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日期:2020-10-14 11:14

FINA 6592 FA Financial Econometrics

Assignment 1

Due Date: October 8, 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 approximations

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?

1

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.P, STDEV.P, SKEW.P 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:

?(?; ?) = ? ??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:

?(?; ?) = ? ?? + 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 NORM.DIST and NORM.INV to answer these questions.

2

8. (2 points) Suppose that ??(?)=1?4 if |?| ? 1 and ??(?)=1?(4?2) 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 ? = 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.

3

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 ? → ∞.

4

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

? =

? |?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?

5

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. Hint: ? has a beta distribution.

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

?(?) = 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 ? ??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.)

7

27. a. (10 points) The daily value-at-risk (VaR) under normal distribution for a bank is

$8,500 (also called daily earnings at risk, DEAR).

i. What is the VaR for a 10-day period?

ii. What is the VaR for a 20-day period?

iii. Why is the VaR for a 20-day period not twice as much as that for a 10-day

period? Explain.

b. (10 points) Assume that the return density has a polynomial left tail, or equivalently

that the loss density has a polynomial right tail. That is, the return density ?

satisfies

?(?) ~ ???(?+1) as ? → ?∞

where ? ? 0 is a constant, ? ? 0 is the tail index, and “~” means that the ratio

of the left-hand to right-hand sides converges to 1.

i. What is the distribution function? That is, find ?(?) = Pr(? ≤ ?).

ii. Suppose an estimate of ? is 3.1. If VaR(0.05) = $252, what is VaR(0.005)?

28. a. (10 points) Let ?1 and ?2 be two portfolios whose returns have a joint normal

distribution with means ?1 and ?2, standard deviations ?1 and ?2, and correlation

?. Suppose the initial investments are ?1 and ?2. Show that ? ??(?1 + ?2) ≤

? ??(?1) + ? ??(?2) under joint normality of the returns.

b. i. (2 points) Suppose that stock ? sells at $85 per share and stock ? at $35 per

share. A portfolio has 300 shares of stock ? and 100 of stock ?. What are the

weight ? and 1 ? ? of stocks ? and ? in this portfolio?

ii. (3 points) More generally, if a portfolio has ? stocks, if the price per share of

the ?th stock is ?? , and if the portfolio has ?? shares of stock ?, then find a

formula for ?? as a function of ?1??????? and ?1??????? .

iii. (5 points) Let R? be a return on some type on a portfolio and let R1????? R?

be the same type of returns on the assets in this portfolio. Is R? = ?1R1 +

··· + ??R? true if R? is a net return? Is this equation true if R? is a gross

return? Is it true if R? is a log return? Justify your answers.

29. a. (5 points) Stocks 1 and 2 are selling for $100 and $125, respectively. You own 200

shares of stock 1 and 100 shares of stock 2. The weekly returns on these stocks have

means of 0.001 and 0.0015, respectively, and standard deviations of 0.03 and 0.04,

respectively. Their weekly returns have a correlation of 0.35. Find the covariance

matrix of the weekly returns on the two stocks, the mean and standard deviation of

the weekly returns on the portfolio, and the one-week VaR(0.05) for your portfolio.

b. (15 points) Obtain the data set Stock_Bond.csv from the website

https://people.orie.cornell.edu/davidr/SDAFE2/index.html in which any

variable whose name ends with “AC” is an adjusted closing price. As the name

suggests, these prices have been adjusted for dividends and stock splits, so that

returns can be calculated without further adjustments.

i. Compute the returns for six stocks, {GM, F, UTX, CAT, MRK, IBM}, create

a scatterplot matrix of these returns, and compute the mean vector, covariance

matrix, and vector of standard deviations of the returns. Does this data set

include Black Monday?

8

ii. Find the efficient frontier, the tangency portfolio, and the minimum variance

portfolio, and plot on “reward-risk space” the location of each of the six stocks,

the efficient frontier, the tangency portfolio, and the line of efficient portfolios.

Use the constraints that ?0?1 ≤ ?? ≤ 0?5 for each stock. The first constraint

limits short sales but does not rule them out completely. The second constraint

prohibits more than 50% of the investment in any single stock. Assume that

the annual risk-free rate is 3% and convert this to a daily rate by dividing by

365, since interest is earned on trading as well as nontrading days.

iii. If an investor wants an efficient portfolio with an expected daily return of

0.07%, how should the investor allocate his or her capital to the six stocks and

to the risk-free asset? Assume that the investor wishes to use the tangency

portfolio computed with the constraints ?0?1 ≤ ?? ≤ 0?5, not the unconstrained

tangency portfolio.

