FINA 4130 A Empirical Finance

Posted 2020-11-22 blogjava2

Department of Finance
FINA 4130 A Empirical Finance
Assignment 3
Due Date: November 12, 2019
Another assignment where (hopefully) you will appreciate mathematics/statistics in
finance and why finance professors may be smart but not necessarily rich. Answer the
following questions for a total of 300 points (questions are equally weighted unless
stated otherwise) and show all your work carefully. You do not have to use Microsoft
Excel for the assignment since all computations can also be done using programming
environments such as EViews, GAUSS, MATLAB, Octave, R, Python, or SAS. 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 as Microsoft Word
or LATEX while you work, and add any comments and answer all questions in this
document.
1. The precise definition of the volatility of an asset is an annualized measure of dispersion
in the stochastic process that is used to model the log returns (continuously compounded
returns). The most common measure of dispersion about the mean of the distribution
is the standard deviation ?. It is a sufficient risk metric for dispersion when returns are
normally distributed. The annualized standard deviation is called the annual volatility,
or simply the volatility.
a. Assume log returns are generated by an i.i.d. process.
i. The variance of daily returns is 0.001. Assuming 250 risk days per year, what
is the volatility?
ii. The volatility is 36%. Assuming 52 weeks per year, what is the standard
deviation of weekly returns?
iii. The volatilities and correlation between returns on three assets are given as
follows: ?1 = 20%, ?2 = 10%, ?3 = 15%, ?1?2 = 0?8, ?1?3 = 0?5, ?2?3 = 0?3.
As usual, the volatilities are quoted as annualized percentages. Calculate the
annual covariance matrix. Then assuming the returns are multivariate normal
i.i.d. and assuming 250 trading days per year, derive from this the 10-day
covariance matrix, i.e., the matrix of covariance of 10-day returns.
b. Monthly log returns on a hedge fund over the last three years have a standard
deviation of 5%.
i. Assume the log returns are i.i.d. What is your volatility estimate?
ii. Now suppose you discover that the log returns have been smoothed before
reporting them to the investors. In fact, the log returns are autocorrelated
with autocorrelation 0.25, i.e., they have the stationary AR(1) representation, i.e., the
correlation between adjacent returns. What is your volatility estimate now?
2. Assume log returns are normally and independently distributed with mean 0 and variance
FINA 4130作业代写、Empirical Finance作业代做
?2. Then an equally weighted estimate of the variance of the log return at time ?,
based on the ? most recent log returns,
Note that the usual degrees-of-freedom correction does not apply since we have assumed
throughout that returns have zero mean. If the mean return is not assumed to be zero
then we will replace ? by ? − 1.
a. An equally weighted volatility estimate based on a sample of 30 observations is
20%. Find a two-sided 95% confidence interval for this estimate.
b. Show that the standard error of variance estimator is ?2?
p??2 in this case.
c. Show that the standard error of volatility estimator is approximately ??√
2? in
this case.
d. An equally weighted volatility estimate based on a sample of 100 observations is
20%. Estimate the standard error of the estimator and find an interval for the
estimate based on one-standard-error bounds.
3. a. Suppose a stock’s continuously compounded (cc) rate of return has annual mean
and variance of ? and ?2. To estimate these quantities, we divide one year into
? equal periods and record the return for each period. Let ?? and ?2
? be the
mean and variance for the cc rate of return for each period. Specifically. Assume the cc returns are independent random variables with a
normal distribution. Note the remark on degrees-of-freedom correction in previous
exercise.
i. Show that ?(?b) is independent of ?.
ii. Show how ?( b2) depends on ?. Are more data helpful?
b. A record of monthly continuously compounded (cc) return of the stock ? is shown
2
in the following table:
Month Percent rate of return Month Percent rate of return
Assume the returns are independent random variables with a normal distribution.
Note the remark on degrees-of-freedom correction in previous exercise.
i. Estimate the mean rate of return, expressed in percent per year.
ii. Estimate the variance and standard deviation of these returns, expressed in
percent squared and percent per year.
iii. Estimate the accuracy of the estimates found in parts (a) and (b).
iv. How do you think answers to (c) would change if you had 2 years of weekly
data instead of monthly data? (See previous exercise.)
4. Gavin Jones figured out a clever way to get 24 samples of monthly returns in just over
one year instead of only 12 samples; he takes overlapping samples; that is, the first
sample covers Jan. 1 to Feb. 1, and the second sample covers Jan. 15 to Feb. 15, and so
forth. He figures that the error in his estimate of ?, the mean monthly return, will be
reduced by this method. Analyze Gavin’s idea. How does the variance of his estimate
compare with that of the usual method of using 12 non-overlapping monthly returns?
