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 = (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
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 −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 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:
(; ) = 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 questions.
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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 :
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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) = 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 (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. 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 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-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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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?
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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 −01 ≤ ≤ 05 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 −01 ≤ ≤ 05, 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
22
, with a degree of risk
aversion of 2.0 (i.e., = 20).
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?
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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
22
, with a degree of risk
aversion of 4.0 (i.e., = 40).
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
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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.
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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( ) − 00052
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 0022. 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 u