Unformatted text preview: Ch 03 Tool Kit 5/26/2002 Chapter 3. Tool Kit for Risk and Return
The relationship between risk and return is a fundamental axiom in finance. Generally speaking, it is totally logical to assume
that investors are only willing to assume additional risk if they are adequately compensated with additional return. This idea is
rather fundamental, but the difficulty in finance arises from interpreting the exact nature of this relationship (accepting that risk
aversion differs from investor to investor). Risk and return interact to determine security prices, hence its paramount importance
in finance.
PROBABILITY DISTRIBUTION
The probability distribution is a listing of all possible outcomes and the corresponding probability.
Demand for the
company's products Probability of this
demand occurring Strong
Normal
Weak 0.30
0.40
0.30
1.00 Rate of Return on stock
if this demand occurs
Martin Products
U.S. Water
100%
20%
15%
15%
70%
10% EXPECTED RATE OF RETURN
The expected rate of return is the rate of return that is expected to be realized from an investment. It is determined as the weighted
average of the probability distribution of returns.
Demand for the
company's products
Strong
Normal
Weak Probability of this
demand occurring Martin Products
Rate of Return
Product 0.3
100%
0.4
15%
0.3
70%
1.0
EXPECTED RATE OF RETURN, r hat 30%
6%
21%
= U.S. Water
Rate of Return
20%
15%
10% 15% Product
6%
6%
3%
15% MEASURING STANDALONE RISK: THE STANDARD DEVIATION
To calculate the standard deviation, there are a few steps. First find the differences of all the possible returns from the expected
return. Second, square that difference. Third, multiply the squared number by the probability of its occurrence. Fourth, find the sum
of all the weighted squares. And lastly, take the square root of that number. Let us apply this procedure to find the standard deviation
of Martin Products' returns.
Demand for the
company's products
Strong
Normal
Weak Probability of this
demand occurring
0.3
0.4
0.3 Deviation from r hat Squared deviation
Martin Products
85%
72.25%
0%
0.00%
85%
72.25%
Sum:
Std. Dev. = Square root of sum Sq Dev * Prob.
21.68%
0.00%
21.68%
43.35%
65.84%
65.84% Sq. root can be
found in two ways Strong
Normal
Weak Probability of this
demand occurring
0.3
0.4
0.3 5%
0%
5% U.S. Water
0.25%
0.00%
0.25% Std. Dev. = Square root of sum 0.08%
0.00%
0.07%
0.15%
3.87%
3.87% Sq. root can be
found in two ways MEASURING STANDALONE RISK: THE COEFFICIENT OF VARIATION
The coefficient of variation indicates the risk per unit of return, and is calculated by dividing the standard deviation by the expected
return.
Std. Dev.
Expected return
CV
Martin Products
65.84%
15%
4.39
U.S. Water
3.87%
15%
0.26 PORTFOLIO RETURNS
The expected return on a portfolio is simply a weighted average of the expected returns of the individual assets in the portfolio.
Consider the following portfolio.
Stock
Microsoft
General Electric
Pfizer
CocaCola
Portfolio's Expected Return Portfolio weight
0.25
0.25
0.25
0.25 Expected Return
12.0%
11.5%
10.0%
9.5%
10.75% PORTFOLIO RISK
Perfect Negative Correlation. The standard deviation of a portfolio is generally not a weighted average of individual standard
deviationsusually, it is much lower than the weighted average. The portfolio's SD is a weighted average only if all the securities in it
are perfectly positively correlated, which is almost never the case. In the equally rare case where the stocks in a portfolio are perfectly
negatively correlated, we can create a portfolio with absolutely no risk. Such is the case for the next example of Portfolio WM, a
portfolio composed equally of Stocks W and M.
Portfolio WM
Year
Stock W returns
Stock M returns (Equally weighted avg.)
1998
40%
10%
15%
1999
10%
40%
15%
2000
35%
5%
15%
2001
5%
35%
15%
2002
15%
15%
15%
Average return
15%
15%
15%
Standard deviation
22.64%
22.64%
0.00%
Correlation Coefficient
1.00
These two stocks are perfectly negatively correlatedwhen one goes up, the other goes down by the same amount. We could use
Excel's correlation function to find the correlation, but when exact positive or negative correlation occurs, an error message is given.
