A Multi-item Risk-Averse Newsvendor with Law Invariant Coherent Measures of Risk

I consider a multi-product risk-averse newsvendor under the law-invariant coherent measures of risk. I first establish a few fundamental properties of the model regarding the convexity of the problem and the symmetry of the solution, and study the impacts of risk aversion and shift in mean demand to the optimal solution with independent demands. Specifically, I show that for identical products with independent demands, increased risk aversion leads to decreased orders. For a large but finite number of heterogenous products with independent demands, I derive closed-form approximations for the optimal order quantities. The approximations are as simple to compute as the classical risk-neutral solutions. I also show that the risk-neutral solution is asymptotically optimal as the number of products tends to be infinity, and thus risk aversion has no impact in the limit. For a risk-averse newsvendor with dependent demands, I show that positively (negatively) dependent demands lead to a lower (higher) optimal order quantities than independent demands. Using a numerical study, I examine the convergence rates of the approximations and develop additional insights on the interplay between dependent demands and risk aversion.