ChapterTool KitChapter 6112018Risk and Return6-1 Investment Retu

ChapterTool KitChapter 611/20/18Risk and Return6-1
Investment Returns and RiskAmount invested$1,000Amount
received in one year$1,100Dollar return (Profit)$100Rate of
return = Profit/Investment =10%6-2 Measuring Risk for
Discrete DistributionsThe 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.A listing of possible outcomes and their probabilities is
called a probability distribution, as shown
below.ScenarioProbability of ScenarioRate of Return in
ScenarioBest Case0.3037%Most Likely0.4011%Worst
Case0.30−15%1.00Figure 6-1Discrete Probability Distribution
for Three ScenariosGiven the probabilities and the outcomes for
possible returns, it is possible to calculate the expected return
and standard deviation.Figure 6-2Calculating Expected Returns
and Standard Deviations: Discrete
ProbabilitiesINPUTS:Expected ReturnStandard
DeviationScenarioProbability of Scenario
(1)Market Rate of Return
(2)Product of Probability and Return
(3) = (1) × (2) Deviation from Expected Return
(4) = (2) − D66Squared Deviation
(5) = (4)2 Prob. × Sq. Dev.
(6) = (1) × (5)Best
Case0.3037%11.1%0.26000.06760.0203Excel functions for
finding expected return and standard deviation of discrete
eventsMost Likely0.4011%4.4%0.00000.00000.0000Use
SUMPRODUCT to find expected return by putting probabilities

in first argument array and rates of return in the second
argument array.11%Worst Case0.30−15%−4.5%-
0.26000.06760.0203=SUMPRODUCT(B63:B65,C63:C65)1.00E
xp. ret. = Sum = 11.0% Sum = Variance =0.0406Use
SUMPRODUCT to find variance. If the first array has
probabilities and the second array subtracts the mean from the
array of outcomes and is squared, then this is exactly the
calculation shown in the step-by-step manner above to find
variance:0.0406Std. Dev. = Square
root of variance =20.1%=SUMPRODUCT( B63:B65,(C63:C65-
D66)^2)Note: Calculations are not rounded in intermediate
steps.6-3 Risk in a Continuous DistributionIt is possible to add
more scenarios.ScenarioPanel A: Probability of Market Return
ScenarioPanel B: Probability of Stock Return Scenario Rate of
Return in Scenario10.00020.0198-66%20.00110.0307-
55%30.00540.0452-44%40.02050.0625-33%50.05750.0806-
22%60.12010.0969-
11%70.18700.10820%80.21670.112311%90.18700.108222%100
.12010.096933%110.05750.080644%120.02050.062555%130.00
540.045266%140.00110.030777%150.00020.019888%1.00001.0
000Average =11.0%11.0%Std. dev. =20.2%36.2%Figure 6-
3Discrete Probability Distributions for 15 ScenariosPanel A:
Market Return for 15 Scenarios: Standard Devation =
20.2%Panel B: Single Company's Stock Return for 15
Scenarios: Standard Devation = 36.2%At some point, it becomes
impractical to keep adding scenarios. Many analysts use the
normal distribution to estimate stock returns.Here is an example
of a normal distribution with a similar mean and standard
deviation as the discrete distribution shown above.6-4 Using
Historical Data to Estimate RiskInvestors often use historical
data to estimate risk. This is quite easy in Excel by using the
AVERAGE and STDEV functions.Standard Deviation Based On
a Sample of Historical
DataInputs:RealizedYearreturn201715.0%2018−5.0%201920.0%
Calculations:=AVERAGE(E183:E185)10.0%=STDEV(E183:E18
5)13.2%Use STDEV, the function for a sample.Measuring the

