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- 1. 5/12/2010 Investment Banking Investment Banking Project Overview The Problem: One of the primary responsibilities of financial advisors is to help their clientele in developing a stock portfolio in which the returns are maximized and risk is minimized. You will need to develop an algorithm that will provide good estimates for stock return and use these estimates to build an optimal portfolio. Areas of Application: • Elementary probability and statistics • Algebra • Geometer’s Sketchpad • Understanding and use of Excel Alignment with Standards: Massachusetts Framework NCTM Mathematic Standards Data Analysis, Statistics, and Data Analysis and Probability Probability 8.D.2-4 8.D.3 12.D.4-5 10.D.1-2 D.3-5 Material Included: • Excel spreadsheet containing the daily price and return data for five stocks o ADM – Archer Daniels Midland Co. o IBM – International Business Machines o MAY – May Department Stores o KO – The Coca-Cola Company o XOM – Exxon Mobil Corporation (See Excel file FiveStocks2000.xls) 1
- 2. 5/12/2010 Investment Banking Background Information & Assumptions: Harry M. Markowitz developed the subject known as Modern Portfolio Theory in the 1950’s. Markowitz built an objective function that combines maximum return and minimum risk. He defined risk as standard deviation in return and defined as efficient a portfolio that maximized return for a given level of risk. One of the reasons that this type of information is important to stock brokerage houses is that they can use this information to assist in purchasing stocks. This project makes extensive use of Excel, as there are many data points that must be analyzed on a large scale. While Excel has the ability to calculate Means, Covariance, Correlation, and Standard Deviation easily, it is good to understand the process that goes into these calculations. 2
- 3. 5/12/2010 Investment Banking Sample Problems Question 1 To begin to create and study investment portfolios, you will need to gather at least one year’s worth of daily closing prices for at least five stocks using Yahoo Finance. To gather this data, go to yahoo.com and click the Finance link at the left. Enter a company’s stock ticker symbol (the two- to four-letter abbreviation that is used to represent the stock, for example DIS is Disney Corp.) and click “Go”. On the next page, click the link for “Historical Prices.” Use the “Set Date Range” box to choose a recent one-year period, and click “Get Prices”. Price data will appear sorted by date. At the bottom of the page, click “Download to Spreadsheet.” Save the spreadsheet to your computer as an .xls file. For the analysis that follows, the price data must be sorted in ascending order by date. To do so, select the price data and click “Sort” under the “Data” menu. Repeat this process until you have gathered daily price data for at least five different stocks. With each set of prices, perform the following calculations: • Compute the daily return for each day. Daily return is the percent change in the price of a stock compared to the previous day. • Compute the annualized return for each stock. Annualized return is the percent change in the price of a stock compared to the price one year ago. • Graph the daily closing price of each stock as a function of the date. Find the line of best fit for the price data, and find the correlation coefficient. Comment on the meaning of the line of best fit and the correlation coefficient in the context of stocks and investing. Which of your stocks seems that it would earn the greatest return on an investment? Which investment seems the riskiest and has the greatest fluctuation in price? Explain. The Excel spreadsheet “FiveStocks2005.xls” contains daily closing prices and returns for five sample companies. However, students are encouraged to gather price data for stocks of their own choosing. 3
