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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.
## Explanation Translation invariance is one of the four properties of coherent risk measures. The translation invariance property states that: **W(X + c) = W(X) - c** Where: - W is the risk measure - X is the portfolio - c is a risk-free cash amount This means that when you add cash to a portfolio, the risk measure should decrease by exactly that amount of cash. This makes intuitive sense because cash reduces the overall risk of the portfolio. Let's analyze the options: **Option A**: Correctly describes translation invariance - "When cash is added to a portfolio, the value of W for that portfolio should decrease by the amount of cash that is added." **Option B**: Describes subadditivity - W(A) + W(B) ≤ W(A+B), which is a different coherence property. **Option C**: Describes monotonicity - if portfolio A is riskier than portfolio B, then W(A) should be greater than W(B). Therefore, Option A is the correct test for translation invariance.
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The CRO of a major bank is reviewing a new risk measure, W, with the risk team. The CRO runs a test on the new risk measure to determine if the measure is coherent and satisfies the property of translation invariance. Which of the following tests would correctly determine that the risk measure W exhibits 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).
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