次可加性

问题如下：

A one-year project has a 3% chance of losing USD 10million, a 7% chance of losing USD 3 million, and a 90% chance of gaining USD 1 million.

Situation 1：We can calculate that the VaR is USD 3 million and the expected shortfall (USD) is 7.2 when the confidence level is 95% and the time horizon is one year.

Situation 2：Suppose that there are two independent identical investments with the properties.We can calculate：

The VaR is 9 and ES is 9.534.

Q：Check whether (a) VaR or (b) expected shortfall satisfy the subadditivity axiom for a coherent risk measure for the investments.

解释：

VaR does not satisfy the subadditivity condition because the VaR for two portfolios combined (9) is greater than the sum of the VaR for each portfolio individually(i.e.,9＞3+3). Meanwhile, expected shortfall does satisfy the condition because its value for the two portfolios combined is less than the sum of each portfolio’s expected shortfall (i.e.,9.534 < 7.2 + 7.2).