VIEWS: 25 PAGES: 14 CATEGORY: Business POSTED ON: 9/19/2011 Public Domain
D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.1/14 Part 3: Portfolio models (chapters 11-16) Reading: ch 11: What is risk? Read section 11.1 & skim the rest. ch 12: Statistics for portfolios. Read. ch 13: Efficient frontiers. Skip first 18 pages & read. ch 14: CAPM. Read. ch 15: SML & performance measurement. Read lecture note first. If you need more details, then read relevant sections in this chapter. ch 16: SML & cost of capital: skip. Ch 12: Statistics for portfolio 12.1 Basic statistics for asset returns (sections 12.1 and 12.3) Calculation of rate of return: r(t) = [P(t) + d(t)]/P(t-1) – 1. Open ch12.xls:GM calculate returns, average, variance and standard deviation. A B C D E 1 Price and dividend data for General Motors (GM) Closing Annual 2 Date Price Dividend return 3 29-Dec-89 42.2500 - 4 31-Dec-90 34.3750 3.00 -11.54% <-- =(C4+B4)/B3-1 5 31-Dec-91 28.8750 1.60 -11.35% <-- =(C5+B5)/B4-1 6 31-Dec-92 32.2500 1.40 16.54% 7 31-Dec-93 54.8750 0.80 72.64% 8 30-Dec-94 42.1250 0.80 -21.78% 9 29-Dec-95 52.8750 1.10 28.13% 10 31-Dec-96 55.7500 1.60 8.46% 11 31-Dec-97 60.7500 5.59 19.00% 12 31-Dec-98 71.5625 2.00 21.09% 13 31-Dec-99 72.6875 14.15 21.34% 14 15 Average return 14.25% <-- =AVERAGE(D4:D13) 16 Variance of return 0.0638 <-- =VARP(D4:D13) 17 Standard deviation of return 25.25% <-- =STDEVP(D4:D13) Note: 1. Which one to use: varp( ) or var( )? For practical purposes, it does not matter. Following the textbook, we use varp( ) and stdevp( ). 2. Of course, you can use the definition of variance to calculate it: var(x) = [ (x – E(x))2 ] / N D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.2/14 Adjusting returns for stock-splits: (section 12.2) Example: You bought a share of Microsoft for $87 on 12/29/89. Microsoft split its stock 2 for 1 in 1990, meaning your one share became two shares. You sold your shares on 12/31/90 for $75.25 per share. What was your rate of return? $75.25*2/$87 – 1 = 72.99% Open ch12.xls:MSFT. See how to adjust returns for a stock-split. A B C D E F G 1 PRICE AND STOCK SPLIT DATA FOR MICROSOFT (MSFT) Closing Stock 2 Date Price split 3 29-Dec-89 87.0000 4 31-Dec-90 75.2500 2.0 for 1 5 31-Dec-91 111.2500 1.5 for 1 6 31-Dec-92 85.3750 1.5 for 1 7 31-Dec-93 80.6250 no 8 30-Dec-94 61.1250 2.0 for 1 9 29-Dec-95 87.7500 no 10 31-Dec-96 82.6250 2.0 for 1 11 31-Dec-97 129.2500 no 12 31-Dec-98 138.6875 2.0 for 1 13 31-Dec-99 116.7500 2.0 for 1 Cumulative Closing Stock adjustment Adjusted Annual 14 Date Price split factor price return 15 29-Dec-89 87.0000 1 87.00 16 31-Dec-90 75.2500 2.0 for 1 2 150.50 72.99% <-- =E16/E15-1 17 31-Dec-91 111.2500 1.5 for 1 3 333.75 121.76% <-- =E17/E16-1 18 31-Dec-92 85.3750 1.5 for 1 4.5 384.19 15.11% 19 31-Dec-93 80.6250 no 4.5 362.81 -5.56% 20 30-Dec-94 61.1250 2.0 for 1 9 550.13 51.63% 21 29-Dec-95 87.7500 no 9 789.75 43.56% 22 31-Dec-96 82.6250 2.0 for 1 18 1,487.25 88.32% 23 31-Dec-97 129.2500 no 18 2,326.50 56.43% 24 31-Dec-98 138.6875 2.0 for 1 36 4,992.75 114.60% 25 31-Dec-99 116.7500 2.0 for 1 72 8,406.00 68.36% 26 The cumulative adjustment factor is 27 Average return 62.72% <-- =AVERAGE(F16:F25) the product of all the splits: 28 Variance of return 72 = 2*1.5*1.5*2*2*2*2 14.43% <-- =VARP(F16:F25) 29 Standard deviation of return 37.99% <-- =SQRT(F28) Note: The prices downloaded from Yahoo are adjusted for stock splits and dividend. The percentage difference of prices will be the rate of return between two dates. D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.3/14 So, prices in Yahoo are not actual trade prices. CRSP prices are actual trade prices, and CRSP returns are adjusted for splits and other corporate events. 