- 4. 5/12/2010 Investment Banking Question 2 Using the stock data you have gathered, you will now prepare simple stock portfolios based on your data. For each portfolio in the list below, perform the following tasks: • Compute the expected return of your portfolio using the annualized return of each stock in the portfolio. • For each day of the year, compute the total value of the portfolio as the price of each stock fluctuates. • Graph the total portfolio value as a function of the date. Find the line of best fit and the correlation coefficient. • Comment on your results. Is this particular portfolio efficient – does it provide a high return at low risk? Portfolios: 1. From your group of stocks, choose the one with the highest annualized return. Invest $1000 in this stock. 2. Invest $200 in each of your five stocks. (If you have gathered data for more than five stocks, invest in five stocks of your choice.) 3. Initially, invest $200 in each of your five stocks. Halfway through the one-year period, evaluate the performance of each stock so far and rebalance your portfolio. That is, change the amount of money invested in each stock in the hopes of achieving a higher return. 4. Distribute a $1000 investment among your stocks according to your own choosing. Answer the following questions: • Of the portfolios above, which gave the highest return? Which seemed the least risky/which fluctuated the least? Which portfolio was the most efficient investment? • When rebalancing your portfolio halfway through the year, how did you decide what amount of money to invest in each stock? Did rebalancing increase your return? Why or why not? 4
- 5. 5/12/2010 Investment Banking Question 3 Economists have developed mathematical models that reduce the guesswork and subjectivity of choosing a portfolio of investments. The first and most influential mathematical approach to investing was published by Harold M. Markowitz in 1952. Markowitz’s paper focused on finding efficient portfolios, which have the maximum possible expected return for a given level of risk. This investment approach formed the foundation the subject known as Modern Portfolio Theory. You will use the mathematical tools of Modern Portfolio Theory to create an efficient portfolio. What percentage of your wealth should you invest in each of your stocks to maximize expected return of the portfolio while minimizing its risk? This question requires you to research portfolio optimization in the context of Modern Portfolio Theory. The following are questions that will guide your research. How is the expected return of a portfolio computed? How is the risk of a portfolio computed? How does covariance relate to investment banking? What is an efficient frontier? 5
- 6. 5/12/2010 Investment Banking Solutions to Sample Problems These solutions are based on stock price data from the Excel spreadsheet “FiveStocks2005.xls”. Student answers will therefore vary based on the companies chosen for analysis, but the approach to the problem should be similar to that of the sample solutions. Question 1 Solution Closing prices and daily return: Stock price data was gathered for the following companies: Oakley Inc., Apple Computer Inc., General Electric Co., The Coca-Cola Co., and General Mills Inc. The partial table below shows daily closing prices for and daily returns for each company. Daily Closing Prices Daily Return Gen. Oakley Apple GE Coke Oakley Apple GE Coke Gen. Mills Date Mills 29-Jul-04 10.77 16.32 32.41 42 43.48 30-Jul-04 10.80 16.17 32.45 42.78 43.42 0.00279 -0.00919 0.00123 0.01857 -0.00138 2-Aug-04 10.55 15.79 32.46 43.12 43.86 -0.02315 -0.02350 0.00031 0.00795 0.01013 3-Aug-04 10.57 15.65 32.08 43.02 43.43 0.00190 -0.00887 -0.01171 -0.00232 -0.00980 4-Aug-04 10.49 15.90 32.09 43.11 43.42 -0.00757 0.01597 0.00031 0.00209 -0.00023 5-Aug-04 10.40 15.69 31.45 42.77 43.45 -0.00858 -0.01321 -0.01994 -0.00789 0.00069 6-Aug-04 10.48 14.89 30.76 42.42 43.58 0.00769 -0.05099 -0.02194 -0.00818 0.00299 … … … … … … … … … … … 20-Jul-05 16.95 43.63 35.3 43.33 45.35 0.01497 0.01019 -0.00085 0.00000 -0.00483 21-Jul-05 18.18 43.29 35 43.95 45.58 0.07257 -0.00779 -0.00850 0.01431 0.00507 22-Jul-05 18.9 44 35.07 44.03 45.71 0.03960 0.01640 0.00200 0.00182 0.00285 25-Jul-05 18.51 43.81 34.77 43.65 45.94 -0.02063 -0.00432 -0.00855 -0.00863 0.00503 26-Jul-05 18.59 43.63 34.7 43.69 45.9 0.00432 -0.00411 -0.00201 0.00092 -0.00087 27-Jul-05 18.86 43.99 34.8 44.01 45.97 0.01452 0.00825 0.00288 0.00732 0.00153 28-Jul-05 18.9 43.8 34.88 44.39 46.58 0.00212 -0.00432 0.00230 0.00863 0.01327 Daily return is calculated using the following formula: daily return = ( price today − price yesterday) price yesterday 6