30. (20 points) A U.S. portfolio manager is trying to allocate a client’s portfolio among various

asset classes. There are six potential asset classes: stocks of U.S. large capitalization

companies, stocks of U.S. small capitalization companies, U.S. corporate bonds, foreign

stocks, foreign bonds, and cash. Expected returns, standard deviations of return, and

correlations between the returns of different pairs of asset classes are as follows:

Foreign Foreign

US large US small US bonds stocks bonds Cash

Expected return 0.12 0.13 0.08 0.12 0.08 0.06

Standard deviation 0.17 0.21 0.09 0.20 0.07 0.005

CORRELATIONS

US large 1.0

US small 0.9 1.0

US bonds 0.3 0.2 1.0

Foreign stocks 0.5 0.4 0.2 1.0

Foreign bonds 0.4 0.4 0.7 0.55 1.0

Cash 0.0 0.0 0.2 0.0 0.0 1.0

The portfolio manager has been informed that the client can be assumed to have a

mean-variance utility function of the form ?(??) = ?? ? 1

2??2

?, with a degree of risk

aversion of 2.0 (i.e., ? = 2?0).

a. If the portfolio manager faces no constraints other than that the portfolio weights

must add up to one, what weights should he choose for the six asset classes? (Hint:

The portfolio manager is assumed to maximize the client’s utility function subject

to the restriction on the portfolio weights.)

b. Suppose now that the client stipulates there should be no short positions in the

portfolio. What portfolio weights should the portfolio manager choose in this case?

How much is this constraint (i.e., no short positions) costing the client in terms of

the objective function?

c. Suppose, instead, that the client now allows short positions but refuses to own any

foreign securities. What portfolio weights should the portfolio manager choose in

this case? How much is this constraint (i.e., no foreign securities) costing the client

in terms of the objective function?

9

31. (20 points) A U.S. portfolio manager is trying to allocate a client’s portfolio among

various asset classes. There are seven potential asset classes: convertible bonds, stocks

of large capitalization companies, stocks of small capitalization companies, long-term

Treasury bonds, Treasury bills, mortgage-backed securities, and real estate. Expected

returns, standard deviations of return, and correlations between the returns of different

pairs of asset classes are as follows:

Large Small Long-term Mortgage- Real

Converts stocks stocks Treasury T-bills backed estate

Expected return 0.102 0.147 0.198 0.073 0.061 0.081 0.106

Standard deviation 0.131 0.208 0.353 0.104 0.031 0.112 0.154

CORRELATIONS

Converts 1.0

Large stocks 0.9 1.0

Small stocks 0.86 0.85 1.0

Long-term Treasury 0.44 0.26 0.16 1.0

T-bills —0.07 —0.08 0.1 0.13 1.0

Mortgage-backed 0.4 0.31 0.19 0.9 0.08 1.0

Real estate 0.14 0.03 0.23 —0.08 0.19 —0.03 1.0

The portfolio manager has been informed that the client can be assumed to have a

mean-variance utility function of the form ?(??) = ?? ? 1

2??2

?, with a degree of risk

aversion of 4.0 (i.e., ? = 4?0).

a. If the portfolio manager faces no constraints other than that the portfolio weights

must add up to one, what weights should he choose for the seven asset classes?

(Hint: The portfolio manager is assumed to maximize the client’s utility function

subject to the restriction on the portfolio weights.)

b. What are the weights for the seven asset classes in the minimum-variance portfolio?

Briefly explain the major differences between the minimum-variance portfolio and

the optimal portfolio you found in (a).

c. Suppose that, in addition to constraining the portfolio weights to sum to one, the

client stipulates that no more than 40% of the portfolio should be allocated to large

capitalization stocks, no more than 40% to small capitalization stocks, and no more

than 10% to real estate. What is the optimal portfolio for the client in the face

of these constraints? Do you think it is a good idea to impose constraints on the

portfolio weights? Briefly explain. What criteria might be used in choosing these

constraints?

32. (20 points) Go to any publicly available source such as Yahoo!Finance and download

monthly data on Amazon.com, Inc. (ticker symbol AMZN) over the period June 1997 to

June 2020. Do your analysis on the monthly adjusted close price data (which should be

adjusted for dividends and stock splits). If you are using Excel, name the spreadsheet

tab with the data “AMZN-Data”.

a. Make a time plot of the price data over the period June 1997 to June 2020. Please

put informative titles and labels on the graph. If you are using Excel, place this

10

graph (line plot) in a separate tab (chart) from the data and name this tab “AMZNChart1”.

