Department of FinanceFINA 4130 A Empirical FinanceAssignment 3Due Date: November 12, 2019Another assignment where (hopefully) you will appreciate mathematics/statistics infinance and why finance professors may be smart but not necessarily rich. Answer thefollowing questions for a total of 300 points (questions are equally weighted unlessstated otherwise) and show all your work carefully. You do not have to use MicrosoftExcel for the assignment since all computations can also be done using programmingenvironments such as EViews, GAUSS, MATLAB, Octave, R, Python, or SAS. Pleasedo not turn in the assignment in reams of unformatted computer output and withoutcomments! Make little tables of the numbers that matter, copy and paste all resultsand graphs into a document prepared by typesetting system such as Microsoft Wordor LATEX while you work, and add any comments and answer all questions in thisdocument.1. The precise definition of the volatility of an asset is an annualized measure of dispersionin the stochastic process that is used to model the log returns (continuously compoundedreturns). The most common measure of dispersion about the mean of the distributionis the standard deviation . It is a sufficient risk metric for dispersion when returns arenormally 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, whatis the volatility?ii. The volatility is 36%. Assuming 52 weeks per year, what is the standarddeviation of weekly returns?iii. The volatilities and correlation between returns on three assets are given asfollows: 1 = 20%, 2 = 10%, 3 = 15%, 12 = 08, 13 = 05, 23 = 03.As usual, the volatilities are quoted as annualized percentages. Calculate theannual covariance matrix. Then assuming the returns are multivariate normali.i.d. and assuming 250 trading days per year, derive from this the 10-daycovariance 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 standarddeviation 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 beforereporting them to the investors. In fact, the log returns are autocorrelatedwith autocorrelation 0.25, i.e., they have the stationary AR(1) representation, i.e., thecorrelation between adjacent returns. What is your volatility estimate now?2. Assume log returns are normally and independently distributed with mean 0 and variance2. 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 assumedthroughout that returns have zero mean. If the mean return is not assumed to be zerothen we will replace by − 1.a. An equally weighted volatility estimate based on a sample of 30 observations is20%. Find a two-sided 95% confidence interval for this estimate.b. Show that the standard error of variance estimator is 2p2 in this case.c. Show that the standard error of volatility estimator is approximately √2 inthis case.d. An equally weighted volatility estimate based on a sample of 100 observations is20%. Estimate the standard error of the estimator and find an interval for theestimate based on one-standard-error bounds.3. a. Suppose a stock’s continuously compounded (cc) rate of return has annual meanand 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 themean and variance for the cc rate of return for each period. Specifically. Assume the cc returns are independent random variables with anormal distribution. Note the remark on degrees-of-freedom correction in previousexercise.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 shown2in the following table:Month Percent rate of return Month Percent rate of returnAssume 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 inpercent 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 weeklydata instead of monthly data? (See previous exercise.)4. Gavin Jones figured out a clever way to get 24 samples of monthly returns in just overone year instead of only 12 samples; he takes overlapping samples; that is, the firstsample covers Jan. 1 to Feb. 1, and the second sample covers Jan. 15 to Feb. 15, and soforth. He figures that the error in his estimate of , the mean monthly return, will bereduced by this method. Analyze Gavin’s idea. How does the variance of his estimatecompare 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 useQQ plots to assess the normality of the data.b. Create a sample of size = 128 from the exponential distribution with parameter1, and use QQ plots to assess the normality of the data. Describe and explain thedifferences with the results of part (a).c. Use computers or calculators to generate 36 random numbers from the uniformdistribution 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( − 05) = 1 2 100. Now make two frequency tables of withthe length of each interval 0.01 and 0.1 respectively. Plot the two histograms andcomment.6. Suppose you have programmed a computer to do the following:i. Draw randomly 100 values from a standard normal distribution.3ii. 