- 7. 5/12/2010 Investment Banking Annualized return: The annualized return for each stock during the time period studied was as follows: Expected Annualized Return Oakley Apple GE Coke Gen. Mills 75.49% 168.38% 7.62% 5.69% 7.13% In this case, the annualized return can be calculated in one of two ways. The simplest method is to find the percent change in each stock’s price for the one-year period studied. That is, use the formula annualized return = ( price on last day − price on first day ) price on first day Because a one-year period is being studied, this formula produces the annualized return. The second, more general, method of calculating the annualized return is as follows: • Add 1 to every daily return you have calculated • Find the product of these values • Raise the result to the power of 1 Number of Years • Subtract 1 from the result. This is the annualized return. The above sequence of steps can be summarized in a formula: 1 n NumYears ∏ 1 + ri ) ( −1 i =1 Where ri is the ith daily return, and the uppercase Greek letter pi (Π) denotes a repeated product. (Zephyr StyleADVISOR, http://www.styleadvisor.com/support/statistics/annualized_return.html) 7
- 8. 5/12/2010 Investment Banking Graphing price vs. date: Stock Prices $60 $50 Oakley Price per share $40 Apple GE $30 Coke $20 Gen. Mills $10 $0 29-Jul-04 17-Sep-04 6-Nov-04 26-Dec-04 14-Feb-05 5-Apr-05 25-May-05 14-Jul-05 Date Lines of best fit: In the graph above, a line of best fit has been added to each company’s stock prices. The equations of the lines of best fit are as follows: Oakley: y = 0.0142x – 11.5374 [i.e. dollars/share = (0.0142 dollars/day)(x days since 7/29/04) – $10.77] Apple: y = 0.0747x – 18.0159 GE: y = 0.0105x – 34.4885 Coke: y = 0.008x – 39.0860 General Mills: y = 0.0152x – 43.9944 (Note: “Zero” on the horizontal axis was defined to be the first date studied: July 29, 2004) 8
- 9. 5/12/2010 Investment Banking To add a line of best fit to a graph using Excel: Right click the plot and click “Add Trendline.” Under “Trend/Regression type,” select “Linear.” Under the “Options” tab, click the checkboxes for “Display equation on chart” and “Display R-squared value on chart.” Click “OK” to display the line, its equation, and the R-squared value (the coefficient of determination) below the equation. Correlation coefficients: The correlation coefficient between two variables is a statistic that measures the strength of the linear relationship between them, on a unitless scale of -1 to +1. That is, it measures the extent to which a linear model can be used to predict the deviation of one variable from its mean given knowledge of the other's deviation from its mean at the same point in time. The correlation coefficients for this stock data are given below. They measure the extent to which the stock price is correlated with the date; that is, they measure the linearity of a stock’s daily price fluctuations. Correlation Coefficients Oakley Apple GE Coke Gen. Mills 0.8527 0.8644 0.7410 0.5016 0.5708 Discussion: The lines of best fit summarize the behavior of the stock prices over the course of the time period studied. The slope of the line indicates the magnitude of the stock’s change in price. The size of the correlation indicates how well a linear model approximates the fluctuation in stock prices (the closer the value of the correlation coefficient to 1, the better the linear fit). Stocks that fluctuate a great deal and have lower correlations could be considered “risky” investments. Because it has the highest annualized return at 168.38%, Apple Computer would be expected to yield the greatest return. The line of best fit for Apple also has the largest slope, another indication of a high rate of return. However, Apple also appears to fluctuate in price to a greater extent than the other companies, so Apple may also represent the riskiest investment despite its high rate of return. 9
- 10. 5/12/2010 Investment Banking Question 2 Solution Portfolio 1: Invest $1000 in Apple Computer, the company with the highest expected return. Expected return: 168.38% (Equal to the annualized return of Apple) Portfolio Value: $1000 in Apple Computer $3,000.00 $2,500.00 Total Portfolio Value $2,000.00 $1,500.00 $1,000.00 $500.00 $0.00 29-Jul-04 17-Sep-04 6-Nov-04 26-Dec-04 14-Feb-05 5-Apr-05 25-May-05 14-Jul-05 Date Line of best fit: y = 4.5783x – 1142.3251 Correlation coefficient: r = 0.8643 (Note: “Zero” on the horizontal axis was defined to be the first date studied: July 29, 2004) As expected, this graph exactly follows the fluctuations in the price of Apple’s stock. This portfolio has a high expected return. However, the stock price appears to be fairly volatile; there are a number of sudden increases/drops in price. Investing entirely in Apple probably represents a risky investment. 10