5. a. Create a sample of size ? = 128 from the standard normal distribution and use
QQ plots to assess the normality of the data.
b. Create a sample of size ? = 128 from the exponential distribution with parameter
1, and use QQ plots to assess the normality of the data. Describe and explain the
differences with the results of part (a).
c. Use computers or calculators to generate 36 random numbers from the uniform
distribution U[0? 1], calculate the sample mean, and repeat this procedure 100 times.
So you will have 100 sample means in hand, say, ?1? ?2????? ?100. Define a variable
?? = √36(?? − 0?5)? ? = 1? 2? ???? 100. Now make two frequency tables of ?? with
the length of each interval 0.01 and 0.1 respectively. Plot the two histograms and
comment.
6. Suppose you have programmed a computer to do the following:
i. Draw randomly 100 values from a standard normal distribution.
3
ii. Multiply each of these values by 5 and add 1.
iii. Average the resulting 100 values.
iv. Call the average ?1 and save it.
v. Repeat the procedure above to produce 2000 averages ?1 through ?2000.
vi. Order the 2000 values from the smallest to the largest.
a. What is your best guess of the 1900th ordered value? Explain your logic.
b. How many of these values should be negative? Explain your logic.
7. Program a computer to do the following:
i. Let ??? be a counter and initialize it as zero, i.e., set ??? = 0.
ii. Draw 60 ? values from a standard normal distribution.
iii. Compute 60 ? values as ?? = ??−1 + ?? with ?0 = 0.
iv. Draw 60 ? values from a standard normal distribution.
v. Compute 60 ? values as ?? = ??−1 + ?? with ?0 = 0.
vi. Regress ? on ?, save the slope estimate as ?1 and the standard error of ?1 as ?b?1 .
vii. Compute |?| = |?1| ??b?1 and save it.
viii. Add one to ??? if |?| is greater than 2.
ix. Repeat from (ii) to obtain 1000 |?| values.
x. Divide ??? by 1000.
a. What is this Monte Carlo study designed to investigate.
b. What number should ??? be close to? Explain your logic.
c. Does the ??? you find confirm your expectation? Why or why not?
8. Download (and compute) the monthly returns on Vanguard’s Long-Term Bond Index
Fund (VBLTX), Emerging Markets Stock Index Fund (VEIEX), and Small-Cap Index
Fund Investor Shares (NAESX) from July 2014 to June 2019 through CRSP in WRDS.
Consider the constant expected return (CER) model
where ?e??? denotes the return on asset ?, ? = VBLTX, VEIEX, and NAESX.
a. Estimate the parameters ??, ?2
? , ??, ???? and ???? using sample descriptive statistics.
Arrange these estimates nicely in a table. Briefly comment.
b. For each estimate of the above parameters (except ???? ) compute the estimated
standard error. That is, compute ??c (?b?), ??c (?b2
?), ??c (?b?) and ??c (b???? ). Show the
estimates with the corresponding SE values underneath. Briefly comment on the
precision of the estimates. Hint: The formulas for these standard errors were given
in class and are given in the lecture notes on the CER model.
4
c. For each parameter ??, ?2
? , ?? and ???? compute 95% and 99% confidence intervals.
Briefly comment on the width of these intervals.
d. Using the estimates values of ?? and ?2
? for each mutual fund, compute the 1% and
5% monthly value-at-risk (VaR) based on an initial $100,000 investment. Which
fund has the lowest VaR?
e. Using the technique of Monte Carlo simulation, create a simulated data set from
the CER model for three assets using the CER model estimates as the parameters
(true values). Use seed = 123 to initialize the random number generator.
i. Plot the simulated data (line plot), and create a pairs plot showing all pair-wise
scatterplots. Does the simulated data look the actual return data for the three
assets?
ii. Compute estimates of the pair-wise covariances and correlations. Also compute
estimated standard errors for the correlations. Are these correlation estimates
close to the true values?
iii. Create 1000 simulated data sets and compare your results to the above.
