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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 **A is correct.** Translation invariance is one of the four properties of coherent risk measures. It states that adding cash (a risk-free asset) to a portfolio should reduce the risk measure by exactly the amount of cash added. Formally: W(P + K) = W(P) - K, where K is the amount of cash added. **B is incorrect.** This describes the property of **subadditivity**, which states that the risk of a combined portfolio should be less than or equal to the sum of the individual portfolio risks: W(A+B) ≤ W(A) + W(B). **C is incorrect.** This describes the property of **monotonicity**, which states that if portfolio A always has worse outcomes than portfolio B, then the risk measure for A should be greater than or equal to that for B: If A ≤ B in all states, then W(A) ≥ W(B). **D is incorrect.** This describes the property of **positive homogeneity**, which states that scaling all positions by a constant factor λ should scale the risk measure by the same factor: W(λP) = λW(P). Translation invariance is particularly important because it ensures that holding cash reduces risk exposure, which aligns with practical risk management intuition.
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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).
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.