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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1981Three-Book Split With a Fixed Spread 11The first and third books must keep a pre-agreed spread while total size stays fixed. Minimize 1x 2 + 1y 2 + 1z 2 subject to x+y+z=9 and x-z=1.数学简单数值题未尝试免费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1987Derive the Alpha-Maximizing Point Under a Quadratic Risk Budget 17Derive the maximizer of mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2 for positive a,b.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1993Three-Book Allocation With Uneven Penalties 23The quadratic penalties differ across books, so the minimum-cost total-size allocation will be visibly asymmetric. Minimize Q(x,y,z) = 2x 2 + 3y 2 + 6z 2 subject to x+y+z = 22.数学简单数值题未尝试免费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1996Convex Execution Cost With a Capacity Barrier 1A schedule pays a quadratic cost but also faces a blow-up term as it nears a hard capacity cap. Show that f(q) = 1 q 2 + 2/(1-q) is strictly convex on q<1.数学简单derivation未尝试免费1997Strict Convexity of a Barrier-Regularized Cost 2The desk wants a direct curvature argument, not a vague appeal to 'it looks bowl-shaped'. Show that f(q) = 3 q 2 + 1/(1-q) is strictly convex on q<1.数学简单derivation未尝试免费1998Convex Portfolio Penalty With Aggregate Exposure 3Each sleeve has its own quadratic penalty, and the whole book also pays for aggregate balance-sheet usage. Prove that F(w 1,w 2,w 3) = 2w 1 2 + 3w 2 2 + 5w 3 2 + 1(w 1+w 2+w 3) 2 is convex.数学中等derivation未尝试免费1999When the Portfolio Penalty Is Strictly Convex 4For F(w)=sum i a i w i 2 + gamma(sum i w i) 2 with all a i>0 and gamma>=0, is F strictly convex?数学中等derivation未尝试免费2000Ridge Needed to Repair Local PnL Curvature 5A research model has a locally non-convex quartic approximation, and risk wants the smallest ridge that fixes it everywhere. A local PnL model is h(q)=q 4-6q 2+lambda q 2. What is the smallest lambda that makes h globally convex?数学困难数值题未尝试免费2001Smooth Worst-Case Loss 6A desk smooths the max of two affine stress losses with a log-sum-exp surrogate. Show that g(x) = ln(exp(1x) + exp(2x + -1)) is convex on R.数学简单derivation未尝试免费2002Shifted Log-Sum-Exp Convexity 7One stress term slopes down and the other slopes up, but the smooth envelope remains convex. Show that g(x) = ln(exp(-1x) + exp(3x + 0)) is convex on R.数学中等derivation未尝试免费2003Execution Cost With Stronger Capacity Wall 8The blow-up term is steeper, but the same convexity logic should still go through. Show that f(q) = 2 q 2 + 4/(1-q) is strictly convex on q<1.数学中等derivation未尝试免费2004Perspective Penalty 9A schedule trades size x over time t and pays x 2/t plus a linear time charge. Show that P(x,t)=x 2/t + 2 t is convex on the domain t>0.数学中等derivation未尝试免费2005Convexity of a Two-Asset Quadratic With a Balance-Sheet Term 10Show that H(w 1,w 2)=2w 1 2+5w 2 2+3(w 1+w 2) 2 is convex.数学困难derivation未尝试免费2006Minimum Ridge for a Sharper Quartic Approximation 11A local approximation is even more non-convex, so the repair parameter must rise accordingly. A local PnL model is h(q)=q 4-10q 2+lambda q 2. What is the smallest lambda that makes h globally convex?数学简单数值题未尝试免费2007Three-Sleeve Convex Balance-Sheet Penalty 12The third sleeve is much more expensive on its own, but the aggregate term still preserves convexity. Prove that F(w 1,w 2,w 3) = 1w 1 2 + 4w 2 2 + 9w 3 2 + 2(w 1+w 2+w 3) 2 is convex.数学简单derivation未尝试免费2008Barrier Cost With Strong Quadratic Curvature 13Both the quadratic inventory term and the capacity wall are active sources of curvature. Show that f(q) = 4 q 2 + 3/(1-q) is strictly convex on q<1.数学中等derivation未尝试免费