FINA 6592 FA Financial Econometrics

Posted 2022-11-05 toyma

Department of Finance
FINA 6592 FA Financial Econometrics
Assignment 1
Due Date: September 26, 2019
An assignment where you should begin to appreciate mathematics/statistics and why
finance professors may be smart but not necessarily rich. Answer the following questions
for a total of 300 points and show all your work carefully. You do not have
to use Microsoft Excel for the assignments since all computations can also be done
using programming environments such as EViews, GAUSS, MATLAB, Octave, Ox, R,
SAS, or S-PLUS. Please do not turn in the assignment in reams of unformatted computer
output and without comments! Make little tables of the numbers that matter,
copy and paste all results and graphs into a document prepared by typesetting system
such Microsoft Word or LATEX while you work, and add any comments and answer all
questions in this document.
1. a. (1 point) By computing the necessary derivatives and evaluating them at ? = 0,
expand the functions ?? and ln(1 + ?) in a Taylor series about the point ? = 0.
b. (1 point) For each of the functions ?? and ln(1 + ?) compute the function values
at ? = 1, 0.1, and 0.01. Keeping only up to second order terms in the Taylor
expansions of these functions from previous part, compute the Taylor series approximations

代写FINA 6592作业、代写Financial Econometrics作业
to the functions at those ? values and determine the approximation
error (absolute and relative) in each case. Try to maintain as many decimal places
in your answer as possible, e.g., 5 or 6 places. Note the improvement in the approximation
due to the presence of the second order terms.
2. (4 points) Given (?1? ?1)?????(??? ??) from R2, find the values of ?1, ?2 and ?3 that will
maximize the function:
Verify your solution with the second derivative test.
3. Consider the matrix M = (???? ) :
a. (2 points) Find the Cholesky decomposition of M. Show your steps or the algorithm.
Then use the Cholesky decomposition to solve Mx = b for x when
b = (2?49? 0?566? 0?787? −2?209)>.
b. (3 points) Find the eigenvalues of M and the corresponding eigenvectors with
unit length. Show your steps or the algorithm. Is M positive definite? Are the
eigenvectors orthogonal?
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4. (2 points) Let ?, ?, ? and ? be random variables describing next year’s annual return
on Weyerhauser, Xerox, Yahoo and Zymogenetics stock. The table below gives a discrete
probability distribution for these random variables based on the state of the economy:
State of Economy ? Pr(?) ? Pr(?) ? Pr(? ) ? Pr(?)
Depression −0?3 0?05 −0?5 0?05 −0?5 0?15 −0?8 0?05
Recession 0?0 0?20 −0?2 0?10 −0?2 0?50 0?0 0?20
Normal 0?1 0?50 0?0 0?20 0?0 0?20 0?1 0?50
Mild Boom 0?2 0?20 0?2 0?50 0?2 0?10 0?2 0?20
Major Boom 0?5 0?05 0?5 0?15 0?5 0?05 1?0 0?05
a. Plot the distributions for each random variable (make a bar chart). Comment on
any difference or similarities between the distributions.
b. For each random variable, compute the expected value, variance, standard deviation,
skewness and kurtosis and briefly comment. Note: You cannot use the
Excel functions AVERAGE, VAR, STDEV, SKEW and KURT for this problem.
These functions compute sample statistics which are different from the population
moment calculations required for this problem.
5. (3 points) Suppose a continuous random variable ? has density function:
?(?; ?) = ½ ??2(1 − ?)3 for 0 ??? 1?
0 otherwise.
a. Find value(s) of ? such that ?(?; ?) is a density function.
b. Find the mean and median of ?.
c. Find Pr(0?25 ≤ ? ≤ 0?75).
6. (3 points) Suppose a continuous random variable ? has density function:
?(?; ?) = ½ ?? + 0?5 for − 1 ≤ ? ≤ 1?
0 otherwise.
a. Find value(s) of ? such that ?(?; ?) is a density function.
b. Find the mean and median of ?.
c. For what value of ? is the variance of ? maximized?
7. (1 points) Suppose ? is a normally distributed random variable with mean 0.05 and
variance (0?10)2, i.e., ? ∼ N (0?05?(0?10)2). Compute the following:
a. Pr(? ? 0?10)?
b. Pr(? ? −0?10)?
c. Pr(−0?05 ??? 0?15)?
d. Determine the 1%, 5%, 10%, 25%, 50%, 75%, 90%, 95% and 99% quantiles of the
distribution of ?.
Hint: you can use the Excel functions NORMDIST and NORMINV to answer these
questions.
b. (1 point) Does ? follow a normal distribution or a skewed distribution?
