Optimal Risk Sharing with Time-Inconsistency and Long-Run Risk


Book Description

I examine the role of the experimentally documented bias time-inconsistency for the dynamics of asset prices and wealth distribution between agents with recursive preferences. In a general equilibrium model with two types of investors, time-consistent and time-inconsistent, I show that the wealth share of the time-inconsistent agent is strictly lower than the one of the time-consistent agent, all else equal. The time-inconsistent investor, however, can dominate in the long-run despite her bias, in case she incorrectly believes that in the future she will save more than the time-consistent investor. In the presence of long-run risk accounting for time-inconsistency allows to study and endogenously match asset pricing dynamics such as the countercyclical feature of the equity premium that we observe in reality. These dynamics stem from the fact that the time-inconsistent investor who is less averse to persistent shocks tends to sell insurance against them to the time-consistent agent.




The Dynamics of Optimal Risk Sharing


Book Description

We study a dynamic-contracting problem involving risk sharing between two parties - the Proposer and the Responder - who invest in a risky asset until an exogenous but random termination time. In any time period they must invest all their wealth in the risky asset, but they can share the underlying investment and termination risk. When the project ends they consume their final accumulated wealth. The Proposer and the Responder have constant relative risk aversion R and r respectively, with R>r>0. We show that the optimal contract has three components: a non-contingent flow payment, a share in investment risk and a termination payment. We derive approximations for the optimal share in investment risk and the optimal termination payment, and we use numerical simulations to show that these approximations offer a close fit to the exact rules. The approximations take the form of a myopic benchmark plus a dynamic correction. In the case of the approximation for the optimal share in investment risk, the myopic benchmark is simply the classical formula for optimal risk sharing. This benchmark is endogenous because it depends on the wealths of the two parties. The dynamic correction is driven by counterparty risk. If both parties are fairly risk tolerant, in the sense that 2>R>r, then the Proposer takes on more risk than she would under the myopic benchmark. If both parties are fairly risk averse, in the sense that R>r>2, then the Proposer takes on less risk than she would under the myopic benchmark. In the mixed case, in which R>2>r, the Proposer takes on more risk when the Responder's share in total wealth is low and less risk when the Responder's share in total wealth is high. In the case of the approximation for the optimal termination payment, the myopic benchmark is zero. The dynamic correction tells us, among other things, that: (i) if the asset has a high return then, following termination, the Responder compensates the Proposer for the loss of a valuable investment opportunity; and (ii) if the asset has a low return then, prior to termination, the Responder compensates the Proposer for the low returns obtained. Finally, we exploit our representation of the optimal contract to derive simple and easily interpretable sufficient conditions for the existence of an optimal contract -- National Bureau of Economic Research web site.







Optimal Risk Sharing for Law Invariant Monetary Utility Functions


Book Description

We consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law-invariant risk measures. We first prove existence of an optimal risk sharing allocation which is in addition increasing in terms of the total risk. We next provide an explicit characterization in the case where both agents' utility functions are comonotone. The general form of the optimal contracts turns out to be given by a sum of options (stop-loss contracts, in the language of insurance) on the total risk. In order to show the robustness of this type of contracts to more general utility functions, we introduce a new notion of strict risk aversion conditionally on lower tail events, which is typically satisfied by the semi-deviation and the entropic risk measures. Then, in the context of an AV@R-agent facing an agent with strict monotone preferences and exhibiting strict risk aversion conditional on lower tail events, we prove that optimal contracts again are European options on the total risk.




Optimal Risk Exchanges


Book Description

The determination of optimal rules for sharing risks and constructing reinsurance treaties has important practical and theoretical interest. Medolaghi, de Finetti, and Ottaviani developed the first linear reciprocal reinsurnace treaties based upon minimizing individual and aggregate variance of risk. Borch then used the economic concept of utility to justify choosing Pareto-optimal forms of risk exchange; in many cases, this leads to familiar linear quota-sharing of total pooled losses, or to stop-loss arrangements. However, this approach does not give a unique, risk-sharing agreement, and may lead to substantial fixed side payments. Gerber showed how to constrain a Pareto-optimal risk exchange to avoid invasion of reserves. To these ideas, the authors have added the actuarial concept of long-run fairness to each participant in the risk exchange; the result is a unique, Pareto-optimal risk pool, with 'quota-sharing-by-layers' of the total losses. There are many interesting special cases, especially when all individual utility functions are of exponential form, giving linear quota-sharing-by-layers. Algorithms and numerical examples are given. (Author).




Financial Markets and the Real Economy


Book Description

Financial Markets and the Real Economy reviews the current academic literature on the macroeconomics of finance.




Protecting All


Book Description

"This white paper focusses on the policy interventions made to help people manage risk, uncertainty and the losses from events whose impacts are channeled primarily through the labor market. The objectives of the white paper are: to scrutinize the relevance and effects of prevailing risk-sharing policies in low- and middle-income countries; take account of how global drivers of disruption shape and diversify how people work; in light of this diversity, propose alternative risk-sharing policies, or ways to augment and improve current policies to be more relevant and responsive to peoples' needs; and map a reasonable transition path from the current to an alternative policy approach that substantially extends protection to a greater portion of working people and their families. This white paper is a contribution to the broader, global discussion of the changing nature of work and how policy can shape its implications for the wellbeing of people. We use the term risk-sharing policies broadly in reference to the set of institutions, regulations and interventions that societies put in place to help households manage shocks to their livelihoods. These policies include formal rules and structures that regulate market interactions (worker protections and other labor market institutions) that help people pool risks (social assistance and social insurance), to save and insure affordably and effectively (mandatory and incentivized individual savings and other financial instruments) and to recover from losses in the wake of livelihood shocks ('active' reemployment measures). Effective risk-sharing policies are foundational to building equity, resilience and opportunity, the strategic objectives of the World Bank's Social Protection and Jobs Global Practice. Given failures of factor markets and the market for risk in particular the rationale for policy intervention to augment the options that people have to manage shocks to their livelihoods is well-understood and accepted. By helping to prevent vulnerable people from falling into poverty --and people in the poorest households from falling deeper into poverty-- effective risk-sharing interventions dramatically reduce poverty. Households and communities with access to effective risk-sharing instruments can better maintain and continue to invest in these vital assets, first and foremost, their human capital, and in doing so can reduce the likelihood that poverty and vulnerability will be transmitted from one generation to the next. Risk-sharing policies foster enterprise and development by ensuring that people can take appropriate risks required to grasp opportunities and secure their stake in a growing economy."--




Optimal Investment


Book Description

Readers of this book will learn how to solve a wide range of optimal investment problems arising in finance and economics. Starting from the fundamental Merton problem, many variants are presented and solved, often using numerical techniques that the book also covers. The final chapter assesses the relevance of many of the models in common use when applied to data.







Modeling, Stochastic Control, Optimization, and Applications


Book Description

This volume collects papers, based on invited talks given at the IMA workshop in Modeling, Stochastic Control, Optimization, and Related Applications, held at the Institute for Mathematics and Its Applications, University of Minnesota, during May and June, 2018. There were four week-long workshops during the conference. They are (1) stochastic control, computation methods, and applications, (2) queueing theory and networked systems, (3) ecological and biological applications, and (4) finance and economics applications. For broader impacts, researchers from different fields covering both theoretically oriented and application intensive areas were invited to participate in the conference. It brought together researchers from multi-disciplinary communities in applied mathematics, applied probability, engineering, biology, ecology, and networked science, to review, and substantially update most recent progress. As an archive, this volume presents some of the highlights of the workshops, and collect papers covering a broad range of topics.