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Questions 1 - 9 refer to the following matrices and vectors:

A = ⎛ ⎝ 126441783 ⎞ ⎠, B = ⎛ ⎝ 452492015 ⎞ ⎠, x = ⎛ ⎝ 123 ⎞ ⎠, y = ⎛ ⎝ 527 ⎞ ⎠

$A = \left( \begin{array}{ccc} 1 & 4 & 7 \\ 2 & 4 & 8 \\ 6 & 1 & 3 \end{array} \right), B = \left( \begin{array}{ccc} 4 & 4 & 0 \\ 5 & 9 & 1 \\ 2 & 2 & 5 \end{array} \right), x = \left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right), y = \left( \begin{array}{c} 5 \\ 2 \\ 7 \end{array} \right)$

In accordance with the Coursera Honor Code, I (Anonymous Student) certify that the answers here are my own work.

Question 1

(10) Compute the transpose of

A $A$ .

A' = ⎛ ⎝ 347248611 ⎞ ⎠

$A' = \left( \begin{array}{ccc} 3 & 2 & 6 \\ 4 & 4 & 1 \\ 7 & 8 & 1 \end{array} \right)$

A' = ⎛ ⎝ 147248613 ⎞ ⎠

$A' = \left( \begin{array}{ccc} 1 & 2 & 6 \\ 4 & 4 & 1 \\ 7 & 8 & 3 \end{array} \right)$

A' = ⎛ ⎝ 621144387 ⎞ ⎠

$A' = \left( \begin{array}{ccc} 6 & 1 & 3 \\ 2 & 4 & 8 \\ 1 & 4 & 7 \end{array} \right)$

A' = ⎛ ⎝ 783441126 ⎞ ⎠

$A' = \left( \begin{array}{ccc} 7 & 4 & 1 \\ 8 & 4 & 2 \\ 3 & 1 & 6 \end{array} \right)$

Question 2

(10) Compute the transpose of

B $B$ .

B' = ⎛ ⎝ 015492452 ⎞ ⎠

$B' = \left( \begin{array}{ccc} 0 & 4 & 4 \\ 1 & 9 & 5 \\ 5 & 2 & 2 \end{array} \right)$

B' = ⎛ ⎝ 254294510 ⎞ ⎠

$B' = \left( \begin{array}{ccc} 2 & 2 & 5 \\ 5 & 9 & 1 \\ 4 & 4 & 0 \end{array} \right)$

B' = ⎛ ⎝ 440591225 ⎞ ⎠

$B' = \left( \begin{array}{ccc} 4 & 5 & 2 \\ 4 & 9 & 2 \\ 0 & 1 & 5 \end{array} \right)$

B' = ⎛ ⎝ 540591224 ⎞ ⎠

$B' = \left( \begin{array}{ccc} 5 & 5 & 2 \\ 4 & 9 & 2 \\ 0 & 1 & 4 \end{array} \right)$

Question 3

(5) Compute the transpose of

x $x$ .

x' = (123)

$x' = \left( \begin{array}{ccc} 1 & 2 & 3 \end{array} \right)$

x' = ⎛ ⎝ 321 ⎞ ⎠

$x' = \left( \begin{array}{c} 3 \\ 2 \\ 1 \end{array} \right)$

x' = (321)

$x' = \left( \begin{array}{ccc} 3 & 2 & 1 \end{array} \right)$

Question 4

(5) Compute the transpose of

y $y$ .

y' = (527)

$y' = \left( \begin{array}{ccc} 5 & 2 & 7 \end{array} \right)$

y' = ⎛ ⎝ 725 ⎞ ⎠

$y' = \left( \begin{array}{c} 7 \\ 2 \\ 5 \end{array} \right)$

y' = (725)

$y' = \left( \begin{array}{ccc} 7 & 2 & 5 \end{array} \right)$

Question 5

(10) Compute

A+B $A+B$ .

A + B = ⎛ ⎝ 241288214166 ⎞ ⎠

$A+B = \left( \begin{array}{ccc} 2 & 8 & 14 \\ 4 & 8 & 16 \\ 12 & 2 & 6 \end{array} \right)$

A + B = ⎛ ⎝ - 3 - 3 4 0 - 5 - 1 77 - 2 ⎞ ⎠

$A+B = \left( \begin{array}{ccc} -3 & 0 & 7 \\ -3 & -5 & 7 \\ 4 & -1 & -2 \end{array} \right)$

A + B = ⎛ ⎝ 5788133798 ⎞ ⎠

$A+B = \left( \begin{array}{ccc} 5 & 8 & 7 \\ 7 & 13 & 9 \\ 8 & 3 & 8 \end{array} \right)$

Question 6

(10) Compute

A−B $A-B$ .

