Published on November 2, 2007
Fin 525 Week 9: Fin 525 Week 9 Volatility and Portfolio Risk Two “Adjustments” to the Google Project Rules: Two “Adjustments” to the Google Project Rules You now have a total of $8,000,000 in capital available for your portfolio (up from $5,000,000) The minimum capital requirement of $500,000 remains in force You will continue to receive 90% of the proceeds of short positions The total “notional amount” of your portfolio cannot exceed $25,000,000 This is the sum of the absolute values of each position (see next slide for an example) Why the Changes?: Why the Changes? The price of Google has risen faster since the semester began than anticipated You could expand your portfolio virtually without limit by financing purchases with short sales. In reality, there would be a limit on your ability to do so and the $25,000,000 notional limit is one way to prevent such behavior The Google Project In Three Parts: The Google Project In Three Parts Predict the closing price of Google: (Nasdaq ticker: GOOG) on December 9, 2005 (5 pts.) Create a portfolio that hedges as completely as possible the risk from 10,000 shares of Google for the period from November 14 through December 9, 2005. Portfolios are graded based on the standard deviations they achieve (20 pts.) Write up an explanation of what you did in 1. and 2. in 1,000 words or less (75 pts.) Key Dates and Times for the Google Project: Key Dates and Times for the Google Project Both your prediction and your portfolio are due Sunday, November 13, 2005 at 6 p.m. via e-mail in a standard format to [email protected] You can use [email protected] as a fallback if the first address does not work for you. Clicking on the above links on a computer running Outlook will create an e-mail message this is properly formatting for submission The write-up is due at the beginning of the Week 10 class meeting (Nov 14 and 16) Part 1: Google Price Prediction: Part 1: Google Price Prediction Prediction is for the price of a single share in $US at the closing of normal Nasdaq trading (usually 4 p.m. or slightly after) on December 9, 2005 as reported on http://finance.yahoo.com. The object is to come as close as possible. Guaranteed grades for price prediction accuracy Within 4% - 5 pts. Within 8% - 4 pts. Within 12% - 3 pts. Grade intervals may be widened at the professor’s discretion Part 2: Hedge Construction: Part 2: Hedge Construction You must hold 10,000 shares of Google You can supplement these shares with either short or long holdings (in multiples of 100 shares) chosen from any or all of the following six securities: Nasdaq 100 Trust (QQQQ) S&P Depository Receipts (SPY) Yahoo! (YHOO) Microsoft (MSFT) Sirius Satellite Radio (SIRI) Wal-Mart (WMT) Part 2: Hedge Construction (Cash Considerations): Part 2: Hedge Construction (Cash Considerations) All long positions must be paid for in cash No purchases on margin Note that the Google shares alone will nearly $4 million at current stock prices All short positions will generate 90% of their sale price in cash This is the typically the best deal any institutional client can get from a broker Your initial cash investment must be between $500,000 and $8,000,000 (up from $5,000,000) Part 2: Hedge Construction (Overall Position Limit): Part 2: Hedge Construction (Overall Position Limit) The total “notional amount” of your portfolio cannot exceed $25,000,000 See next slides for details on how the notional amount is computed The basic idea is that the full dollar value of short positions counts against your $25,000,000 Notional Amount Limit of $25,000,000:An Example: Notional Amount Limit of $25,000,000: An Example Suppose that Google is worth $400/share at the close on Friday, November 11, 2005 Your mandatory 10,000 shares will generate a notional amount of $4,000,000, leaving you with $21,000,000 If you were to sell 500,000 shares of WMT short at $48/share, that would generate a notional amount of $24,000,000—more than the $21,000,000 available to you How Your Hedge are Evaluated: How Your Hedge are Evaluated Lower volatilities earn higher grades The portfolio volatility the daily standard deviation of its return over the four-week period A spreadsheet containing a useful example will be available after the midterm exam How Hedges are Evaluated (continued): How Hedges are Evaluated (continued) The six lowest volatilities are guaranteed 20 out of 20 points The next ten lowest volatilities are guaranteed at least 19 out of 20 points E-mail Submission Details: E-mail Submission Details Enter your last name after “Last Name: ” (leave at least one space) Give the price for Google stock in dollars and cents directly after GOOG (leave at least one space) Give the quantities for your portfolio (the 10000 shares of GOOG are automatically in it) Again, leave at least one space after the symbol Quantities must be a multiple of 100 shares and should not include commas If the automated form does not work for you, create your own version and place “Fin525 GOOG” in the subject of your e-mail addressed to [email protected] Sample E-Mail Submission: Sample E-Mail Submission Last Name: SquarePants GOOG 300.50 QQQQ 700 SPY YHOO MSFT 1200 