- 11. 5/12/2010 Investment Banking Portfolio 2: Invest $200 in each of five stocks. Expected return: 52.86% Portfolio Value: $200 in each stock $1,800.00 $1,600.00 $1,400.00 Total Portfolio Value $1,200.00 $1,000.00 $800.00 $600.00 $400.00 $200.00 $0.00 29-Jul-04 17-Sep-04 6-Nov-04 26-Dec-04 14-Feb-05 5-Apr-05 25-May-05 14-Jul-05 Date Line of best fit: y = 1.3517x – 1026.8849 Correlation coefficient: r = 0.9224 This portfolio, in which money is distributed evenly between five stocks, has an expected return of 52.86%. This is a lower return than the previous portfolio, which invests entirely in Apple Computer. However, the more diversified portfolio is much more stable; the value of the portfolio changes much more gradually and predictably. The correlation coefficient is also higher for Portfolio 2. This portfolio is less risky than investing entirely in Apple. 11
- 12. 5/12/2010 Investment Banking Portfolio 3: Invest $200 in each of five stocks. Rebalance the portfolio after a half-year. Rebalancing was done on January 27, 2005. The following rebalancing scheme was used: Initial % % Return at Rebalanced % % Return from half- Company invested half-year invested year to year’s end Oakley 20% 16.99% 20% 50.96% Apple 20% 122.55% 40% 18.41% GE 20% 8.52% 15% -1.22% Coke 20% -1.86% 5% 8.37% General Mills 20% 18.70% 20% -10.13% Portfolio Value Initial: $1000.00 Half-year: $1329.80 Final: $1539.45 Expected return (overall): 53.95% Portfolio Value: $200 in each stock, rebalance at half-year $1,800.00 $1,600.00 $1,400.00 Total Portfolio Value $1,200.00 $1,000.00 $800.00 $600.00 $400.00 $200.00 $0.00 29-Jul-04 17-Sep-04 6-Nov-04 26-Dec-04 14-Feb-05 5-Apr-05 25-May-05 14-Jul-05 Date Line of best fit: y = 1.3345x – 1026.8965 Correlation coefficient: r = 0.9144 In this particular case, rebalancing did not greatly increase or decrease the return that was obtained. When rebalancing, more money was invested in companies that had performed well so far. However, this did not always help the investor – General Mills stock increased by 18.70% in the first half of the year and decreased by 10.13% in the second half. This illustrates a fundamental limitation of Modern Portfolio Theory: past performance is not necessarily indicative of future performance. (The red line represents the time of rebalancing) 12
- 13. 5/12/2010 Investment Banking Portfolio 4: Distribute a $1000 investment among your stocks according to your own choosing. Money was distributed as follows: Initial % Company invested Oakley 23% Apple 43% GE 0% Coke 11% General Mills 23% Expected return: 92.03% Portfolio Value: Portfolio of your choice $2,500.00 $2,000.00 Total Portfolio Value $1,500.00 $1,000.00 $500.00 $0.00 29-Jul-04 17-Sep-04 6-Nov-04 26-Dec-04 14-Feb-05 5-Apr-05 25-May-05 14-Jul-05 Date Line of best fit: y = 2.3729x – 1058.6613 Correlation coefficient: r = 0.9012 This portfolio strikes a balance between risk and return: while more diversified than investing completely in Apple, this portfolio has a high return at 92.03%. This is actually a portfolio from the “efficient frontier,” which will be discussed in the solution to Question 3. 13
- 14. 5/12/2010 Investment Banking Discussion: Answer the following questions: • Of the portfolios above, which gave the highest return? Which seemed the least risky/which fluctuated the least? Which portfolio was the most efficient investment? Investing 100% of your money in Apple Computer gave the highest return (168.38%). The portfolio with the least amount of fluctuation and the least risk appears to be the portfolio in which money is divided evenly among the five stocks. The correlation coefficient for that portfolio (r = 0.9224) is the highest of the five. The most efficient portfolio appears to be Portfolio 4, in which a balance is struck between risk and return. (Note to teachers: Question 2 is designed to provide a qualitative understanding of investment banking with a rather simple introduction to the mathematics involved. While students should incorporate numerical values such as expected return and the correlation coefficient into their discussions, the additional use of qualitative judgment is also acceptable. A deeper exploration of the precise mathematics of investment banking occurs in Question 3.) • When rebalancing your portfolio halfway through the year, how did you decide what amount of money to invest in each stock? Did rebalancing increase your return? Why or why not? Rebalancing should generally be done in light of a stock’s past performance – invest more money in stocks that have performed well. In the example given, rebalancing did not affect the return appreciably because past success did not indicate future success in all cases. 14
