Equilibrium in an ambiguity-averse mean-variance investors market

Keywords: Robust optimization Mean–variance portfolio theory Ellipsoidal uncertainty Equilibrium price system a b s t r a c t In a financial market composed of n risky assets and a riskless asset, where short sales are allowed and mean–variance investors can be ambiguity averse, i.e., diffident about mean return estimates where confidence is represented using ellipsoidal uncertainty sets, we derive a closed form portfolio rule based on a worst case max–min criterion. Then, in a market where all investors are ambiguity-averse mean–vari-ance investors with access to given mean return and variance–covariance estimates, we investigate conditions regarding the existence of an equilibrium price system and give an explicit formula for the equilibrium prices. In addition to the usual equilibrium properties that continue to hold in our case, we show that the diffidence of investors in a homogeneously diffident (with bounded diffidence) mean–variance investors' market has a deflationary effect on equilibrium prices with respect to a pure mean–variance investors' market in equilibrium. Deflationary pressure on prices may also occur if one of the investors (in an ambiguity-neutral market) with no initial short position decides to adopt an ambiguity averse attitude. We also establish a CAPM-like property that reduces to the classical CAPM in case all investors are ambiguity-neutral. A major theme in mathematical finance is the study of inves-tors' portfolio decisions using the well-established theory of mean–variance that began with the seminal work of Markowitz (1987). The mean–variance portfolio theory then formed the basis of the celebrated Capital Asset Pricing Model (CAPM) (Sharpe, 1964), the most commonly used equilibrium and pricing model in the financial literature. However, it is a well-known fact that the investors' portfolio holdings in the mean–variance portfolio theory are very sensitive to the estimated mean returns of the risky assets; see e.g., Best and Grauer (1991a), Best and Grauer (1991b), Black and Litterman (1992). The purpose of the present paper is to investigate equilibrium relations in a financial market composed of n risky assets and a riskless asset using an approach that takes into account the imprecision in the mean return estimates. In our model , investors act as mean–variance investors with a degree of diffidence (or confidence) towards the mean return estimates of risky assets. We refer to this attitude of diffidence as ambiguity aversion to distinguish it from risk aversion quantified by a mean–variance objective function. Decision making under ambiguity aversion is an active research area …