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RETURNS ON INVESTMENTS (Section 6.1)

Amount invested $1,000

Amount received in one year $1,100

Dollar return (Profit) $100

Rate of return = Profit/Investment = 10%

STAND-ALONE RISK (Section 6.2)

PROBABILITY DISTRIBUTION

A probability distribution is a listing of all possible outcomes and their corresponding probabilities.

Figure 6-1. Probability Distributions for Sale.Com and Basic Foods Inc.

Demand for the Probability of this

Company's Products Demand Occurring

Sale.com Basic Foods

Strong 0.30 90% 45%

Normal 0.40 15% 15%

Weak 0.30 60% 15%

1.00

Chapter 6. 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 it is of paramount importance in finance.

Rate of Return on Stock

if this Demand Occurs

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7576777879

A B C D E F

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 found as the weighted average of the probability distribution of returns.

Figure 6-2. Calculation of Expected Rates of Return: Payoff Matrix

Demand for the Probability of this

Company's Products

(1)

Demand Occurring

(2)

Rate of Return

(3)

Product

(2) x (3) = (4)

Rate of Return

(5)

Product

(2) x (5) = (6)

Strong 0.3 90% 27.0% 45% 13.5%

Normal 0.4 15% 6.0% 15% 6.0%

Weak 0.3 60% 18.0% 15% 4.5%

1.0

15.0% 15.0%

MEASURING STAND-ALONE RISK: THE STANDARD DEVIATION

The standard deviation is a measure of a distribution's dispersion.

Figure 4. Probability Distributions of Sale.com's and Basic Foods' Rates of Return

Expected Rate of Return =

Sum of Products =

Sale.com Basic Foods

0.00

0.10

0.20

0.30

0.40

-75 -60 -45 -30 -15 0 15 30 45 60 75 90

Probability of

Occurrence

Rate ofReturn

(%)

Panel a. Sale.com

Expected Rateof Return

0.00

0.10

0.20

0.30

0.40

-75 -60 -45 -30 -15 0 15

Probability of

Occurrence

Panel b. Basic Foods

Expectedof Retur

rr

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105

106107108109110

111112113

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A B C D E F

Calculating Standard Deviation

Figure 6-5. Calculating Sale.com's and Basic Foods' Standard Deviations

Panel a.

Probability of

Occurring

(1)

Rate of Return on

Stock

(2)

Expected Return

(3)

Deviation from

Expected

Return

(2) (3) = (4)

Squared

Deviation

(4)2= (5)

Sq. Dev. Prob.

(5) x (1) = (6)

0.3 90% 15% 75.0% 56.25% 16.88%

0.4 15% 15% 0.0% 0.00% 0.00%

0.3 60% 15% 75.0% 56.25% 16.88%

1.0 Sum = Variance = 33.75%

Std. Dev. = Square root of variance = 58.09%

Panel b.

Probability of

Occurring

(1)

Rate of Return on

Stock

(2)

Expected Return

(3)

Deviation from

Expected

Return

(2) (3) = (4)

Squared

Deviation

(4)2= (5)

Sq. Dev. Prob.

(5) x (1) = (6)

0.3 45% 15% 30.0% 9.00% 2.70%

0.4 15% 15% 0.0% 0.00% 0.00%

0.3 15% 15% 30.0% 9.00% 2.70%

1.0 Sum = Variance = 5.40%

Std. Dev. = Square root of variance = 23.24%

If Sales.com's and Basic Foods' stock return distributions are from normal distributions, then we can find confiden

0.6826

Expected Return Std. DeviationSale.com 15% 58.09% -43.09% to 73.09%

Basic Foods 15% 23.24% -8.24% to 38.24%

USING HISTORICAL DATA TO MEASURE RISK

Basic Foods

Sale.com

Here are the steps used to calculate the standard deviation. First, find the differences of all the possible returns

from the expected return. Second, square those differences. Third, multiply the squared numbers by the

probability of their occurrence. Fourth, find the sum of all the weighted squares. Finally, take the square root of

that number. Here are the calculations for Sale.com and Basic Foods.

