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Chapter 8: What is Risk? Chapter 9: Portfolio Statistics FIN 450 Dr. Anthony May Working with stock prices in excel Computing realized returns Realized return on an asset during a given period is defined as follows: P1 P0 D1 R P0 R = Realized Return P0 = Price at beginning of period P1 = Price at end of period D1 = Total amount of dividends (or coupon payment) paid during the period. Example: Computing Returns Kellogg stock Example Working with stock prices Stock price data from most commercial sources like Yahoo! Finance are already adjusted for dividends (and stock splits). This means that you can simply compute the return as the percentage change in the adjusted price – this will give you the correct return AP1 AP0 R AP0 R = Realized Return AP0 = Adjusted Price at beginning of period AP1 = Adjusted Price at end of period Getting Stock Price Data from Yahoo! Finance www.finance.yahoo.com Enter the ticker of your stock in the “Quotes” box and hit Enter. Click on Historical Prices Choose frequency (daily, weekly, monthly) and then choose date range Click “Get Prices” button Click “Download to Spreadsheet” at bottom Getting Stock Price Data from Yahoo! Finance Excel file should open Data is in reverse chronological order (most recent date first) – but we usually work with stock prices in chronological order Click on Data tab, Sort. Next to “Sort By” choose date “Adjusted Close” is the data that we want – this is the stock price adjusted for dividends and stock splits Example: Getting Stock Price Data from Yahoo! Finance Get adjusted stock price data for Microsoft (ticker MSFT) Get monthly prices between Jan. 1, 1990 and Jan. 3, 2012. Compute monthly returns using the adjusted prices Compute the average monthly return, using the Average function What is risk? Most important and problematic concept in finance In finance we define risk as the degree of uncertainty about what an asset’s future returns (or cash flows) will be. How is risk measured? Measuring Investment Risk: Variability of Stock Returns Variance and Standard Deviation Variance of returns: The variance of an asset’s realized returns over a given period of time is a measure of how volatile or how spread out those returns are around the average. Variance and Standard Deviation (2) - the expected value of squared deviations from the average Variance N ( R R) 2 t Variance 2 t 1 N 1 • The unit of the variance is percentage-squared, which is difficult to interpret. • Standard deviation () =square root of the variance. Taking the square root gets the units back to percentage terms. • Standard deviation and variance are the most common measures of a stock’s risk. Variance and Standard Deviation In Excel, we will use the functions VAR and STDEV to compute variance and standard deviation. Example: Variance and Standard Deviation Compare the monthly returns of Wal-Mart and Arena Pharmaceuticals. Which stock had a higher average monthly return? But…which stock was more risky Making a frequency distribution (histogram) Chapter 8, page 270 Frequency function counts number of data points in specific ranges Frequency is an array function – works differently than a normal function Fist Mark the range for the function, then type =Frequency( Mark the data array (stock returns in this case) and type a comma Mark the bins array, then type ) DON’T HIT ENTER!!!! Instead, hold down Ctrl and Shift and then hit Enter Making a frequency distribution (histogram) WHEN GRAPHING A FREQUENCY DISTRIBUTION, ALWAYS USE A SCATTER CHART Use one that connects the markers with a line If you want, you can use one without markers (just lines). DO NOT USE A LINE GRAPH TO MAKE A FREQUENCY DISTRIBUTION Example: Frequency Distribution of Daily Returns Compare the distribution of Wal-Mart’s daily returns to those of Arena Pharmaceuticals by making a frequency distribution and graphing it Excel functions used Average Sqrt Var Stdev Covariance.s Correl Frequency Count Ln (advanced) Additional Issues: Graphing Stock prices (or Returns) with Dates on the X-Axis Graph on following slide: McDonald’s adjusted stock price during 1998-2008 Dates are on the X-axis If dates are on X-axis, USE A LINE CHART If dates are not on the x-axis, use a scatter chart McDonald's Stock Prices 31 Dec 1998 - 31 Dec 2008 70 60 50 40 30 20 10 0 28-Dec-08 29-Dec-07 29-Dec-06 29-Dec-05 29-Dec-04 30-Dec-03 30-Dec-02 30-Dec-01 30-Dec-00 31-Dec-99 31-Dec-98 Portfolio Returns Most investors hold a portfolio of multiple assets Return on a portfolio over a given period is the weighted