12.3 Covariance and correlation Open ch12.xls: GM&MSFT and calculate mean, variance, standard deviations, covariance and correlation. A B C D 1 GM and MSFT, annual return data 2 Date GM return MSFT return 3 31-Dec-90 -11.54% 72.99% 4 31-Dec-91 -11.35% 121.76% 5 31-Dec-92 16.54% 15.11% 6 31-Dec-93 72.64% -5.56% 7 30-Dec-94 -21.78% 51.63% 8 29-Dec-95 28.13% 43.56% 9 31-Dec-96 8.46% 88.32% 10 31-Dec-97 19.00% 56.43% 11 31-Dec-98 21.09% 114.60% 12 31-Dec-99 21.34% 68.36% 13 14 Average return 14.25% 62.72% <-- =AVERAGE(C3:C12) 15 Variance of return 6.38% 14.43% <-- =VARP(C3:C12) 16 Standard deviation of return 25.25% 37.99% <-- =STDEVP(C3:C12) 17 Covariance of returns -0.0552 <-- =COVAR(B3:B12,C3:C12) 18 Correlation of returns -0.5755 <-- =CORREL(B3:B12,C3:C12) 19 Correlation using formula -0.5755 <-- =B17/(B16*C16) Note: 1. You can use the definition of covariance to calculate it: cov (x,y) = [ (x – E(x))(y – E(y)) ] / N 2. Correlation () is always between –1 (perfectly negative) and +1 (perfectly positive). When = -1 or 1, x and y has a linear relation: y = a + bx. (Details are in p. 17-21 of ch. 12.) 12.4 Portfolio mean and variance Mean and variance of returns of a portfolio We consider a portfolio of two risky assets, GM and Microsoft. Portfolio’s return in month t is equal to the weighted average of returns of GM and Microsoft in month t. rp = xGM rGM + (1 - xGM)rMSFT. D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.4/14 Once you find out portfolio’s return for each month in your sample, you can easily calculate the mean and variance of portfolio’s return. Open ch12.xls: 2-asset port. Calculate portfolio’s return each month, and mean and variance of a portfolio’s returns. A B C D E F 1 CALCULATING PORTFOLIO RETURNS AND THEIR STATISTICS 2 Proportion of GM 0.5 3 Proportion of MSFT 0.5 <-- =1-B2 Portfolio 4 Date GM MSFT return 5 Dec-90 -11.54% 72.99% 30.73% <-- =$B$2*B5+$B$3*C5 6 Dec-91 -11.35% 121.76% 55.21% 7 Dec-92 16.54% 15.11% 15.82% 8 Dec-93 72.64% -5.56% 33.54% 9 Dec-94 -21.78% 51.63% 14.93% 10 Dec-95 28.13% 43.56% 35.84% 11 Dec-96 8.46% 88.32% 48.39% 12 Dec-97 19.00% 56.43% 37.71% 13 Dec-98 21.09% 114.60% 67.85% 14 Dec-99 21.34% 68.36% 44.85% 15 16 Mean 14.25% 62.72% 38.49% <-- =AVERAGE(E5:E14) 17 Variance 6.38% 14.43% 2.44% <-- =VARP(E5:E14) 18 St. dev. 25.25% 37.99% 15.62% <-- =STDEVP(E5:E14) 19 Covariance -0.0552 20 Correlation -0.5755 21 22 Direct calculation of portfolio mean and variance 23 Portfolio mean 38.49% <-- =B2*B16+B3*C16 24 Portfolio variance 2.44% <-- =B2^2*B17+B3^2*C17+2*B2*B3*C19 25 Portfolio st. dev. 15.62% <-- =SQRT(B24) Note: (1) This is the long-way. (2) You can use the following formula to calculate directly portfolio mean and variance: Do this calculation now. E(rp) = xGM E(rGM) + (1 - xGM) E(rMSFT), [xMSFT = 1 - xGM ] var(rp) = x2GM var(rGM) + x2MSFT var(rMSFT) + 2 xGMxMSFT cov(rGM, rMSFT). Portfolio frontier: You just calculated the mean and variance of a portfolio. The calculation is based on a portfolio weight, i.e., 50% for GM, and 50% for