1- range around expected return

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122123

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129130131132133134

135136137

138139

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144145146147

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151152

153

154155156

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159

A B C D E F

Figure 6-7. Standard Deviation Based On a Sample of Historical Data

Realized

Year return

2008 15.0%

2009 5.0%

2010 20.0%

Average =AVERAGE(D122:D124) = 10.0%

Standard deviation =STDEV(D122:D124) = 13.2%

MEASURING STAND-ALONE RISK: THE COEFFICIENT OF VARIATION

The coefficient of variation indicates the risk per unit of return, and it is calculated by dividing the

standard deviation by the expected return.

Std. Dev. Expected return CV

Sale.com 58.09% 15% 3.87

Basic Foods 23.24% 15% 1.55

RISK IN A PORTFOLIO CONTEXT (Section 6.3)

Portfolio Expected Return

Figure 6-8. Expected Returns on a Portfolio of Stocks

StockAmount of

Investment

Portfolio

Weight

Expected

Return

Weighted

Expected

Return

Southwest Airlines $300,000 0.3 15.0% 4.5%

Starbucks $100,000 0.1 12.0% 1.2%

FedEx $200,000 0.2 10.0% 2.0%

Dell $400,000 0.4 9.0% 3.6%Total investment = $1,000,000 1.0

Portfolio's Expected Return = 11.3%

Portfolio Standard Deviation

The expected return on a portfolio is simply a weighted average of the expected returns of the individual assets inthe portfolio. The weights are the percentage of total portfolio funds invested in each asset. Consider the following

portfolio and the hypothetical illustrative returns data.

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Figure 6-9. Portfolio Risk: Perfect Negative Correlation

Stock W Stock M

Weights 0.5 0.5

Year Stock W Stock M Portfolio WM

2006 40% -10% 15%

2007 -10% 40% 15%

2008 35% -5% 15%

2009 -5% 35% 15%

2010 15% 15% 15%

Average return = 15.00% 15.00% 15.00%

Standard deviation = 22.64% 22.64% 0.00%

Correlation coefficient = -1.00

Figure 6-10. Portfolio Risk: Perfect Positive Correlation

Portfolios of stocks are created to diversify investors from unnecessary risk. The diversifiable, or idiosyncratic,

risk is eliminated as more stocks are added. Diversification effects are strongest when combining uncorrelated

assets. The following figures illustrate how creating two-stock portfolios with different correlations between the

stocks affects the expected return and risk of various fictional portfolios.

CONCLUSION: When two stocks are perfectly negatively correlated, diversification is its strongest, and in this

case the portfolio return is a certain (no risk) 15%. Of course, this situation is very rare.

-10%

0%

10%

20%

30%

40%

Return

2010

Stock W

-10%

0%

10%

20%

30%

40%

Return

2010

Stock M

-10%

0%

10%

20%

30%

40%

Return Portfolio WM

Stock W Stock W' Portfolio WW'

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Stock W Stock W'

Weights 0.5 0.5

Year Stock W Stock W' Portfolio WW'

2006 40% 40% 40%

2007 -10% -10% -10%

2008 35% 35% 35%

2009 -5% -5% -5%

2010 15% 15% 15%

Average return = 15.00% 15.00% 15.00%


Correlation coefficient = 1.00

Figure 6-11. Portfolio Risk: Imperfect (Partial) Correlation

CONCLUSION: When two stocks are perfectly positively correlated, diversification has no effect, and the portfoli

is a weighted average of its individual stocks' risks. Note that in this graph only the portfolio returns are visible, b

realize that the stocks' returns follow an identical path.

