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中文题目A trading sleeve has realized return 12% and market beta 1.4. The risk committee wants the reported beta reduced to exactly 1.0 by parking the rest of the capital in T-bills earning 2%. If the market returned 8%, what alpha will the combined position report relative to CAPM?
打开 →The second sleeve has both larger alpha and a different risk unit, so the optimal point must balance both effects. Maximize 4x + 6y subject to 4x^2 + 9y^2 = 225.
打开 →A 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?
打开 →A signal is expected to decay within the next 20 minutes, and waiting is more expensive than crossing a bit more spread now. Should the schedule become more or less front-loaded?
打开 →Two sleeves carry different expected edges but must lie on one fixed quadratic risk budget. Maximize 3x + 4y subject to 1x^2 + 1y^2 = 25.
打开 →The 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?
打开 →A PM runs a stock with beta 0.7. After the quarter, a 0.8% fee is deducted from the stock's gross return before alpha is reported. If the risk-free rate is 1.5% and the market returned 5.5%, what gross stock return is needed so the reported post-fee alpha is +1.0%?
打开 →A PM says a stock with beta 1.5 and realized return -1% still generated exactly zero alpha this month. If the risk-free rate was 2%, what market return would make that statement true?
打开 →A 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?
打开 →Sleeve A has weight 40% and alpha 2%, sleeve B has weight 60% and alpha -0.5%. What is the portfolio alpha?
打开 →Why can measured alpha disappear once you change the factor model used to benchmark a portfolio?
打开 →A portfolio mixes a market-neutral arbitrage sleeve with realized return 4% and beta 0.2, and a directional sleeve with realized return 14% and beta 1.4. Let w be the weight in the directional sleeve and 1-w in the arbitrage sleeve. If the risk-free rate is 2% and the market retu
打开 →A 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?
打开 →A desk forms a composite alpha A = 0.6 A_fast + 0.4 A_slow. The expected gross daily alpha of A_fast is 8 bps and of A_slow is 5 bps. Their daily turnover is 90% and 20%, and every 1% of turnover costs 0.02 bps. What is the composite's expected net daily alpha?
打开 →In a GARCH-style volatility recursion, a deviation from long-run variance decays approximately by the factor $\rho=\alpha+\beta=\frac{4}{5}$ each step. What is the half-life of the deviation?
打开 →A node has leaf error 18 if pruned into a single leaf. Its current subtree has training error 10 and 3 leaves. What is the weakest-link alpha for pruning this subtree?
打开 →A fast signal has expected alpha 9 bps and a slow signal has expected alpha 3 bps. In a composite C = w fast + (1-w) slow, what weight on the fast signal produces expected alpha 6.6 bps?
打开 →Replacing a single leaf by a 3-leaf subtree reduces validation loss by 4.5. If the complexity charge is alpha = 1.2 per extra leaf, should you keep the subtree?
打开 →A desk has 80 highly correlated alphas that all measure similar value exposure. Why can Ridge be preferable to pure Lasso if the goal is stable prediction rather than sparse interpretation?
打开 →If the target alpha level A in mu_1 x + mu_2 y = A is zero, what is the minimum of a x^2 + b y^2?
打开 →Derive the maximizer of mu_1 x + mu_2 y subject to a x^2 + b y^2 = R^2 for positive a,b.
打开 →A regime label takes values A, B, C with probabilities 0.2, 0.5, 0.3. Conditional on A the source is deterministic, conditional on B it is uniform over 4 symbols, and conditional on C it is uniform over 2 symbols. Assuming the symbol sets are disjoint across regimes, what is the
打开 →What is the maximum value of mu_1 x + mu_2 y subject to a x^2 + b y^2 = R^2?
打开 →If a desk needs mu_1 x + mu_2 y = A with minimum quadratic risk a x^2 + b y^2, what minimum risk level is required?
打开 →A desk wants to minimize a x^2 + b y^2 subject to mu_1 x + mu_2 y = A. What is the minimum achievable value?
打开 →Only 2% of days contain a true dislocation worth trading. A classifier catches 65% of those days but fires falsely on 4% of normal days. What is the precision of a positive alert?
打开 →A 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?
打开 →Why can two testing systems with the same maximum branch count per question have very different identification power once branch-specific follow-up rules are imposed?
打开 →Why does a fixed-length binary code necessarily waste some average code length when the number of equally likely symbols is not a power of two?
打开 →A manager sleeve has beta 1.25. It is mixed with cash of beta 0. What weight on the manager sleeve makes the overall beta exactly 1?
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