9. Download (and compute) the monthly returns on Vanguard’s Short-Term Bond Index
Fund Investor Shares (VBISX), Extended Market Index Fund Investor Shares (VEXMX),
and 500 Index Fund Investor Shares (VFINX) from July 2014 to June 2019 through
CRSP in WRDS. Consider the constant expected return (CER) model
where ?e??? denotes the return on asset ?, ? = VBISX, VEXMX and VFINX.
a. Estimate the parameters ??, ?2
? , ??, ???? and ???? using sample descriptive statistics.
Arrange these estimates nicely in a table. Briefly comment.
b. For each estimate of the above parameters (except ???? ) compute the estimated
standard error. That is, compute ??c (?b?), ??c (?b2
?), ??c (?b?) and ??c (b???? ). Briefly
comment on the precision of the estimates. Hint: The formulas for these standard
errors were given in class and are given in the lecture notes on the CER model.
c. For each estimate of the above parameters (except ???? ) compute the estimated
standard error using the bootstrap with 1000 bootstrap replications. That is compute,
??c ????(?b?), ??c ????(?b2
?), ??c ????(?b?) and ??c ????(b???? ). Compare the bootstrap
standard errors to the analytic standard errors. Hint: If you insist on doing bootstrapping
in Excel, you can visit the following sites for more information:
• http://www.anthony-vba.kefra.com/vba/vba10.htm (for a sample VBA code
on bootstrap)
• http://people.revoledu.com/kardi/tutorial/Bootstrap/examples.htm
• http://www.stat.auckland.ac.nz/~iase/publications/13/Carr-Salzman.pdf
d. For each estimate of the above parameters (except ???? ), plot the histogram and QQ
plot of the bootstrap distribution. Do the bootstrap distributions look normal?
e. For each asset, compute estimates of the 5% value-at-risk. Use the bootstrap to
compute the ??c (? ?? [0?05) values as well as the 95% confidence intervals. Briefly
comment on the accuracy of the 5% VaR estimates.
5
10. Assuming perfect capital markets, you will estimate expected returns, variances and
covariances to be used as inputs to the Markowitz algorithm, then compute efficient
portfolios allowing for short-sales and plot the frontier. Download the monthly returns
on the “5 Industry Portfolios” from July 2014 to June 2019 through Kenneth French’s
web site at Dartmouth. Note the returns are in percent.
a. Estimate the parameters ??, ?2
? , ??, ???? and ???? of the constant expected return
(CER) model:????
where ?e??? denotes the return on asset ? at time ?. Arrange these estimates nicely
in a table. Briefly comment. Give time plots of the data as well as a pairs plot.
Comment on any relationships you see in the data.
b. Compute the global minimum variance portfolio allowing short-sales. The minimization
problem is
where w??? is the vector of portfolio weights and Σ is the covariance matrix. Are
there any negative weights in this portfolio? If so, interpret them. Compute the
expected return, variance and standard deviation of this portfolio.
c. Determine the asset with the highest average historical return. Use this average
return as the target return for the computation of an efficient portfolio allowing
for short-sales. That is, find the minimum variance portfolio that has an expected
return equal to this target return. The minimization problem is
min
where w? is the vector of portfolio weights, μ is the vector of expected returns and
?? is the target expected return. Are there any negative weights in this portfolio?
Compute the expected return, variance and standard deviation of this portfolio.
Finally, compute the covariance between the global minimum variance portfolio
and the above efficient portfolio using the formula
Cov(?e?? ?e?) = w0
?Σw?
d. Using the fact that all efficient portfolios can be written as a convex combination
of two efficient portfolios, compute efficient portfolios as convex combinations of
the global minimum variance portfolio and the efficient portfolio computed in part
c. That is, compute
w? = ? · w? + (1 − ?) · w???
for values of ? between 0 and 1 (make a grid for ? = 0? 0?1????? 0?9? 1). Compute
the expected return, variance and standard deviation of these portfolios.
e. Plot the Markowitz bullet based on the efficient portfolios you computed in part
(d). On the plot, indicate the location of the minimum variance portfolio and the
location of the efficient portfolio found in part (c).
6
f. Compute the tangency portfolio assuming the risk-free rate is 0.0003 (?? = 0?03%)
per month. That is,
where wtan denotes the portfolio weights in the tangency portfolio. Are there any
negative weights in the tangency portfolio? If so, interpret them.
g. On the graph with the Markowitz bullet, plot the efficient portfolios that are combinations
of T-bills and the tangency portfolio. Indicate the location of the tangency
portfolio on the graph.
h. Suppose you have $100,000 to invest over one month. Compare 5% value-at-risk
(VaR) for the stock with the largest average historical return and the efficient
portfolio you got from part (c).

