Optimal risk sharing and borrowing constraints in a continuous-time model with limited commitment

We study a continuous-time version of the optimal risk-sharing problem with one-sided commitment. In the optimal contract, the agent?s consumption is a time-invariant, strictly increasing function of a single state variable: the maximal level of the agent?s income realized to date. We characterize this function in terms of the agent?s outside option value function and the discounted amount of time in which the agent?s income process is expected to reach a new to-date maximum. Under constant relative risk aversion we solve the model in closed-form: optimal consumption of the agent equals a constant fraction of his maximal income realized to date. In the complete-markets implementation of the optimal contract, the Alvarez–Jermann solvency constraints take the form of a simple borrowing constraint familiar from the Bewley–Aiyagari incomplete-markets models.     

Diverse beliefs and time variability of risk premia

Why do risk premia vary over time? We examine this problem theoretically and empirically by studying the effect of market belief on risk premia. Individual belief is taken as a fundamental primitive state variable. Market belief is observable; it is central to the empirical evaluation and we show how to measure it. Our asset pricing model is familiar from the noisy REE literature but we adapt it to an economy with diverse beliefs. We derive equilibrium asset prices and implied risk premium. Our approach permits a closed form solution of prices; hence we trace the exact effect of market belief on the time variability of asset prices and risk premia. We test empirically the theoretical conclusions. Our main result is that, above the effect of business cycles on risk premia, fluctuations in market belief have significant independent effect on the time variability of risk premia. We study the premia on long positions in Federal Funds Futures, 3- and 6-month Treasury Bills (T-Bills). The annual mean risk premium on holding such assets for 1?C12?months is about 40?C60 basis points and we find that, on average, the component of market belief in the risk premium exceeds 50% of the mean. Since time variability of market belief is large, this component frequently exceeds 50% of the mean premium. This component is larger the shorter is the holding period of an asset and it dominates the premium for very short holding returns of less than 2?months. As to the structure of the premium we show that when the market holds abnormally favorable belief about the future payoff of an asset the market views the long position as less risky hence the risk premium on that asset declines. More generally, periods of market optimism (i.e. ??bull?? markets) are shown to be periods when the market risk premium is low while in periods of pessimism (i.e. ??bear?? markets) the market??s risk premium is high. Fluctuations in risk premia are thus inversely related to the degree of market optimism about future prospects of asset payoffs. This effect is strong and economically very significant.     

EMS exchange rate expectations and time-varying risk premia

In this paper we examine exchange risk premia employing a survey dataset of EMS exchange rates. We are able to test a risk premium model directly, i.e. without having to rely on the rational expectations assumption. Our results indicate that time-varying risk premia are present in almost all cases and that a GARCH-in-mean specification for the premium is often appropriate.     

Financial amplification of foreign exchange risk premia

Theories of financial frictions in international capital markets suggest that financial intermediaries' balance sheet constraints amplify fundamental shocks. We present empirical evidence for such theories by decomposing the U.S. dollar risk premium into components associated with macroeconomic fundamentals, and a component associated with financial intermediary balance sheets. Relative to the benchmark model with only macroeconomic state variables, balance sheets amplify the U.S. dollar risk premium. We discuss applications to financial stability monitoring.     

Money and credit with limited commitment and theft

We study the interplay among imperfect memory, limited commitment, and theft, in an environment that can support monetary exchange and credit. Imperfect memory makes money useful, but it also permits theft to go undetected, and therefore provides lucrative opportunities for thieves. Limited commitment constrains credit arrangements, and the constraints tend to tighten with imperfect memory, as this mitigates punishment for bad behavior in the credit market. Theft matters for optimal monetary policy, but at the optimum theft will not be observed in the model. The Friedman rule is in general not optimal with theft, and the optimal money growth rate tends to rise as the cost of theft falls.     

Fiber investment and access under uncertainty: long-term contracts,risk premia,and access options

Journal of Regulatory Economics - Regulated access schemes shape incentives for both investment and entry in next-generation networks. We study in a general duopoly setting whether and how risk...     

Foreign currency debt, risk premia and macroeconomic volatility

This paper studies the relationships between foreign currency debt, macroeconomic volatility, and risk premia in a model of a small open emerging market economy. The external value of the local currency is counter-cyclical, so that foreign currency debt requires larger repayments than local currency debt in bad states of nature. The level of foreign currency-denominated debts, therefore, affects the volatility of aggregate demand and by extension of the exchange rate. Exchange rate volatility is in turn an important determinant of the risk premium on local currency debt. Finally, this risk premium is a major factor in the choice of local versus foreign currency for emerging market borrowers. The mutual endogeneity of foreign currency debt, risk premia, and macroeconomic volatility creates important feedback effects in the economy: small increases in international risk aversion may entail large amplification effects on macroeconomic volatility since domestic borrowers substitute towards cheaper but riskier foreign currency debt finance.     

