Fin 406 – Fall 2022 Security Analysis & Portfolio Management
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Fin 406 – Fall 2022
Security Analysis & Portfolio Management
PROJECT 1
(Due October 23, 2022, Sunday)
As indicated on the syllabus, students are strongly encouraged to work on this project as a group (up to six students in each group) and can form a group with students from other sections. Each group needs to upload a single file for the project. The report should be a single well-organized document with clearly labeled answers to each question, with included tables and graphs where necessary. Please submit your report to Canvas by the due date. Please also include the names of all the group members and their section numbers on the title page of the report.
This project consists of five questions as follows.
1. Estimate the correlation (coefficient) between two assets based on real data. Note that for our purpose, correlation means return correlation not correlation between prices of two assets.
(a) Briefly describe the investment objective of each of the following ETFs: SPY, LQD, SH, SDS, SSO, and SHV
(b) Download monthly prices of these six ETFs over the period 08/01/2010−07/01/2018 from Yahoo or other sources you like.
(c) Compute monthly returns for Aug. 2010, Sep. 2010, …, May 2018, and June 2018 using the adj. close prices, for each of the six ETFs. For example, to compute the return for Aug. 2010, you use the prices for 09/01/2010 and 08/01/2010. Note that while the monthly price series begins on 8/1/2010, the monthly return series begins on 9/1/2010.
Note: Return = (Final Price – Initial Price)/Initial Price.
(d) Estimate the correlation coefficients between SPY and each of the other five ETFs using Excel function CORREL. See Example 6.1 in Chapter 6 in the textbook (p. 153 in the 12th edition) on how to use excel function CORREL to compute the correlation.
If necessary, you may refer to the following files posted on Canvas under “Projects” on how to get data from Yahoo and calculate returns using downloaded data:
• “howToGetHistPrcFromYahoo.pdf” has instructions on how to download historical prices from Yahoo and then calculate returns using the prices.
• “example-correlation-beta-slope.xls” illustrates how to calculate monthly returns of XOM and S&P500 from historical prices and then estimate the correlation between the returns of XOM and S&P500 using function CORREL.
Things to turn in (Please organize your work appropriately and label each tab clearly):
(i) a brief description of investment objective for each of the six ETFs (one or two sentences per ETF would be sufficient)
(ii) the monthly returns (not prices) of the six ETFs (it is better to place all six monthly return series in one spreadsheet)
(iii) the estimates of the six correlation coefficients specified in part (d) of Question 1; and
(iv) a brief discussion of whether your estimates of the six correlations are consistent with your intuition. E.g., should SPY and SSO be highly correlated? Is your estimate of their correlation consistent with your expectation?
2. Construct the efficient frontier of SPY and LQD using real data.
(a) You need to estimate five input parameters in this case: the expected returns and standard deviations of SPY and LQD, and their correlation. Suppose you use monthly returns of the SPY and LQD over the period 09/01/2010−07/01/2018 – that are obtained in Question 1 – to estimate those five parameters.
Note: The main reason for choosing this particular period (09/01/2010−07/01/2018) is that both SPY and LQD have a positive average return over the period, which makes it possible to estimate
the expected return using the average return.
(b) Estimate the average monthly return and standard deviation of the monthly returns for SPY. Then estimate the average monthly return and standard deviation of the monthly returns for LQD. Recall that the correlation between SPY and LQD is already estimated in Question 1.
(c) Annualize the average returns and standard deviations of SPY and LQD obtained in part (b) using monthly data. You can use the following formulas to do that
See also “example-correlation-beta-slope.xls” on how to annualize the average monthly returns and monthly standard deviations. Correlation need not be annualized.
(d) Assume that both the expected returns and standard deviations of SPY and LQD going forward are the same as their annualized average returns and standard deviations obtained in part (c), respectively. Tabulate the investment opportunity set of these two ETFs. Namely, select, say, 20 different portfolios (of SPY and LQD), with SPY’s weight ranging from -0.5 to 1.5 with an incremental of 0.1. Then calculate the expected return and standard deviation for each of the 20 portfolios. Spreadsheet 6.5 in Chapter 6 in the textbook (p.156 in the 12e) provides a similar example.
(e) Plot the expected returns of 20 portfolios (the y-variable) vs. these portfolios’ standard deviations
(the x-variable) to generate a curve (whose upper branch is the efficient frontier). If you want, you can use “example-efficientFrontier.xls” posted on Canvas as a template.
