiple mechanisms of power competition leads to uncertainty in the risk-sharing rules of actual operations.

Mutual peer-to-peer insurance: The allocation of risk

• We analyze the conditional mean sharing rule in peer-to-peer insurance. • We increase the conditional mean contribution with a safety loading term. • We show our model’s effectiveness by two examples: dependent and independent risks. This work focuses on a modern typology through which mutual solidarity in the insurance sector finds application: peer-to-peer insurance. This cooperative insurance model arises from the translation of the sharing economy concept into insurance risk management, and it is realized thanks to the use of digital technology connecting policyholders and sharing risks. The participants to a peer-to-peer insurance scheme share the first layer of their cumulative losses, while it is possible to transfer to a third part the higher layer. To enter the mutual group, each participant has to pay an initial contribution based on a risk-sharing rule that has to be intuitive and transparent. According to the most considered conditional mean risk-sharing rule, the participant has to contribute with an amount equal to the expected value of the risk he brings to the pool given the total loss distribution. We propose to modify the conditional mean risk-sharing rule with an ex-ante contribution that takes into account a safety loading to hedge the possible fluctuations of total losses.

Double Moral Hazard and Risk-Sharing in Construction Projects

Construction projects are riddled with risks that should be shared by both owners and contractors alike. The principal-agent theory has proven to be an influential theoretical apparatus in designing optimal risk-sharing rules and provides contractors with proper incentives to improve project efficiencies. According to the principal-agent theory, it is optimal to let the contractor assume all of the risk if it is risk-neutral. However, this article argues that this theoretical model may lead to a biased incentive structure. To demonstrate this point, this article analyzes the necessity of risk-sharing between the owner and risk-neutral contractor in construction projects. Taking into account cases where the respective planning efforts of the owner and contractor are supposed to be unverifiable and mutually complementary in terms of cost overrun risk, we find that either double moral hazard (i.e., both parties take opportunistic strategies) or inefficient breach of contract by the contractor occurs if the contractor assumes the entire cost overrun risk. An optimal risk-sharing strategy, therefore, plays an important role in coordinating both parties to choose efficient strategies.

The economics of sharing macro-longevity risk

Pension funds face macro-longevity risk or uncertainty about future mortality rates. We analyze macro-longevity risk sharing between cohorts in a pension scheme as a risk management tool. We show that the optimal risk-sharing rule and the welfare gains from risk sharing largely depend on the retirement age policy. In case of a fixed retirement age welfare gains from sharing macro-longevity risk measured on a 10-year horizon are between 0.1 and 0.3 percent of certainty equivalent consumption after retirement for each cohort. By contrast, if the retirement age is fully linked to changes in life expectancy, welfare gains are substantially higher. In this case the risk bearing capacity of workers is particularly large because their labor supply acts as a hedge against macro-longevity risk. As a result, workers absorb risk from retirees in the optimal risk-sharing rule, thereby increasing the welfare gain for each cohort up to 1.8 percent in the Lee–Carter mortality model and up to 2.9 percent in the Cairns–Blake–Dowd model.

Optimal Commissions and Subscriptions in Mutual Aid Platforms

This paper investigates an operation mechanism for mutual aid platforms to develop more sustainably and profitably. A mutual aid platform is an online risk-sharing platform for risk-heterogeneous participants, and the platform extracts revenues by charging participants commission and subscription fees. A modeling framework is proposed to identify the optimal commissions and subscriptions for mutual aid platforms. Participants are divided into different types based on their loss probabilities and values derived from the platform. We present how these commissions and subscriptions should be set in a mutual aid plan to maximize the platform's revenues. Our analysis emphasized the importance of accounting for risk heterogeneity in mutual aid platforms. Specifically, different types of participants should be charged different commissions/subscriptions depending on their loss probabilities and values on the platform. Participants' shared costs should be determined based on their loss probabilities. Adverse selection occurs on the platform if participants with different risks pay the same shared costs. Our results also show that the platform's maximum revenue will be lower if the platform charges the same fee to all participants. The numerical results of a practical example illustrate that the optimal commission/subscription scheme and risk-sharing rule result in considerable improvements in platform revenue over the current scheme implemented by the platform.