Bond risk premia, macroeconomic fundamentals and the exchange rate

We propose a two-country no-arbitrage term-structure model to analyze the joint dynamics of bond yields, macroeconomic variables and the exchange rate. The model allows to understand how exogenous shocks to the exchange rate affect the yield curves, how bond yields co-move in different countries and how the exchange rate is influenced by interest rates, macro-economic variables and time-varying bond risk premia.Estimating the model with US and German data, we find that time-varying bond risk premia account for a significant portion of the variability of the exchange rate: apparently, a currency tends to appreciate when investors expect large capital gains on long-term bonds denominated in that currency. A number of other novel empirical findings emerge.     

Imperfect credibility and risk premia in the European Monetary System

The paper deals with the estiamation of models for the expected rate of depreciation within the currency bands of the French franc and the Italian lira against the deutshmark, both unconditional and conditional upon no realignment, as well as the estimastion of models for risk premia. Using these estimates,estimates are constructed for the expected rate of depreciation, the expected rate of realignment and the expected rate of devaluation of these exchange rates during their EMS period by appropriate adjustment of interest rate differentials. It is found that these adjustments are of non-trivial magnitude.     

Optimal risk‐sharing under adverse selection and imperfect risk perception

Abstract.  The present paper thoroughly explores second‐best efficient allocations in an insurance economy with adverse selection. We start with a natural extension of the classical model, assuming less than perfect risk perception. We characterize the constraints on efficient redistribution, and we summarize the incidence of incentives on the economy with the notions of weak and strong adverse selection. Finally, we show in what sense improving risk perception enhances welfare.     

Competition, human capital and income inequality with limited commitment

We develop a dynamic model with two-sided limited commitment to study how barriers to competition, such as restrictions to business start-up and non-competitive covenants, affect the incentive to accumulate human capital. When contracts are not enforceable, high barriers lower the outside value of ‘skilled workers’ and reduce the incentive to accumulate human capital. In contrast, low barriers can result in over-accumulation of human capital. This can be socially optimal if there are positive spillovers. A calibration exercise shows that this mechanism can account for a sizable portion of cross-country income inequality.     

Social security and risk sharing

In this paper we identify conditions under which the introduction of a pay-as-you-go social security system is ex ante Pareto-improving in a stochastic OLG economy with capital accumulation and land. We argue that these conditions are consistent with realistic specifications of the parameters of the economy. In our model financial markets are complete and competitive equilibria interim Pareto efficient. Therefore, a welfare improvement can only be obtained if agents? welfare is evaluated ex ante, and arises from an improvement in intergenerational risk sharing. We also examine the optimal size of a given social security system as well as its optimal reform.     

Optimal risk sharing with background risk

This paper examines qualitative properties of efficient insurance contracts in the presence of background risk. In order to get results for all strictly risk-averse expected utility maximizers, the concept of “stochastic increasingness” is used. Different assumptions on the stochastic dependence between the insurable and uninsurable risk lead to different qualitative properties of the efficient contracts. The new results obtained under hypotheses of dependent risks are compared to classical results in the absence of background risk or to the case of independent risks. The theory is further generalized to nonexpected utility maximizers.     

International risk sharing and commodity prices

Abstract Cole and Obstfeld (1991) exposited a classic result where equilibrium movements in the terms of trade could make ex ante risk‐sharing arrangements unnecessary: a unity elasticity of substitution across goods and production specialization. This paper extends their model to N countries and M commodities (N > M). Here the terms of trade provides insurance against commodity‐specific shocks, not country‐specific shocks. Using commodity‐level production data at the national level and world commodity prices, we document significant terms of trade variability and positive responses of nation‐specific production to terms of trade improvements. The endogenous terms of trade insurance mechanism highlighted in CO is virtually non‐existent.     

Robust estimation and control under commitment

In a Markov decision problem with hidden state variables, a decision maker expresses fear that his model is misspecified by surrounding it with a set of alternatives that are nearby as measured by their expected log likelihood ratios (entropies). Sets of martingales represent alternative models. Within a two-player zero-sum game under commitment, a minimizing player chooses a martingale at time 0. Probability distributions that solve distorted filtering problems serve as state variables, much like the posterior in problems without concerns about misspecification. We state conditions under which an equilibrium of the zero-sum game with commitment has a recursive representation that can be cast in terms of two risk-sensitivity operators. We apply our results to a linear quadratic example that makes contact with findings of T. Ba?ar and P. Bernhard [H^∞-Optimal Control and Related Minimax Design Problems, second ed., Birkhauser, Basel, 1995] and P. Whittle [Risk-sensitive Optimal Control, Wiley, New York, 1990].     