-10%

0%

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20%

30%

40%

e urn

2010

-10%

0%

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20%

30%

40%

e urn

2010

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20%

30%

40%

10%

20%

30%

40%

Return Stock W

10%

20%

30%

40%

Return Stock Y

10.00%

20.00%

30.00%

40.00%

Return Portfolio WY

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Stock W Stock Y

Weights 0.5 0.5

Year Stock W Stock Y Portfolio WY

2006 40% 40% 40.00%

2007 -10% 15% 2.50%

2008 35% -5% 15.00%

2009 -5% -10% -7.50%

2010 15% 35% 25.00%

Average return = 15.00% 15.00% 15.00%


Correlation coefficient = 0.35

Contribution to Market Risk: Beta

The beta coefficient measures the amount of risk that a stock contributes to a well-diversified portfolio. It also

reflects the tendency of a stock to move up and down with the market. Shown below in the chart and in the table

are the returns for three stocks and for the stock market.

CONCLUSION: In the case where two stocks are somewhat correlated, diversification is effective in lowering

portfolio risk. Here, the portfolio return is an average of the stock returns and risk is reduced from 22.64% for

the individual stocks to 18.62% for the portfolio. Notice that the portfolio's return is always between that of the

two stocks. If more similarly-correlated stocks were added, risk would continue to fall, but as we shall see, there is

a limit to how low risk (the portfolio's SD) can go.

-10%

0%

2010

-10%

0%

2010

-10.00%

0.00%

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Figure 6-13. Relative Returns of Stocks H, A, and L

Year Market Stock H Stock A Stock L

1 19.0% 26.0% 19.0% 12.0%

2 25.0% 35.0% 25.0% 15.0%

3 -15.0% -25.0% -15.0% -5.0%

Average = 9.7% 12.0% 9.7% 7.3%

Standard deviation = 21.6% 32.4% 21.6% 10.8%

Beta = 1.5 1.0 0.5

Probability Distributions for H, A, and L

Historical Returns

Notice that Stock L has the lowest average return, but it also has the tightest distribution. On the other hand,

Stock H has the highest average return, but the widest distribution.

Note: These three stocks plot exactly on their regression lines. This indicates that they are

exposed only to market risk. Portfolios that concentrate on stocks with betas of 1.5, 1.0, and

0.5 have patterns similar to those shown in the graph. Standard deviation is calculated with

the ExcelSTDEV function because the data come from an historical sample.

-40%

0%

40%

-40% 0% 40%

Returns on Stocks

H, A, and L

Return on the Market

Stock H: b = 1.5

Stock A: b = 1.0

Stock L: b = 0.5

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Calculating Beta for H, A, and L

First, calculate correlation and covariance.

Correlation of stockwith Market, i,M 1.00 1.00 1.00

Covariance of stock

with Market, COVi,M 6.98% 4.65% 2.33%

Method 1:

bi= i,M(i/ M) 1.5 1.0 0.5

Method 2:

bi= COVi,M/ (M)2

1.5 1.0 0.5

Method 3:

Slope of regression

Beta =1.5 1.0 0.5

-80.0% -60.0% -40.0% -20.0% 0.0% 20.0% 40.0% 60.0% 80.0% 100.0%

Stock L

Stock H

Stock A

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407408409410411412413414415

A B C D E F

CALCULATING BETA COEFFICIENTS (Section 6.4)

Now we show how to calculate beta for an actual company, General Electric.