SIRI WMT -7500 Google Project Write-Up: Google Project Write-Up 1,000-word limit is strictly enforced Up to 3 optional tables or figures of reasonable size are allowed in addition to the 1,000 words Formatting guidelines Single-spaced within paragraphs and double-spaced between paragraphs 12-point standard proportional typeface Left-hand justified Margins of at least 1 inch on all sides Google Project Write-Up (continued): Google Project Write-Up (continued) Should be readable with Microsoft Word or Adobe Acrobat Restrained use of hyperlinks is fine Do not include macros in your spreadsheets or files in formats other than DOC, PDF, and XLS The write-up may be either e-mailed or submitted as hard copy in class or both The Standard Deviation of Returns: The Standard Deviation of Returns This is also known as the stock’s volatility; however, there are other methods of measuring volatility Volatility is a standard measure of the stock’s risk—higher volatility means more risk Annualizing volatility is somewhat tricky: For weekly returns, we multiply by 52 to annualize Some Numbers from the Last 60 Weeks: Some Numbers from the Last 60 Weeks Correlation: Correlation A function that connects two parallel series of numbers, such as stock returns (Excel uses CORREL) Varies from -1 to 1 1 is perfectly correlated 0 is uncorrelated (as if the two series were chosen independently of one another -1 is perfectly uncorrelated (when one series zigs, the other series zags) A Scary Halloween Slide from Equation 8.10 in BKM: A Scary Halloween Slide from Equation 8.10 in BKM How Finance Theory Makes Things Simpler: How Finance Theory Makes Things Simpler In most financial models, all correlations are accounted for through common “factors” Everyone’s favorite factor is known as the “market” factor From a practical standpoint, the S&P 500 is usually chosen as the market factor Many reason people (like your professor) think that broader indexes, like the Russell 3000, are a better choice for the market Two Ways to Reduce Portfolio Volatility: Two Ways to Reduce Portfolio Volatility Add more assets Even though most assets are to some degree correlated with one another though common factors, as long as that correlation is less than 1, added assets tends to reduce portfolio volatility Sell the appropriate assets short to hedge Obvious choice: SPY Less obvious choices: Related stocks—for example, these in the same industry A Simple Experiment: A Simple Experiment Go to the What-If for the Past 60 Weeks worksheet in GoogleStats.xls Add 1,000 shares of any stock to the 10,000 shares of Google Notice that the portfolio volatility (standard deviation) goes down in every case Why Does This Work?: Why Does This Work? While the additional stocks may amplify risk factors they have in common with Google, their “specific risks” promote diversifiable, which lower portfolio volatility Because this risk is easy to diversify away, the market does not reward anyone for bearing it The observation that only holding undiversifiable risk can increase one’s expected returns is at the heart of the Capital Asset Pricing Model (CAPM) How Do We Get Rid of the Common Risk Factors?: How Do We Get Rid of the Common Risk Factors? Buying more only adds common risk, so that will not work Instead, we have to sell short either EFTs or related stocks to shed this risk Short Selling : Short Selling Selling a stock short involves borrowing shares, selling them, and buying them back at a future date The short seller must pay dividends Only institutional short sellers get to use to the some of the proceeds of the short sale Short selling is one of those things that works easy in theory, but may be fraught with difficulties (and margin calls) in practice Fortunately, selling ETFs (SPY, QQQQ, etc.) short is not difficult, which is one reason that they are so actively traded An Example of How Short Selling Works: An Example of How Short Selling Works Suppose SPY is at $121/share and you sell 1,000 shares short Your brokerage account is credited with the proceeds of $121,000 An individual customer would not get to use the proceeds and would have to post 30% margin The terms for institutional customers are more complicated and often determined based on the entirety of the customer’s holdings If SPY goes down to $120/share, you can buy the shares back at $120,000 and make $1,000 (ignoring commissions, taxes, etc.) How Do We Find Common Risk Factors?: How Do We Find Common Risk Factors? Regression analysis and related statistical tools (discriminant analysis, factor analysis, neural networks, etc.) These tools perform what is known as variance decomposition The variance of the stock or portfolio that we are interested in is decomposed into two parts Specific (or idiosyncratic) variance/risk/volatility Market (or systematic) variance/risk/volatility Capital Asset Pricing Model (CAPM): Capital Asset Pricing Model (CAPM) Risk comes in two varieties Market or systematic risk Diversifiable (or specific) risk You are stuck with market risk You can diversify away diversifiable or specific risk CAPM is based on the notion that the only kind of risk that the market will reward you for bearing