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1869RW Versus MR Diagnosis 4If a spread's conditional mean always equals today's level, regardless of horizon, which model is the closer description?统计中等derivation未尝试面试订阅1870RW Versus MR Diagnosis 5A desk notices that expected residual carry from holding a spread for longer horizons quickly saturates instead of growing linearly forever. Is that more consistent with random walk or mean reversion?统计困难derivation未尝试面试订阅1947Log Utility Versus Impact Budget 2A desk chooses a participation tilt with diminishing marginal benefit and quadratic implementation drag. The desk chooses a scalar exposure x > -1 to maximize F(x) = 4 ln(1+x) - 1 x 2. What exposure x maximizes F?数学简单数值题未尝试免费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1958Verify the Designed Minimizer in a Quadratic-Reciprocal Objective 13Let r>0 and a>0. Show that x=r uniquely minimizes H(x)=a x 2 + 2 a r(1+r) 2 /(1+x) over x>-1.数学中等derivation未尝试免费1964Negative Exponential Tilt 19Minimize J(x)=16 e x + 4 e -2x . What x minimizes J?数学困难derivation未尝试免费1967Positive Softplus Tilt 22The linear reward is strong relative to the saturation penalty, so the optimum should be positive. The desk maximizes K(x) = 3 x - 4 ln(1+e x). What x is optimal?数学简单derivation未尝试免费1968Negative Softplus Tilt 23The saturation penalty dominates until x turns sufficiently negative. The desk maximizes K(x) = 1 x - 3 ln(1+e x). What x is optimal?数学中等derivation未尝试免费1970Maximum Value at the Interior Utility-Impact Optimum 25For F(x)=4 ln(1+x)-x 2 on x>-1, what x maximizes F and what is the maximum value?数学困难derivation未尝试免费1973How the Cheap Book Gets More Size 3In the problem min a x 2 + b y 2 subject to x+y=c, which book gets the larger allocation when a<b, and why?数学中等derivation未尝试免费1977Derive the Three-Book Total-Size Allocation 7Derive the minimizer of a x 2 + b y 2 + c z 2 subject to x+y+z=N for positive a,b,c.数学中等derivation未尝试免费1983Derive the Total-and-Spread Solution 13For positive a,b,c, derive the minimizer of a x 2 + b y 2 + c z 2 subject to x+y+z=N and x-z=d.数学中等derivation未尝试免费1988How the Optimizer Scales With the Risk Radius 18In max mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2, how does the optimizer change when R is doubled?数学中等derivation未尝试免费1990Minimum Risk Needed for a Chosen Alpha Level 20If 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?数学困难derivation未尝试面试订阅1991Zero Alpha Target Implies Zero Minimum Risk 21If 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?数学中等derivation未尝试免费2024Square-Root Impact Gap for a Two-Scenario Slice 4A child-order size V is 0 with probability 1/2 and 3 with probability 1/2. Compute E[sqrt(1+V)] and sqrt(1+E[V]).数学中等数值题未尝试免费2029Exact Utilization Penalty Gap With Unequal Weights 9A utilization surcharge uses phi(q)=q/(1-q) on 0<=q<1. Suppose Q equals 0 with probability 1/3 and 3/4 with probability 2/3. Compute E[phi(Q)] and phi(E[Q]).数学困难数值题未尝试面试订阅2035Funding Buffer Gap With Unequal Scenario Weights 15The high-leverage state is rarer, but still materially affects the convex average. A funding-buffer model uses phi(L)=1/(1+L). Suppose L takes values 1 and 4 with probabilities 1/4 and 3/4. Compute E[phi(L)] and phi(E[L]).数学困难数值题未尝试面试订阅2036Quadratic Jensen Gap From Mean and Variance 16Let phi(x)=x 2. If a random variable X has E[X]=2 and Var(X)=5, what is E[phi(X)] - phi(E[X])?数学简单数值题未尝试免费2039Convex Penalty for Mixed Schedules 19Two execution schedules have penalties phi(q 1) and phi(q 2) under a convex phi. What does Jensen say about a random 50-50 mix versus the penalty at the average size?数学困难derivation未尝试面试订阅