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Answer: When cash is added to a portfolio, the value of W for that portfolio should decrease by the amount of cash that is added.
The correct answer is A. According to the property of translation invariance, when cash is added to a portfolio, the value of the risk measure W for that portfolio should decrease by the amount of cash that is added. This is because translation invariance implies that adding a fixed amount of cash to a portfolio should reduce the risk measure by the same amount, without changing the relative risk of the portfolio's other assets. The other options describe different properties: B is a test for subadditivity, C is a test for monotonicity, and D is a test for homogeneity, none of which are related to translation invariance.
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The Chief Risk Officer (CRO) of a leading financial institution is collaborating with the risk management team to evaluate a newly developed risk metric, referred to as W. The purpose of this evaluation is to determine if this new risk metric is coherent and specifically if it satisfies the principle of translation invariance. Which of the following tests would correctly verify that the risk metric W possesses the property of translation invariance?
A
When cash is added to a portfolio, the value of W for that portfolio should decrease by the amount of cash that is added.
B
When W is used to measure the risk of two portfolios A and B, then W(A) + W(B) should be less than or equal to W(A+B).
C
When W is used to measure the risk of two portfolios A and B, and if portfolio A always produces a worse outcome than portfolio B, then W(A) should always be higher than W(B).
D
When W is used to measure the risk of portfolio A, and if all exposures in portfolio A are increased by a constant factor, then W(A) should increase proportionally by that factor.