Fin603 Fall2005 Week9

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Published on November 2, 2007

Author: Sharck

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Fin 603 Week 9:  Fin 603 Week 9 Portfolios and CAPM 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) 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 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 From Last Time:  From Last Time We looked at the volatility of a single asset Next up: Portfolio volatility Easy first case: One risky asset and one risk-free (zero volatility) asset This week is all analytics, so the cute icon has been eliminated so that it will not become monotonous BKM Chapter 7 in Two Graphs Graph 1: Capital Allocation Line:  BKM Chapter 7 in Two Graphs Graph 1: Capital Allocation Line Comments on the Capital Allocation Line:  Comments on the Capital Allocation Line Movements along it reflect varying degrees of leverage Leverage does not change the line’s slope That slope is the reward-to-variability ratio, and is also the most common form of the Sharpe ratio (which, for some unknown reason, BKM call Sharpe’s measure) It is equal to the excess return on the risky portfolio divided by its standard deviation Utility Alert:  Utility Alert The next two slides concern the concept of utility You can do the Google project without knowing a thing about utility Why? Utility “falls out” of much of portfolio theory, including CAPM and the Sharpe ratio Portfolio managers don’t worry about utility BKM Chapter 7 in Two Graphs Graph 2: Maximize Utility:  BKM Chapter 7 in Two Graphs Graph 2: Maximize Utility Comments on Utility Maximization:  Comments on Utility Maximization Each “indifference curve” represents the same level of utility from different combination of risk and return These curves are derived from a utility function of the form: The graph on the previous slide has A=4 Portfolios with Multiple Risky Assets:  Portfolios with Multiple Risky Assets Expected returns are still the value-weighted average of the expected returns on the assets Example: A portfolio with 20% of its money in an asset with an expected return of 10% and 80% in an asset with an expected return of 15% will have an expected return of .2(10%)+.8(15%)=14% The volatility (standard deviation) of the returns is more complicated It depends on how highly correlated the returns of the assets are—higher correlation means more volatility 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:  A Scary Halloween Slide This equation can be extended to deal with any number of assets and becomes even scarier (unless you are comfortable with matrix notation How Finance Theory Makes Things Simpler:  How Finance Theory Makes Things Simpler In most financial models, all correlations are accounted for through common “factors” (this is what Chapter 11 of BKM is all about) 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 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 Very Important!!!!:  Very Important!!!! The CAPM equation is not an accounting identity It is the result of a useful—but to various degrees flawed—theory In theory, the “market” in CAPM consists of a basket of every capital asset; in practice, the S&P 500 Index (annual return between 9% and 11%) is most often used to represent the market So What?:  So 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 Even if you could hold cash, you goal is to maximize your Sharpe ratio If you really wanted to hedge GOOG risk for whatever reason, SPY is not the best choice Performance:  Performance Alpha is a reasonable measure of performance The slick magazine for the hedge-fund world is called Alpha The goal of most hedge funds is to capture alpha for their clients The problem with alpha is that it does not take into account the risk incurred to capture the excess return The measurement that does take risk into account is the Sharpe ratio Introduction to Sharpe Ratios:  Introduction to Sharpe Ratios Sharpe ratio = Portfolio excess return/Portfolio volatility The simple version is what BKM first call the return-to-variability ratio The Sharpe Ratio is the standard way that many types of hedge funds are evaluated A Sharpe Ratio of at least 0.5 is you get with S&P over long period of time Hedge funds aim for at least 1.0 and the top ones get 3.0 to 4.0 or more 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 maximize the Sharpe ratio over the historical 60-week time period Your problem is how to adapt this to come with a forward-looking portfolio

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