average of returns on the individual assets in the portfolio, where the weights reflect the amount of money invested in each asset Return on a portfolio of two stocks Rp wa Ra wb Rb Rp= Portfolio Return Ra = Return on Stock A Rb = Return on Stock B wa = weight of Stock A in the portfolio wb = weight of Stock B in the portfolio Example: Portfolio Returns Average, variance, and standard deviation of portfolio returns computed the same way as before In-Class Example: Compute the returns for a portfolio with 50% invested in Exxon and 50% invested in Kellogg Additional statistical concepts Covariance and Correlation Both are measures of how the returns of two assets move together. The Covariance between two stocks, denoted i and j, is defined as: N (R i ,t Covariance Ri )( R j ,t R j ) t 1 N 1 Additional statistical concepts: Covariance and Correlation Covariance: Do the returns on two stocks tend to move in same direction? Meaning: When one stock goes up (down), does the other tend to go up (down) also? If “on average, yes,” then Covariance > 0 If “on average, no, they tend to move in opposite directions,” then Covariance < 0. Sign of covariance is interpretable, but magnitude of covariance is not directly interpretable without additional info, i.e., generally we cannot use covariance alone to determine how closely two assets returns move together Additional statistical concepts: Covariance and Correlation Correlation: Generally a more useful measure of how two stocks “move” together. For two stock’s, denoted i and j, correlation is defined as: Correlationi,j Covariancei,j σiσ j From the formula above, you can see that another way to express the Covariance is: Covariancei,j Correlationi,jσ iσ j Additional statistical concepts: Covariance and Correlation Correlation always > -1 and < +1 Correlation > 0: returns of two assets tend to move in same direction Correlation ≤ 0: returns of two assets tend to move in opposite direction Higher (more positive) the correlation, more closely the two assets’ returns move together Correlation +1 or -1 means perfect linear relation between two assets Covariance and Correlation in Excel Use COVARIANCE.S function to compute covariance Use CORREL function for correlation Example: Covariance and Correlation of Exxon and Kellogg Facts about covariance and correlation Symmetry: Covariance between Exxon and Kellogg = Covariance between Kellogg and Exxon Same for correlation Diversification • Average return of a portfolio = weighted average of average returns on all the stocks in the portfolio. • Standard deviation of the portfolio is not a weighted average of individual stock standard deviations. • As long as the stocks in the portfolio are not perfectly positively correlated (correlation=1), the portfolio standard deviation will be less than the weighted average of standard deviations of individual stocks in the portfolio. Diversification • • • Why? Idiosyncratic fluctuations of the different assets partially cancel each other out. If correlations of the stocks are low enough, you can even achieve a portfolio standard deviation that is lower than the standard deviation of any individual stock in the portfolio Diversification = elimination of some risk. Diversification • • In general, the benefits of diversification (reduction of volatility) increase as you add more and more individual assets to the portfolio, but only up to a point. When investing in risky assets in the real world, generally you cannot completely eliminate all risk, i.e., get a portfolio standard deviation of zero. Diversification See Diversification examples in Lecture Spreadsheets Risk-Return Tradeoff Make a data table that shows how the standard deviation of a portfolio consisting of Exxon and Kellogg changes as you vary the weight of Kellogg from 0 to 1 (use increments of 0.1). Do the same for the portfolio average return. Using your data table values, make a graph that plots the portfolio’s average return on the Y-axis and standard deviation on the X-axis. Make sure to use a SCATTER CHART. DO NOT USE A LINE CHART. Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 If a firm has one or more stock splits during a given period of time, you cannot use the raw (unadjusted) stock prices to directly compute stock returns. You must first adjust the closing stock prices (and dividends) for splits. Then you can compute the proper stock returns using the split-adjusted prices (and dividends). Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 The daily (unadjusted) stock prices and dividends of a firm are given in the lecture spreadsheet. The firm had a 2 for 1 stock split on Feb. 18, 2003. This means that the firm’s shares outstanding were doubled on this date. Firm had a 3 for 1 stock split on Nov. 10, 2003. Shares outstanding were tripled (each share was converted into 3 shares). Adjust the stock prices and dividends for splits in such a way that the adjusted prices and dividends can be used to compute the daily stock returns Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 What to do? Make an “adjustment factor.” We will multiply the raw stock prices and dividends by this factor to get the split-adjusted prices and dividends Start with the last (latest) date and set the adjustment factor to 1. Work your way up and change the adjustment factor on every date preceding a split. The split factor will determine how the adjustment factor changes. See next few slides for details Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 Multiply the stock prices and dividends by the adjustment factor Use the split-adjusted prices and dividends to compute returns. Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 1 Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 2 If you have annual or monthly stock prices (and dividends), the process is the same. The annual closing prices and dividends for a firm are in the lecture spreadsheet. The firm had a 2 for 1 stock split in May 1997. The firm had a 3 for 2 stock split in April 1999. Adjusting Raw Stock Prices (and dividends) for Stock Splits: Example 2 Adjusting Raw Stock Prices (and dividends) for Reverse Splits: Example 1 Whereas Stock Splits increase the number of outstanding shares, Reverse Splits decrease the number of outstanding shares. The annual stock returns and dividends of a firm are in the lecture spreadsheet. In 1996, the firm had a 1 for 3 reverse stock split. Means that every three shares were converted into one share In 1998, the firm had 2 for 1 stock split. Each share was converted into two shares (same as before). Adjusting Raw Stock Prices (and dividends) for Reverse Splits: Example 1 Chapter 10 Portfolio Returns and the Efficient Frontier FIN 450 Dr. Anthony May This chapter Formula definitions for expected return and standard deviation of portfolio of two assets Portfolio risk and return Finding the minimum variance portfolio and other efficient portfolios Three-Asset Portfolios 48 Excel in Chapter 10 Excel Functions: Average, Var, Stdev, Covariance.s, Data Table and Graphing Solver 49 Refresher: Average, Variance, and St. Deviation of Portfolio of 50% Kellogg and 50% Exxon In previous lecture, we used the AVERAGE, VAR and STDEV functions to compute the average, variance, and standard deviation of historical returns for a portfolio consisting of 50% Kellogg and 50% Exxon. 50 Expected Return on a Portfolio of two assets Often we use the term “expected” synonymously with average. “Expected return” often means the average return we expect for a stock in the future. 51 Expected Return of a Two-Asset Portfolio Expected (or average) return of a portfolio of two assets Rportfolio= waRa + wbRb wa = weight of asset a wb = weight of asset b Ra = expected return of Asset a Rb = expected return of Asset b Rportfolio = expected return of portfolio 52 Variance of a Two-Asset Portfolio VARportfolio= wa2VARa + wb2VARb + 2wawbCOVa,b SDportfolio=√ VARportfolio VARportfolio = Variance of portfolio wa = weight (or proportion) of asset a wb = weight of asset b VARa = Variance of Asset a VARb = Variance of Asset b COVa,b = Covariance of Asset a with Asset b 53 Variance of a Two-Asset Portfolio COVa,b= (CORa,b)(SDa)(SDb), formula for variance of a portfolio is sometimes written as: Since VARportfolio= wa2VARa + wb2VARb + 2wawb(CORa,b)(SDa)(SDb), SDa = Standard Deviation of Asset a SDb = Standard Deviation of Asset b CORa,b = Correlation of Asset a with Asset b 54 Implementing the formulas 55 Past (realized) Returns vs. Expected Future Returns In the previous slide, we applied the portfolio formulas to returns realized in the past. Investors are often interested in estimating the future expected return and risk of a portfolio. You can use the portfolio average and variance formulas to do this, as long as you have estimates of the future expected returns, variances, and covariances of stocks in the portfolio. A variety of methods are used in practice to obtain estimates of future expected returns, variances, and covariances, e.g., asset pricing models (CAPM), statistical models, etc. We will talk about the CAPM in a subsequent chapter. 