is market risk CAPM explicit assumes that markets are efficient and that markets are dominated by risk-averse individuals The CAPM Equation: The CAPM Equation Expected return = Risk-free return + Premium for risk Where E(ri) is the expected return for stock i rf is the risk-free rate of return i is the beta for stock i E(rM) is the expected market rate of return An Example in the Current Market: GEQuestion: What Will It Be Worth in 4 weeks?: An Example in the Current Market: GE Question: What Will It Be Worth in 4 weeks? Useful GE facts Closing price on November 4: $34.02/share Beta = 0.98 (according to Reuters Investor) Other necessary parameters rf is roughly 4% in the current market rm is roughly 10%, so the risk premium is 6% Annual return for GE is 4% + 0.98(6%) = 9.88% For 1/13 of a year (4 weeks), that 0.76% So the FV of GE in 4 weeks if $34.02(1.0076) = $34.28 Now What?: Now What? When CAPM works, we can use it to predict stock prices for weeks, months, even years, into the future Problems The predictions, while possibly the best we can do, may not be very accurate CAPM does not handle “event risk” well All the variables in the model are abstractions that have only rough real-world approximations CAPM as a Regression Equation: CAPM as a Regression Equation The independent variable (x in most textbooks) is the excess return on the market index The dependent variable (y in most textbooks) is the excess return on the asset/portfolio Beta (the slope of the regression line) is the amount of market risk in the asset/portfolio Alpha (the intercept of the regression line) is the risk-adjusted performance of the asset/portfolio The CAPM Regression in Graphic Form: The CAPM Regression in Graphic Form Asset Excess Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Market Excess Returns . . . My Favorite Equation in BKM (page 320): My Favorite Equation in BKM (page 320) i can be anything (stock or portfolio) and this equation separates the variance of that stock (or portfolio) into a systematic piece and a specific piece Linear regression or handy functions within Excel can be used to find beta Why is This Equation Useful?: Why is This Equation Useful? The amount of variance captured by the market is known as R2 (often written R-squared) You can (in a statistical sense) get rid of all systematic risk by selling short the market index in an amount indicated by beta You are left with the specific risk, which you can then do what you want with (leverage, diversify, etc.) How Things Tie Together: How Things Tie Together When you have a single market factor (BKM refer to it as an “index”), then the R in R2 is the same as the correlation between the portfolio/asset and the “market” (usually written as r) You do not have to use any form of regression analysis to get beta in Excel, you can use the CORREL function on excess returns to get r, and then use the formula on the previous slide to solve for beta: Regression Analysis in Excel: Regression Analysis in Excel In GoogleStats.xls look at the worksheet called “CAPM Regression for Google” Excel has built-in function for single variable regression Excel has an Analysis ToolPak for doing all kinds of regression (single and multiple variable) One can also “hard-wire” regression into Excel using the matrix math and summation functions Things to Make You Happy: Things to Make You Happy Beta and R2 do not depend on the frequency (daily, weekly, monthly, or whatever) of the data you use All that matters is that the same time period is used consistently Things to Make Your Unhappy: Things to Make Your Unhappy Alpha, sigma, and anything involving returns does depend on the frequency of the data used in the regression To convert a weekly alpha into an annual alpha (approximately), use the same compounding conversion that we used to returns earlier Using Regressions to Hedge Away Market Risk: Using Regressions to Hedge Away Market Risk Here is the regression equation for GOOG relative to SPY: So, alpha = 0.017 (per week) and beta = 0.672 Hence, if for every $1 we have of Google, we sell $0.672 of SPY short (and hold the cash proceeds), we can (in theory) fully remove the market risk (and volatility) from Google Your Problems in Hedging GOOG with SPY: Your Problems in Hedging GOOG with SPY You cannot hold cash, so the proceeding from selling SPY reduce your capital base If you really wanted to hedge GOOG risk for whatever reason, SPY is not the best choice A Final Bit of Useful Advice: A Final Bit of Useful Advice Excel’s Analysis ToolPak comes with a Solver tool. The current version of GoogleStats.xls comes with the Solver already set up to minimize the portfolio volatility over the historical 60-week time period Your problem is how to adapt this spreadsheet to come with a forward-looking portfolio Note that the prices of the stocks are different now than they were back in August of 2004 Before and For Next Time: Before and For Next Time Send a practice Google submission via e-mail before Friday Send real submission via e-mail before 6pm on Sunday, November 13, 2005 Turn in your write-up either in person or via e-mail on or before the beginning of the Week 10 class (November 14 or 16).