CO 372: Portfolio Optimization Models Fall Practice Midterm Solutions All answers verified

CO 372: Portfolio Optimization Models Fall 2025 Practice Midterm Solutions S. Vavasis Handed out: . 1. If a symmetric matrix is positive definite, then all of its entries must be positive. Solution. F 2. The ‘Eva’ algorithm has two QR-factorizations, and the second one is used to solve a certain least-squares problem. Solution. T 3. The parameter t in Pr3 controls the weight placed by the investor on return. Solution. T 4. Portfolios on the efficient frontier correspond to nonnegative values of the parameter rp in Pr1. Solution. F 5. The risk-free return rate rf in Pr3′ must be positive else Pr3′ has no solution. Solution. F 6. Suppose H, the covariance matrix, has H(5, 5) = 3. This means that security 5 cannot be risk-free. By considering the variance of a portfolio that invests only in security 5, explain the previous sentence. Solution. Consider a portfolio with x5 = 1, xi = 0 for i ̸= 5. Then the variance of this portfolio x T Hx = 3, meaning that there is risk associated with security 5. 7. Short-selling is commonly considered riskier than ordinary purchasing of securities. Why? Solution. The loss from short-selling is unbounded if the security price goes up, whereas the loss from ordinary purchasing is bounded by the amount of the original investment. 8. Portfolio xmr, the min-risk portfolio, whose return is denoted rp,min and whose risk is denoted σ 2 p,min, can be recovered as a particular solution to any one of Pr1, Pr2, Pr3 for the appropriate parameter choice. Write down Pr2 with its parameter chosen so that xmr is the optimizer. Solution. The Pr2 problem that recovers xmr is: min −r¯ Tx subject to Ax = b, x T Hx ≤ σ 2 p,min. 1 This study source was downloaded by from CourseH on :23:12 GMT -06:00 9. Suppose in a universe of 6 securities whose covariance matrix H ∈ S 6 is positive definite and whose expected return vector is r¯ ∈ R 6 , that the investor wants 50% of the portfolio in securities 1, . . . , 4, wants the amount invested in security 5 to be the same as the amount in security 6, has a total budget of exactly $1 to invest, and wants a return of $1.2. Subject to these constraints, the investor wants to minimize risk. Write down the optimization problem that describes this situation. Gather the constraints into a single matrix and right-hand side vector. Solution. The constraints are x1 + · · · + x6