CO 372: Portfolio Optimization Fall Practice Midterm 100% correct

CO 372: Portfolio Optimization Fall 2019 Practice Midterm S. Vavasis Handed out: 2019-Oct-11. This was an 80 minute exam. Students were permitted one 8.5-by-11-inch sheet of paper with notes written on both sides. The following are true-false questions. Each one is 5 marks. 1. If A ∈ R n×n is symmetric and positive definite, then A is invertible. 2. In QR-factorization A = Q  R1 0  , the Q and the R1 factors are both orthogonal matrices. 3. Suppose symmetric n×n matrix H is positive semidefinite. Then the (n+ 1)×(n+ 1) matrix H+ that results from appending a row and column of zeros to H is also positive semidefinite. 4. In solving Lx = b with forward substitution, where L is an n × n lower triangular matrix, the entries of x are computed in the order xn, xn−1, . . . , x1. 5. The variance of the return of a risk-free investment instrument is always positive. Short answers (a phrase, sentence or sketch) are required for the following questions. Each question is 10 marks. 6. Let A be an n × n symmetric positive definite matrix. Write down the formula that describes the Cholesky factorization of A, and indicate with a brief phrase what types of matrices are present in the factorization. 7. Consider the constraint Ax = b where A is an m × n matrix and b is an m-vector. Suppose x0 is a particular solution for this constraint and suppose the columns of Z make up a basis for the null space of A. Write the general form for all solutions to Ax = b. 8. In the capital-asset pricing model, the efficient frontier (plotted as r¯ Tx-versus-√ xT Hx) could be either (a) entirely a line or (b) partly a line and partly a hyperbolic curve. What assumption in the model distinguishes case (a) from case (b)? Longer answers are required for the following questions, which are each 15 marks. 1 This study source was downloaded by from CourseH on :24:47 GMT -06:00 9. Write the function kx − b0k 2 − Pn i=1 xi in the standard form x T Hx/2 + g Tx + const for quadratics. Here, b0 is a given (fixed) n-vector. Explain whether the resulting H is positive definite. 10. Suppose Kane wants to invest in 4 securities with the constraints (1) exactly one currency unit is available for investing, (2) no short-selling in any security, (3) at most 30% of the portfolio is invested in securities 1 and 2, (4) Exactly 30% of the portfolio is invested in securities 2 and 3. (a) Write down the resulting constraints formally