On double value at risk

Value at Risk (VaR) is used to illustrate the maximum potential loss under a given confidence level, and is just a single indicator to evaluate risk ignoring any information about income. The present paper will generalize one-dimensional VaR to two-dimensional VaR with income-risk double indicators. We first construct a double-VaR with (μ,σ 2 ) (or (μ,VaR 2 ) indicators, and deduce the joint confidence region of (μ,σ 2 ) (or (μ,VaR 2 ) by virtue of the two-dimensional likelihood ratio method. Finally, an example to cover the empirical analysis of two double-VaR models is stated.