56 Expected Return and St. Deviation of Different Portfolio Choices Data Table 57 Some portfolios are obviously suboptimal Portfolios on the bottom (downward sloping) portion of the curve are suboptimal. For the same level of risk, you could achieve a higher portfolio return by moving to a portfolio on the top (upward sloping) portion of the curve. Expected return E(rp) 12% 11% Portfolio Returns: Expected Return and Standard Deviation 10% 9% 8% 7% 6% 5% 10% 11% 12% 13% 14% 15% 16% 17% Standard deviation of return, p 58 Some are more interesting! The upward sloping part of the curve is called the “efficient frontier.” Every portfolio on the efficient frontier is “efficient” because it provides the highest portfolio return, given a specific level of risk. Expected return E(rp) 12% 11% Portfolio Returns: Expected Return and Standard Deviation 10% 9% 8% 7% 6% 5% 10% 11% 12% 13% 14% 15% 16% 17% Standard deviation of return, p 59 The minimum variance portfolio The Minimum Variance Portfolio Expected return E(rp) 12% 11% 10% 9% 8% 7% 6% 5% 10% 11% 12% 13% 14% 15% 16% 17% Standard deviation of return, p Minimum variance portfolio can be located using: Trial and error A formula Excel’s Solver tool 60 The Efficient Frontier Expected Return and Standard Deviation of Portfolio Return--Showing Efficient Frontier Expected portfolio return, E(rp) 12% 11% 10% 9% 8% 7% 6% 5% 10% 11% 12% 13% 14% 15% 16% 17% Standard deviation of portfolio return, p 61 The meaning of the efficient frontier The efficient frontier: includes all portfolio choices that maximize the expected return subject to a given level of risk (standard deviation). All the portfolios represent a tradeoff between expected return and risk On the efficient frontier Higher expected return is accompanied by higher risk 62 Solver We will use Solver to find the minimum-variance portfolio and portfolios along the efficient frontier. 63 Install Solver Click the File button (or Microsoft Office Button in 2007) Click Options (Excel Options in 2007) Click Add-Ins In the Manage box, select Excel Add-ins. Click Go. In the Add-Ins available box, select the Solver Add-in check box, and then click OK. If Solver Add-in is not listed in the Add-Ins available box, click Browse to locate it. If you get prompted that the Solver Add-in is not currently installed on your computer, click Yes to install it. After you load the Solver Add-in, the Solver command is available on the Data tab, in the Analysis group 64 Solver: Example 1 Your portfolio consists of K and XOM stock. Find the minimum variance portfolio. 65 Solver: Example 2 Find the weights of K and XOM that maximize the portfolio’s expected return, subject to the constraint that the portfolio’s standard deviation is less than or equal to 13%. $B$14 < = 0.13 66 Solver: Example 3 Stock A has expected return of 7% and a standard deviation of 12.25%. Stock B has expected return of 9.2% and a standard deviation of 17.89%. The correlation between A and B is 0.55. Find the minimum variance portfolio that consists of Stocks A and B (i.e., find the weights on A and B that minimize portfolio variance). 67 Solver: Example 3 68 Solver: Example 4 You have a portfolio with 40% invested in Stock 1 and 60% invested in Stock 2. Stock 1 has expected return of 10% and variance of 0.015. Stock 2 has variance of 0.032. Your portfolio has an expected return of 11.8% and standard deviation of 14.84% What is the Expected Return of Stock 2? What is the Correlation between Stock 1 and Stock 2? 69 Solver: Example 4 See next slide for Solver Parameters 70 Solver: Example 4 71 Expected Return of a Portfolio of Three assets Rportfolio= waRa + wbRb + wcRc 72 Variance and Standard Deviation of a Three-Asset Portfolio VARportfolio= wa2VARa + wb2VARb + wc2VARc + 2wawbCOVa,b + 2wawcCOVa,c + 2wbwcCOVb,c SDportfolio=√ VARportfolio 73 Expected Return and Variance of a Three Asset Portfolio Calculate the expected return, variance, and standard deviation of a portfolio that consists of 25% Stock A, 35% Stock B, and 40% Stock C. 74 Three-Asset Portfolio: Solver Example 1 Find the weights that minimize the risk of a portfolio consisting of Stocks A, B, and C. 75 Three-Asset Portfolio: Solver Example 2 Find the weights that maximize the expected return of the three-asset portfolio, subject to the constraint that the portfolio’s standard deviation is less than or equal to 45%. 76 Notes on Ch. 10 Practice Exercises #12 The question asks you to compute the “return statistics” of the second stock. This means you need to compute the expected return, variance, and standard deviation of the second stock A1, A2 Located in the Appendix of Ch. 10 (Page 343). 77