- 15. 5/12/2010 Investment Banking Question 3 Solution Solutions will vary, since students are given considerable freedom to choose stocks and to implement an optimization method. The solution depends on which were chosen, their return for the year, and their volatility over the course of the year. The set of portfolios that maximize return for a given level of risk is called the efficient frontier; this is an integral concept of portfolio optimization. The efficient frontier was first defined by Harold Markowitz in his groundbreaking (1952) paper that launched modern portfolio theory. That theory considers a universe of risky investments and explores what might be an optimal portfolio based upon those possible investments. Consider an interval of time. It starts today. It can be any length, but a one-year interval is typically assumed. Today's values for all the risky investments in the universe are known. Their accumulated values (reflecting price changes, coupon payments, dividends, stock splits, etc.) at the end of the horizon are random. As random quantities, we may assign them expected returns and volatilities. We may also assign each pair a correlation for their returns. Generally, these quantities are based upon simple returns. We can use these inputs to calculate the expected return and volatility of any portfolio that can be constructed using the instruments that comprise the universe. The notion of "optimal" portfolio can be defined in one of two ways: 1. For any level of market risk (volatility), consider all the portfolios which have that volatility. From among them all, select the one which has the highest expected return. 2. For any expected return, consider all the portfolios which have that expected return. From among them all, select the one which has the lowest volatility. Each definition produces a set of optimal portfolios. Definition (1) produces an optimal portfolio for each possible level of risk. Definition (2) produces an optimal portfolio for each expected return. Actually, the two definitions are equivalent. The set of optimal portfolios obtained using one definition is exactly the same set which is obtained from the other. That set of optimal portfolios is called the efficient frontier. This is illustrated in Exhibit 1 below: 15
- 16. 5/12/2010 Investment Banking Efficient Frontier Exhibit 1 The green region corresponds to the achievable risk-return space. For every point in that region, there will be at least one portfolio that can be constructed and has the risk and return corresponding to that point. The efficient frontier is the gold curve that runs along the top of the achievable region. Portfolios on the efficient frontier are optimal in both the sense that they offer maximal expected return for some given level of risk and minimal risk for some given level of expected return. In Exhibit 1, the yellow region is the unachievable risk-return space. No portfolios can be constructed corresponding to the points in this region. The gold curve running along the top of the achievable region is the efficient frontier. The portfolios that correspond to points on that curve are optimal according to both definitions (1) and (2) above. Typically, the portfolios which comprise the efficient frontier are the ones which are most highly diversified. Less diversified portfolios tend to be closer to the middle of the achievable region. Reference: http://www.riskglossary.com/articles/efficient_frontier.htm 16
- 17. 5/12/2010 Investment Banking Portfolio optimization with Modern Portfolio Theory can be accomplished with the following general procedure: • Collect historical closing price data for the stocks in which you want to invest. • Compute the daily returns for each stock on each day. • Use the daily returns to compute the expected annualized return of each stock. • Use the expected return of each stock to compute the expected return of the portfolio. • Compute the annualized standard deviation in closing price data (risk) for each stock. • Compute the covariance matrix of the stocks. • Use the covariance matrix to compute the standard deviation (risk) of the portfolio. • Use an optimization algorithm to generate the efficient frontier – the set of portfolios that