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4898Translation-Invariant Capital Recovery 8A desk starts with coherent capital rho(L)=4.4. After Treasury injects exactly enough deterministic cash to leave residual capital 0.9, how much cash was injected? If Treasury had injected only 3.0 instead, what residual capital would have remained?数理金融简单数值题未尝试面试订阅4899Translation-Invariant Capital Recovery 9The amount of deterministic capital needed to make a position acceptable is 3.9. After adding a guaranteed scenario-by-scenario recovery h, the coherent capital drops to 1.6. What recovery h was added?数理金融简单数值题未尝试面试订阅4900Translation-Invariant Capital Recovery 10A loss book has coherent capital 9.6. One deterministic hedge of size 1.8 is already in place. What residual capital remains? If the desk wants residual capital 5.0 instead, how much larger must the deterministic hedge be in total?数理金融简单数值题未尝试面试订阅4903Worst-Case Hedge Design 13Desk A has scenario losses [3, 2, 7, 1] and desk B has [1, 0, 2, 5]. Under rho(L)=max scenario loss, a hedge pays h only in scenario 3. What minimum h is needed so the combined capital becomes 6?数理金融中等数值题未尝试面试订阅4905Worst-Case Hedge Design 15Desk A has scenario losses [5, 3, 1, 4] and desk B has [1, 2, 0, 3]. Under rho(L)=max scenario loss, a hedge pays h only in scenario 4. What minimum h makes the combined capital drop from 7 to 6?数理金融中等数值题未尝试面试订阅4906Infer Worst Tail Loss From a Spectral Quote 16A spectral risk measure applies weights [0.1, 0.2, 0.3, 0.4] to ordered losses [1, 3, 5, x] from best to worst. If the reported spectral risk is 6.2, what worst ordered loss x is implied?数理金融中等数值题未尝试面试订阅4907Infer Worst Tail Loss From a Spectral Quote 17A spectral risk measure applies weights [0.05, 0.15, 0.3, 0.5] to ordered losses [0, 2, 4, x]. If the reported spectral risk is 7.1, what x is implied?数理金融中等数值题未尝试面试订阅4908Infer Worst Tail Loss From a Spectral Quote 18A spectral risk measure applies weights [0.2, 0.2, 0.25, 0.35] to ordered losses [1, 2, 6, x]. If the risk value is 5.6, what x is implied?数理金融中等数值题未尝试面试订阅4911Acceptance-Set Geometry 21Why is a convex acceptance set operationally useful when two desks ask whether they can partially blend books without blowing up capital?数理金融困难essay未尝试面试订阅4912Firmwide Aggregation Intuition 22Why is subadditivity the coherence axiom desks care about most when the firm aggregates capital across books?数理金融困难essay未尝试面试订阅4913Liquidity Blind Spot 23Why can a perfectly coherent risk measure still miss a major liquidity problem on a stressed desk?数理金融困难essay未尝试面试订阅4914VaR Aggregation Critique 24Why can a VaR number look reasonable for one desk in isolation yet still be awkward as a firmwide capital aggregator?数理金融困难essay未尝试面试订阅4915Spectral Weight Design 25Why should a spectral risk measure place weakly larger weights on worse outcomes than on mild ones?数理金融困难essay未尝试面试订阅4916Infer Normal Parameters From VaR And ES 1A desk assumes losses are normal with mean mu and standard deviation sigma. At alpha=0.95 it uses VaR = mu + z*sigma and ES = mu + k*sigma with z=1.645 and k=2.063. If the reported VaR is 7.58 and ES is 9.252, what are mu and sigma?数理金融中等数值题未尝试面试订阅4917Infer Normal Parameters From VaR And ES 2A normal-loss desk uses alpha=0.99 with VaR = mu + z*sigma and ES = mu + k*sigma, where z=2.326 and k=2.665. If VaR is 7.478 and ES is 8.495, what are mu and sigma?数理金融中等数值题未尝试面试订阅4918Infer Normal Parameters From VaR And ES 3A risk report uses alpha=0.975 with z=1.96 and k=2.338 in the formulas VaR = mu + z*sigma and ES = mu + k*sigma. If VaR is 11.8 and ES is 13.69, what are mu and sigma?数理金融中等数值题未尝试面试订阅4919Infer Normal Parameters From VaR And ES 4A normal-loss desk uses alpha=0.95 with z=1.645 and k=2.063. If the reported VaR is 4.1125 and ES is 5.1575, what are mu and sigma?数理金融中等数值题未尝试面试订阅4920Infer Normal Parameters From VaR And ES 5A desk uses alpha=0.99 with z=2.326 and k=2.665 in the formulas VaR = mu + z*sigma and ES = mu + k*sigma. If VaR is 15.156 and ES is 17.19, what are mu and sigma?数理金融中等数值题未尝试面试订阅4921Infer Missing Tail Observation 6Sorted empirical losses are [1, 2, 3, 4, 7, 9, 12, x]. Using alpha=0.75, define empirical VaR as the ceil(alpha*n)-th order statistic and empirical ES as the average of losses at and beyond that cutoff. If empirical ES is 11.5, what is x?数理金融中等数值题未尝试面试订阅4922Infer Missing Tail Observation 7Sorted empirical losses are [0, 1, 1, 3, 5, 6, 8, x]. Using alpha=0.875 with the same empirical VaR and ES conventions, if empirical ES is 9.5, what is x?数理金融中等数值题未尝试面试订阅