The indeterminacy of financial markets leads investors to face different types of security returns. Usually, security returns are assumed to be random variables when sufficient transaction data are available. If data are missing, they can be regarded as uncertain variables. However, uncertainty and randomness coexist. In this situation, chance theory is the main tool to deal with this complex phenomenon. This paper investigates the conditional value at risk (CVaR) of uncertain random variables and its application to portfolio selection. First, we define the CVaR of uncertain random variables and discuss some of its mathematical properties. Then, we propose an uncertain random simulation to approximate the CVaR. Next, we define the inverse function of the CVaR of uncertain random variables, as well as a computational procedure. As an application in finance, we establish uncertain random mean-CVaR portfolio selection models. We also perform a numerical example to illustrate the applicability of the proposed models. Finally, we numerically compare the mean-CVaR models with the mean-variance models with respect to the optimal investment strategy.

金融市场的不确定性导致投资者面临不同类型的证券回报。通常，当有足够的交易数据可用时，证券回报被假定为随机变量。如果数据缺失，则可以将其视为不确定变量。然而，不确定性和随机性并存。在这种情况下，机会理论是处理这种复杂现象的主要工具。本文研究了不确定随机变量的条件风险价值（CVaR）及其在投资组合选择中的应用。首先，我们定义不确定随机变量的 CVaR 并讨论其一些数学性质。然后，我们提出不确定随机模拟来近似 CVaR。接下来，我们定义不确定随机变量的 CVaR 的反函数，以及计算过程。作为金融领域的应用，我们建立了不确定随机均值-CVaR 投资组合选择模型。我们还通过一个数值示例来说明所提出模型的适用性。最后，我们在最优投资策略方面对均值CVaR模型与均值方差模型进行了数值比较。