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1、Section 12.3 Portfolio OptimizationThe Two Risky Asset Problem. The riskless rate is 6.0%. Risky Asset 1 has a mean return of 14.0% and a standard deviation of 20.0%. Risky Asset 2 has a mean return of 8.0% and a standard deviation of 15.0%. The correlation between Risky Asset 1 and 2 is 0.0%. Graph
2、 the Efficient Trade-Off Line and the Risky Asset Trade-Off Curve.Solution Strategy. Determine the Risky Asset Trade-Off Curve for two-asset portfolios by varying the proportion in the first asset and calculating the resulting portfolios standard deviation and expected return. Then, determine the Op
3、timal Combination of Risky Assets by calculating the optimal proportion in the first asset and calculating the corresponding standard deviation and expected return. Finally, determine the Efficient Trade-Off Line by varying the amount in the Optimal Combination and calculating the corresponding stan
4、dard deviation and expected return. Then graph everything.FIGURE 12.3.1 Spreadsheet for a Two Risky Asset Example of Portfolio OptimizationHow To Build Your Own Spreadsheet Model.1. Inputs. Enter the inputs described above into the ranges B6:B8 and C5:C7.2. Expected Return Riskless Rate. Calculate t
5、he Expected Return minus the Riskless Rate by entering =C5-$C$5 in cell D5 and copying it down to the range D6:D7.FIGURE 12.3.2 Spreadsheet Details for a Two Risky Asset Example of Portfolio Optimization.3. Proportion in Risky Asset 1. In order graph the Risky Asset Trade-off Curve, we need to evalu
6、ate a wide range of values (-60.0% to 140%) for the Proportion in Risky Asset 1. Enter 60.0% in cell B14, 50.0% in cell B15, and highlight the range B14:B15. Then hover the cursor over the lower right corner and it turns to a “fill handle” (which looks like a “+” sign). Drag the fill handle down to
7、B34.4. Standard Deviation. The x-axis of our graph is the portfolios standard deviation, which is calculated by the formula . Enter =SQRT(B142*$B$62+(1-B14)2*$B$72+2*B14*(1-B14)*$B$8*$B$6*$B$7) in cell C14 and copy the cell C14 to the range C15:C34.5. Expected Return. The formula for a portfolios ex
8、pected return is . Enter =B14*$C$6+(1-B14)*$C$7 in cell D14 and copy the cell D14 to the range D15:D34.6. Optimal Combination of Risky Assets. Using the notation that and , then the formula for the optimal proportion in the first asset is . In cell B35, enter =(D6*B72-D7*B8*B6*B7)/(D6*B72+D7*B62-(D6
9、+D7)*B8*B6*B7) Calculate the corresponding Mean and Standard Deviation by copying the range C34:D34 to the range D34:D35. We want to create a separate column for the Efficient Trade-Off Line, so cut the cell D35 and paste in it in cell E35.7. Efficient Trade-Off Line. The Efficient Trade-Off Line is
10、 a combination of the Riskless Asset and the Risky Asset Optimal Combination. It can be calculated as follows: Enter 0.0% in cell B36, 100.0% in cell B37, and 200.0% in cell B38. Since the Riskless Asset has a standard deviation of zero, the standard deviation formula simplifies to , where = standar
11、d deviation of the Optimal Combination of Risky Assets (or Tangent Portfolio). Enter =B36*$C$35 in cell C36 and copy to the range C37:C38. The Expected Return formula is , were expected return of the Tangent Portfolio. Enter =$E$35*B36+$C$5*(1-B36) in cell E36 and copy to the range E37:E38.8. Create
12、 And Locate The Graph. Highlight the range C14:E48 and then choose Insert Chart from the main menu. Select an XY(Scatter) chart type and make other selections to complete the Chart Wizard. Place the cursor in cell A11, click on Window Freeze Panes, and then scroll down so that the Graph is just belo
13、w the input area.9. (Optional) Formatting The Graph. Here are some tips to make the chart look attractive: Click on one of the Chart curves, then click on Format Selected Data Series. In the Format Data Series dialog box under the Patterns tab, select None for the Marker and click on OK. Repeat for
14、the other curve. Highlight individual points, such as the Riskless Asset, Tangent Portfolio, and Risky Assets 1 and 2, by clicking on a chart curve, then click a second time on an individual point (the four-way arrows symbol appears), then click on Format Selected Data Point. In the Format Data Poin
15、t dialog box under the Patterns tab under the Marker, select a market Style, Foreground Color, Background Color, and increase the size to 8 pts and click on OK. Click on the x-axis, then click on Format Selected Axis. In the Format Axis dialog box under the Scale tab, enter 0.25 for the Maximum and
16、click on OK.Investors prefer points on the graph that yield higher mean returns (further “North”) and lower standard deviations (further “West”). The graph shows that best combinations of high return and low risk (furthest in the “Northwest” direction) are given by the Efficient Trade-Off Line. Bett