11. (4 points; Normal mixture models)
a. What is the kurtosis of a normal mixture distribution that is 95% N (0? 1)
b. Find a formula for the kurtosis of a normal mixture that is 100?% ?(0? 1) and 100(1 − ?)% ?(0? ?2) where and are parameter. Your formula should give the
kurtosis as a function of and .
c. Show that the kurtosis of the normal mixtures in part (b) can be made arbitrarily
large by choosing and appropriately. Find values of ? and ? so that the kurtosis
is 10,000 or larger.
d. Let ? ? 0 be arbitrarily large. Show that for any ?0 ? 1, no matter how close
to 1, there is a ???0 and a ? such that the normal mixture with these values
of ? and ? has a kurtosis at least ?. This shows that there is a normal mixture
arbitrarily close to a normal distribution with a kurtosis above any ?.
12. (2 points) Let ? be N(0? ?2). Show that the CDF of the conditional distribution of ?
given that ??? is
Φ(???) − Φ(???)
1 − Φ(???)
where ???, and that the PDF of this distribution is
?(???)
?(1 − Φ(???))
where ???. Also show that if ? = 0?25 and ? = 0?3113, then at ? = 0?25 this PDF
equals the PDF of a Pareto distribution with parameters ? = 1?1 and ? = 0?25. Note:
The value of ? = 0?3113 was originally found by interpolation.
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13. (7 points) Consider the following joint distribution of ? and ? :
?
123
1 0.1 0.2 0
? 2 0.1 0 0.2
3 0 0.1 0.3
a. Find the marginal distributions of ? and ? . Using these distributions, compute
E(?), Var(?), ??(?), E(? ), Var(? ) and ??(? ).
b. Determine the conditional distribution of ? given that ? equals 1, 2 and 3. Plot
the marginal distribution of ? along with the conditional distributions of ? and
briefly comment.
c. Determine the conditional distribution of ? given that ? equals 1, 2 and 3. Plot
the marginal distribution of ? along with the conditional distributions of ? and
briefly comment.
d. Compute E(?|? = 1), E(?|? = 2), E(?|? = 3) and compare to E(?). Compute
E(? |? = 1), E(? |? = 2), E(? |? = 3) and compare to E(? ).
e. Plot E(?|? = ?) versus ? and E(? |? = ?) versus ? and briefly comment.
f. Are ? and ? independent? Fully justify your answer.
g. Compute Cov(?? ? ) and Corr(?? ? ).
14. (3 points) Let ?, ?, ? , ? be random variables and ?, ?, ?, ? be constants. Show that:
a. Var(?? + ?) = Var(−?? − ?)?
b. Cov(??? ?? ) = ?? Cov(?? ? )?
c. Cov(?? ?) = Var(?)?
d. Cov(??+??? ??+??) = ?? Cov(???)+?? Cov(?? ?)+?? Cov(???)+?? Cov(???)?
e. Suppose ? =3+5? and ? = 4 − 8??
i. Is ???? = 1? Prove or disprove.
ii. Is ???? = ???? ? Prove or disprove.
15. (4 points) Let ? and ? be two random variables.
a. If Cov(?2? ? 2)=0, then Cov(?? ? )=0. True/False/Uncertain. Explain.
b. If ? and ? are independent, then Cov(?2? ? 2) ? Cov(?? ? ). True/False/Uncertain.
Explain.
c. If ? and ? are independent and E( ?
? ) ? 1, then E(?)
E(? ) ? 1. True/False/Uncertain.
Explain.
d. Prove that (Cov(?? ? ))2 ≤ Var(?) Var(? ) and thus −1 ≤ ???? ≤ 1.
16. (4 points) Let ? and ? be independent U(−?? ?) random variables. Find (a) the probability
that the quadratic equation ?
2 + ?? + ? = 0 has real roots, and (b) the limit of
this probability as ? → ∞.
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17. (4 points) Let us assume that ?1 and ?2 are independent N (0? 1) random variables and
let us define the random variable ? by
? =
½ |?2| if ?1 ? 0;
− |?2| otherwise.
a. Prove that ? ∼ N (0? 1).
b. Say if (?1? ? ) is bivariate Gaussian, and explain why.
18. (6 points) The purpose of this problem is to show that lack of correlation does not
imply independence, even when the two random variables are Gaussian!!! We assume
that ?, ?1 and ?2 are independent random variables, that ? ∼ N (0? 1), and that
Pr(?? = −1) = Pr(?? = +1) = 1?2 for ? = 1? 2. We define the random variables ?1 and
?2 by ?1 = ?1? and ?2 = ?2?.
a. Prove that ?1 ∼ N (0? 1), ?2 ∼ N (0? 1) and that ?(?1? ?2)=0.
b. Show that ?1 and ?2 are not independent.
19. The goal of this problem is to prove rigorously a couple of useful results for normal and
log-normal random variables.
a. (2 points) Use the chain rule to differentiate
with respect to ? and hence find the density function of the random variable ?
such that ? = ?−?