FINA 4130作业代写、Empirical Finance作业代做、代做R, Python程序语言作业、R, Python作业代写
Department of Finance
FINA 4130 A Empirical Finance
Assignment 3
Due Date: November 12, 2019
Another assignment where (hopefully) you will appreciate mathematics/statistics in
finance and why finance professors may be smart but not necessarily rich. Answer the
following questions for a total of 300 points (questions are equally weighted unless
stated otherwise) and show all your work carefully. You do not have to use Microsoft
Excel for the assignment since all computations can also be done using programming
environments such as EViews, GAUSS, MATLAB, Octave, R, Python, or SAS. 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 as Microsoft Word
or LATEX while you work, and add any comments and answer all questions in this
document.
1. The precise definition of the volatility of an asset is an annualized measure of dispersion
in the stochastic process that is used to model the log returns (continuously compounded
returns). The most common measure of dispersion about the mean of the distribution
is the standard deviation ?. It is a sufficient risk metric for dispersion when returns are
normally distributed. The annualized standard deviation is called the annual volatility,
or simply the volatility.
a. Assume log returns are generated by an i.i.d. process.
i. The variance of daily returns is 0.001. Assuming 250 risk days per year, what
is the volatility?
ii. The volatility is 36%. Assuming 52 weeks per year, what is the standard
deviation of weekly returns?
iii. The volatilities and correlation between returns on three assets are given as
follows: ?1 = 20%, ?2 = 10%, ?3 = 15%, ?1?2 = 0?8, ?1?3 = 0?5, ?2?3 = 0?3.
As usual, the volatilities are quoted as annualized percentages. Calculate the
annual covariance matrix. Then assuming the returns are multivariate normal
i.i.d. and assuming 250 trading days per year, derive from this the 10-day
covariance matrix, i.e., the matrix of covariance of 10-day returns.
b. Monthly log returns on a hedge fund over the last three years have a standard
deviation of 5%.
i. Assume the log returns are i.i.d. What is your volatility estimate?
ii. Now suppose you discover that the log returns have been smoothed before
reporting them to the investors. In fact, the log returns are autocorrelated
with autocorrelation 0.25, i.e., they have the stationary AR(1) representation, i.e., the
correlation between adjacent returns. What is your volatility estimate now?
2. Assume log returns are normally and independently distributed with mean 0 and variance
?2. Then an equally weighted estimate of the variance of the log return at time ?,
based on the ? most recent log returns,
Note that the usual degrees-of-freedom correction does not apply since we have assumed
throughout that returns have zero mean. If the mean return is not assumed to be zero
then we will replace ? by ? − 1.
a. An equally weighted volatility estimate based on a sample of 30 observations is
20%. Find a two-sided 95% confidence interval for this estimate.
b. Show that the standard error of variance estimator is ?2?
p??2 in this case.
c. Show that the standard error of volatility estimator is approximately ??√
2? in
this case.
d. An equally weighted volatility estimate based on a sample of 100 observations is
20%. Estimate the standard error of the estimator and find an interval for the
estimate based on one-standard-error bounds.
3. a. Suppose a stock’s continuously compounded (cc) rate of return has annual mean
and variance of ? and ?2. To estimate these quantities, we divide one year into
? equal periods and record the return for each period. Let ?? and ?2
? be the
mean and variance for the cc rate of return for each period. Specifically. Assume the cc returns are independent random variables with a
normal distribution. Note the remark on degrees-of-freedom correction in previous
exercise.
i. Show that ?(?b) is independent of ?.
ii. Show how ?( b2) depends on ?. Are more data helpful?
b. A record of monthly continuously compounded (cc) return of the stock ? is shown
2
in the following table:
Month Percent rate of return Month Percent rate of return
Assume the returns are independent random variables with a normal distribution.
Note the remark on degrees-of-freedom correction in previous exercise.
i. Estimate the mean rate of return, expressed in percent per year.
ii. Estimate the variance and standard deviation of these returns, expressed in
percent squared and percent per year.
iii. Estimate the accuracy of the estimates found in parts (a) and (b).
iv. How do you think answers to (c) would change if you had 2 years of weekly
data instead of monthly data? (See previous exercise.)
4. Gavin Jones figured out a clever way to get 24 samples of monthly returns in just over
one year instead of only 12 samples; he takes overlapping samples; that is, the first
sample covers Jan. 1 to Feb. 1, and the second sample covers Jan. 15 to Feb. 15, and so
forth. He figures that the error in his estimate of ?, the mean monthly return, will be
reduced by this method. Analyze Gavin’s idea. How does the variance of his estimate
compare with that of the usual method of using 12 non-overlapping monthly returns?
