Multidimensional Coherent Risk Measures and Their Properties

Coherent risk measures are widely used in practice to calculate risk. Multidimensional coherent risk measures are important to use for companies and banks operating in different international markets. Here we introduce an example which clearly shows that multidimensional coherent risk measures reduce capital requirements on multi-asset or multi-currency markets. In this paper we consider two different approaches to define multidimensional coherent risk measures. The first approach is based on a cone-based exchange rate set, whereas the second employs random sets to account for liquidity constraints and other market frictions. Also constructive approach for multidimensional risk measures using one-dimensional law invariant ones is considered. We introduce the example of coherent risk measure which cannot be represented by the constructive approach. Two important properties of multidimensional risk measures such as law invariance and space consistency are investigated and their equivalent properties in terms of determining set and acceptance set are given. Then we consider a multidimensional generalization of Tail VaR and show that it satisfies space consistency and law invariance properties and provide an example which shows that multidimensional portfolio risk estimation using this risk measure gives an adequate result. JEL Classification: C39, D81. UDC: 519.25. For reference: Kulikov A. V., Volkov N. V. (2025). Multidimensional coherent risk measures and their properties. Economics and Mathematical Methods, 61, 3, 116–125.