A - B = ⎛ ⎝ 5788133798 ⎞ ⎠

$A-B = \left( \begin{array}{ccc} 5 & 8 & 7 \\ 7 & 13 & 9 \\ 8 & 3 & 8 \end{array} \right)$

A - B = ⎛ ⎝ 241288214166 ⎞ ⎠

$A-B = \left( \begin{array}{ccc} 2 & 8 & 14 \\ 4 & 8 & 16 \\ 12 & 2 & 6 \end{array} \right)$

A - B = ⎛ ⎝ - 3 - 3 4 0 - 5 - 1 77 - 2 ⎞ ⎠

$A - B = \left( \begin{array}{ccc} -3 & 0 & 7 \\ -3 & -5 & 7 \\ 4 & -1 & -2 \end{array} \right)$

Question 7

(10) Compute

2∗A $2*A$ .

2 * A = ⎛ ⎝ 5788133798 ⎞ ⎠

$2*A = \left( \begin{array}{ccc} 5 & 8 & 7 \\ 7 & 13 & 9 \\ 8 & 3 & 8 \end{array} \right)$

2 * A = ⎛ ⎝ 241288214166 ⎞ ⎠

$2*A = \left( \begin{array}{ccc} 2 & 8 & 14 \\ 4 & 8 & 16 \\ 12 & 2 & 6 \end{array} \right)$

2 * A = ⎛ ⎝ - 3 - 3 4 0 - 5 - 1 77 - 2 ⎞ ⎠

$2*A = \left( \begin{array}{ccc} -3 & 0 & 7 \\ -3 & -5 & 7 \\ 4 & -1 & -2 \end{array} \right)$

Question 8

(10) Compute

Ax $Ax$ .

A x = ⎛ ⎝ 122830 ⎞ ⎠

$Ax = \left( \begin{array}{c} 12 \\ 28 \\ 30 \end{array} \right)$

A x = (122830)

$Ax = \left( \begin{array}{ccc} 12 & 28 & 30 \end{array} \right)$

A x = ⎛ ⎝ 303417 ⎞ ⎠

$Ax = \left( \begin{array}{c} 30 \\ 34 \\ 17 \end{array} \right)$

A x = (303417)

$Ax = \left( \begin{array}{ccc} 30 & 34 & 17 \end{array} \right)$

Question 9

(10) Compute

y′Ax $y'Ax$ .

337

224

158

Question 10

(10) Consider the system of equations:

z 1 + z 2 = 1,

$z_1 + z_2 = 1,$

2 z 1 + 4 z 2 = 2

$2z_1+4z_2 = 2$
Write the system using matrix notation as

Az=b $Az = b$ and solve for

z $z$ .

z = (0.5 0.5)

$z = \left( \begin{array}{c} 0.5 \\ 0.5 \end{array} \right)$

z = (10)

$z = \left( \begin{array}{c} 1 \\ 0 \end{array} \right)$

z = (0 0.5)

$z = \left( \begin{array}{c} 0 \\ 0.5 \end{array} \right)$

z = (01)

$z = \left( \begin{array}{c} 0 \\ 1 \end{array} \right)$

Question 11

Consider creating a portfolio of three assets denoted

A $A$ ,

B $B$ and

C $C$ . Assume the following information:

μ = ⎛ ⎝ 0.01 0.04 0.02 ⎞ ⎠, Σ = ⎛ ⎝ 0.10 0.30 0.10 0.30 0.15 - 0.20 0.10 - 0.20 0.08 ⎞ ⎠

$\mu = \left( \begin{array}{c} 0.01 \\ 0.04 \\ 0.02 \end{array} \right), \Sigma = \left( \begin{array}{ccc} 0.10 & 0.30 & 0.10 \\ 0.30 & 0.15 & -0.20 \\ 0.10 & -0.20 & 0.08 \end{array} \right)$

(10) Compute the expected return for an equally weighted portfolio (i.e.,

xA=xB=xC= $x_A = x_B = x_C =$ 1/3).

0.070

0.023

0.018

0.034

Question 12

(10) Continuing from the previous question, what is the variance for an equally weighted portfolio?

0.076

0.096

0.081

0.110

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You cannot submit your work until you agree to the Honor Code. Thanks!

Introduction to Computational Finance and Financial Econometrics

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Introduction to Computational Finance and Financial Econometrics

Introduction to Computational Finance and Financial Econometricsby Eric Zivot

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Introduction to Computational Finance and Financial Econometrics
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