Estimates of foreign exchange risk premia: a pricing kernel approach

The goal of this study is to measure market prices of risk and foreign exchange risk premia. Estimations of minimum variance pricing kernels permit to determine market prices of risk, which, in an international no-arbitrage framework, allow to measure foreign exchange risk premia. Market prices of risk are time-varying and surge during financial turmoils. Foreign exchange risk premia are on average small in comparison to interest rate differentials and exhibit significant variation from the early 1970s onwards, when the Bretton Woods exchange rate system collapsed. At times, foreign exchange risk premia dominate interest rate differentials. We are indebted to Baldev Raj, Robert Kunst, the associate editor of Empirical Economics and two anonymous referees for their valuable comments. We also thank seminar participants at the European Central Bank, the Bank of England and Queen Mary University of London. The views expressed in this paper are those of the authors and do not necessarily reflect those of the European Central Bank or the Eurosystem.     

Public versus private risk sharing

Can public income insurance through progressive income taxation improve the allocation of risk in an economy where private risk sharing is incomplete? The answer depends crucially on the fundamental friction that limits private risk sharing in the first place. If risk sharing is limited because insurance markets are missing for model-exogenous reasons (as in Bewley (1986) [8]) publicly provided risk sharing improves on the allocation of risk. If instead private insurance markets exist but their use is limited by limited enforcement (as in Kehoe and Levine (1993) [23]) then the provision of public insurance interacts with equilibrium private insurance, as, by providing risk sharing, the government affects the value of exclusion from private insurance markets and thus the enforcement mechanism of these contracts. We characterize consumption allocations in an economy with limited enforcement and a continuum of agents facing plausible income risk and tax systems with various degrees of progressivity (public risk sharing). We provide conditions under which more publicly provided insurance actually reduces total insurance for agents (excess crowding-out), or under which more public insurance increases total insurance (partial crowding-out).     

Fiscal policy under loose commitment

Due to time-inconsistency or political turnover, policymakers' promises are not always fulfilled. We analyze an optimal fiscal policy problem where the plans made by the benevolent government are periodically revised. In this loose commitment setting, the properties of labor and capital income taxes are significantly different than under the full-commitment and no-commitment assumptions. Because of the occasional reoptimizations, the average capital income tax is positive even in the long-run. Also, the autocorrelation of taxes is lower, their volatility with respect to output increases and the correlation between capital income taxes and output changes sign. Our method can be used to analyze the plausibility and the importance of commitment in a wide-class of dynamic problems.     

Modelling risk premia in CO2 allowances spot and futures prices

This article analyzes the modelling of risk premia in CO₂ allowances spot and futures prices, valid for compliance under the EU Emissions Trading Scheme (EU ETS). Similarly to electricity markets, a salient characteristic of CO₂ allowances is that the theory of storage does not hold, as CO₂ allowances only exist on the balance sheets of companies regulated by the scheme. The main result features positive time-varying risk premia in CO₂ spot and futures prices, which are strictly higher for post-2012 contracts (€6–9/ton of CO₂) than for Phase II contracts (€0–6/ton of CO₂). Contrary to Benth et al.'s (2008) for electricity markets, a positive relationship between risk premia and time-to-maturity is found in the EU ETS. As for relative differences between CO₂ futures and spot prices, CO₂ futures traded between + 1% (December 2008 contract) and + 33% (December 2014 contract) above spot prices during February 2008–April 2009. Contrary to Bessembinder and Lemmon (2002) for the electricity market, a positive relationship between risk premia and the variance/skewness of CO₂ spot prices is found. The futures-spot bias to the EU ETS explains around 1–6% of the variance of CO₂ futures premia.     

On optimality in intergenerational risk sharing

Summary. This paper defines and studies optimality in a dynamic stochastic economy with finitely lived agents, and investigates the optimality properties of an equilibrium with or without sequentially complete markets. Various Pareto optimality concepts are considered, including interim and ex ante optimality. We show that, at an equilibrium with a productive asset (land) and sequentially complete markets, the intervention of a government may be justified, but only to improve risk sharing between generations. If markets are incomplete, constrained interim optimality is investigated in two-period lived OLG economies. We extend the optimality properties of an equilibrium with land and give conditions under which introducing a pay-as-you-go system at an equilibrium would not lead to any Pareto improvement. Received: October 5, 1998; revised version: April 3, 2001