17、er combinations are simply not feasible. Since the Efficient Trade-Off Line is a combination of the Riskless Asset and a Tangent Portfolio, then all investors prefer to invest only in the Riskless Asset and a Tangent Portfolio.Using The Power Of Your Spreadsheet Model.Suppose that you had N risky as
18、sets, rather than just two risky asset. How would you calculate the Efficient Trade-Off Line and the Risky Asset Trade-Off Curve in this case? It turns out that it is much easier to handle N risky assets in a spreadsheet than any other way. The figure below shows the results of the N=5 risky assets
19、case, including a bar chart of the portfolio weights of the optimal (tangent) portfolio.FIGURE 12.3.3 Spreadsheet for a Five Risky Asset Example of Portfolio Optimization.1. Inputs. Enter the standard deviation inputs (as shown in the figure above) into the range B6:B10, the expected return inputs i
20、nto the range C5:C10, 100% into the range E6:E10, and the correlation inputs in the triangular range from H7 to H10 to K10. 2. One plus the Expected Return. It will be useful to have a column based on . Enter =1+C5 in cell D5 and copy it to D6:D10.3. Fill Out the Correlations Table (Matrix). The cor
21、relations table (matrix) from H6:L10 has a simple structure. All of the elements on the diagonal represent the correlation of an asset return with itself. For example, H6 is the correlation of the Asset 1 return with the Asset 1 return, which is one. I7 is the correlation of Asset 2 with 2, and so o
22、n. Enter 100.0% into the diagonal cells from H6 to L10. The off-diagonal cells in the upper triangular range from I6 to L6 to L9 are the “mirror image” of the lower triangular range from H7 to H10 to K10. In other words, the correlation of Asset 2 with Asset 1 in I6 is equal to the correlation of As
23、set 1 with Asset 2 in H7. Enter =H7 in cell I6, =H8 in J6, =I8 in J7, etc. Each cell of the upper triangular range from I6 to L6 to L9 should be set equal to its mirror image cell in the lower triangular range from H7 to H10 to K10.FIGURE 12.3.4 Spreadsheet Details for a Five Risky Asset Example of
24、Portfolio Optimization.4. Transposed Standard Deviations. In addition to the standard deviation input range which runs vertically from top to bottom, it will be useful to have a range of standard deviations that runs horizontally from left-to-right. This can be done easily by using the matrix comman
25、d to transpose a range. Highlight the range H14:L14 and type =TRANSPOSE(B6:B10). Then, hold down the Shift and Control buttons simultaneously, and while continuing to hold them down, press Enter.5. Variances and Covariances Table (Matrix). The Variances and Covariances Table (Matrix) in the range H1
26、8:L22 has a simple structure. All of the elements on the diagonal represent the covariance of an asset return with itself, which equals the variance. For example, H18 is the covariance of the Asset 1 return with the Asset 1 return, which equals the variance of asset 1. I19 is the variance of Asset 2
27、, and so on. The off-diagonal cells are covariances. For example, H19 is the covariance of the Asset 1 return with the Asset 2 return and is calculated with the formula for the Covariance(Asset 1, Asset 2) = (Std Dev 1) * (Std Dev 2) * Correlation(Asset 1, Asset 2). Enter =H$14*$B7*H7 in cell H19. B
28、e very careful to enter the $ absolute references exactly right. Then copy H19 to the range H18:L22.6. Hyperbola Coefficients. In a Mean vs. Standard Deviation graph, the Efficient Frontier is a hyperbola. The exact location of the hyperbola is uniquely determined by three coefficients, unimaginativ
29、ely called A, B, and C. The derivation of the formulas can be found in Merton (1972). See Robert C. Merton, An Analytic Derivation of the Efficient Portfolio Frontier, Journal of Financial and Quantitative Analysis, September 1972, pp. 1851-72. The article uses slightly different notation. They are
30、easy to implement using the matrix functions of Excel. In each case, you type the formula and then, hold down the Shift and Control buttons simultaneously, and while continuing to hold them down, press Enter. For A: =MMULT(MMULT(TRANSPOSE(E6:E10),MINVERSE(H18:L22),E6:E10) in cell I24. For B: =MMULT(
31、MMULT(TRANSPOSE(E6:E10),MINVERSE(H18:L22),D6:D10) in cell I25. For C: =MMULT(MMULT(TRANSPOSE(D6:D10),MINVERSE(H18:L22),D6:D10) in cell I267. Miscellaneous. It will simplify matters to create range names for various cells. Put the cursor in cell I24, click on Insert Name Define, enter the name “A” an
32、d click on OK. Repeat this procedure to give cell I25 the name “B”, give cell I26 the name “C.” (Excel does not accept plain “C”), give cell I27 the name “Delta”, give cell I28 the name “Gamma”, and give cell D5 the name “R.” (again, Excel does not accept plain “R”). Enter =A*C.-(B2) in cell I27 and
