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<p>In this paper, we consider <i>n</i> agents who invest in a general financial market that is free of arbitrage and complete. The aim of each investor is to maximize her expected utility while ensuring, with a specified probability, that her terminal wealth exceeds a benchmark defined by her competitors’ performance. This setup introduces an interdependence between agents, leading to a search for Nash equilibria. In the case of two agents and CRRA utility, we are able to derive all Nash equilibria in terms of terminal wealth. For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n&gt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&gt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> agents and logarithmic utility we distinguish two cases. In the first case, the probabilities in the constraint are small and we can characterize all Nash equilibria. In the second case, the probabilities are larger and we look for Nash equilibria in a certain set. We also discuss the impact of the competition using some numerical examples. As a by-product, we solve some portfolio optimization problems with probability constraints.</p>

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Relative portfolio optimization via a value at risk based constraint

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月23日

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  • Nicole Bäuerle,
  • Tamara Göll

摘要

In this paper, we consider n agents who invest in a general financial market that is free of arbitrage and complete. The aim of each investor is to maximize her expected utility while ensuring, with a specified probability, that her terminal wealth exceeds a benchmark defined by her competitors’ performance. This setup introduces an interdependence between agents, leading to a search for Nash equilibria. In the case of two agents and CRRA utility, we are able to derive all Nash equilibria in terms of terminal wealth. For \(n>2\) n > 2 agents and logarithmic utility we distinguish two cases. In the first case, the probabilities in the constraint are small and we can characterize all Nash equilibria. In the second case, the probabilities are larger and we look for Nash equilibria in a certain set. We also discuss the impact of the competition using some numerical examples. As a by-product, we solve some portfolio optimization problems with probability constraints.