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Introduction to Computational Finance and Financial Econometrics

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Introduction to Computational Finance and Financial Econometrics
by Eric Zivot

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Assignment 2 Help Center

Warning: The hard deadline has passed. You can attempt it, but you will not get credit for it. You are welcome to try it as a learning exercise.

In this lab you will become more familiar with random variables and probability distributions. Try to do all of the calculations and plots in R. You can also do everything in Excel too. You will find the examples in probReview.R and probReview.xls (available on the course webpage) to be helpful for some of the exercises that follow.

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

Question 1

Hint: you can use the R functions pnorm and qnorm to answer these questions.

(5) Suppose

X $X$ is a normally distributed random variable with mean 0.05 and variance

(0.10)2. $(0.10)^2.$ Compute the following.

a)

Pr(X>.10) $Pr(X\gt.10)$

Question 2

(5) Suppose

X $X$ is a normally distributed random variable with mean 0.05 and variance

(0.10)2. $(0.10)^2.$ Compute the following.

b)

Pr(X<−.10) $Pr(X\lt-.10)$

Question 3

(5) Suppose

X $X$ is a normally distributed random variable with mean 0.05 and variance

(0.10)2. $(0.10)^2.$ Compute the following.

c)

Pr(−0.05<X<0.15) $Pr(-0.05\lt X\lt 0.15)$

Question 4

(5) Suppose

X $X$ is a normally distributed random variable with mean 0.05 and variance

(0.10)2. $(0.10)^2.$ Compute the following.

d) 1% quantile,

q.01 $q_{.01}$

Question 5

(5) Suppose

X $X$ is a normally distributed random variable with mean 0.05 and variance

(0.10)2. $(0.10)^2.$ Compute the following.

e) 5% quantile,

q.05 $q_{.05}$

Question 6

(5) Suppose

X $X$ is a normally distributed random variable with mean 0.05 and variance

(0.10)2. $(0.10)^2.$ Compute the following.

f) 95% quantile,

q.95 $q_{.95}$

Question 7

(5) Suppose

X $X$ is a normally distributed random variable with mean 0.05 and variance

(0.10)2. $(0.10)^2.$ Compute the following.

g) 99% quantile,

q.99 $q_{.99}$

Question 8

Let

X $X$ denote the monthly return on Microsoft Stock and let

Y $Y$ denote the monthly return on Starbucks stock. Assume that

X∼N(0.05,(0.10)2) $X \sim N(0.05, (0.10)^2)$ and

Y∼N(0.025,(0.05)2). $Y \sim N(0.025, (0.05)^2).$

Using a grid of values between –0.25 and 0.35, plot the normal curves for

X $X$ and

Y $Y$ . Make sure that both normal curves are on the same plot.

(5) Which stock has the higher risk?

Starbucks

Microsoft

Question 9

Let

R $R$ denote the simple monthly return on Microsoft stock and let

W0 $W_0$ denote the initial wealth to be invested over the month. Assume that

R∼N(0.04,(0.09)2)) $R \sim N(0.04,(0.09)^2))$ and that

W0=$100,000 $W_0=\$100,000$

(10) Determine the 1% and 5% value-at-risk (VaR) over the month on the investment. That is, determine the loss in investment value that may occur over the next month with 1% probability and with 5% probability.

1% Var = -$10,241
5% Var = -$15,580

1% Var = -$16,937
5% Var = -$10,804

1% Var = -$10,804
5% Var = -$16,937

1% Var = -$3,228
5% Var = -$21,752

1% Var = -$15,580
5% Var = -$10,241

1% Var = -$21,752
5% Var = -$3,228

Question 10

Let the

r $r$ denote the continuously compounded monthly return on Microsoft stock and let

W0 $W_0$ denote the initial wealth to be invested over the month. Assume that

r∼ $r \sim$ iid

N(0.04,(0.09)2) $N(0.04, (0.09)^2)$ and that

W0=$100,000. $W_0 = \$100,000.$

(10) Determine the 1% and 5% value-at-risk (VaR) over the month on the investment. That is, determine the loss in investment value that may occur over the next month with 1% probability and with 5% probability. (Hint: compute the 1% and 5% quantile from the Normal distribution for

r $r$ and then convert continuously compounded return quantile to a simple return quantile using the transformation

R=er−1 $R=e^r -1$ )

1% Var = -$3,228
5% Var = -$21,752

1% Var = -$16,937
5% Var = -$10,804

1% Var = -$21,752
5% Var = -$3,228

1% Var = -$10,241
5% Var = -$15,580

1% Var = -$15,580
5% Var = -$10,241

1% Var = -$10,804
5% Var = -$16,937

Question 11

For the following questions:

Consider a one month investment in two Northwest stocks: Amazon and Costco. Suppose you buy Amazon and Costco at the end of September at

PA,t−1=$38.23 $P_{A,t-1} = \$ 38.23$ ,

PC,t−1=$41.11 $P_{C,t-1} = \$ 41.11$ and then sell at the end of October for

PA,t=$41.29 $P_{A,t} = \$ 41.29$ ,

PC,t=$41.74 $P_{C,t} = \$41.74$ . (Note: these are actual closing prices for 2004 taken from Yahoo!)

(10) What are the simple monthly returns for the two stocks?

Amazon: 7.7%
Costco: 1.5%

Amazon: -7.4%
Costco: -1.5%

Amazon: 8.0%
Costco: 1.5%

Amazon: -7.7%
Costo: -1.5%

Question 12

(10) What are the continuously compounded returns for the two stocks?

Amazon: -7.4%
Costco: -1.5%

Amazon: 8.0%
Costco: 1.5%

Amazon: -7.7%
Costo: -1.5%

Amazon: 7.7%
Costco: 1.5%

Question 13

(10) Suppose Amazon paid a $0.10 per share cash dividend at the end of October. What is the monthly simple total return on Amazon? What is the monthly dividend yield?

Total Return: 8.3%
Div Yield: 0.24%

Total Return: 8.3%
Div Yield: 0.26%

Total Return: 8.0%
Div Yield: 0.24%

Total Return: 8.0%
Div Yield: 0.26%

Question 14

(10) Suppose the monthly returns on Amazon from question 12 above are the same every month for 1 year. Compute the simple annual returns, as well as the continuously compounded annual returns.

Simple Annual: 92.4%
CC Annual: 151.9%

Simple Annual: 151.9%
CC Annual: 92.4%

Question 15

(10) At the end of September 2004, supposed you have $10,000 to invest in Amazon and Costco over the next month. If you invest $8,000 in Amazon and $2,000 in Costco, what are your portfolio shares,

xA $x_A$ and

xC $x_C$ .

xA=20% $x_A = 20\%$

xC=80% $x_C = 80\%$

xA=81% $x_A = 81\%$

xC=19% $x_C = 19\%$

xA=80% $x_A = 80\%$

xC=20% $x_C = 20\%$

xA=19% $x_A = 19\%$

xC=81% $x_C = 81\%$

Question 16

(10) Continuing with the previous question, compute the monthly simple return on the portfolio. Assume than no dividends are paid.

6.7%

9.2%

5.4%

8.1%

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

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

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