? is a standard normal random variable with the distribution
b. (2 points) Compute the density of a random variable ? whose logarithm is N (?? ?2).
Such a random variable is usually called a log-normal random variable with mean
? and variance ?2. Hint: You can use the previous method to find the density function
of the random variable ? such that ? = ln ?−?
? is a standard normal random
variable.
c. (4 points) Suppose the random vector (?1? ?2) follows a bivariate normal distribution,
where ?1 ∼ N (0? 1), ?2 ∼ N (0? ?2), and the correlation coefficient of ?1 and
?2 is ?. Throughout the rest of the problem we assume that (ln ?? ln ? )=(?1? ?2),
in other words, ? is a log-normal random variable with parameters 0 and 1 (i.e.,
? is the exponential of a N (0? 1) random variable) and that ? is a log-normal
random variable with parameters 0 and ?2 (i.e., ? is the exponential of a N(0? ?2)
random variable). We will show the possible values of the correlation coefficient
between ? and ? are limited to an interval [?min? ?max], which is not the whole
interval [−1? +1]. Specifically, show that:
i. ?min = (?−? − 1)?
p(? − 1)(??2 − 1).
ii. ?max = (?? − 1)?
p(? − 1)(??2 − 1).
iii. lim?→∞ ?min = lim?→∞ ?max = 0.
Do we have a problem interpreting the correlation between log-normal random
variables as to their normal counterparts?
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20. Suppose that ?1? ?2??????? are independent real-valued random variables and that ??
has distribution function ?? for each ?. The maximum and minimum transformations are
very important in a number of applications. Specifically, let ? = max?1? ?2???????,
? = min?1? ?2???????, and let ? and ? denote the distribution functions of ? and
? respectively.
a. (2 points) Show that:
i. ?(?) = ?1(?)?2(?)··· ??(?) for ? ∈ R.
ii. ?(?)=1 − [1 − ?1(?)][1 − ?2(?)] ··· [1 − ??(?)] for ? ∈ R.
b. (4 points) If ?? has a continuous distribution with density function ?? for each ?,
then ? and ? also have continuous distributions, and the densities can be obtained
by differentiating the distribution functions above. Suppose that ?1? ?2???????
are independent random variables, each uniformly distributed on (0? 1).
i. Find the distribution function, density function, expected value and variance
of ?. Hint: ? has a beta distribution.
ii. Find the distribution function, density function, expected value and variance
of ? . Hint: ? has a beta distribution.
21. (4 points) Let ? and ? be two independent N (0? 1) random variables. Find:
a. Cov(?? max[?? ? ]) and Cov(?? min[?? ? ])
b. Cov(max[?? ? ]? min[?? ? ]), Var(max[?? ? ]), and Var(min[?? ? ])
22. a. (3 points) Suppose that ?1 and ?2 are independent random variables each uniformly
distributed over the interval (0? 1). Define two random variables as ?1 =
?1 + ?2 and ?2 = ?1??2. Find the joint density function of ?1 and ?2.
b. (3 points) If ? is uniform on (0? 2?) and ? , independent of ?, is exponential with
rate 1.
standardized kurtosis of ?? based on such sample moments? How does it compare
with that of the above?
25. a. (2 points) Suppose that a random variable ? has the uniform distribution on the
interval [0? 5] and the random variable ? is defined by ? = 0 if ? ≤ 1, ? = 5 if
? ≥ 3, and ? = ? otherwise. Sketch the cumulative distribution function of ? .
b. (2 points) Suppose ? has a continuous distribution with probability density function
?. Let ? = ?2, show that the probability density function of is
c. (2 points) Suppose that one can simulate as many i.i.d. Bernoulli random variables
with parameter ? as one wishes. Explain how to use these to approximate the mean
of the geometric distribution with parameter ?.
26. a. (10 points) Let ?1 and ?2 be random variables with CDF ?1(?) and ?2(?) with
?1(?) ≤ ?2(?) for all values of ?.
i. Which of these two distributions has the heavier lower tail? Explain.
ii. Which of these two distributions has the heavier upper tail? Explain.
iii. If these two distributions are proposed as models for the return of a given
portfolio over the next month, and if you are asked to compute ? ??0?01 for
this portfolio over that period, which of these two distributions will give the
larger value at risk?
b. (10 points) Let ?0 denote initial wealth to be invested over the month and assume
?0 = $100? 000.
i. Let ? denote the monthly simple return on Microsoft stock and assume that
? ∼ N (0?04?(0?09)2). 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.
ii. Let ? denote the monthly continuously compounded return on Microsoft stock
and assume that ? ∼ N (0?04?(0?09)2). Determine the 1% and 5% value-atrisk
(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 ? and then convert continuously compounded return quantile
to a simple return quantile using the transformation ? = ?? − 1.)

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