05model 4/17/2023 16:52 12/2/2002

Chapter 5. Model for Analyzing Risk and Rates of Return

The relationship between risk and return is a fundamental axiom in finance. Generally speaking, it is totally logical to assumethat investors are only willing to assume additional risk if they are adequately compensated with additional return. This ideais rather fundamental, but the difficulty in finance arises from interpreting the exact nature of this relationship (accepting thatrisk aversion differs from investor to investor). Risk and return interact to determine security prices, hence its paramountimportance in finance.

PROBABILITY DISTRIBUTION

The probability distribution is a listing of all possible outcomes and the corresponding probability.

Demand for the Probability of this Rate of Return on stockcompany's products demand occurring if this demand occurs

Martin Products U.S. WaterStrong 30% 100% 20%Normal 40% 15% 15%Weak 30% -70% 10%

100%

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 theweighted average of the probability distribution of returns.

Demand for the Probability of this Martin Products U.S. Electriccompany's products demand occurring Rate of Return Product Rate of Return Product

Strong 30% 100% 30% 20% 6%Normal 40% 15% 6% 15% 6%Weak 30% -70% -21% 10% 3%

100% EXPECTED RATE OF RETURN, k hat 15% 15%

MEASURING STAND-ALONE RISK: THE STANDARD DEVIATION

To calculate the standard deviation, there are a few steps. First find the differences of all the possible returns from theexpected 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 tofind the standard deviation of Martin Products' returns.

Demand for the Probability of this Deviation from k hat Squared deviation Sq Dev * Prob.

company's products demand occurring Martin ProductsStrong 30% 85% 72.25% 21.67%Normal 40% 0% 0.00% 0.00%Weak 30% -85% 72.25% 21.67%

Sum: 43.35%Std. Dev. = Square root of sum 65.84% Sq. root can be

65.84% found in two waysProbability of thisdemand occurring U.S. Electric

Strong 30% 5% 0.25% 0.08%Normal 40% 0% 0.00% 0.00%Weak 30% -5% 0.25% 0.07%

0.15%Std. Dev. = Square root of sum 3.87% Sq. root can be

3.87% found in two ways

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MEASURING STAND-ALONE 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 theexpected return.

Std. Dev. Expected return CVMartin Products 65.84% 15% 4.39 U.S. Electric 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 Portfolio weight Expected ReturnMicrosoft 0.25 12.0%General Electric 0.25 11.5%Pfizer 0.25 10.0%Coca-Cola 0.25 9.5%

Portfolio's Expected Return 10.75%

PORTFOLIO RISK

deviations--usually, it is much lower than the weighted average. The portfolio's SD is a weighted average only if all thesecurities in it are perfectly positively correlated, which is almost never the case. In the equally rare case where the stocks ina portfolio are perfectly negatively correlated, we can create a portfolio with absolutely no risk. Such is the case for the nextexample of Portfolio WM, a portfolio composed equally of Stocks W and M.

Portfolio WMYear Stock W returns Stock M returns (Equally weighted avg.)1997 40% -10% 15%1998 -10% 40% 15%1999 35% -5% 15%2000 -5% 35% 15%2001 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 correlated--when one goes up, the other goes down by the same amount. We canuse Excel's correlation function to find the correlation.

Year Stock M returns Stock M' returns Portfolio MM'1997 -10% -10% -10%1998 40% 40% 40%1999 -5% -5% -5%2000 35% 35% 35%2001 15% 15% 15%

Average return 15% 15% 15%Standard deviation 22.64% 22.64% 22.64%Correlation Coefficient 1.00

With perfect positive correlation, the portfolio is exactly as risky as the individual stocks.

Perfect Negative Correlation. The standard deviation of a portfolio is generally not a weighted average of individual standard

Perfect Positive Correlation. Now suppose the stocks were perfectly positively correlated, as in the following example:

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portfolio's expected return, standard deviation, and correlation coefficient?

Year Stock W returns Stock Y returns Portfolio WY1997 40% 28% 34%1998 -10% 20% 5%1999 35% 41% 38%2000 -5% -17% -11%2001 15% 3% 9%

Average return 15% 15% 15%Standard deviation 22.64% 22.57% 20.63%Correlation coefficient 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 expectfor two randomly selected stocks. Stocks in the same industry would tend to be more highly correlated than stocks in differentindustries.

THE CONCEPT OF BETA

The beta coefficient reflects the tendency of a stock to move up and down with the market. An average-risk stock moves equallyup and down with the market and has a beta of 1.0. Beta is found by regressing the stock's returns against returns on somemarket index. It is also useful to show graphs with individual stocks' returns on the vertical axis and market returns on thehorizontal axis. The slopes of the lines represent the stocks betas. We show a graph of the illustrative stocks in the screen tothe right, and we use regression to calculate betas below.

