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1599Optimizer of a Coupled Positive-Definite Penalty 4Minimize f(x,y) = 0.5(3x 2 + 4xy + 5y 2) - (6x + 8y). What is the optimizer?数学中等数值题未尝试免费1600Coupling Needed for Equal Optimizer Coordinates 5For f(x) = 0.5 x T A(t) x - b T x with A(t) = [[5,t],[t,4]] and b = (6,2), for what t does the optimizer satisfy x1 = x2?数学困难derivation未尝试面试订阅1601Why Every Covariance Matrix Is PSDWhy must every valid covariance matrix be positive semidefinite?数学简单essay未尝试免费1602Equal-Weight Two-Asset Variance 1Two assets have variances 4 and 9, with covariance 2. What is the variance of the portfolio with weights (1/2, 1/2)?数学简单数值题未尝试免费1605Hedged Pair Variance 4Two assets have variances 1 and 9, with covariance -1. What is the variance of the portfolio with weights (1/3, 2/3)?数学困难数值题未尝试面试订阅1607Why Sample Covariance Can Be Rank DeficientWhy can a sample covariance matrix become rank deficient when the number of observations is smaller than the number of assets?数学中等essay未尝试免费1608Minimum-Variance Hedge Ratio 1You hedge X with h units of Y and want to minimize Var(X - hY). If Cov(X, Y) = 8 and Var(Y) = 4, what is the optimal hedge ratio h?数学中等derivation未尝试免费1611Equicorrelation Validity Threshold 1For an n=4 equicorrelation matrix with 1s on the diagonal and rho off the diagonal, what is the lowest rho that still keeps the matrix positive semidefinite?数学简单数值题未尝试免费1614One-Factor Covariance Entry 1Under a one-factor model X = bF + epsilon with factor variance 3, asset loadings (1, 2), and idiosyncratic variances (4, 9), what are Var(X 1), Var(X 2), and Cov(X 1, X 2)?数学困难derivation未尝试面试订阅1615Signed-Factor Covariance Entry 2Under a one-factor model X = bF + epsilon with factor variance 5, asset loadings (2, -1), and idiosyncratic variances (1, 4), what are Var(X 1), Var(X 2), and Cov(X 1, X 2)?数学困难derivation未尝试面试订阅1616How to Read Diagonal and Off-Diagonal EntriesIn a covariance matrix, what do the diagonal entries and off-diagonal entries mean?数学简单essay未尝试免费1617Correlation from Covariance 1Two assets have variances 9 and 16, and covariance 6. What is their correlation?数学中等derivation未尝试免费1618Negative Correlation Extraction 2Two assets have variances 4 and 25, and covariance -6. What is their correlation?数学中等derivation未尝试免费1619Why Large Condition Number Destabilizes WeightsWhy does a covariance matrix with a very large condition number make optimized portfolio weights unstable?数学困难essay未尝试面试订阅3266Best Rank-One Reconstruction Error From Singular ValuesA centered data matrix has singular values \sigma 1\ge\sigma 2\ge\cdots\ge\sigma r. What is the Frobenius-norm reconstruction error of the best rank-one approximation?数学中等derivation未尝试面试订阅3268Recovering Correlation Strength from a Known First PCA covariance matrix has the form \Sigma=egin pmatrix 5&c\c&5\end pmatrix . You are told that the first principal component points along (1,1) and has variance 8. What is c?数学中等derivation未尝试面试订阅3269One-PC Reconstruction of a Two-Asset MoveLet \Sigma=egin pmatrix 9&-3\-3&9\end pmatrix , whose first principal direction is along (1,-1). For the observed move x=(2,-1), what are the rank-1 reconstruction using only the first principal component and the residual?数学中等derivation未尝试面试订阅3270How Standardization Changes the PCA SpectrumA raw covariance matrix is \Sigma=egin pmatrix 9&6\6&9\end pmatrix . After standardizing each coordinate to unit variance, what are the eigenvalues of the correlation matrix, and what fraction of standardized variance does the first component explain?数学中等derivation未尝试面试订阅3272How Large Can the Third Singular Value Be?A data matrix has singular values 7, 4, and s. You plan to keep rank 2 and want to retain at least 95\% of total squared Frobenius energy. What is the largest allowable value of s?数学困难derivation未尝试面试订阅3273Which Portfolio Is Better Aligned with the Low-Variance PC?A covariance matrix has eigenvalues 25 and 1, with first eigenvector proportional to (2,1) and second eigenvector proportional to (1,-2). Compare the variances of portfolios p 1=(1,-2) and p 2=(2,1).数学中等derivation未尝试面试订阅