Sharing of Longevity Basis Risk in Pension Schemes With Income-Drawdown Guarantees

This work studies a stochastic optimal control problem for a pension scheme which provides an income-drawdown policy to its members after their retirement. To manage the scheme efficiently, the manager and members agree to share the investment risk based on a pre-decided risk-sharing rule. The objective is to maximise both sides' utilities by controlling the manager's investment in risky assets and members' benefit withdrawals. We use stochastic affine class models to describe the force of mortality of the members' population and consider a longevity bond whose coupon payment is linked to a survival index. In our framework, we also investigate the longevity basis risk, which arises when the members' and the longevity bond's reference populations show different mortality behaviours. By applying the dynamic programming principle to solve the corresponding HJB equations, we derive optimal solutions for the single- and sub-population cases. Our numerical results show that by sharing the risk, both manager and members increase their utility. Moreover, even in the presence of longevity basis risk, we demonstrate that the longevity bond acts as an effective hedging instrument.

The Economics of Sharing Macro-longevity Risk

Pension funds face macro-longevity risk or uncertainty about future mortality rates. We analyze macro-longevity risk sharing between cohorts in a pension scheme as a risk management tool. We show that the optimal risk-sharing rule and the welfare gains from risk sharing largely depend on the retirement age policy. In case of a xed retirement age welfare gains from sharing macro-longevity risk measured on a 10-year horizon are between 0.1 and 0.3 percent of certainty equivalent consumption after retirement for each cohort. By contrast, if the retirement age is fully linked to changes in life expectancy, welfare gains are substantially higher. In this case the risk bearing capacity of workers is particularly large because their labor supply acts as a hedge against macro-longevity risk. As a result, workers absorb risk from retirees in the optimal risk-sharing rule, thereby increasing the welfare gain for each cohort up to 1.8 percent in the Lee-Carter mortality model and up to 2.9 percent in the Cairns-Blake-Dowd model.

Product options for enhanced retirement income

AbstractA new stream of research proposes how people can increase their income in retirement by pooling their mortality risk. How one of these mortality risk-sharing rules could be implemented in practice, as part of a retirement income scheme, is considered. A potential advantage of the scheme is that a retiree’s housing wealth can be monetised to provide an income stream. This would mean that retirees can continue living in their home, without needing to downsize. It may be most attractive to the millions of single pensioners, particularly those who are “asset-rich and cash-poor”. Other types of assets that could be included and how to mitigate selection risks are assessed. A way of smoothing the raw mortality credits in order to make the scheme more appealing to potential members is proposed. An illustrative premium calculation suggests that the cost of the smoothing is very small compared to the potential attractiveness of an enhanced, smoothed income.

The composite iteration algorithm for finding efficient and financially fair risk-sharing rules

We consider the problem of finding an efficient and fair ex-ante rule for division of an uncertain monetary outcome among a finite number of von Neumann–Morgenstern agents. Efficiency is understood here, as usual, in the sense of Pareto efficiency subject to the feasibility constraint. Fairness is defined as financial fairness with respect to a predetermined pricing functional. We show that efficient and financially fair allocation rules are in one-to-one correspondence with positive eigenvectors of a nonlinear homogeneous and monotone mapping associated to the risk sharing problem. We establish relevant properties of this mapping. On the basis of this, we obtain a proof of existence and uniqueness of solutions via nonlinear Perron–Frobenius theory, as well as a proof of global convergence of the natural iterative algorithm. We argue that this algorithm is computationally attractive, and discuss its rate of convergence.