Step 1. Retrive Data

Step 2. Calculate Returns

Figure 6-14. Stock Return Data for GE and the S&P 500 Index

Month

Market Level

(S&P 500 Index) at

Month End

Market's

Return

GE Adjusted

Stock Price at

Month End

GE's

Return

March 2009 797.87 8.5% $10.11 18.8%

February 2009 735.09 -11.0% $8.51 -27.8%

January 2009 825.88 -8.6% $11.78 -25.2%

December 2008 903.25 0.8% $15.74 -3.8%

November 2008 896.24 -7.5% $16.37 -12.0%

October 2008 968.75 -16.8% $18.60 -23.5%

September 2008 1,164.74 -9.2% $24.30 -8.1%

August 2008 1,282.83 1.2% $26.43 -0.7%

July 2008 1,267.38 -1.0% $26.61 6.0%

June 2008 1,280.00 -8.6% $25.10 -12.1%

May 2008 1,400.38 1.1% $28.57 -6.1%

April 2008 1,385.59 4.8% $30.42 -11.6%March 2008 1,322.70 -0.6% $34.43 11.7%

February 2008 1,330.63 -3.5% $30.83 -5.4%

January 2008 1,378.55 -6.1% $32.59 -4.6%

December 2007 1,468.36 -0.9% $34.17 -2.4%

November 2007 1,481.14 -4.4% $35.00 -7.0%

October 2007 1,549.38 1.5% $37.62 -0.6%

September 2007 1,526.75 3.6% $37.84 7.2%

August 2007 1,473.99 1.3% $35.29 0.3%

We downloaded stock prices and dividends from http://finance.yahoo.com for General Electric, using its ticker

symbol GE, and for the S&P 500 Index ( symbol SPX), which contains 500 actively traded large stocks. For

example, to download the GE data, enter its ticker symbol in the upper left section and click Go. Then select

Historical Prices from the upper left side of the new page. After the daily prices come up, click monthly prices,

enter a start and stop date, and click "Get Prices." When presenting monthly data, the date shown is for the fi rst

date in the month, but the data are actually for the last day of trading in the month, so be alert for this. Note that

these prices are "adjusted" to reflect any dividends or stock splits. Scroll to the bottom of the page and click

"Download to Spreadsheet."

The downloaded data are in csv format. Convert to xls by opening a new Excel worksheet, copying the date and

adjusted index price data to it, and saving as an xls file. Then repeat the process to get the S&P index data. At

this point you have returns data for GE and the S&P Index, as we show below.

Next, calculate the percentage change in adjusted prices (which already reflect dividends) for GE and the S&P to

obtain returns, with the spreadsheet set up as shown below. At this point, we are ready to calculate some statistics

and to find GE's beta coefficient. This is shown below the data.

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464465466467468469470471472

473474475476477478479480481482483

484485486487488489490491492493494495

496497

498499500501502503

A B C D E F

Step 3. Examine the Data

Step 4. Plot the Data and Calculate BetaUsing the Chart Wizard, we plotted the GE returns on the y-axis and the market returns on

the x-axis. We also used the menu Chart > Options to add a trend line, and to display the

regression equation and R2on the chart. The chart is shown below.

Using the AVERAGE function and the STDEV function, we found the average historical

return and standard deviation for GE and the market. (We converted these from monthlyfigures 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 shown in the

rows above. We also used the CORREL function to find the correlation between GE and the

market. We used the SLOPE, INTERCEPT, and RSQ functions to estimate the regression for

y = 1.3744x - 0.0094R = 0.5719

-30.0%

0.0%

30.0%

-30% 0% 30%

y-axis: HistoricalGE Returns

x-axis: HistoricalMarket Returns

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As of March 2009 As of July 2008

Market GE Market GE

Average return (annual) -8.45% -22.94% 3.83% -0.14%

Standard deviation (annual) 15.92% 28.93% 9.60% 15.66%

Beta (using the SLOPE function)

Correlation between GE and the market.

R2(using the Excel function)

Average returns for GE and the market both fell.

SDs rose, indicating higher stand-alone risk.

GE's beta rose, indicating greater sensitivity to changes in the market.

GE's correlation with the market rose, indicating that much of GE's decline was due to the market drop.

These changes are all logical, but perhaps the most interesting one, for our purposes is the change in beta.

We will use beta when we estimate a firm's cost of capital, and the change in beta indicates a significant

change in the cost of equity. Based on its low beta (well below 1.0) in July, GE appeared to have a low cost

of equity. It's sharper-than-average price drop indicated that it was, ex post, really more risky than

average. That indicates that its "true risk" in July was risker than the average investor thought. People

have tried to forecast beta, and if they can do so, they can earn abnormal returns in the market. At any rate,

forecasting betas by adjusting historical betas for changes in leverage and other factors is widely

practiced.