Things to turn in (Please organize your work appropriately and label each tab clearly):
(i) a brief description of investment objectives of SPY and LQD (one or two sentences per ETF would be sufficient);
(ii) the estimates of average monthly returns and monthly standard deviations of these two
ETFs;
(iii) the estimates of annualized average returns and standard deviations of these two ETFs; (iv) the estimate of the correlation between these two ETFs;
(v) the tabulated investment opportunity set (say, 20 different portfolios of SPY and LQD) and the corresponding portfolio expected returns and standard deviations; and
(vi) a properly labeled graph of the investment opportunity set (the curve) – namely,
labeling the x-axis and y-axis of your plot at least.
3. Consider an investor who plans to allocate $10,000 to SPY, LQD, and cash, using the asset allocation model introduced in our Fin406 class. Use the annualized average returns, annualized standard deviations, and the correlation estimated earlier in Question 2 (part C). The risk-free rate is assumed to be 2% for this question.
(a) Determine the tangency portfolio using Eq. (6.10) in Chapter 6 in the textbook.
(b) Consider three investors whose risk aversion coefficients (coefficient A) are 8.0, 12.0, and 16.0, respectively. Determine these three investors’ optimal complete portfolios.
You can use “example-assetAllocator.xls” posted on Canvas as a template for this exercise.
Things to turn in (Please organize your work appropriately and label each tab clearly):
(i) the annualized average returns and annualized standard deviations of SPY and LQD; the correlation coefficient between SPY and LQD; and the risk-free interest;
(ii) the portfolio weights of SPY and LQD in the tangency portfolio;
(iii) the expected return and standard deviation of the tangency portfolio; and (iv) the three investors’ optimal complete portfolios (including both portfolio weights and
dollar amounts invested on the three asset classes).
4. You are asked to estimate betas of the following four funds against the S&P 500 index (e.g., Vanguard
500 index ETF) in this question: the Vanguard Market Neutral Fund (ticker: VMNFX), the ProShares Ultra S&P500 ETF (SSO), Consumer Staples Select Sector SPDR ETF (ticker: XLP), and Consumer Discretionary Select Sector SPDR ETF (ticker: XLY). Intuitively, the ranking of these four funds by beta should look like this: VMNFX < XLP < XLY < SSO (why?). Now estimate these funds’ betas and see if your estimates are consistent with the ranking.
(a) Download monthly prices of these four funds and the Vanguard 500 Index ETF (ticker VOO) over the past five years from Yahoo and then compute their monthly returns using the adj. close prices.
(b) Beta can be estimated using either returns or excess returns. As such, please estimate each fund’s beta (against the S&P 500) using the Excel function SLOPE (see below on how to use this function).
Note:
(i) Function SLOPE has two arguments: slope(y-array, x-array), where the 1st argument
represents the y-variable (returns of a fund) and the 2nd argument represents the x- variable (returns of the S&P 500).
For example, XLP’s beta = slope(XLP returns, VOO returns).
(ii) You may also refer to “example-correlation-beta-slope.xls” posted on Canvas on how to
estimate beta.
(iii) See Question 1 of this project on how to download historical prices from Yahoo and
calculate monthly returns
(iv) Formula for return: Return = (Final Price – Initial Price)/Initial Price.
Things to turn in (Please organize your work appropriately and label each tab clearly):
(i) a brief description of investment objectives ofVMNFX, SSO, XLP, and XLY (one or two sentences per ETF would be sufficient);
(ii) monthly returns of VMNFX, SSO, XLP, XLY and the Vanguard 500 index ETF over the
past 5 years;
(iii) your estimates of the four funds’ betas against the Vanguard 500 index ETF; and (iv) the ranking of these four beta estimates
5. Analyze the performance of the mutual fund LMVTX (once managed by the well-known fund manager Bill Miller), discussed in Example 7 in notes lecs05-11-portfolioTheory-part2.
(a) Download monthly prices of the LMVTX and the SPDR S&P 500 ETF (ticker SPY) over the period 10/02/2000-09/30/2003 from Yahoo and then compute their monthly returns using the adj. close prices.
Note:
(i) See Question 1 of this project on how to download historical prices from Yahoo and calculate monthly returns.
(ii) Formula for return: Return = (Final Price – Initial Price)/Initial Price.
(iii) The monthly return series of the fund (or the index) will start from November 1, 2000, although the fund price on 10/02/2000 is needed for the calculation of the fund return for
October 2000.
(b) Calculate the monthly excess returns of LMVTX and the S&P 500 ETF, where the excess return = return – risk free rate. Although the risk-free rate is assumed to be constant in portfolio theory covered in Fin 406, the risk-free rate itself is not a constant and varies over time in the real world. The time-series data on the risk-free rate for this question are available in “risk-free-rate.xls” posted on Canvas under the folder “Projects.”
(c) Estimate the beta of LMVTX using the Excel function SLOPE.