Cooperation under uncertainty: Assessing the value of risk sharing and determining the optimal risk-sharing rule for agents with pre-existing business and diverging risk attitudes

The allocation of risk among the cooperating parties in a shared project is an important decision. This is especially true in the case of large infrastructure investments. Existing risk allocation methods are either simplistic or do not consider the effect of the agents' pre-existing businesses. In this paper, we model and analyse the effect of risk sharing when two agents want to co-develop an energy infrastructure project in an uncertain environment. The cooperating agents have a pre-existing risky business, and the new common project has a deterministic initial cost but random revenue potential. Our analysis shows that the optimal risk-sharing rule depends not only on the agents' risk aversions but also on the volatility of the common project profit, the volatilities of the agents' pre-existing businesses and the correlation of each agent's pre-existing business with the common project. An illustrative example based on energy infrastructure is used to show the implications of the sharing rule for partners.

Optimal multi-period consumption and investment with short-sale constraints

This article examines agents’ consumption-investment problem in a multi-period pure exchange economy where agents are constrained with the short-sale of state-dependent risky contingent claims. In equilibrum, agents hold options written on aggregate consumption in their optimal portfolios. Furthermore, under the specific case of quadratic utility, the optimal risk-sharing rule derived for the pricing agent leads to a multifactor conditional consumption-based capital asset pricing model (CCAPM), where excess option returns appear as factors.

Intra-Group Risk Sharing Under Financial Fairness

We examine intra-group risk sharing under the assumption that prices are given exogenously. Two situations are considered: one in which the entities that constitute the group choose to trade only among each other, and one in which the entities have limited access to the external market. In the first situation we prove that, if agents are expected utility maximizers, there is a unique risk-sharing rule that preserves value according to the exogenous rule and that is Pareto optimal among feasible allocations. This provides an extension of a result by Balasko (1979). In the second situation we give necessary and sufficient conditions for the internal market to be completable in the sense that all agents can reach all positions, subject only to the budget constraint, by combining their own partial access to the external market with internal trades under the given pricing rule.

Short-Sale Constraints, Options, and CCAPM

This article examines agents' consumption-investment problem in a multi-period pure exchange economy where agents are constrained with the short-sale of state-dependent risky contingent claims. In equilibrum, agents hold options written on aggregate consumption in their optimal portfolios. Furthermore, under the speci fic case of quadratic utility, the optimal risk-sharing rule derived for the pricing agent leads to a multifactor conditional consumption-based capital asset pricing model (CCAPM), where excess option returns appear as factors.

Pareto efficiency for the concave order and multivariate comonotonicity

This paper studies efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson (1994), that efficiency is characterized by a comonotonicity condition. The goal of the paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multidimensional case. The multivariate case is more involved (in particular because there is no immediate extension of the notion of comonotonicity), and it is addressed by using techniques from convex duality and optimal transportation.

Pareto efficiency for the concave order and multivariate comonotonicity

This paper studies efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson (1994) [27], that efficiency is characterized by a comonotonicity condition. The goal of the paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multidimensional case. The multivariate case is more involved (in particular because there is no immediate extension of the notion of comonotonicity), and it is addressed by using techniques from convex duality and optimal transportation.

Bankable emission permits under uncertainty and optimal risk-management rules

This article proposes a theory of banking of emission permits under conditions of regulatory uncertainty. Based on a two-period partial equilibrium framework, we examine the effects of increasing risk–in the sense of a mean preserving spread–regarding a future permit allocation at the firm level. We also examine the role of an agency to pool risks by re-allocating permits for a group of firms. Our results are twofold. First, an increase in risk may lead to changes in a firm’s banking strategy, depending on the third partial derivative of its production function with respect to pollution. Second, we define an optimal risk-sharing rule between agents to respond to political decision changes. Our results overall suggest that the bankability of permits may be used as a risk-management tool.

A numerical approach for a class of risk-sharing problems

Abstract This paper deals with risk-sharing problems between many agents, each of whom having a strictly concave law invariant utility. In the special case where every agent’s utility is given by a concave integral functional of the quantile of her individual endowment, we fully characterize the optimal risk-sharing rules. When there are many agents, these rules cannot be computed analytically. We therefore give a simple convergent algorithm and illustrate it on several examples.