Aumann and Serrano's Economic Index of Risk for Sums of Gambles

We study Aumann and Serrano's (2008) risk index for sums of gambles that are not necessarily independent. We show that if the dependent parts of two gambles are similarly ordered, or more generally positively quadrant dependent, then the risk index of the sum of two gambles is always larger than the minimum of the risk indices of the two gambles. For negative dependence, the risk index of the sum is always smaller than the maximum of the two risk indices. The above results agree with our intuitions well. For example, the result for negative dependence agrees with our intuition of risk diversification. Thus this result can be considered another attractive property of Aumann and Serrano's risk index.