33、 enter =1/(B-A*R.) in cell I28. Some restrictions do apply on the range of permissible input values. The variable Delta must always be positive or else the calculations will blow-up or produce nonsense results. Simply avoid entering large negative correlations for multiple assets and this problem wi
34、ll be taken care of.8. Individual Risky Assets. In order to add the individual risky assets to the graph, reference their individual standard deviations and expected returns. Enter =B6 in cell C17 and copy down to the range C18:C21. Enter =C6 in cell F17 and copy down to the range F18:F21.9. Expecte
35、d Return. Using the three Hyperbola coefficients, we can solve for the expected return on the upper and lower branches of the hyperbola which correspond to a standard deviation of 25% and then fill in intermediate values in order to generate the Efficient Frontier graph. For the upper branch, enter
36、=(2*B-(4*B2-4*A*(C.-(0.252)*Delta)(0.5)/(2*A)-1 in cell D22. For the lower branch, enter =(2*B+(4*B2-4*A*(C.-(0.252)*Delta)(0.5)/(2*A)-1 in cell D42. Fill out in index for expected return by entering 0 in cell B22, entering 1 in cell B23, selecting the range B22:B23, and dragging the fill handle (in
37、 the lower right corner) down the range B24:B42. Fill in the intermediate values by entering =$D$22+($D$42-$D$22)*(B23/20) in cell D23 and copying down the range D24:D41.10. Standard Deviation. Again using the three Hyperbola coefficients, we can solve for the Efficient Frontier standard deviation w
38、hich corresponds to any particular value of expected return. Enter =(A*(1+D22)2-(2*B*(1+D22)+C.)/(A*C.-(B2)(1/2) in cell C22 and copy it down the range C23:C42.11. Tangent Portfolio. The Optimal Combination of Risky Assets (or Tangent Portfolio) can be calculated using the matrix functions of Excel.
39、 In each case, you type the formula and then, hold down the Shift and Control buttons simultaneously, and while continuing to hold them down, press Enter. For the portfolio weights: Select the range H34:H38, then type =Gamma*MMULT(MINVERSE(H18:L22),(D6:D10-R.*E6:E10) For expected returns: enter =MMU
40、LT(TRANSPOSE(H34:H38),D6:D10)-1 in cell E43. For standard deviations: enter =SQRT(MMULT(MMULT(TRANSPOSE(H34:H38),H18:L22),H34:H38) in cell C43.12. Efficient Trade-Off Line. The Efficient Trade-Off Line is a combination of the Riskless Asset and the Risky Asset Optimal Combination. It can be calculat
41、ed as follows: Enter 0.0% in cell B44, 100.0% in cell B45, and 200.0% in cell B46. As before, the standard deviation formula simplifies to . Enter =B44*$C$43 in cell C44 and copy to the range C45:C46. The Expected Return formula is . Enter =$E$43*B44+$C$5*(1-B44) in cell E44 and copy to the range E4
42、5:E46.13. Create And Locate The Graphs. Highlight the range C17:F46 and then choose Insert Chart from the main menu. Select an XY(Scatter) chart type and make other selections to complete the Chart Wizard. Highlight the range H34:H38 and then choose Insert Chart from the main menu. Select a Column c
43、hart type and make other selections to complete the Chart Wizard. Place the cursor in cell A11, click on Window Freeze Panes, and then scroll down so that the Graph is just below the input area. Optionally, one can format the graph as discussed in the previous section.The graphs show several interes
44、ting things. First, look at the Efficient Frontier Curve. At a standard deviation of 20% (same as all of the individual assets), it is possible to achieve a mean return of nearly 13% despite the fact that 12% is the highest mean return offered by any individual asset. How is this possible? The answe
45、r is that it is possible to sell (or short sell) low mean return assets and use the proceeds to invest in high mean return assets. Said differently, put a negative portfolio weight (short sell) in low mean assets and “more than 100%” in high mean assets.Second, the bar chart shows that the optimal (
46、tangent) portfolio represents a trade-off between exploiting higher means vs. lowering risk by diversifying (e.g., spreading the investment across assets). On the one hand it is desirable to put a larger portfolio weight in those assets with higher mean assets (#4 and #5 in this example). On the oth
47、er hand, spreading assets more (getting closer to 20% per each of the five risky assets) would lower the overall risk of the portfolio. Hence, the optimal portfolio does not put 100% in the high mean assets, nor does it put 20% in each asset, but instead finds the best trade-off possible between the
48、ses two goals.Third, you should be delighted to find any mispriced assets, because these are delightful investment opportunities for you. Indeed, many investors spend money to collect information (do security analysis) which identifies mispriced assets. The bar chart shows you how to optimally exploit any mispriced assets that you find. Below is a list of experiments that you might wish to perform. Notice as you perform these experiments that the optimal po
