F Kubler, Karl Schmedders, Approximate versus exact equilibria in dynamic economies, In: Computational aspects of general equilibrium theory: refutable theories of value, Springer, Berlin, p. 135 - 164, 2008. (Book Chapter)
This paper develops theoretical foundations for an error analysis of approximate equilibria in dynamic stochastic general equilibrium models with heterogeneous agents
and incomplete financial markets. While there are several algorithms that compute prices and allocations for which agents’ first-order conditions are approximately satisfied
(“approximate equilibria”), there are few results on how to interpret the errors in these candidate solutions and how to relate the computed allocations and prices to exact equilibrium allocations and prices. We give a simple example to illustrate that approximate equilibria might be very far from exact equilibria. We then interpret approximate equilibria as equilibria for close-by economies; that is, for economies with close-by individual endowments and preferences.
We present an error analysis for two models that are commonly used in applications, an overlapping generations (OLG) model with stochastic production and an asset pricing
model with infinitely lived agents.We provide sufficient conditions that ensure that approximate equilibria are close to exact equilibria of close-by economies. Numerical
examples illustrate the analysis. |
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Tymon Tatur, Robert Earle, Karl Schmedders, On Price Caps under Uncertainty, Review of Economic Studies, 2007. (Journal Article)
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Karl Schmedders, Two-Fund Separation in Dynamic General Equilibrium, Theoretical Economics, 2007. (Journal Article)
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P Jean-Jacques Herings, Karl Schmedders, Computing equilibria in finance economies with incomplete markets and transaction costs, Economic Theory, Vol. 27 (3), 2006. (Journal Article)
Transaction costs on financial markets may have important consequences for volumes of trade, asset pricing, and welfare. This paper introduces an algorithm for the computation of equilibria in the general equilibrium model with incomplete asset markets and transaction costs. We show that economies with transaction costs can be analyzed with differentiable homotopy techniques and thus in the same framework as frictionless economies despite the existence of non-differentiabilities of agents’ asset demand functions and the existence of locally non-unique equilibria. We introduce an equilibrium selection concept into the computation of economic equilibria that picks out a specific equilibrium in the presence of a continuum of equilibria. |
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János Mayer, Alexandra Künzi-Bay, Computational aspects of minimizing conditional value-at-risk, Computational Management Science (CMS), Vol. 3 (1), 2006. (Journal Article)
We consider optimization problems for minimizing conditional value-at-risk (CVaR) from a computational point of view, with an emphasis on financial applications. As a general solution approach, we suggest to reformulate these CVaR optimization problems as two-stage recourse problems of stochastic programming. Specializing the L-shaped method leads to a new algorithm for minimizing conditional value-at-risk. We implemented the algorithm as the solver CVaRMin. For illustrating the performance of this algorithm, we present some comparative computational results with two kinds of test problems. Firstly, we consider portfolio optimization problems with 5 random variables. Such problems involving conditional value at risk play an important role in financial risk management. Therefore, besides testing the performance of the proposed algorithm, we also present computational results of interest in finance. Secondly, with the explicit aim of testing algorithm performance, we also present comparative computational results with randomly generated test problems involving 50 random variables. In all our tests, the experimental solver, based on the new approach, outperformed by at least one order of magnitude all general-purpose solvers, with an accuracy of solution being in the same range as that with the LP solvers. |
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Peter Kall, János Mayer, Some insights into the solution algorithms for SLP problems, Annals of Operations Research, Vol. 142 (1), 2006. (Journal Article)
We consider classes of stochastic linear programming problems which can be efficiently solved by deterministic algorithms. For two–stage recourse problems we identify two such classes. The first one consists of problems where the number of stochastically independent random variables is relatively low; the second class is the class of simple recourse problems. The proposed deterministic algorithm is successive discrete approximation. We also illustrate the impact of required accuracy on the efficiency of this algorithm. For jointly chance constrained problems with a random right–hand–side and multivariate normal distribution we demonstrate the increase in efficiency when lower accuracy is required, for a central cutting plane method. We support our argumentation and findings with computational results. |
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Karl Schmedders, Kenneth Judd, Felix Kübler, Reply to 'Asset Trading Volume in Innite-Horizon Economies with Dynamically Complete Markets and Heterogeneous Agents: Comment', Finance Research Letters, Vol. 3 (2), 2006. (Journal Article)
In a comment, Peter Bossaerts and William R. Zame [2006. Finance Research Letters. This issue] claim that the main result of our paper [Judd, K.L., Kubler, F., Schmedders, K., 2003. The Journal of Finance 58, 2203–2217], namely the no-trade theorem for the dynamic Lucas infinite horizon economy with heterogeneous agents, is an artifact of the assumption that asset dividends and individual endowments follow the same stationary finite-state Markov process. In this reply, we clarify our assumptions and contrast them with the examples in Bossaerts and Zame. |
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Kenneth L Judd, Felix Kübler, Karl Schmedders, Asset Trading Volume with Dynamically Complete Markets and Heterogeneous Agents, Journal of Finance, Vol. LVIII (5), 2003. (Journal Article)
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Maximilian Adelmann, Karl Schmedders, János Mayer, A Large-Scale Optimization Model for Replicating Portfolios in the Life Insurance Industry, In: Swiss Finance Inst, No. 16-04, . (Working Paper)
Replicating portfolios have recently emerged as an important tool in the life insurance industry, used for the valuation of companies' liabilities. This paper presents a replicating portfolio (RP) model for approximating life insurance liabilities as closely as possible. We minimize the L1 error between the discounted life insurance liability cash flows and the discounted RP cash flows over a multi-period time horizon for a broad range of different future economic scenarios. We apply two different linear reformulations of the L1 problem to solve large-scale RP optimization problems and also present several out-of-sample tests for assessing the quality of RPs. A numerical application of our RP model to empirical data sets demonstrates that the model delivers RPs that match the liabilities rather closely. The numerical analysis demonstrates that our model delivers RPs with excellent practical properties in a reasonable amount of time. We complete the paper with a description of an implementation of the RP model at a global insurance company. |
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