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4876Paired-Difference Standard Error 11Two pricing algorithms are compared with common random numbers. The paired difference sample has standard deviation s D=3 across n=100 shared paths. What is the standard error of the estimated mean difference?数理金融中等数值题未尝试面试订阅4877Paired-Difference Standard Error 12A common-random-numbers comparison produces paired differences with s D=2.4 over n=64 paths. What is the standard error of the sample mean difference?数理金融中等数值题未尝试面试订阅4878Paired-Difference Standard Error 13A common-random-numbers run yields paired differences with s D=5 over n=25 paths. What is the standard error of the mean difference?数理金融中等数值题未尝试面试订阅4880Variance Improvement From Pairing 15Without common random numbers, two independent estimators have standard deviations 4 and 5, so the variance of their difference would be 4 2+5 2. With pairing, the observed paired-difference standard deviation is 3. What fraction of the unpaired difference variance remains?数理金融中等数值题未尝试面试订阅4886Infer Second-Stratum Volatility From Target Neyman Share 21Two strata have population weights N1=0.6 and N2=0.4. Their standard deviations are sigma1=2 and sigma2=unknown. Under equal-cost Neyman allocation, the desk wants stratum 2 to receive 50% of the samples. What sigma2 is implied?数理金融中等数值题未尝试面试订阅4887Updated Neyman Share After a Volatility Change 22Two equal-size strata each have population weight 0.5. After a shock, sigma1 rises to 2 while sigma2 stays at 1.5. Under equal-cost Neyman allocation, what sample share should stratum 1 now receive?数理金融中等数值题未尝试面试订阅4888Infer Variance Ratio From Equal Neyman Shares 23Two strata have population weights N1=0.7 and N2=0.3. Under equal-cost Neyman allocation they end up with equal sample shares. What ratio sigma2/sigma1 is implied?数理金融中等数值题未尝试面试订阅4889Three-Stratum Neyman Share 24Three strata have population weights 0.5, 0.3, and 0.2, with standard deviations 1, 2, and 3. Under equal-cost Neyman allocation, what sample share should the third stratum receive?数理金融中等数值题未尝试面试订阅4890Updated Small-Stratum Share After Tail Risk Doubles 25Two strata have population weights 0.8 and 0.2. The first stratum has sigma1=1. The second stratum's standard deviation jumps from 4 to 8. Under equal-cost Neyman allocation, what sample share should the second stratum now receive?数理金融中等数值题未尝试面试订阅5306Three-Factor Return Attribution 1A portfolio has alpha 0.01, market beta 1.1 with market factor move 0.02, value exposure 0.4 with value-factor move -0.01, and size exposure 0.3 with size-factor move 0.015. What return does this linear factor model attribute to the portfolio?金融与交易中等数值题未尝试面试订阅5307Three-Factor Return Attribution 2A portfolio has alpha 0.005, market beta 0.9 with market factor move 0.015, value exposure -0.2 with value-factor move 0.01, and size exposure 0.2 with size-factor move 0.012. What return does this linear factor model attribute to the portfolio?金融与交易中等数值题未尝试面试订阅5309Three-Factor Return Attribution 4A portfolio has alpha 0.008, market beta 0.7 with market factor move 0.01, value exposure 0.3 with value-factor move 0.005, and size exposure -0.1 with size-factor move 0.011. What return does this linear factor model attribute to the portfolio?金融与交易中等数值题未尝试面试订阅5310Three-Factor Return Attribution 5A portfolio has alpha 0.009, market beta 1 with market factor move 0.017, value exposure -0.4 with value-factor move 0.008, and size exposure 0.25 with size-factor move 0.014. What return does this linear factor model attribute to the portfolio?金融与交易中等数值题未尝试面试订阅5311Why Beta Is Not Total RiskWhy can a stock have a low beta but still be risky in an absolute sense?金融与交易困难essay未尝试面试订阅5312Why Alpha Is FragileWhy can measured alpha disappear once you change the factor model used to benchmark a portfolio?金融与交易困难essay未尝试面试订阅5313Why CAPM Is Still UsedEven though markets are more complex than one-factor CAPM, why is CAPM still used in practice?金融与交易困难essay未尝试面试订阅5314Why Factor Exposures Help HedgingWhy is it often more informative to hedge factor exposures than to hedge names one by one?金融与交易困难essay未尝试面试订阅5315Why Expected Return And Attribution DifferWhy is estimating an expected return from a factor model different from explaining a realized return after the fact?金融与交易困难essay未尝试面试订阅5322Maximum Scale Of Second Sleeve Under VaR CapTwo independent sleeves have zero-mean normal daily PnL. Sleeve A has standard deviation 6 million and cannot be changed. Sleeve B has standard deviation 8 million but can be scaled by a factor lambda. Using 1-day 99% delta-normal VaR with z=2.326, what is the largest lambda allowed if total VaR must not exceed 20 million?金融与交易简单数值题未尝试面试订阅5331Covariance Contribution Implied By VaR BudgetA Gaussian portfolio uses z=2.0 and has total volatility 0.25. A position currently has weight 0.4, and the risk committee gives it a component VaR budget of 0.08. What covariance contribution (Sigma w) i would exactly exhaust that budget?金融与交易困难数值题未尝试面试订阅