arXiv subjects

Nazem Khan

Publications and source records attributed to Nazem Khan.

4 recordsLinked to original sources

arXiv2025

The Interplay between Utility and Risk in Portfolio Selection

We revisit the problem of portfolio selection, where an investor maximizes utility subject to a risk constraint. Our framework is very general and accommodates a wide range of utility and risk functionals, including non-concave utilities such as S-shaped utilities from prospect theory and non-convex risk measures such as Value at Risk. Our main contribution is a novel and complete characterization of well-posedness for utility-risk portfolio selection in one period that takes the interplay between the utility and the risk objectives fully into account. We show that under mild regularity conditions the minimal necessary and sufficient condition for well-posedness is given by a very simple either-or criterion: either the utility functional or the risk functional need to satisfy the axiom of sensitivity to large losses. This allows to easily describe well-posedness or ill-posedness for many utility-risk pairs, which we illustrate by a large number of examples. In the special case of expected utility maximization without a risk constraint (but including non-concave utilities), we show that well-posedness is fully characterised by the asymptotic loss-gain ratio, a simple and interpretable quantity that describes the investor's asymptotic relative weighting of large losses versus large gains.

q-fin.MF↗

arXiv2024

Risk, utility and sensitivity to large losses

Risk and utility functionals are fundamental building blocks in economics and finance. In this paper we investigate under which conditions a risk or utility functional is sensitive to the accumulation of losses in the sense that any sufficiently large multiple of a position that exposes an agent to future losses has positive risk or negative utility. We call this property sensitivity to large losses and provide necessary and sufficient conditions thereof that are easy to check for a very large class of risk and utility functionals. In particular, our results do not rely on convexity and can therefore also be applied to most examples discussed in the recent literature, including (non-convex) star-shaped risk measures or S-shaped utility functions encountered in prospect theory. As expected, Value at Risk generally fails to be sensitive to large losses. More surprisingly, this is also true of Expected Shortfall. By contrast, expected utility functionals as well as (optimized) certainty equivalents are proved to be sensitive to large losses for many standard choices of concave and nonconcave utility functions, including $S$-shaped utility functions. We also show that Value at Risk and Expected Shortfall become sensitive to large losses if they are either properly adjusted or if the property is suitably localized.

q-fin.RM↗

arXiv2022

$\rho$-arbitrage and $\rho$-consistent pricing for star-shaped risk measures

This paper revisits mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure $\rho$. We make three contributions. First, we introduce the new axiom of sensitivity to large expected losses and show that it is key to ensure the existence of optimal portfolios. Second, we give primal and dual characterisations of (strong) $\rho$-arbitrage. Finally, we use our conditions for the absence of (strong) $\rho$-arbitrage to explicitly derive the (strong) $\rho$-consistent price interval for an external financial contract.

q-fin.MF↗

arXiv2021

Mean-$ρ$ portfolio selection and $ρ$-arbitrage for coherent risk measures

We revisit mean-risk portfolio selection in a one-period financial market where risk is quantified by a positively homogeneous risk measure $ρ$. We first show that under mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical mean-variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation $ρ$-arbitrage, and prove that it cannot be excluded -- unless $ρ$ is as conservative as the worst-case risk measure. After providing a primal characterisation of $ρ$-arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterisation of $ρ$-arbitrage. We show that the absence of $ρ$-arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual representation of $ρ$. A special case of our result shows that the market does not admit $ρ$-arbitrage for Expected Shortfall at level $α$ if and only if there exists an EMM $\mathbb{Q} \approx \mathbb{P}$ such that $\Vert \frac{\text{d}\mathbb{Q}}{\text{d}\mathbb{P}} \Vert_\infty < \frac{1}α$.

q-fin.MF↗