We demonstrate correlation in a later section. Perfect Positive Correlation. Now suppose the stocks were perfectly positively correlated, as in the following example:
Year
1998
1999
2000
2001
2002
Average return
Standard deviation
Correlation Coefficient Stock M returns
10%
40%
5%
35%
15%
15%
22.64% Stock M' returns
10%
40%
5%
35%
15%
15%
22.64% Portfolio MM'
10%
40%
5%
35%
15%
15%
22.64%
1.00 With perfect positive correlation, the portfolio is exactly as risky as the individual stocks.
Partial Correlation. Now suppose the stocks are positively but not perfectly so, with the following returns. What is the portfolio's
expected return, standard deviation, and correlation coefficient?
Year
1998
1999
2000
2001
2002
Average return
Standard deviation
Correlation coefficient Stock W returns
40%
10%
35%
5%
15%
15%
22.64% Stock Y returns
28%
20%
41%
17%
3%
15%
22.57% Portfolio WY
34%
5%
38%
11%
9%
15%
20.63%
0.67 Here the portfolio is less risky than the individual stocks contained in it.
We found the correlation coefficient by using Excel's "CORREL" function. Click the wizard, then Statistical, then CORREL, and
then use the mouse to select the ranges for stocks W and Y's returns. The correlation here is about what we would expect for two
randomly selected stocks. Stocks in the same industry would tend to be more highly correlated than stocks in different industries.
THE CONCEPT OF BETA
The beta coefficient reflects the tendency of a stock to move up and down with the market. An averagerisk stock moves equally up
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.
It is also useful to show graphs with individual stocks' returns on the vertical axis and market returns on the horizontal axis. The
slopes of the lines represent the stocks betas. We show a graph of the illustrative stocks in the screen to the right, and we use
Beta Graphs
regression to calculate betas below. We used the scatt
and stock returns Returns on The Market and on Stocks L (for Low), A (for Average), and H (for High)
rM
rL
rA
rH
10%
10%
10%
10%
20%
15%
20%
30%
10%
0%
10%
30% Regression analysis is performed by following the command path: Tools => Data Analysis => Regression. This will yield the
Regression input box. If Data Analysis is not an option in your Tools menu, you will have to load that program. Click on the AddIns
option in the Tools menu. When the AddIns box appears, click on Analysis ToolPak and a check mark will appear next to the
Analysis ToolPak. Then, click OK and you will now be able to access Data Analysis. From this point, you must designate the Y input
range (stock returns) and the X input range (market returns). You can have the summary output placed in a new worksheet, or you
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: 30% Stocks returns Year
2000
2001
2002 0% 30%
10% Note: When you get the menu box on the
screen, and the cursor blinking in the Y
Range slot, use the mouse to select the Y
range, and then click on the X range
box. Then fill in the X range the same
way. Regression Output of Stock H Returns
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations Beta Coefficient for Stock H = 2.00
1
1
1
0
3 ANOVA
df
Regression
Residual
Total SS
1
1
2 Coefficients
Intercept
X Variable 1 MS
0.19
0
0.19 Standard Error
0.1
2.00 0
0 F
0.19
0 t Stat
1.10E+016
3.11E+016 Significance F
9.69E+032 Pvalue
0
0 0 Lower 95%
Upper 95%
0.1
0.1
2
2 Regression Output of Stock A Returns
SUMMARY OUTPUT
Beta Coefficient for Stock A is 1
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 1
1
1
0
3 ANOVA
df
Regression
Residual
Total SS
1
1
2 Coefficients
Intercept
X Variable 1 MS
0.05
0
0.05 Standard Error
0
1.00 F
0.05
0 t Stat
0
0 #NUM! Pvalue
65535
65535 Significance F
#NUM! Lower 95%
#NUM!
#NUM! 0
1 Upper 95%
0
1 Regression Output of Stock L Returns
SUMMARY OUTPUT
The beta coefficient for Stock L is .5
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 1
1
1
0
3 ANOVA
df
Regression
Residual
Total SS
1
1
2 Coefficients
Intercept
X Variable 1 MS
0.01
0
0.01 Standard Error
0.05
0.5 0
0 F
0.01
0 t Stat
3.67E+015
5.19E+015 Significance F
2.69E+031 Pvalue Lower 95%
0
0 CALCULATING THE BETA COEFFICIENT FOR AN ACTUAL COMPANY
Now we show how to calculate beta for an actual company, WalMart Stores.
Step 1. Acquire Data
We downloaded stock prices and dividends from http://finance.yahoo.com for WalMart, using its ticker symbol
WMT. We also downloaded data for the S&P 500 Index (^SPX), which contains 500 actively traded large stocks.