5. a. Create a sample of size ? = 128 from the standard normal distribution and use
QQ plots to assess the normality of the data.
b. Create a sample of size ? = 128 from the exponential distribution with parameter
1, and use QQ plots to assess the normality of the data. Describe and explain the
differences with the results of part (a).
c. Use computers or calculators to generate 36 random numbers from the uniform
distribution U[0? 1], calculate the sample mean, and repeat this procedure 100 times.
So you will have 100 sample means in hand, say, ?1? ?2????? ?100. Define a variable
?? = √36(?? − 0?5)? ? = 1? 2? ???? 100. Now make two frequency tables of ?? with
the length of each interval 0.01 and 0.1 respectively. Plot the two histograms and
comment.
6. Suppose you have programmed a computer to do the following:
i. Draw randomly 100 values from a standard normal distribution.
3
ii. Multiply each of these values by 5 and add 1.
iii. Average the resulting 100 values.
iv. Call the average ?1 and save it.
v. Repeat the procedure above to produce 2000 averages ?1 through ?2000.
vi. Order the 2000 values from the smallest to the largest.
a. What is your best guess of the 1900th ordered value? Explain your logic.
b. How many of these values should be negative? Explain your logic.
7. Program a computer to do the following:
i. Let ??? be a counter and initialize it as zero, i.e., set ??? = 0.
ii. Draw 60 ? values from a standard normal distribution.
iii. Compute 60 ? values as ?? = ??−1 + ?? with ?0 = 0.
iv. Draw 60 ? values from a standard normal distribution.
v. Compute 60 ? values as ?? = ??−1 + ?? with ?0 = 0.
vi. Regress ? on ?, save the slope estimate as ?1 and the standard error of ?1 as ?b?1 .
vii. Compute |?| = |?1| ??b?1 and save it.
viii. Add one to ??? if |?| is greater than 2.
ix. Repeat from (ii) to obtain 1000 |?| values.
x. Divide ??? by 1000.
a. What is this Monte Carlo study designed to investigate.
b. What number should ??? be close to? Explain your logic.
c. Does the ??? you find confirm your expectation? Why or why not?
8. Download (and compute) the monthly returns on Vanguard’s Long-Term Bond Index
Fund (VBLTX), Emerging Markets Stock Index Fund (VEIEX), and Small-Cap Index
Fund Investor Shares (NAESX) from July 2014 to June 2019 through CRSP in WRDS.
Consider the constant expected return (CER) model
where ?e??? denotes the return on asset ?, ? = VBLTX, VEIEX, and NAESX.
a. Estimate the parameters ??, ?2
? , ??, ???? and ???? using sample descriptive statistics.
Arrange these estimates nicely in a table. Briefly comment.
b. For each estimate of the above parameters (except ???? ) compute the estimated
standard error. That is, compute ??c (?b?), ??c (?b2
?), ??c (?b?) and ??c (b???? ). Show the
estimates with the corresponding SE values underneath. Briefly comment on the
precision of the estimates. Hint: The formulas for these standard errors were given
in class and are given in the lecture notes on the CER model.
4
c. For each parameter ??, ?2
? , ?? and ???? compute 95% and 99% confidence intervals.
Briefly comment on the width of these intervals.
d. Using the estimates values of ?? and ?2
? for each mutual fund, compute the 1% and
5% monthly value-at-risk (VaR) based on an initial $100,000 investment. Which
fund has the lowest VaR?
e. Using the technique of Monte Carlo simulation, create a simulated data set from
the CER model for three assets using the CER model estimates as the parameters
(true values). Use seed = 123 to initialize the random number generator.
i. Plot the simulated data (line plot), and create a pairs plot showing all pair-wise
scatterplots. Does the simulated data look the actual return data for the three
assets?
ii. Compute estimates of the pair-wise covariances and correlations. Also compute
estimated standard errors for the correlations. Are these correlation estimates
close to the true values?
iii. Create 1000 simulated data sets and compare your results to the above.
9. Download (and compute) the monthly returns on Vanguard’s Short-Term Bond Index
Fund Investor Shares (VBISX), Extended Market Index Fund Investor Shares (VEXMX),
and 500 Index Fund Investor Shares (VFINX) from July 2014 to June 2019 through
CRSP in WRDS. Consider the constant expected return (CER) model
where ?e??? denotes the return on asset ?, ? = VBISX, VEXMX and VFINX.
a. Estimate the parameters ??, ?2
? , ??, ???? and ???? using sample descriptive statistics.