Note: use slope(y-array, x-array), where the 1st argument represents the y-variable (the fund LMVTX’s excess returns) and the 2nd argument represents the x-variable (the S&P’s excess returns). Also, refer to Question 4 of this project on how to estimate beta using function Slope.
(d) Estimate LMVTX's beta as well as alpha against the S&P500 using regression in Excel. By the way, the fund beta estimated using regression should be close to the estimate obtained in part (c).
Here are two methods for running a regression in Excel:
(1) Following Example 6.3 in Chapter 6 in the textbook (p. 173 in the 12th edition);
(2) Following the instructions given below in the appendix on the next page
(e) Replace the S&P500 index by the Russell 2000 index (Yahoo ticker ^RUT), and then redo the regression analysis to obtain the alpha and beta of LMVTX against the Russell 2000 index.
Note:
• “example-correlation-beta-slope.xls” posted on Canvas illustrates how to estimate XOM’s beta against the S&P500 using Function SLOPE.
• If you do it correctly, your estimates of alpha and beta should be close to (not necessarily the same as) those shown in Example 7 in notes lecs05-11-PortfolioTheory-part2 (see also the example in the appendix below on page 7).
Things to turn in (Please organize your work appropriately and label each tab clearly):
(i) the monthly returns of LMVTX, the S&P500, and the Russell 2000, and the monthly risk- free interest;
(ii) a graph of LMVTX vs. the S&P 500 that includes the regression line, the regression
equation and the R-squared (similar to the example shown on page 7 of this document); and
(iii) a graph of LMVTX vs. the Russell 2000 that includes the regression line, the regression
equation and the R-squared.
Appendix: How to run a regression in Excel?
Consider the following equation: Y = α + β X, where Y denotes the excess return of a security or a mutual fund, and X the excess return of the market index. The objective here is to estimate a mutual fund’s α and β using past returns of the fund and those of the benchmark.
The estimation can be done in the following steps:
Obtain historical prices for the fund and the benchmark for a specific sample period from Yahoo
o Say, obtain the monthly prices of the mutual fund and the market index for the past 3 years
o Calculate monthly excess returns of the mutual fund and the market index for the past 3 years. Note
that excess returns need to be used to estimate alpha.
Generate two columns of data in excel:
o one for X (excess returns of the market index) on the left column;
o one for Y (excess returns of the mutual fund) on the right column
Plot the Y variable vs. the X variable using the scatter plot in excel
o To generate this plot, you highlight the two columns of excess return data >click on tab “Insert” > different options for generating a plot >scatter plot
Run regressions based on the plot generated above
o How to run a regression in this way depends on the specific version of excel you use (see below for detailed instructions on this).
o Regardless of the versions of the spreadsheet used, if it is done correctly, a straight line (called regression line) and an equation (called regression Equation) along with the R-squared value will appear on the chart. The y-intercept and slope of the regression equation are the estimates of alpha and beta, respectively. See the example on the next page.
Instructions on how to run regressions based on a plot:
Under the Office 365 Version (the current version) of Excel:
o Click on the scatter chart generated above >click on tab “Design” on menu bar >click on Add Chart Element (the top-left corner of the screen) >Trendline (on the drop-down menu) >More Trendline Options (on the bottom) to get a small window “Format Trendline”
o Then
- Check "Linear”
- Check "Display Equation on chart" (on the bottom)
- Check "Display R-squared value on chart” (on the bottom)
- You are done!!!
Under the previous version of Excel:
o Click on the chart >select Chart Tools (on menu bar) >Layout >Analysis >Trendline >More Trendline Options
o Then
- Select “Linear” under "Trend/Regression Type"
- Check “display equation on chart"
- Check “Display R-squared value on chart”
- Click on “close” >You are done!!!
Under Apple Numbers:
o Chart >2D Line >generate the plot of the fund vs. the market
o Click on the chart >select View >Inspector >Show Inspector
o From the tab: select Series >Trendline >Linear
o Check “show equation” and “show R2 value”
o You are done!!!
Example: Estimating alpha and beta of LMVTX using regressions
It follows from the regression equation displayed on the following graph that
LMVTX’s β (slope) against the S&P 500 = 1.1875
LMVTX’s α (y-intercept) against the S&P 500 = 0.0052 (monthly)
LMVTX’s R2 against the S&P 500 is 79.91%
Note: The estimates of α and β shown here are based on the data for LMVTX returns that were downloaded some years ago. These estimates α and β may be different from the estimates (of α and β) obtained using the fund data downloaded more recently due to fund data revisions.
LMVTX excess return |
Fund excess return vs. Index excess return |
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S&P 500 excess return |
2022-10-20