Step 2. Calculate Returns
We used the percentage change in adjusted prices (which already reflect dividends) for WalMart to calculate
returns. We used the percentage change for the S&P 500 Index as the market return.
Now go to the bottom of the data, row 310.
Date
August01
July01
June01
May01
April01
March01
February01
January01
December00
November00
October00
September00
August00
July00
June00
May00
April00
March00
February00
January00
December99 Market Level (S&P 500
Index)
1,294.0
1,341.0
1,386.8
1,424.2
1,250.3
1,257.4
1,414.5
1,476.0
1,454.7
1,532.3
1,566.8
1,696.0
1,739.6
1,731.7
1,746.7
1,663.3
1,663.8
1,712.0
1,674.2
1,677.2
1,715.0 Market Return
3.5%
3.3%
2.6%
13.9%
0.6%
11.1%
4.2%
1.5%
5.1%
2.2%
7.6%
2.5%
0.5%
0.9%
5.0%
0.0%
2.8%
2.3%
0.2%
2.2%
4.9% WalMart Adjusted
Stock Price
47.976
55.814
48.725
51.596
51.586
50.350
49.868
56.548
52.890
51.891
45.117
47.852
47.302
54.875
57.234
57.172
54.940
56.056
48.306
54.251
68.495 0 WalMart Return
14.0%
14.5%
5.6%
0.0%
2.5%
1.0%
11.8%
6.9%
1.9%
15.0%
5.7%
1.2%
13.8%
4.1%
0.1%
4.1%
2.0%
16.0%
11.0%
20.8%
20.0% 0.05
0.5 Upper 95%
0.05
0.5 November99
October99
September99
August99
July99
June99
May99
April99
March99
February99
January99
December98
November98
October98
September98
August98
July98
June98
May98
April98
March98
February98
January98
December97
November97
October97
September97
August97 1,635.2
1,521.5
1,550.6
1,547.4
1,614.8
1,538.7
1,533.4
1,527.6
1,456.2
1,470.6
1,451.7
1,367.6
1,309.7
1,114.9
1,117.3
1,134.7
1,300.8
1,252.7
1,254.6
1,255.3
1,200.9
1,145.7
1,058.7
1,066.7
1,052.4
994.0
1,057.3
1,049.4 7.5%
1.9%
0.2%
4.2%
4.9%
0.3%
0.4%
4.9%
1.0%
1.3%
6.2%
4.4%
17.5%
0.2%
1.5%
12.8%
3.8%
0.2%
0.1%
4.5%
4.8%
8.2%
0.7%
1.4%
5.9%
6.0%
0.8%
NA 57.057
55.758
47.094
43.829
41.789
47.724
42.111
45.445
45.538
42.499
42.437
40.185
37.125
34.044
26.927
29.048
31.078
29.909
27.103
24.860
24.983
22.736
19.545
19.361
19.635
17.153
17.950
17.368 2.3%
18.4%
7.4%
4.9%
12.4%
13.3%
7.3%
0.2%
7.2%
0.1%
5.6%
8.2%
9.0%
26.4%
7.3%
6.5%
3.9%
10.4%
9.0%
0.5%
9.9%
16.3%
1.0%
1.4%
14.5%
4.4%
3.4%
NA Average (Annual)
Standard deviation
(Annual)
Correlation between WalMart and the market.
Beta (using the SLOPE function) 6.9% 31.4% 18.7%
27.4%
0.51 34.5% Step 3. Examine the Data Using the AVERAGE function and the STDEV function, we found the average historical return and standard
deviation for WalMart and the market. (We converted these from monthly figures to annual figures. Notice that you
must multiply the monthly standard deviation by the square root of 12, and not 12, to convert it to an annual basis.)
These are show in the rows above. Notice that WalMart has a standard deviation about twice that of the market. We
also used the CORREL function to find the correlation between WalMart and the market.
Step 4. Plot the Data and Calculate Beta
Using the Chart Wizard, we plotted the WalMart returns on the yaxis and the market returns on the xaxis. We
also used the menu Chart > Options to add a trend line, and to display the regression equation and R2 on the chart.
The chart is shown below.
