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1921Required Observation Count 1A research lead uses the approximation n >= K * sigma 2 / delta 2 with K = 7.84, signal standard deviation 24 bp, and target detectable mean shift 3 bp. What is the minimum integer sample size?统计简单数值题未尝试免费1923Required Observation Count 3A research lead uses the approximation n >= K * sigma 2 / delta 2 with K = 9, signal standard deviation 30 bp, and target detectable mean shift 4 bp. What is the minimum integer sample size?统计中等数值题未尝试免费1928Minimum Detectable Edge 3Using the same approximation n >= K * sigma 2 / delta 2, suppose K = 9, sigma = 30 bp, and you can collect only n = 900 observations. What is the smallest detectable mean edge delta, in basis points?统计中等数值题未尝试免费1949Derive the Optimizer for Log Utility Minus Quadratic Cost 4For parameters a>0 and b>0, derive the unique maximizer of F(x)=a ln(1+x)-b x 2 on x>-1.数学中等derivation未尝试免费1950Quarter-Size Utility Balance 5A PM uses a very steep quadratic penalty, so the optimal tilt should stay small even with positive edge. The desk chooses a scalar exposure x > -1 to maximize F(x) = 5 ln(1+x) - 8 x 2. What exposure x maximizes F?数学困难derivation未尝试免费1953Mild Positive Skew From Asymmetric Flow 8Order-flow asymmetry is present but not extreme, so the optimal skew should stay close to flat. A market maker chooses a skew x in (-1,1) to maximize G(x) = 7 ln(1+x) + 5 ln(1-x). What skew is optimal?数学中等derivation未尝试免费1954Symmetric Flow Implies Flat Skew 9If a market maker maximizes G(x)=a ln(1+x)+a ln(1-x) on x in (-1,1), what skew is optimal?数学中等derivation未尝试免费1955Large Buy-Side Imbalance Skew 10A quoting engine values upside fill opportunities much more than downside ones, so the optimal skew should be meaningfully positive. A market maker chooses a skew x in (-1,1) to maximize G(x) = 9 ln(1+x) + 3 ln(1-x). What skew is optimal?数学困难derivation未尝试免费1959Large Cushion Interior Minimum 14The reciprocal term is strong enough that the optimum sits well away from the boundary. Minimize H(x) = 1 x 2 + 36/(1+x) over x > -1.数学困难derivation未尝试免费1960Derive the Optimizer for an Exponential Asymmetry Objective 15For A>0 and B>0, derive the unique minimizer of J(x)=A e x + B e -2x .数学困难derivation未尝试免费1963How the Exponential Optimum Moves With the Back-End Penalty 18For J(x)=A e x + B e -2x , suppose A stays fixed and B is quadrupled. How does the optimizer x* move?数学中等derivation未尝试免费1965Derive the Optimizer for a Linear-Softplus Objective 20For 0<m<n, derive the unique maximizer of K(x)=m x - n ln(1+e x).数学困难derivation未尝试免费1974Weighted Exposure Hedge Split 4The first hedge line loads twice as much on the required exposure as the second, but also has its own quadratic penalty. Minimize L(x,y) = 2x 2 + 3y 2 subject to 2x + 1y = 12.数学中等数值题未尝试免费1975Three-to-One Loading Tradeoff 5One book moves the target much faster per unit notional, so the optimizer must trade off load efficiency and cost. Minimize L(x,y) = 3x 2 + 1y 2 subject to 1x + 3y = 9.数学困难derivation未尝试面试订阅1978Why the Three-Book Solution Is Inverse-Coefficient Weighted 8In min a x 2 + b y 2 + c z 2 subject to x+y+z=N, why do larger quadratic coefficients receive smaller allocations?数学中等derivation未尝试免费1980Minimum Risk Needed for a Target Alpha 10A 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?数学困难derivation未尝试面试订阅1984Why the Total-and-Spread Problem Has a Unique Solution 14Why does the problem min a x 2 + b y 2 + c z 2 subject to x+y+z=N and x-z=d have a unique optimizer when a,b,c>0?数学困难derivation未尝试面试订阅1989Maximum Alpha Under a Quadratic Risk Budget 19What is the maximum value of mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2?数学困难derivation未尝试面试订阅1994Spread-Constrained Allocation With Uneven Penalties 24The spread target fixes how far apart the outer books must sit, while the middle book absorbs the rest. Minimize 1x 2 + 2y 2 + 3z 2 subject to x+y+z=11 and x-z=1.数学中等derivation未尝试免费1995Alpha Maximization With Uneven Risk Units 25The 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.数学困难derivation未尝试面试订阅