Answer-first summary for fast verification
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 Statement A is CORRECT because: - **Translation invariance** is one of the four coherence properties for risk measures - It states that adding a risk-free amount (cash) to a portfolio should reduce the risk measure by exactly that amount - Mathematically: W(X + c) = W(X) - c, where c is the cash amount - This makes intuitive sense - adding cash reduces risk exposure For the other statements: - **B describes Subadditivity**: W(A+B) ≤ W(A) + W(B) - diversification benefit - **C describes Monotonicity**: If A is always worse than B, then W(A) ≥ W(B) The four coherence properties are: 1. **Translation invariance**: W(X + c) = W(X) - c 2. **Subadditivity**: W(A+B) ≤ W(A) + W(B) 3. **Positive homogeneity**: W(λX) = λW(X) for λ ≥ 0 4. **Monotonicity**: If X ≤ Y, then W(X) ≥ W(Y) Value at Risk (VaR) fails subadditivity, while Expected Shortfall (ES) satisfies all four properties.
Author: LeetQuiz Editorial Team
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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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