calculus
alanhon2005
Calculus.pdf
Prof. Chow Ying-Foon, Ph.D. Room 1215, Cheng Yu Tung Building Department of Finance The Chinese University of Hong Kong
2020—2021 First Term Phone: 2609 7638 Fax: 2603 6586 Email: [email protected]
FINA 6592 FA Financial Econometrics Assignment 1
Due Date: October 22, 2020
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, Python, 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 ap- proximations 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 ap- proximation 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:
(1 2 3) = Y =1
1
(23) 12 exp
µ − 1 23
( − 1 − 2)2 ¶
Verify your solution with the second derivative test.
3. Consider the matrixM = () :
M =
⎛⎜⎜⎝ 9 06 −03 15 06 1604 118 −15 −03 118 41 −057 15 −15 −057 2545
⎞⎟⎟⎠ a. (2 points) Find the Cholesky decomposition of M. Show your steps or the al- gorithm. Then use the Cholesky decomposition to solve Mx = b for x when b = (249 0566 0787−2209)>.
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 onWeyerhauser, 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 −03 005 −05 005 −05 015 −08 005 Recession 00 020 −02 010 −02 050 00 020 Normal 01 050 00 020 00 020 01 050 Mild Boom 02 020 02 050 02 010 02 020 Major Boom 05 005 05 015 05 005 10 005
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 devia- tion, skewness and kurtosis and briefly comment. Note: You cannot use the Excel functions AVERAGE, VAR.P, STDEV.P, SKEW.P and KURT for this problem. These func- tions 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(025 ≤ ≤ 075).
6. (3 points) Suppose a continuous random variable has density function:
(; ) =
½ + 05 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 (010)2, i.e., ∼ N (005 (010)2). Compute the following:
a. Pr( 010)
b. Pr( −010) c. Pr(−005 015) 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 NORM.DIST and NORM.INV to answer these ques- tions.
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8. (2 points) Suppose that () = 14 if || 1 and () = 1(42) if || ≥ 1. Show that
R∞ −∞ () = 1 so that really is a density, but that
R 0 −∞ () = −∞
and R∞ 0
() = ∞ so that a random variable with this density does not have an expected value.
9. (3 points) Let be a standard normal random variable, and let be a differentiable function with derivative 0. Note: Assume that () ∈ (exp(2)) with E(|0()|) ∞, where () ∈ (()) means lim→∞ ()() = 0.
a. Show that E(0()) = E( ()).
b. Show that E(+1) = E(−1).
c. Find E(4).
10. Suppose ∼ N ( 2) with 0 for = 1 . Define =Q=1.
a. (2 points) Find E() and Var().
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) and 5% N (0 10)̇?
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 = 025 and = 03113, then at = 025 this PDF equals the PDF of a Pareto distribution with parameters = 11 and = 025. Note: The value of = 03113 was originally found by interpolation.
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13. (7 points) Consider the following joint distribution of and :
1 2 3
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, thenCov(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 prob- ability 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) = 12 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 (12) = 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 differentiateZ (−) −∞
1√ 2 exp
µ −1 2 2 ¶
with respect to and hence find the density function of the random variable such that = −
is a standard normal random variable with the distribution
Φ() = 1√ 2
R −∞ exp(−122).
b. (2 points) Compute the density of a random variable whose logarithm isN ( 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 func- tion 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 distribu- tion, 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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24. (4 points; stochastic volatility) Consider a sequence of random variables:
= +
for = 1 2 , where are serially independent and identically distributed random variables with a mean 0, variance 1, and kurtosis . In general, can follow a stochastic process and, if so, is independent of . However, for now, assume that future values of are known (deterministic). Define as the de-meaned version of so that ≡ −. a. Find the standardized kurtosis of , i.e., E(4 )(E(
2 ) 2).
b. Suppose you compute the -th sample moment (denoted as ) using the series of realized values of as follows: =
P =1
. What will be the (theoretical)
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 func- tion . Let = 2, show that the probability density function of is
() = 1
2 √ (( √ ) + (−√))
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 001 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 (004 (009)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 (004 (009)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. (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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