MSFT. Let us change the portfolio weight for GM from 0% to 100%, and calculate the mean D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.5/14 and variance of each portfolio. When you plot the mean (on Y-axis) and standard deviation (on x-axis) of all portfolios, you get the portfolio frontier. Open ch12.xls:port frontier. (i) Fill up the table of portfolio variance, standard deviation and mean. (ii) Plot portfolio mean and standard deviation. A B C D E F G H 1 Portfolio frontier 2 GM MSFT Portfolio Mean and Standard Deviation 3 Mean 14.25% 62.72% 4 Variance 6.38% 14.43% 70% p) Portfolio return mean, E(r 5 St. dev. 25.25% 37.99% 60% 6 correlation -0.5755 50% 7 0.0000 40% portfolio Portfolio weight of Portfolio standard Portfolio 30% 8 GM Variance deviation mean 20% 9 0% 14.43% 37.99% 62.72% 10% 10 10% 10.76% 32.80% 57.87% 0% 11 20% 7.72% 27.79% 53.03% 12% 17% 22% 27% 32% 37% 42% 12 30% 5.33% 23.08% 48.18% Portfolio return standard deviation, p 13 40% 3.56% 18.88% 43.33% 14 50% 2.44% 15.62% 38.49% 15 60% 1.95% 13.98% 33.64% 16 70% 2.11% 14.51% 28.79% 17 80% 2.89% 17.01% 23.95% 18 90% 4.32% 20.78% 19.10% 19 100% 6.38% 25.25% 14.25% 20 =A19*$B$3+(1-A19)*$C$3 =SQRT(C19) 21 =A19^2*$B$4+(1-A19)^2*$C$4+2*A19*(1-A19)*B$5*C$5*$C$6 22 Note: In filling up the table in above workbook (ch11.xls:port frontier), you can use data|table function. Open ch12.xls: data table. Fill up the table by using the data|table function. Steps: (i) Input portfolio weights of GM in column A (from A13-A23). (ii) Input “=B9” in cell B12; Input “=C9” in cell C12; Input “=D9” in cell D12. (iii) Highlight entire table (A12:D23). (iv) Click data|table. (v) Input “=B6” in column input cell box, and click ctrl+Alt+Enter. D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.6/14 A B C D 1 Using data|table: Portfolio frontier 2 GM MSFT 3 Mean 14.25% 62.72% 4 St. dev. 25.25% 37.99% 5 Correlation -0.5755 6 portfolio weight 0.1 90.00% 7 8 port var port std port mean 9 0.1076 0.3280 0.5787 10 portfolio Portfolio Portfolio Portfolio 11 weight: GM Variance st dev mean 12 0.1076 0.3280 0.5787 13 0% 14.43% 37.99% 62.72% 14 10% 10.76% 32.80% 57.87% 15 20% 7.72% 27.79% 53.03% 16 30% 5.33% 23.08% 48.18% 17 40% 3.56% 18.88% 43.33% 18 50% 2.44% 15.62% 38.49% 19 60% 1.95% 13.98% 33.64% 20 70% 2.11% 14.51% 28.79% 21 80% 2.89% 17.01% 23.95% 22 90% 4.32% 20.78% 19.10% 23 100% 6.38% 25.25% 14.25% D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.7/14 Ch. 13 Efficient frontier 13.1 The advantage of diversification (13.1 (skip) and 13.5) Open ch13.xls:port frontier. Change correlation between GM and MSFT from +1, 0.5, 0.0, -1.0. What happens to the portfolio frontier? corr=1 corr=0.5 corr=0 corr=-1 70% 60% 50% 40% 30% 20% 10% 0% 0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 Summary: The lower the correlation, the greater the diversification benefits. 13.2 Efficient frontier and minimum variance portfolio (p.20-21, and 13.4) Expected Return and Standard Deviation of Portfolio Return 0.70 The portfolios on the top 0.60 are the efficient frontier-- portfolios with a positive Expected portfolio return, E(r p) risk-return tradeoff 0.50 0.40 This curve is called The minimum portfolio frontier-- portfolios variance portfolio with the lowerest risk 0.30 among portfolios that give the same expected return. 