Comment on what you see (e.g., price trends, etc.). If you invested

US$1,000 at the end of June 1997, what would your investment be worth at the

end of June 2020? What is the annual rate of return over this 23-year period

assuming annual compounding?

b. Make a time plot of the natural logarithm of price data over the period June 1997

through June 2020. If you are using Excel, place it in a new “AMZN-Chart2” tab.

Comment on what you see and compare with the plot of the raw price data. Why

is a plot of the log of prices informative?

c. Using the price data over the period June 1997 through June 2020, compute simple

monthly returns, make a time plot of the returns and comment. If you are using

Excel, place the returns in the “AMZN-Data” tab and the graph in a new “AMZNChart3”

tab. Note: When computing returns, use the convention that ?? is the end

of month (adjusted) closing price. Keep in mind that the returns are percent per

month, so how would you compare the returns to a risk-free asset like U.S. T-bill?

d. Using the simple monthly returns computed above, compute simple annual returns

for the years 1997 through 2020 (from June to June), make a time plot of the returns

and comment. If you are using Excel, put the graph in a new “AMZN-Chart4” tab.

Note: You may compute annual returns using overlapping data or non-overlapping

data. With overlapping data you get a series of annual returns for every month

(sounds weird, I know). That is, the first month’s annual return is from the end of

June 1997 to the end of June 1998. Then second month’s annual return is from the

end of July 1997 to the end of July 1998, etc. With non-overlapping data you get

a series of 23 annual returns for the 23-year period 1997—2020. That is, the annual

return for 1997-98 is computed from the end of June 1997 through the end of June

1998. The second annual return is computed from the end of June 1998 through

the end of June 1999, etc.

e. Using the price data over the period June 1997 through June 2020, compute continuously

compounded monthly returns, make a time plot of the returns and comment.

If you are using Excel, place the returns in the “AMZN-Data” tab and the graph in

a new “AMZN-Chart5” tab. Briefly compare the continuously compounded returns

to the simple returns.

f. Using the continuously compounded monthly returns, compute continuously compounded

annual returns for the years 1997 through 2020 (from June to June), make

a time plot of the returns and comment. If you are using Excel, put the graph in a

new “AMZN-Chart6” tab. Briefly compare the continuously compounded returns

to the simple returns.

33. (20 points) Download monthly data on the prices of five financial sector stocks in Hang

Seng Index for the period June 2010 through June 2020.

a. Graph the log-prices and comment on any notable fluctuations in the graphs obtained.

b. Compute the sample means and sample covariance matrix for the log-returns and

simple returns, then briefly discuss your results.

11

c. For each return series compute the sample skewness, sample kurtosis and the

Jarque—Bera test and briefly discuss your results.

d. Using an annual risk-free rate of 0.6%, compute the optimal risky portfolio (ORP)

and the optimal final portfolio (OFP) assuming a one-year investment horizon, a

quadratic utility function of the form

? = E(?? ) ? 0?005??2

?

with ? set to 2, 3, or 4, no borrowing and no short sales. Graph your results.

e. Investigate the impact on the impact of ORP and OFP of the following:

i. Reducing the sample size used to compute the inputs.

ii. Allowing for borrowing.

iii. Allowing for borrowing and short sales.

34. (20 points) Do either (a) or (b).

a. This question illustrates the effects of diversification. In order to proceed, you need

the Microsoft Excel file 6592_A1_data.xlsx.

i. Using data in the spreadsheet, calculate the standard deviation of returns for

each of the seven equally-weighted portfolios. Plot estimated standard deviations

as a function of the number of stocks in the equally-weighted portfolio.

“Eyeballing” the chart, does it look like adding more and more stocks will

diversify away all the standard deviation?

ii. Now calculate the standard deviations of returns for each of the value-weighted

portfolios. Plot the estimated standard deviations as a function of the number

of stocks in the value-weight portfolio, and compare this to the graph you

created in part (i). Why are they different?

iii. Assume that the average variance of individual stocks’ daily return is 0?022. For

each equally-weighted portfolio, decompose the estimated portfolio variance

into its two components, the contributions of security return variances and

covariances. If we keep adding more securities to the portfolios, what happens

to the contribution of the variances of individual security returns to the variance

of portfolio returns?

b. The objective of this question is to examine how the variances of portfolios may

be reduced as a result of diversification. Download through Bloomberg or Yahoo!Finance

monthly data of constituent stocks of Hang Seng Index covering the

period June 2015 to June 2020.

i. Form equal-weight and value-weight portfolios using 5, 10, 25, and all 50 stocks.

