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5271Minimum-Variance Weight 1Two risky assets have variances 0.04 and 0.09 and covariance 0.015. What weight on asset 1 gives the two-asset minimum-variance portfolio?金融与交易中等数值题未尝试面试订阅5278Variance Needed For One-Third Tangency WeightTwo uncorrelated risky assets have expected returns 0.10 and 0.14, risk-free rate 0.05, and asset 1 variance 0.05. What variance would asset 2 need so that the tangency portfolio puts exactly one-third of the risky allocation in asset 2?金融与交易困难数值题未尝试面试订阅5279Fee That Shrinks Tangency Weight To 20%Two uncorrelated risky assets have expected returns 0.06 and 0.09, risk-free rate 0.02, and variances 0.025 and 0.07. Suppose asset 2 carries an annual implementation fee c that reduces its expected return from 0.09 to 0.09-c. What fee c would make the tangency portfolio weight on asset 2 equal to 20%?金融与交易困难数值题未尝试面试订阅5280Variance Penalty For Equal Tangency WeightsTwo uncorrelated risky assets have expected returns 0.09 and 0.13, risk-free rate 0.04, and variances 0.035 and 0.10. Risk control adds a variance penalty x to asset 1, so its effective variance becomes 0.035 + x. What x would make the tangency portfolio split the risky allocation equally across the two assets?金融与交易困难数值题未尝试面试订阅5281Correlation Shock And Volatility Repricing 1An equal-weight portfolio holds two assets with volatilities 0.04 and 0.09. The current correlation estimate is -0.2, but stress testing revises it to +0.3. With weights unchanged, what is the new portfolio volatility, and by how many volatility points does it rise versus the original estimate?金融与交易中等数值题未尝试面试订阅5283Correlation Break And Diversification Loss 3A two-asset portfolio has weights 0.3 and 0.7, volatilities 0.03 and 0.10, and current correlation 0.25. After a macro shock, correlation is revised to 0.75 with all else unchanged. What is the new portfolio volatility, and how many volatility points of diversification are lost relative to the original estimate?金融与交易中等数值题未尝试面试订阅5284Correlation Normalization And Variance Jump 4A portfolio holds weights 0.4 and 0.6 in two assets with volatilities 0.06 and 0.12 and current correlation -0.4. If correlation normalizes to 0 while weights and volatilities stay fixed, by what percentage does portfolio variance increase?金融与交易中等数值题未尝试面试订阅5286PM Dismisses A Low-Sharpe Hedge SleeveA PM says, "This market-neutral sleeve has a poor standalone Sharpe, so it should never receive weight in a Markowitz optimizer." The sleeve is mildly negatively correlated with the core book. How would you respond in two or three sentences?金融与交易困难essay未尝试面试订阅5289Why Long-Only Can Reshape The Entire SolutionAn unconstrained optimizer wants a large short in one equity sleeve to hedge two crowded longs, but mandate rules require long-only weights. Explain why the long-only solution can look qualitatively different rather than just a clipped version of the unconstrained one.金融与交易困难essay未尝试面试订阅5290Tiny Alpha Changes, Huge Weight FlipsA Markowitz backtest flips from +18% to -12% in one sleeve after a tiny revision to expected-return estimates, while the covariance matrix barely changes. What does this tell you, and what would you do before trading the result live?金融与交易困难essay未尝试面试订阅5291Benchmark Mix to Hit a Beta Target 1A sleeve with beta 1.4 is blended with a hedge sleeve of beta -0.2. What weight on the first sleeve gives portfolio beta 0.5?金融与交易中等数值题未尝试面试订阅5292Implied Market Premium From a Hurdle Return 2A stock has beta 1.2 and the risk-free rate is 2%. If CAPM says its expected return should be 9.2%, what market risk premium is implied?金融与交易中等数值题未尝试面试订阅5293Alpha Relative to a Benchmark Move 3A stock returns 1.8% while the benchmark returns 0.9%. If the stock beta is 1.1 and the risk-free rate is negligible over the horizon, what single-period alpha do you attribute to the stock?金融与交易中等数值题未尝试面试订阅5294Index Futures Overlay To Reach A Beta TargetA cash equity book has beta 0.60 to the market. The desk can short index futures with beta 1.00 per unit notional. What short futures notional, as a fraction of NAV, is needed to bring the portfolio beta down to 0.15?金融与交易中等数值题未尝试面试订阅5295Overlay Notional for Beta Neutrality 4A core book has beta 0.9 and notional 200. An index future has beta 1.0. What short futures notional makes the combined beta exposure zero?金融与交易中等数值题未尝试面试订阅5296Idiosyncratic Variance From Total and Market Pieces 5A stock follows a one-factor model with beta 1.5. Market variance is 0.04 and the stock's total variance is 0.16. What idiosyncratic variance is implied?金融与交易中等数值题未尝试面试订阅5297Portfolio Alpha Across Two Sleeves 6Sleeve A has weight 40% and alpha 2%, sleeve B has weight 60% and alpha -0.5%. What is the portfolio alpha?金融与交易中等数值题未尝试面试订阅5298Benchmark Change and Alpha Revision 7A stock returned 1.3%. Under the old benchmark, the benchmark return was 0.8% and beta was 1.0. Under a new benchmark, the benchmark return is 0.5% with the same beta. By how much does the stock's measured alpha increase?金融与交易中等数值题未尝试面试订阅5299Minimum Expected Return To Clear An Alpha HurdleA PM requires at least 1.0% expected alpha over CAPM before buying a stock. If the risk-free rate is 1%, the market expected return is 7%, and the stock's beta is 0.6, what minimum expected return must the stock have to qualify?金融与交易中等数值题未尝试面试订阅5300Compare Two Stocks Under CAPM Plus AlphaThe risk-free rate is 2.5% and the market expected return is 9.5%. Stock A has beta 1.1 and zero alpha forecast. Stock B has beta 0.4 and a forecast alpha of 1.5%. Under a CAPM-plus-alpha view, which stock has the higher expected return, and by how much?金融与交易中等数值题未尝试面试订阅