# Chapter 11, Return, Risk and the Security Market Line - Download as Excel

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```					            A                B                 C                D                 E              F
29   EXPECTED AND REALIZED RATES OF RETURN (PAGE 120)
30
31   The probability distribution is a listing of all possible outcomes and their corresponding probabilities.
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33     State of the   Probability of             Rate of Return if
34      Economy        Occurrence                 State Occurs
35                                            MRI             Clinic
36     Very poor            0.10              -10%            -20%         Change the probability inputs
37       Poor               0.20               0%              0%          to see the effect on expected rate
38      Average             0.40               10%             15%         of return. Note that the probabilities
39       Good               0.20               20%             30%         must sum to 1.00.
40     Very good            0.10               30%             50%
41                          1.00
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43
44   The expected rate of return is the weighted average of the returns estimated for each state of the economy.
45
46                         E(RMRI) =               10.0%
47                         E(RClinic) =            15.0%
48
49
50   STAND-ALONE RISK (PAGE 122)
51
52   Several steps are required to calculate standard deviation. First, find the differences of all the possible returns from the expected
53   return. Second, square that difference. Third, multiply the squared number by the probability of its occurrence. Fourth, find the sum
54   of all the weighted squares--this is the variance. Fifth, take the square root of the variance to find the standard deviation.
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56
57     State of the   Probability of                         MRI
58     Economy          Occurrence        Dev from E(R)   Deviation2       Dev2 * Prob.
59     Very poor           0.10               -20%           400                40         Change the probabilities to see
60       Poor              0.20               -10%           100                20         the effect on risk. For example,
61      Average            0.40                0%             0                  0         "stretch" them by having higher
62       Good              0.20                10%           100                20         probabilities at the extremes, or
63     Very good           0.10                20%           400                40         "close" them by increasing the
64                         1.00                       Sum = Variance =         120         middle values.
65
66
67                    Standard deviation = Square root of variance =         11.0%
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69
70
71     State of the   Probability of                          Clinic
2      2
72     Economy          Occurrence        Dev from E(R)   Deviation        Dev * Prob.
73     Very poor           0.10               -35%           1225             123          Change the probabilities to see
74       Poor              0.20               -15%            225              45          the effect on risk. For example,
75      Average            0.40                0%              0                0          "stretch" them by having higher
76       Good              0.20                15%            225              45          probabilities at the extremes, or
77     Very good           0.10                35%           1225             123          "close" them by increasing the
78                         1.00                       Sum = Variance =        335          middle values.
79
80
81                    Standard deviation = Square root of variance =         18.3%
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83
A                  B               C                D                 E               F
84    The coefficient of variation (CV) indicates the risk per unit of return.
85
86                          Std. Dev.         E(R)              CV
87    MRI                    11.0%           10.0%             1.10
88    Clinic                 18.3%           15.0%             1.22
89
90
91    PORTFOLIO RISK AND RETURN (PAGE 124)
92
93    Consider the following individual investments and portfolios.
94
95      State of the     Probability of
96       Economy          Occurrence                                                Rate of Return If State Occurs
97                                             A                B                  C              D
98      Very poor             0.10           -10%              30%               -25%            15%
99        Poor                0.20            0%               20%                -5%            10%
100      Average              0.40            10%              10%                15%            0%
101       Good                0.20            20%              0%                 35%            25%
102     Very good             0.10            30%             -10%                55%            35%
103                           1.00
104
105            Expected rate of return =     10.0%            10.0%              15.0%          12.0%
106
107                Standard deviation =      11.0%            11.0%              21.9%          12.1%
108
109
110   Note that the standard deviation of a portfolio is generally not a weighted average of individual component standard deviations.
111   It is a weighted average only in the rare case of perfect positive correlation, which occurs in Portfolio AC.
112
113   In the equally rare case where the returns are perfectly negatively correlated, such as in Portfolio AB, the portfolio has zero risk.
114
115   We can use the Function Wizard to find the correlation coefficient between two returns distributions. We have copied the returns
116   into the rows below to make the illustration easier. Here is the dialog box for the results given in Cell F132.
117   Click the Function Wizard, then Statistical, then CORREL, and then use the mouse to mark the ranges for the returns on A and B.
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131
132          A                 B
133        -10%               30%
134         0%                20%                                Correlation coefficient =       -1.00
135         10%               10%
136         20%               0%                         These two investments are perfectly negatively correlated.