0.20 0.10 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Standard deviation of portfolio return, p Note: We have considered only portfolios of two risky assets, but the portfolio can contain any number of risky assets. D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.8/14 Finding the minimum variance portfolio: Solver or algebra (1) Algebra (Appendix 1 of ch. 13) Let x denote portfolio weight for GM. Choose x to minimize the portfolio variance. Min (1/2) [ x2var(rGM) + (1-x)2 var(rMSFT) + 2 x(1-x) cov(rGM, rMSFT) ] {x} The first order condition for this optimization problem is: x var(rGM) + (1-x) var(rMSFT) + (1-2x) cov(rGM, rMSFT) = 0. Solving the first order condition for x yields, x = [var(rMSFT) - cov(rGM, rMSFT)] / [var(rGM) + var(rMSFT) -2 cov(rGM, rMSFT)]. (2) Using Excel Solver Open ch13.xls:Min-var solver. A B C D E F 1 CALCULATING THE MINIMUM-VARIANCE PORTFOLIO WITH SOLVER 2 GM MSFT 3 Average 14.25% 62.72% 4 Variance 6.38% 14.43% 5 Sigma 25.25% 37.99% 6 Covariance of returns -5.52% 7 8 Portfolio return and risk 9 Percentage in GM 62.64% 10 Percentage in MSFT 37.36% <-- =1-B9 11 12 Expected portfolio return 32.36% <-- =B9*C3+B10*D3 13 Portfolio variance 0.0193 <-- =B9^2*C4+B10^2*D4+2*B9*B10*C6 14 Portfolio standard deviation 13.90% <-- =SQRT(B13) Solver: Target cell (=B13); By changing cell (=B9). Open ch13.xls:Min-var formula. Calculate mean and variance of minimum variance portfolio, using formula. D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.9/14 Ch. 14 CAPM 14.1 Combining a portfolio of risky assets with a riskless asset Once we know the means, variances and correlations among assets, we can draw the portfolio frontier of risky assets. You pick a portfolio from the portfolio frontier, and call it k. Suppose there is a riskless asset, and you want to combine the riskless asset with portfolio k. What are the mean and variance of the portfolio of riskless and portfolio k? Start with the following observation: k: portfolio of risky asset riskless asset mean E(rk) rf standard deviation (rk) 0.0 correlation (rk, rf) 0.0 Let rp denote return on the portfolio of k and riskless asset. Let x denote portfolio weight of k. Then we have E(rp) = x E(rk) + (1-x) rf, and var(rp) = x2 var(rk) + (1-x)2 var(rf) + 2 x (1-x) cov(rk, rf) = x2 var(rk) + (1-x)2 0.0 + 2 x (1-x) 0.0, so (rp) = x k. When you plot the mean and standard deviation of portfolio p, you will get a straight line connecting the riskless rate and the portfolio p. Open ch14.xls:2stocks, 1.5: Calculate the mean and standard deviation of the portfolio of a risky portfolio and riskless asset. Fill up the shaded area, and see what you get on the chart. You should get a straight line. [Optional material] Proof for getting a straight line. Since (rp) = x k, x =(rp)/k. Substitute this into E(rp): E(rp) = x E(rk) + (1-x) rf = rf + x [E(rk) - rf ] = rf + [(rp)/k] [E(rk) - rf ]. Recall Y = a + b*X plots a straight line with slope=b, and intercept=a. Treat E(rp) as Y variable, (rp) as X variable. Then, we have a straight line with slope=[E(rk) - rf ]/k, and intercept= rf. [End of optional material]. D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.10/14 A B C D E F G H 1 TWO STOCKS AND A RISK-FREE ASSET % in k st dev mean 2 Stock 1 Stock 2 0.0 0.00% 2.00% 3 Average return 7.00% 15.00% 0.1 0.75% 2.58% 4 Sigma 8.00% 14.00% 0.2 1.49% 3.16% 5 Correlation 0.10 0.3 2.24% 3.74% 6 Risk-free rate 2% 0.4 2.99% 4.32% 