Beta Graphs

Returns on The Market and on Stocks L (for Low), A (for Average), and H (for High)

Year1999 10% 10% 10% 10%2000 20% 15% 20% 30%2001 -10% 0% -10% -30%

Regression analysis is performed by following the command path: Tools => Data Analysis => Regression. This will yield theRegression input box. If Data Analysis is not an option in your Tools menu, you will have to load that program. Click on theAdd-Ins option in the Tools menu. When the Add-Ins box appears, click on Analysis ToolPak and a check mark will appear nextto the Analysis ToolPak. Then, click OK and you will now be able to access Data Analysis. From this point, you must designatethe Y input range (stock returns) and the X input range (market returns). You can have the summary output placed in a newworksheet, or you can have it shown directly in the worksheet, as we did here. The Regression dialog box for the regression ofStock H is as follows:

Note: When you get the menu box onthe screen, and the cursor blinking inthe Y Range slot, use the mouse toselect the Y range, and then click onthe X range box. Then fill in the Xrange the same way.

Partial Correlation. Now suppose the stocks are positively but not perfectly so, with the following returns. What is the

kM kL kA kH

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Regression Output of Stock H Returns

SUMMARY OUTPUT

Regression Statistics Beta Coefficient for Stock H = 2.00Multiple R 1R Square 1Adjusted R Square 1Standard Error 1.38777878078145E-17Observations 3

ANOVAdf SS MS F Significance F

Regression 1 0.186666666666667 0.186666666666667 9.69228746927E+32 2.04487600835E-17Residual 1 1.92592994439E-34 1.92592994439E-34Total 2 0.186666666666667

Coefficients Standard Error t Stat P-value Lower 95%Intercept -0.1 9.08514472969E-18 -1.1006979303E+16 5.78378276875E-17 -0.1

X Variable 1 2.00 6.42416744643E-17 3.11324388207E+16 2.04487600835E-17 2

Regression Output of Stock A Returns

SUMMARY OUTPUTThe beta coefficient for Stock A = 1.00

Regression StatisticsMultiple R 1R Square 1Adjusted R Square 1Standard Error 0Observations 3

ANOVAdf SS MS F Significance F

Regression 1 0.046666666666667 0.046666666666667 #NUM! #NUM!Residual 1 0 0Total 2 0.046666666666667

Coefficients Standard Error t Stat P-value Lower 95%Intercept 0 0 65535 #NUM! 0

X Variable 1 1.00 0 65535 #NUM! 1

Regression Output of Stock L Returns

SUMMARY OUTPUTThe beta coefficient for Stock L = 0.50

Regression StatisticsMultiple R 1R Square 1Adjusted R Square 1Standard Error 2.08166817117217E-17Observations 3

ANOVAdf SS MS F Significance F

Regression 1 0.011666666666667 0.011666666666667 2.6923020748E+31 1.22692560501E-16Residual 1 4.33334237487E-34 4.33334237487E-34Total 2 0.011666666666667

Coefficients Standard Error t Stat P-value Lower 95%Intercept 0.05 1.36277170945E-17 3.66899310084E+15 1.73513483063E-16 0.05

X Variable 1 0.50 9.63625116964E-17 5188739803455012 1.22692560501E-16 0.499999999999999

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THE SECURITY MARKET LINE

The Security Market Line shows the relationship between a stock's beta and its expected return.

Risk-free rate (Varies over time) 6%Market return (Also varies over time) 11% Required Return when beta = 0.50 8.5%Beta (Varies by company) 0.5

With the above data, we can generate a Security Market Line that will be flexible enough to allow for changes in any of theinput factors. We generate a table of values for beta and expected returns, and then plot the graph as a scatter diagram.

Required Return Beta 8.5%0.00 6.0%0.50 8.5%1.00 11.0%1.50 13.5%2.00 16.0%

The Security Market Line shows the projected changes in expected return, due to changes in the beta coefficient. However, wecan also look at the potential changes in the required return due to variation of other factors, namely the market return andrisk-free rate. In other words, we can see how required returns can be influenced by changing inflation and risk aversion. The

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:

ORScenario 1. Inflation Increases: Scenario 2. Investors become more risk averse:Risk-free Rate 6% Risk-free Rate 6%Change in inflation 2% Old Market Return 11%

Old Market Return 11% 2.5%New Market Return 13% New Market Return 13.5%Beta 0.50 Beta 0.50

Required Return 10.5% Required Return 9.75%

level of investor risk aversion is measured by the market risk premium (kM-kRF), which is also the slope of the SML. Hence, an

Increase in RPM

0.00 0.50 1.00 1.50 2.00 2.50

0%

6%

12%

18%

Security Market Line

Beta

Re

qu

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d R

etu

rn

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Now, we can see how these two factors can affect a Security Market Line, by creating a data table for the required return withdifferent 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 graph shows that as risk as measured by beta increases, so does the required rate of return on securities. However, therequired return for any given beta varies depending on the position and slope of the SML.

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.000%

5%

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20%

25%

The SML Under Different Conditions

OriginalScenario #1Scenario #2

Beta

Req

uir

ed R

etu

rns

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