137         30%              -10%
138
139
A               B               C                D                E                F
140
141
142          A               C                                Correlation coefficient =       1.00
143        -10%            -25%
144         0%              -5%                        These two investments are perfectly positively correlated.
145         10%             15%
146         20%             35%
147         30%             55%
148
149
150          A              D                                 Correlation coefficient =       0.64
151        -10%            15%
152         0%             10%                         These two investments are positively correlated, but not perfectly so. This is about
153         10%            0%                          what would be expected on two randomly chosen investments.
154         20%            25%
155         30%            35%
156
157
158
159   THE CALCULATION OF BETA (NO MATCHING TEXT SECTION)
160
161   The beta coefficient reflects the tendency of a stock to move up and down with the market. An average-risk stock moves equally up
162   and down with the market and has a beta of 1.0. Beta is found by regressing the stock's returns against returns on some market index.
163   It is also useful to show graphs with individual stocks' returns on the vertical axis and market returns on the horizontal axis. The
164   slopes of the lines represent the stock betas. We will illustrate the calculation with the data below:
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166
A                B                C                D             E                F
167                                            Historical Returns
168        Year            Market           Stock L         Stock M        Stock H
169        1998             10%              10%               10%           10%         Changing these data will have no effect
170        1999             20%              15%               20%           30%         on the results unless the regressions
171        2000            -10%               0%              -10%          -30%         are rerun.
172
173   Regression analysis is performed by following the command path: Tools => Data Analysis => Regression. This will yield the
174   Regression input box. If Data Analysis is not an option in your Tools menu, you will have to load that program. Click on the Add-Ins
175   option in the Tools menu. When the Add-Ins box appears, click on Analysis ToolPak, and a check mark will appear next to the
176   Analysis ToolPak. Then, click OK and you will now be able to access Data Analysis. From this point, you must designate the Y input
177   range (stock returns) and the X input range (market returns). You can have the summary output placed in a new worksheet, or you
178   can have it shown directly in the worksheet, as we did here. The Regression dialog box for the regression of Stock H is as follows:
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199
200   Here are the results of the three regressions:
201
202   SUMMARY OUTPUT
203
204        Regression Statistics                       Beta Coefficient for Stock H = 2.00   2.00
205   Multiple R                  1
206   R Square                    1
207   Adjusted R Square           1
208   Standard Error    1.38778E-17
209   Observations                3
210
A              B                   C                 D               E                F
211   ANOVA
212                         df                 SS                MS               F       Significance F
213   Regression                      1     0.186666667       0.186666667     9.69229E+32    2.04488E-17
214   Residual                        1     1.92593E-34       1.92593E-34
215   Total                           2     0.186666667
216
217                    Coefficients Standard Error              t Stat         P-value        Lower 95%
218   Intercept                  -0.1  9.08514E-18            -1.1007E+16      5.78378E-17            -0.1
219   X Variable 1                 2.00     6.42417E-17      3.11324E+16       2.04488E-17                2
220
221
222
223   SUMMARY OUTPUT
224                                                        Beta Coefficient for Stock M =        1.00
225        Regression Statistics
226   Multiple R                      1
227   R Square                        1
228   Adjusted R Square               1
229   Standard Error                  0
230   Observations                    3
231
232   ANOVA
233                         df                 SS                MS               F          Significance F
234   Regression                      1     0.046666667       0.046666667       #NUM!            #NUM!
235   Residual                        1               0                 0
236   Total                           2     0.046666667
237
238                    Coefficients       Standard Error        t Stat         P-value        Lower 95%
239   Intercept                       0                0             65535     #NUM!                      0
240   X Variable 1                 1.00               0               65535     #NUM!                     1
241
242
243
244   SUMMARY OUTPUT
245                                                        Beta Coefficient for Stock L =        0.50
246        Regression Statistics
247   Multiple R                  1
248   R Square                    1
249   Adjusted R Square           1
250   Standard Error    2.08167E-17
251   Observations                3
252
253   ANOVA
254                         df                 SS                MS               F       Significance F
255   Regression                      1     0.011666667       0.011666667      2.6923E+31    1.22693E-16
256   Residual                        1     4.33334E-34       4.33334E-34
257   Total                           2     0.011666667
258
259                    Coefficients Standard Error             t Stat          P-value        Lower 95%
260   Intercept                  0.05  1.36277E-17           3.66899E+15       1.73513E-16            0.05
261   X Variable 1                  0.5     9.63625E-17      5.18874E+15       1.22693E-16              0.5
A                      B               C                   D              E                F
262
263   Here is a graph of the regressions:
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265
266                                                                       REGRESSION RESULTS
267
268                                                                     30%
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271
Stocks returns