Arrange these estimates nicely in a table. Briefly comment.
b. For each estimate of the above parameters (except ???? ) compute the estimated
standard error. That is, compute ??c (?b?), ??c (?b2
?), ??c (?b?) and ??c (b???? ). Briefly
comment on the precision of the estimates. Hint: The formulas for these standard
errors were given in class and are given in the lecture notes on the CER model.
c. For each estimate of the above parameters (except ???? ) compute the estimated
standard error using the bootstrap with 1000 bootstrap replications. That is compute,
??c ????(?b?), ??c ????(?b2
?), ??c ????(?b?) and ??c ????(b???? ). Compare the bootstrap
standard errors to the analytic standard errors. Hint: If you insist on doing bootstrapping
in Excel, you can visit the following sites for more information:
• http://www.anthony-vba.kefra.com/vba/vba10.htm (for a sample VBA code
on bootstrap)
• http://people.revoledu.com/kardi/tutorial/Bootstrap/examples.htm
• http://www.stat.auckland.ac.nz/~iase/publications/13/Carr-Salzman.pdf
d. For each estimate of the above parameters (except ???? ), plot the histogram and QQ
plot of the bootstrap distribution. Do the bootstrap distributions look normal?
e. For each asset, compute estimates of the 5% value-at-risk. Use the bootstrap to
compute the ??c (? ?? [0?05) values as well as the 95% confidence intervals. Briefly
comment on the accuracy of the 5% VaR estimates.
5
10. Assuming perfect capital markets, you will estimate expected returns, variances and
covariances to be used as inputs to the Markowitz algorithm, then compute efficient
portfolios allowing for short-sales and plot the frontier. Download the monthly returns
on the “5 Industry Portfolios” from July 2014 to June 2019 through Kenneth French’s
web site at Dartmouth. Note the returns are in percent.
a. Estimate the parameters ??, ?2
? , ??, ???? and ???? of the constant expected return
(CER) model:????
where ?e??? denotes the return on asset ? at time ?. Arrange these estimates nicely
in a table. Briefly comment. Give time plots of the data as well as a pairs plot.
Comment on any relationships you see in the data.
b. Compute the global minimum variance portfolio allowing short-sales. The minimization
problem is
where w??? is the vector of portfolio weights and Σ is the covariance matrix. Are
there any negative weights in this portfolio? If so, interpret them. Compute the
expected return, variance and standard deviation of this portfolio.
c. Determine the asset with the highest average historical return. Use this average
return as the target return for the computation of an efficient portfolio allowing
for short-sales. That is, find the minimum variance portfolio that has an expected
return equal to this target return. The minimization problem is
min
where w? is the vector of portfolio weights, μ is the vector of expected returns and
?? is the target expected return. Are there any negative weights in this portfolio?
Compute the expected return, variance and standard deviation of this portfolio.
Finally, compute the covariance between the global minimum variance portfolio
and the above efficient portfolio using the formula
Cov(?e?? ?e?) = w0
?Σw?
d. Using the fact that all efficient portfolios can be written as a convex combination
of two efficient portfolios, compute efficient portfolios as convex combinations of
the global minimum variance portfolio and the efficient portfolio computed in part
c. That is, compute
w? = ? · w? + (1 − ?) · w???
for values of ? between 0 and 1 (make a grid for ? = 0? 0?1????? 0?9? 1). Compute
the expected return, variance and standard deviation of these portfolios.
e. Plot the Markowitz bullet based on the efficient portfolios you computed in part
(d). On the plot, indicate the location of the minimum variance portfolio and the
location of the efficient portfolio found in part (c).
6
f. Compute the tangency portfolio assuming the risk-free rate is 0.0003 (?? = 0?03%)
per month. That is,
where wtan denotes the portfolio weights in the tangency portfolio. Are there any
negative weights in the tangency portfolio? If so, interpret them.
g. On the graph with the Markowitz bullet, plot the efficient portfolios that are combinations
of T-bills and the tangency portfolio. Indicate the location of the tangency
portfolio on the graph.
h. Suppose you have $100,000 to invest over one month. Compare 5% value-at-risk
(VaR) for the stock with the largest average historical return and the efficient
portfolio you got from part (c).
7

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