THE RELATIONSHIP BETWEEN RISK AND RATES OF RETURN (Section 6.5)

rRF 6%

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A B C D E F

0.00 6.0%

0.50 8.5%

1.00 11.0%

1.50 13.5%

2.00 16.0%

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570571572573574575

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588589590591592593594595

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600

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A B C D E F

We will look at two potential conditions as shown in the following columns:

ORScenario 1. Interest rates increase: Scenario 2. Investors become more risk averse:

Risk-free rate 6% Risk-free rate 6%

Beta 0.50 Beta 0.50

Old market return 11% Old market return 11%

Change in interest rates 2% Increase in MRP 2.5%

New market return 13% New market return 13.5%

Required return 10.5% Required return 9.75%

Now we can see how these two factors can affect a Security Market Line, using a data table for the required

return with different beta coefficients.

Beta Original SituationInterest Rate

Increases

Risk Aversion

Increases

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%

Required Return

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 variations in other factors, for

example the market return and risk-free 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.

0%

6%

12%

18%

0.00 0.50 1.00 1.50 2.00 2.50

RequiredReturn

Beta

Figure 6-11.Security Market Line

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640

641

A B C D E F

2. If iinterest rates increase, the required return on all securities, regardless of risk increases by the increase in

the risk-free rate.

3. If risk aversion increases, the return on all securities except the riskless asset (beta = 0) increase. However, the

higher the beta, the greater the increase in the required return.

1. As beta, which measures risk, increases, the required return on securities increases, given the existence of risk

aversion.

0%

5%

10%

15%

20%

25%

0.00 0.50 1.00 1.50 2.00

Required

Returns

Risk, bi

The SML Under Inflation and Risk Aversion

Increases

Original

Inflation up

Risk aversion up

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SECTION 6.1SOLUTIONS TO SELF-TEST

Amount invested $500

Amount received in on $600

Dollar return $100

Rate of return 20%

Suppose you pay $500 for an investment that returns $600 in one year. What is the

annual rate of return?

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Probability Return Prob x Ret.

20% 25% 5.0%60% 10% 6.0%20% -15% -3.0%

Expec ted retu rn = 8.0%

Probability Return

Deviation from

expected return Deviation2 Prob x Dev.2

20% 25% 17.0% 2.890% 0.578%60% 10% 2.0% 0.040% 0.024%20% -15% -23.0% 5.290% 1.058%

Variance = 1.660%

Stan dard d ev iatio n = 12.9%

RealizedYear return

1 10%2 -15%3 35%

Average = 10.0%

Standard deviation = 25.0%

Expected return 15.0%Standard deviation 30.0%

Coefficient of variation 2.0

An investment has a 20% chance of producing a 25% return, a 60% chance of producing a 10% re

a 20% chance of producing a -15% return. What is its expected return? What is its standard devi

An investment has an expected return of 15% and a standard deviation of 30%. What is its coeffic

variation?

A stocks returns for the past three years are 10%, -15%, and 35%. What is the historical average

return? What is the historical sample standard deviation?

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urn, and

tion?

ient of

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Stock Investment Beta Weight Beta x Weight

Dell $25,000 1.2 0.25 0.30

Ford $50,000 0.8 0.50 0.40Wal-Mart $25,000 1.0 0.25 0.25

Total $100,000

Portfo lio beta = 0.95

An investor has a 3-stock portfolio with $25,000 invested in Dell, $50,000 invested in Ford, and $25,000

invested in Wal-Mart. Dells beta is estimated to be 1.20, Fords beta is estimated to be 0.80, and Wal-

Marts beta is estimated to be 1.0. What is the estimated beta of the investors portfolio?

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Beta 1.4Risk-free rate 5.5%Market risk premium 5.0%

Required rate of return 12.50%

A stock has a beta of 1.4. Assume that the risk-free rate is 5.5% and the market risk premium is 5%.

What is the stocks required rate of return?

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