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274                                                                       0%
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-10%                        0%                    10%                    20%
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280                                                                    -30%
281
282                                                                                 Market returns
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285
286   THE RELATIONSHIP BETWEEN RISK AND RETURN (PAGE 143)
287
288   The Security Market Line (SML) shows the relationship between portfolio risk (as measured by beta) and required rate of return.
289
290   Risk-free rate                                             6%             Change the inputs to the SML
291   Required return on the market                              10%            to see the impact on required
292   Beta of Regis Healthcare                                    1.1           return on equity.
293
294                                                    R(R) =    10.4%
295
296   With the above data, we can generate a Security Market Line that will be flexible enough to allow for changes in
297   any of the input factors. We generate a table of values for beta and required return, and then plot the graph as a scatter diagram.
298
299                        Beta           R(R)                                 To create the data table that appears to the left, we used the "Data Table" feature in Excel.
300                                      10.4%                                 First, you must determine the output that you would like to see calculated for a number of different
301                        0.00          6.0%                                  values. In this case, we want to see the effect of different betas on the required rate of return.
302                        0.50          8.0%                                  of return. Therefore, we set up a column of possible beta values in A301 to A305. In the cell that
303                        1.00          10.0%                                 is diagonally up and to the right of this column (B300), we reference the cell that contains the
304                        1.50          12.0%                                 desired output value (C294 in this case). Excel will fill in values to the right of the beta entries.
305                        2.00          14.0%                                 Now, highlight the entire range of the data table (A300 to B305), including the reference cell.
306                                                                            Then click on "Data" from the menu bar and select "Table" from the menu choices.
307                                                                            A dialog box will appear asking for the column or row input cell. In this case, our variables are
308                                                                            arranged in column format, so we click on the column cell to put the cursor there. Then, we
309                                                                            enter the input cell where the beta values enter the calculation (C292). Finally, click on OK and
310                                                                            Excel will fill in values for required rate of return in B301 to B305.
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312
313                                                Security Market Line
314
315                         18%
316
317
Required Return

318                         12%
Required Return

A               B                     C                  D                E                F
12%
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322                        6%
323
324
325                        0%
326                              0.00   0.50       1.00          1.50        2.00       2.50
327                                                       Beta
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330
331   The Security Market Line shows the relationship between required return and beta. However, we can also examine the impact of
332   other factors, namely the risk-free rate and the required return on the market. In other words, we can see how required returns
333   are influenced by changes in inflation expectations and risk aversion.
334
335   We will look at two potential conditions, as shown in the following columns:
336
337                                                          OR
338   Scenario 1. Inflation increases:                                                         Scenario 2. Investors become more risk averse:
339
340   Change in inflation                                               2%                     Increase market risk prem.
341
342   Old risk-free rate                                               6%                      Old and new risk-free rate
343   Old market return                                               10%                      Old market return
344   New risk-free rate                                               8%                      New market return
345   New market return                                               12%
346                                                                                                                       R(R) =
347                                            R(R) =                12.4%
348
349
A                B                C                D             E   F
350 Now, we can see how these two factors affect the Security Market Line:
351
352                                 Required Return
353      Beta          Original       Scenario 1        Scenario 2
354                     10.4%            12.4%           13.15%
355      0.00           6.00%            8.00%            6.00%
356      0.50           8.00%           10.00%            9.25%
357      1.00          10.00%           12.00%           12.50%
358      1.50          12.00%           14.00%           15.75%
359      2.00          14.00%           16.00%           19.00%
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361
The SML Under Different Conditions
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364        20%
365        18%
366        16%
Required Returns

367        14%
368        12%                                                Original
369        10%                                                Scenario #1
370         8%                                                Scenario #2
371         6%
372         4%
373         2%
374         0%
375            0.00   0.50     1.00    1.50     2.00
376
Beta
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