maximizes return for a given level of risk. • Choose one particular portfolio from the efficient frontier in which to invest. Each of the preceding steps will be explained in detail, with examples drawn from “FiveStocks2005.xls” • Compute the daily returns for each stock on each day. Daily return is calculated using the following formula: daily return = ( price today − price yesterday) price yesterday Daily return is to be calculated for each stock, on each day data was collected. (Note: Daily return on the first day for which data was collected is not defined.) • Use the daily returns to compute the expected annualized return of each stock. Use the following method to compute annualized: o Add 1 to every daily return you have calculated o Find the product of these values o Raise the result to the power of 1 Number of Years o Subtract 1 from the result. This is the annualized return. The above sequence of steps can be summarized in a formula: 1 n NumYears ∏1 + ri ) ( −1 i =1 Where ri is the ith daily return, and the uppercase Greek letter pi (Π) denotes a repeated product. The annualized return denotes the expected return of a single stock. (Zephyr StyleADVISOR, http://www.styleadvisor.com/support/statistics/annualized_return.html) 17
- 18. 5/12/2010 Investment Banking • Use the expected return of each stock to compute the expected return of the portfolio. The expected return of a portfolio is a weighted average of the expected returns of the stocks in the portfolio. It is calculated with the formula: r p = ∑wi ri In this formula, rp represents the expected return of the entire portfolio, ri is the expected return of the ith stock, wi is the weight of the ith stock, i.e. the fraction of your money that is invested in this stock. The Greek letter sigma (Σ) denotes a sum, meaning to add the product of every stock’s return and its weight. • Compute the annualized standard deviation in closing price data (risk) for each stock. Risk can be determined by measuring how much the price of an investment changes over a period of time. This is called volatility. Volatility measures how much variation there is in the price of an investment. In the following example, there are two investments that have a beginning and ending value that are the same. However, the investment on the left has a much greater variation in its price. Example: Two Time Series of Prices http://www.riskglossary.com/articles/volatility.htm#volatility Therefore, risk can be measured as the standard deviation of the value of the investment. 18
- 19. 5/12/2010 Investment Banking Standard deviation is defined graphically below. High vs. Low Standard Deviation These graphs illustrate the difference between a stock price that fluctuates a great deal and one that stays centered around the mean. The stock price on the left is more dispersed than the one on the right; it has a higher standard deviation. http://www.riskglossary.com/articles/standard_deviation.htm Standard deviation, represented by the Greek letter sigma (σ), is defined in general as follows: σ= ∑ ( x - x) n −1 Where σ = the standard deviation, x = one data value, x = the average value of the set of data, Σ(x- x ) = the sum of every data value minus the average value, and n = the number of data values. To annualize this standard deviation, simply multiply the result by the square root of the number of years of data you have collected. • Compute the covariance matrix of the stocks. 19
- 20. 5/12/2010 Investment Banking A covariance matrix measures the degree to which the rates of return for investment assets move together over time. For example, one would expect the stocks of two different airline companies to have a high covariance because they come from the same industry. On the other hand, if two stocks have a low – or even a negative – covariance, then the bad years of one stock will not necessarily coincide with the bad years of another, making for a less risky portfolio. Covariance is defined by the following formula: covariance = ∑ ( x - x) ( y − y) n −1 Where x = the value of one variable x = the average value of that variable y = the value of a second variable y = the average value of the second variable n = the number of data values (must be the same for each variable) In this case, the two variables of interest would be the daily returns of two different stocks. (Note: the covariance of a variable with itself is the variance of that variable, which is equal to the standard deviation