7 0.5 3.74% 4.90% 8 weight in portfolio k 0.9 0.1 0.6 4.48% 5.48% 9 mean of portfolio k 0.0780 0.7 5.23% 6.06% 10 st dev of portfolio k 0.0747 0.8 5.98% 6.64% 11 0.9 6.72% 7.22% 12 1.0 7.47% 7.80% 13 Expected Return and Standard Deviation of 14 Portfolio Return % in k st dev mean 15 -0.3 0.1812 0.1740 20% 16 -0.2 0.1672 0.1660 18% 17 -0.1 0.1534 0.1580 Expected portfolio return, E(r p) 16% 18 0.0 0.1400 0.1500 19 14% 0.1 0.1270 0.1420 20 12% 0.2 0.1147 0.1340 g 21 10% 0.3 0.1032 0.1260 22 8% 0.4 0.0928 0.1180 23 6% 0.5 0.0840 0.1100 24 4% 0.6 0.0773 0.1020 25 2% h 0.7 0.0733 0.0940 26 0% 0.8 0.0724 0.0860 27 0% 5% 10% 15% 20% 0.9 0.0747 0.0780 28 Standard deviation of portfolio return, p 1.0 0.0800 0.0700 29 1.1 0.0877 0.0620 30 1.2 0.0973 0.0540 31 1.3 0.1082 0.0460 32 1.4 0.1201 0.0380 Note: A point on the blue line represents a particular portfolio of portfolio k and riskless asset. Portfolios g and h are indicated on the blue line. For the blue line, note that slope = [E(rp)-rf]/p, and intercept = rf. [E(rp)-rf]/p is known as the Sharpe ratio, which is the reward-risk measure. 14.2 and 14.3 Finding out the tangency portfolio: Sharpe ratio Can you have a better combination of a risky asset portfolio and riskless asset than the ones on blue line? *Open ch13.xls:2stocks,2-4. Yes. The best line is called the Capital Market Line (CML). D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.11/14 Note: The Capital Market Line is tangent to the market portfolio, M. Why? In equilibrium, supply equals to demand. So, the tangency portfolio should contain all risky assets in the economy in a proportion to a stock’s market value. Now, the efficient frontier is equal to the Capital Market Line, whereas previously it was top part of the curve. Open ch14.xls:CML portfolios. If portfolio weight on market portfolio is: (This is section 14.2.) zero: you invest all of your money in riskless asset, one: you invest all of your money in market portfolio, 0-1: your portfolio is somewhere between riskfree rate and point M, > 1: you borrow money at riskfree rate and invest in market portfolio. How do you find out tangency portfolio? It has the highest Sharpe ratio. Use Solver to find out the tangency portfolio. Open ch14.xls: solver Sharpe. A B C D E F PORTFOLIO RETURNS WITH A RISK-FREE ASSET 1 THE SHARPE RATIO: Using Solver to get the tangency portfolio 2 Stock A Stock B Risk-free 3 mean return 7.00% 15.00% 2.00% 4 st dev 8.00% 14.00% 5 covariance 0.0011 6 7 Portfolio % in A % in B 8 24.00% 76.00% 9 10 mean return 13.08% <-- =B8*B3+C8*C3 11 st dev 11.00% <-- =SQRT(B8^2*B4^2+C8^2*C4^2+2*B8*C8*B5) 12 13 Sharpe ratio 1.0073 <-- =(B10-D3)/B11 D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.12/14 Note: You can get the following Answer Report from Solver. After clicking “Solve” button, check “Restore original values” box, and click “Answers”. Microsoft Excel 9.0 Answer Report Worksheet: [ch13.xls]Sharpe ratio Report Created: 2/28/2004 11:33:16 AM Target Cell (Max) Cell Name Original Value Final Value $B$13 Sharpe ratio % in A 1.0073 1.0716 Adjustable Cells Cell Name Original Value Final Value $B$8 % in A 24.00% 51.81% Constraints NONE 14.4 Security Market Line (SML) The Capital Market Line describes risk-return relation for a portfolio on the straight line, which represents the efficient frontier. This risk-return relation holds only for