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 b1 .vii. Compute || = |1| b1 and save it.viii. Add one to if || is greater than 2.ix. ReFINA 4130代写、Empirical Finance代peat 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 IndexFund (VBLTX), Emerging Markets Stock Index Fund (VEIEX), and Small-Cap IndexFund Investor Shares (NAESX) from July 2014 to June 2019 through CRSP in WRDS.Consider the constant expected return (CER) modelwhere 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 estimatedstandard error. That is, compute c (b), c (b2), c (b) and c (b ). Show theestimates with the corresponding SE values underneath. Briefly comment on theprecision of the estimates. Hint: The formulas for these standard errors were givenin class and are given in the lecture notes on the CER model.4c. 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% and5% monthly value-at-risk (VaR) based on an initial $100,000 investment. Whichfund has the lowest VaR?e. Using the technique of Monte Carlo simulation, create a simulated data set fromthe 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-wisescatterplots. Does the simulated data look the actual return data for the threeassets?ii. Compute estimates of the pair-wise covariances and correlations. Also computeestimated standard errors for the correlations. Are these correlation estimatesclose 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 IndexFund Investor Shares (VBISX), Extended Market Index Fund Investor Shares (VEXMX),and 500 Index Fund Investor Shares (VFINX) from July 2014 to June 2019 throughCRSP in WRDS. Consider the constant expected return (CER) modelwhere 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 estimatedstandard error. That is, compute c (b), c (b2), c (b) and c (b ). Brieflycomment on the precision of the estimates. Hint: The formulas for these standarderrors 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 estimatedstandard error using the bootstrap with 1000 bootstrap replications. That is compute,c (b), c (b2), c (b) and c (b ). Compare the bootstrapstandard errors to the analytic standard errors. Hint: If you insist on doing bootstrappingin Excel, you can visit the following sites for more information:• http://www.anthony-vba.kefra.com/vba/vba10.htm (for a sample VBA codeon bootstrap)• http://people.revoledu.com/kardi/tutorial/Bootstrap/examples.htm• http://www.stat.auckland.ac.nz/~iase/publications/13/Carr-Salzman.pdfd. For each estimate of the above parameters (except ), plot the histogram and QQplot 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 tocompute the c ( [005) values as well as the 95% confidence intervals. Brieflycomment on the accuracy of the 5% VaR estimates.510. Assuming perfect capital markets, you will estimate expected returns, variances andcovariances to be used as inputs to the Markowitz algorithm, then compute efficientportfolios allowing for short-sales and plot the frontier. Download the monthly returnson the “5 Industry Portfolios” from July 2014 to June 2019 through Kenneth French’sweb 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 nicelyin 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 minimizationproblem iswhere w is the vector of portfolio weights and Σ is the covariance matrix. Arethere any negative weights in this portfolio? If so, interpret them. Compute theexpected return, variance and standard deviation of this portfolio.c. Determine the asset with the highest average historical return. Use this averagereturn as the target return for the computation of an efficient portfolio allowingfor short-sales. That is, find the minimum variance portfolio that has an expectedreturn equal to this target return. The minimization problem isminwhere 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 portfolioand the above efficient portfolio using the formulaCov(e e) = w0Σwd. Using the fact that all efficient portfolios can be written as a convex combinationof two efficient portfolios, compute efficient portfolios as convex combinations ofthe global minimum variance portfolio and the efficient portfolio computed in partc. That is, computew = · w + (1 − ) · wfor values of between 0 and 1 (make a grid for = 0 01 09 1). Computethe 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 thelocation of the efficient portfolio found in part (c).6f. Compute the tangency portfolio assuming the risk-free rate is 0.0003 ( = 003%)per month. That is, where wtan denotes the portfolio weights in the tangency portfolio. Are there anynegative weights in the tangency portfolio? If so, interpret them.g. On the graph with the Markowitz bullet, plot the efficient portfolios that are combinationsof T-bills and the tangency portfolio. Indicate the location of the tangencyportfolio 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 efficientportfolio you got from part (c).7转自：http://www.daixie0.com/contents/3/4356.html