squared). A covariance matrix is simply a matrix containing the covariance of each stock with every other stock for which data was collected. Thus, five stocks would produce a 5x5 covariance matrix. This is illustrated by example below, which uses Excel (recommended). Covariance can be calculated easily and on a large scale using the capabilities of an Excel spreadsheet. First, create a chart to compare the different stocks: Covariance Matrix Oakley Apple GE Coke Gen. Mills Oakley Apple ☺ GE Coke Gen. Mills For example, the ☺ will represent the calculated covariance of Apple and GE stocks, or the degree to which their rates of return move together over the investigated period. Next, the covariance must be calculated for each cell in the table. As an example, for cell ☺, Apple vs. GE, follow the steps below. 20
- 21. 5/12/2010 Investment Banking 1. Select the cell 2. Click the function (fx) button 3. Select the function COVAR, click OK 4. A box will appear into which you must enter two arrays 5. For Array 1, click the button at the end of the text field, then choose the entire column of daily returns for Apple. 6. For Array 2, use the same process to select the entire column of GE daily returns. 7. Click OK Repeat this process for each cell. The covariance function is commutative, so the order in which the arrays are selected does not matter. Apple vs. GE will have the same covariance as GE vs. Apple. Following this procedure, the following covariance matrix was developed: Covariance Matrix Oakley Apple GE Coke Gen. Mills Oakley 0.00045421 0.00012632 0.00005159 0.00004805 0.00001277 Apple 0.00012632 0.00067822 0.00006129 0.00004063 0.00004870 GE 0.00005159 0.00006129 0.00007698 0.00003199 0.00003033 Coke 0.00004805 0.00004063 0.00003199 0.00008241 0.00002462 Gen. Mills 0.00001277 0.00004870 0.00003033 0.00002462 0.00008435 • Use the covariance matrix to compute the standard deviation (risk) of the portfolio. The risk of a portfolio is defined as the standard deviation of the entire portfolio. The formula for calculating the risk of a portfolio is as follows: σp = σp = 2 ∑∑ w A w B σ A, B Where σp = the standard deviation of the portfolio wA = the weight of investment A (i.e. the fraction of money invested in stock A), wB = the weight of investment B, and σA,B = the covariance of stocks A and B, taken from the covariance matrix. The formula above first multiplies the product of the weights of two investments by the covariance of the two investments. This is done for every possible pair of investments, and the sum of these results is determined. This sum is the portfolio variance; taking the positive square root of the variance gives the standard deviation. The formula for portfolio risk can also be expressed using matrices as follows. 21
- 22. 5/12/2010 Investment Banking σ 11 ... σ 1n w1 ( w1 ... wn ) ... ... ... ... σ ... σ nn wn n1 This is simply another way of expressing the same mathematical operation. A row vector containing the stock weights is multiplied by the covariance matrix, which is then multiplied by a column vector containing the weights. • Use an optimization algorithm to generate the efficient frontier – the set of portfolios that maximizes return for a given level of risk. Now that the expected return and risk of a portfolio have been defined, it remains to choose weights for each stock such that the portfolio maximizes return while minimizing risk. The efficient frontier is the set of portfolios that maximize return for a given level of risk. Equivalently, efficient portfolios minimize risk for a given level of return. Students are free to devise their own method for generating the efficient frontier. Since finding efficient portfolios is a difficult optimization problem, it is highly recommended that students use some type of numerical optimization software. An accessible and user-friendly optimization program is the Solver add-in for Microsoft Excel. To make Solver available in Excel, click the “Add-ins” item in the “Tools” menu. Check the box for Solver and click “OK.” If prompted to install the add-in, click “Yes.” Solver will then be available under the “Tools” menu. An example of the Solver dialog box is shown below. An explanation of each input area follows. “Set Target Cell” represents the cell in the spreadsheet that is to be optimized. The “Equal To” radio buttons allow you to choose to minimize or maximize the target cell, or set the target cell equal to a specific value. 22