portfolios on the efficient frontier. Question: What is risk-return relation for any individual stock or any portfolios (of risky assets, or of risky assets and riskless asset)? Answer: The security market line, or the Capital Asset Pricing Model (CAPM). The CAPM states that the expected return of any individual asset or any portfolio is determined by the asset’s risk (called ), the risk-free rate, and the market portfolio (or, the portfolio which maximizes the Sharpe ratio). The CAPM states: E(rp) = rf + p [E(rM) – rf], where p cov(rp, rM)/var(rM). Note: The portfolio p can be a combination of any number of risky assets, so it can be a single risky asset. The portfolio p can also be a combination of any number of risky assets and the risk-free asset. Note that beta of a portfolio is equal to weighted average of beta of individual stock in the portfolio. For example, beta of a portfolio of GM and MSFT is given by p = xGM GM + xMSFT MSFT. In next chapter, we use the CAPM relation to evaluate investment performance of a fund manager. D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.13/14 Appendix: Multiple assets So far, we have only considered portfolios of two risky assets. How can we deal with more than two assets? It is most convenient to use matrices in this case. We will briefly look at basics in matrix operation. Refer to the hand-out on matrix: definition, matrix addition, transpose, multiplication, inverse, solving a linear equation system. Open Matrix.xls: solver 4 assets. Find out the tangency portfolio Recall how we find out the Capital Market Line: we use Excel Solver to find out portfolio weights, which maximize the Sharpe ratio. We use the same approach here. The only difference is that we use matrix operation to calculate the mean and variance of portfolio returns. A B C D E F G 1 Finding tangency portfolio: Multiple assets 2 Variance-covariance matrix mean returns 3 0.40 0.03 0.02 0.00 0.06 4 0.03 0.20 0.00 -0.06 0.05 5 0.02 0.00 0.30 0.03 0.07 6 0.00 -0.06 0.03 0.10 0.08 7 riskfree riskfree 8 0.02 0.05 9 portfolio A portfolio B 10 0.0506 0.0314 weight #1 11 0.2857 0.2059 weight #2 12 0.0604 0.0597 weight #3 13 0.6034 0.7031 weight #4 14 portfolio 1.00 1.00 sum 15 mean 0.0698 0.0726 <--=MMULT(TRANSPOSE(C10:C13),$E3:$E6) <-- =MMULT(TRANSPOSE(C10:C13), 16 variance 0.0374 0.0450 MMULT($A$3:$D$6,C10:C13)) 17 st dev 0.1933 0.2121 18 Sharpe ratio 0.2577 0.1065 <-- =(C15-C8)/C17 19 20 cov(r(a), r(b)) 0.0395 <-- =MMULT(TRANSPOSE(B10:B13), MMULT(A3:D6,C10:C13)) Note: D:\Docstoc\Working\pdf\956af5a8-ec03-41f8-a290-beca3b0baef8.doc 9/19/2011 p.14/14 Let xA and xB denote portfolio weights for portfolio A and B. Let V denote variance-covariance matrix in A3::D6. Then variance of returns of portfolio A, and covariance of returns of portfolio A and B are given by var(rA) = xAT V xA cov(rA, rB) = xAT V xB Portfolio A maximizes the Sharpe ratio when risk-free rate is 2%, and Portfolio B maximizes the Sharpe ratio when risk-free rate is 5%. Both are on efficient frontier. Suppose you want to draw entire efficient frontier. How can we do this? We only know two portfolios on efficient frontier. There is a very convenient property of efficient frontier: linear combination of two efficient portfolio generates entire efficient frontier. Remember how we draw portfolio frontier in section 12.5. Follow exactly the same procedure.