Calculate the sample mean and standard deviation of the returns for each of

the eight portfolios. Plot estimated standard deviations as a function of the

number of stocks in the equal-weight portfolio. Comment on the shape of the

function. Are the results consistent with what you would expect theoretically?

“Eyeballing” the graph, does it look like adding more and more stocks will

diversify away all the standard deviation? Why?

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ii. For all four equal-weight portfolios, decompose the estimated portfolio variance

into its two components (the contributions of variances and covariances). Plot

the percentage of the portfolio’s variance due to the variances of individual

security returns as a function of the number of stocks in the portfolio. Comment

on the shape of the function. Are the results consistent with what you would

expect theoretically? Use the relevant equations in your explanation. Hint:

you do not have to estimate the pair-wise covariances in order to compute the

decomposition.

iii. Would you expect a 5-stock value-weight portfolio to exhibit more, less, or

about the same variance as an equal-weight portfolio consisting of the same 5

stocks? Describe the factors that influence your decision. What if there were

1000 stocks? (Hint: Are large stocks typically more or less volatile than small

stocks?)

35. (20 points) The objective of this exercise is to see what effect, if any, international

diversification has on portfolio risk. Assume capital markets are perfect so that you can

trade global stock market indexes as portfolios denominated in USD. Download through

Bloomberg or Yahoo!Finance daily data covering the period June 29, 2018, to June 30,

2020, on the values of following stock market indexes:

Canada: S&P TSX Composite

France: CAC 40

Germany: DAX

Japan: Nikkei 225

UK: FTSE 100

US: S&P 500

Australia: All Ordinaries

Hong Kong: Hang Seng

To convert into USD denominated returns, assume you can trade at the exchange rates

available at PACIFIC Exchange Rate Service (http://fx.sauder.ubc.ca/). Alternatively,

you can download the exchange rates through the US Federal Reserve System

(http://www.federalreserve.gov/releases/h10/current/).

a. Using full sample to construct the following measures:

i. Mean, standard deviation, and variance for the daily return on each index.

ii. The variance-covariance matrix of the daily returns across all indexes.

iii. Rank the indexes by their coefficient of variation (CV) for their daily return.

Recall that CV is defined as the ratio of the standard deviation to the absolute

value of mean.

Assume you have $10 million to invest equally across three of the indexes. Which

three would you choose based on the CV measure? Why? What other considerations

are important to determining your choice of three indexes to use in your

portfolio? Why?

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b. Suppose you decide to invest $10 million equally amongst those three indexes with

lowest CV measure.

i. Using observations from first twenty one months on these indexes to construct

the following measures:

A. A covariance-based VaR measure at the 99% confidence level. Compare

this to a covariance-based VaR measure for $10 million invested in the

S&P 500 index exclusively.

B. A historical simulation-based VaR measure at the 99% confidence level.

Compare this to a historical simulation-based VaR measure for $10 million

invested in the S&P 500 index exclusively.

C. Compare your VaR estimates for the equal-weight portfolio from above.

What do these two measures lead you to conclude about the distribution

of returns on these indexes?

Please make sure that your answer demonstrates all the formulas, summary

measures, and explanations necessary for an informed reader to understand

what you are doing.

ii. Using observations from last three months on these indexes to construct the

mean daily return for each index and the variance-covariance matrix for the

daily returns across these indexes. Compare these to the results for the earlier

period. Is there any difference? If so, does this matter for the accuracy of your

VaR estimates above? Did your VaR calculations based on the earlier sample

period provide a good guide to the actual risk your portfolio faced over this

period? Why?

c. The efficient frontier discussed in the mean-variance portfolio theory is generated as

the outer envelope of all “bullets” that would be generated by varying all investment

weights in all possible ways. It represents the best trade-off between risk and return

for a portfolio consisting of all assets.

i. Using results from part (a), graph the relationship between portfolio return

(? -axis) and standard deviation (?-axis). This is known as the “bullet” relationship

between risk and return on a portfolio.

ii. What seems to be the “best” portfolio allocation (roughly) between risk and

return. Why?

iii. What percent of your assets would be invested in each index? Would you

consider this portfolio to be “internationally-oriented”?

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