- 23. 5/12/2010 Investment Banking In the “By Changing Cells” box, select the cells that Solver will change to optimize the target cell. The target cell therefore must be a formula whose value depends on the cells being changed. In the “Subject to Constraints” box, enter any relevant constraints for your optimization. For portfolio optimization, a suitable constraint would be that each stock’s weight must be less than 1. To add a constraint, click “Add.” When the parameters above have been set, click “Solve.” The setup for finding an efficient frontier with Solver would be as follows: • Minimize portfolio risk • By changing the weight of each stock • Subject to the constraints: o Portfolio return is fixed o The weight of each stock must total 1 o Each weight must be less than 1 o Each weight must be greater than or equal to 0 Or, equivalently, • Maximize portfolio return • By changing the weight of each stock • Subject to the constraints: o Portfolio risk is fixed o The weight of each stock must total 1 o Each weight must be less than 1 o Each weight must be greater than or equal to 0 Running Solver thusly a number of times, with different fixed values of risk and return, will generate a number of points on the efficient frontier. 23
- 24. 5/12/2010 Investment Banking Using the stock data from “FiveStocks2005.xls”, Solver was used to generate 25 points on the efficient frontier. The resulting curve is shown below: Return vs. Risk Efficient Frontier 180.00% Return (Portfolio Expected 160.00% 140.00% 120.00% Return) 100.00% 80.00% 60.00% 40.00% 20.00% 0.00% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Risk (Portfolio Standard Deviation) Each point on the curve represents a portfolio that maximizes return for a given risk, and minimizes risk for a given return. Note that some portfolios on the efficient frontier have a risk of about 0.7%, which is less than the risk of any individual stock in the portfolio. This demonstrates why diversification is a sound principle of investment. • Choose one particular portfolio from the efficient frontier in which to invest. The efficient frontier represents a set of efficient portfolios; however, an investor must choose a single portfolio in which to invest. This is largely dependent on the level of risk with which the investor is comfortable. Choosing a level of risk is very much a judgment call for the investor, but mathematical methods exist to aid in this choice. Two common mathematical tools are the Sharpe ratio and utility functions. The Sharpe ratio, derived by economist William Sharpe, is a measurement of the return-to-risk ratio of a portfolio. The Sharpe ratio is defined as follows: rp − r f S= σp Where S is the Sharpe Ratio of portfolio, rx is the expected annualized return of x, σ(x) is the standard deviation of x, and 24
- 25. 5/12/2010 Investment Banking Rx is the annualized return of a risk-free investment, such as a bond. Maximizing the Sharpe Ratio along the efficient frontier yields a single portfolio. This optimization problem can also be performed with Solver. In the example below, the point on the efficient frontier marked with a red dot represents the portfolio with the highest Sharpe Ratio. A risk-free rate of 5% was assumed. Return vs. Risk Efficient Frontier 180.00% Return (Portfolio Expected 160.00% 140.00% 120.00% Return) 100.00% 80.00% 60.00% 40.00% 20.00% 0.00% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% Risk (Portfolio Standard Deviation) Optimal Sharpe ratio portfolio: Company Weight Oakley 29.45% Apple 70.55% GE 0.00% Coke 0.00% General Mills 0.00% Expected Return 139.32% Risk 2.08% Choosing a portfolio can also be accomplished with a utility function, which measures an investor’s satisfaction with a particular investment. 1 U ( p) = rp − Aσ p 2 2 U(p) is the utility of the portfolio p. Rp is the return, σp is the standard deviation for the investment, and A is a positive constant that measures an individual’s aversion to risk. A value of A greater than of equal to 4 would represent a very risk-averse investor, whereas a value of A close to 0 means an investor is very tolerant of risk. The larger the value of U, the more satisfied an investor is with their chosen portfolio. High expected returns tend to increase utility, whereas high risks lower utility. Solver can determine the portfolio with the highest utility for a given value of A. 25
- 26. 5/12/2010 Investment Banking 26

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