Step 4. Interpret the Results
The beta coefficient is about .51, as shown by the slope of the coefficient in the regression equation on the chart. The
R2 of about 0.08 indicates that 8% of the variance in the stock return can be explained by the market. The rest of the
stock's variance is due to factors other than the market. If we had done this regression for a portfolio of 50 well
diversified stocks, we would have gotten an R2 of over 0.9. 30% 20% 10% 30% 20% 10% Historic Realized Returns
on WalMart, rS 0% 10% 20% 30%
30% 20% 10% 0% 10% 20% 30% Historic Realized Returns
on the Market, rM THE SECURITY MARKET LINE
The Security Market Line shows the relationship between a stock's beta and its expected return.
Riskfree rate (Varies over time)
Market return (Also varies over time)
Beta (Varies by company) 6%
11%
0.5 Required Return for a stock with beta = 0.50 8.5% With the above data, we can generate a Security Market Line that will be flexible enough to allow for changes in
any of the input factors. We generate a table of values for beta and expected returns, and then plot the graph as a scatter diagram. Beta
0.00
0.50
1.00
1.50
2.00 Required Return
8.5%
6.0%
8.5%
11.0%
13.5%
16.0% In order to create
To create a data t
variable changing
changes. Therefo
cell that is diagon
is originally perfo Then, we highligh (B371 to C376, in
the menu choices.
of variables is arr
input cell, like thi Security Market Line Required Return 18% 12% 6% 0%
0.00 Click on OK, and
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Beta The Security Market Line shows the projected changes in expected return, due to changes in the beta coefficient. However, we can also
look at the potential changes in the required return due to variation of other factors, namely the market return and riskfree rate. In
other words, we can see how required returns can be influenced by changing inflation and risk aversion. The level of investor risk
aversion is measured by the market risk premium (rmrrf), which is also the slope of the SML. Hence, an increase in the market
return results in an increase in the maturity risk premium, other things held constant. We will look at two potential conditions as shown in the following columns:
OR
Scenario 1. Inflation Increases:
Riskfree Rate
Change in inflation
Old Market Return
New Market Return
Beta
Required Return 6%
2%
11%
13%
0.50
10.5% Scenario 2. Investors become more risk averse:
Riskfree Rate
6%
Old Market Return
11%
Increase in MRP
2.5%
New Market Return
13.5%
Beta
0.50
Required Return 9.75% Now, we can see how these two factors can affect a Security Market Line, by creating a data table for the required return with
different beta coefficients.
Required Return
Beta
Original Situation
New Scenario 1
New Scenario 2
8.5%
10.5%
9.75%
0.00
6.00%
8.00%
6.00%
0.50
8.50%
10.50%
9.75%
1.00
11.00%
13.00%
13.50%
1.50
13.50%
15.50%
17.25%
2.00
16.00%
18.00%
21.00% The SML Under Different Conditions
23%
Required Returns 20%
18%
15% Column B
Column C
Column D 13%
10%
8%
5%
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Beta The graph shows that as risk as measured by beta increases, so does the required rate of return on securities. However, the required
return for any given beta varies depending on the position and slope of the SML. We used the scatter diagram procedure to calculate the graphs, with Market returns on the horizontal axis
and stock returns on the vertical axis. Beta Graph Stocks returns 30% Column C 0% 30%
10% Column D
Column E 0% 10% 20% Market returns
It is easy to see from the graph that H has the steepest slope, hence is
most volatile, which means most risky. This is confirmed by the
regression analysis, where betas are calculated. Lower 95.0%
Upper 95.0%
0.1
0.1
2
2 Lower 95.0%
Upper 95.0%
0
0
1
1 Lower 95.0%
Upper 95.0%
0.05
0.05
0.5
0.5 In order to create the data table that appears to the left, we used the "Data Table" feature in Excel.
To create a data table, you must determine a calculation you would like to see simulated with a
variable changing. In this case, we want to see the different calculations of expected return if beta
changes. Therefore, we set up a vertical column of possible beta values (or, a "beta range"). In the
cell that is diagonally up and to the right of this column, we reference the cell in which the calculation
is originally performed (F364, in this case). Then, we highlight all of the cells that encompass the domain of the beta range and the calculation cell (B371 to C376, in this case). Then we click on "Data" from the menus above, and select "Table" from
the menu choices. A dialog box will appear asking for the column or row input cell. In this case, our
of variables is arranged in column format, so we will enter the variable we want to change into the column
input cell, like this: Click on OK, and Excel will fill out the rest of the data table for you. ...
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 Standard Deviation, The Market, Errors and residuals in statistics, returns

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