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 ∈ Rn×n is symmetric and positive definite, then A is invertible.
R1
2. In QR-factorization A = Q , the Q and the R1 factors are both orthogonal
0
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 r̄ T x-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 100000899606070 from CourseHero.com on 02-08-2026 03:24:47 GMT -06:00
https://www.coursehero.com/file/56067407/CO372-pracmidtermpdf/
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 ∈ Rn×n is symmetric and positive definite, then A is invertible.
R1
2. In QR-factorization A = Q , the Q and the R1 factors are both orthogonal
0
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 r̄ T x-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 100000899606070 from CourseHero.com on 02-08-2026 03:24:47 GMT -06:00
https://www.coursehero.com/file/56067407/CO372-pracmidtermpdf/