Standard Deviation of MicroDriveThe monthly stock returns for
MicroDrive and one of its competitors, SnailDrive, during the
past 48 months are shown in the figure below. The actual data
are below the figure.Figure 6-5Historical Monthly Stock
Returns for MicroDrive and
SnailDriveMicroDriveSnailDriveAverage Return
(annualized)12.0%9.3%Standard Deviation
(annualized)51.8%34.2%Weights are specified in Section 6-
5.Portfolio
weightsSnailDrive:75%MicroDrive:25%PeriodMarketMicroDriv
eSnailDrivePortfolio12.37%2.81%14.93%11.90%212.68%14.79
%−3.26%1.25%3−1.13%0.79%−10.57%−7.73%410.93%8.71%−
10.76%−5.89%5−0.02%0.83%9.71%7.49%6−3.31%−32.42%6.4
0%−3.30%711.89%22.56%0.26%5.83%8−3.96%−24.21%0.52%
−5.66%9−4.90%8.00%−8.67%−4.50%107.10%−1.29%21.62%15
.89%112.94%4.43%3.87%4.01%12−6.52%−6.36%5.00%2.16%1
33.72%11.79%−12.32%−6.29%144.74%21.32%−2.43%3.51%15
−8.21%−10.28%4.87%1.08%16−5.15%3.96%−17.85%−12.40%1
73.92%34.98%−10.89%0.57%181.08%2.56%−16.68%−11.87%1
9−2.48%−10.80%9.02%4.07%203.92%−6.70%22.60%15.28%21
3.13%−2.31%14.25%10.11%220.17%8.26%−7.68%−3.69%235.1
7%0.51%−2.52%−1.76%242.56%−14.61%9.32%3.34%25−5.41
%−4.56%−1.38%−2.17%26−2.09%−12.08%13.92%7.42%271.08
%31.68%3.91%10.85%2810.47%4.43%8.91%7.79%29−3.74%0.
32%−3.76%−2.74%302.94%4.59%−3.95%−1.82%31−9.50%2.08
%−1.43%−0.55%32−3.10%−15.49%−6.38%−8.65%337.95%32.3
9%12.00%17.10%3410.93%15.15%−2.00%2.28%35−1.70%−3.7
2%−12.51%−10.31%36−3.96%−15.40%−0.49%−4.22%375.17%
−12.67%9.91%4.27%38−0.75%−10.43%−8.21%−8.76%39−9.04
%−7.14%−11.27%−10.24%40−9.50%−4.85%−10.32%−8.96%41
4.74%8.15%9.19%8.93%42−0.38%−14.72%−0.43%−4.00%434.
32%32.45%0.99%8.85%44−1.89%−28.34%3.96%−4.12%45−3.9
6%−5.55%−8.50%−7.77%466.58%5.81%16.10%13.52%47−1.32
%4.02%8.86%7.65%484.74%4.38%1.30%2.07%Full 48
MonthsMarketMicroDriveSnailDrivePortfolioAverage monthly
return:0.9%1.00%0.77%0.8%Standard deviation of monthly

returns:5.7%14.94%9.87%7.8%Average return
(annual):10.8%12.0%9.3%10.0%Standard deviation
(annual):19.9%51.8%34.2%27.1%Maximum of monthly
returns:12.7%34.98%22.60%17.1%Minimum of monthly
returns:-9.5%-32.42%-17.85%-12.4%Past 12
MonthsMonthMarketMicroDriveSnailDrivePortfolio375.2%-
12.7%9.9%4.3%38-0.8%-10.4%-8.2%-8.8%39-9.0%-7.1%-
11.3%-10.2%40-9.5%-4.9%-10.3%-
9.0%414.7%8.1%9.2%8.9%42-0.4%-14.7%-0.4%-
4.0%434.3%32.4%1.0%8.9%44-1.9%-28.3%4.0%-4.1%45-4.0%-
5.6%-8.5%-7.8%466.6%5.8%16.1%13.5%47-
1.3%4.0%8.9%7.7%484.7%4.4%1.3%2.1%Past 12
return:-0.11%-2.41%0.97%0.12%Average return (annual):-
1.3%-28.9%11.6%1.5%Standard deviation
(annual):18.9%52.4%31.4%29.1%Total compound return:-2.9%-
34.3%7.3%-2.4%The total compound return is the total return
on $1 invested at the end of month 36. The Excel function
=FVSCHEDULE calculates the ending value given an initial
amount and a series of returns.6-5 Risk in a Portfolio
ContextNow we are going to analyze the risk of a portfolio
instead of the stand-alone risk of individual assets.Creating a
PortfolioLook at the data for MicroDrive and SnailDrive shown
above. The last column shows a portfolio with the weights
shown below. Here are the results for the two companies and for
the portfolio. Notice that the portfolio has a higher return than
SnailDrive and less risk than either of the two stocks.Portfolio
weightsSnailDrive:75%MicroDrive:25%Full 48
(annual):19.9%51.8%34.2%27.1%CorrelationLoosely speaking,
correlation measures the tendency of two variables to move
together.Correlation between MicroDrive and SnailDrive:r =-
0.133=CORREL(E232:E279,F232:F279)6-6 The Relevant Risk

of a Stock: The Capital Asset Pricing Model (CAPM)The
Capital Asset Pricing Model (CAPM) provides a measure of
risk.Contribution to Market Risk: BetaThe relevant risk of an
individual stock as defined by its beta. Beta measures how much
risk a stock contributes to a well-diversified portfolio.Beta for
Stock i = bi = riM(si/sM)A portfolio's beta is the weighted
average of the stock's individual betas. Consider the following
example.Stock Beta:Weight in Portfolio:Contribution of Stock
to Portfolio Beta:biwibi x wi x sMStock 10.625.0%0.150Stock
21.225.0%0.300Stock 31.225.0%0.300Stock
41.425.0%0.350Portfolio beta = 1.100The standard deviation of
a well-diversified portfolio is:Std. Dev. of portfolio = sp = bp
(sM)Note: if the bp is negative, then σp = |bp| (σM).If the
example portfolio had more than 4 stocks and was well -
diversified, then its standard deviation would be:Beta of
portfolio = bp =1.1Std. Dev. of market = sM =20%Std. Dev. of
portfolio = sp =22%Figure 6-7The Contribution of Individual
Stocks to Portfolio Risk: The Effect of BetaMarket standard
deviation = sM =20.0%Stock Beta:Weight in
Portfolio:Contribution of Stock to Portfolio Beta:Contribution
of Stock to Portfolio Risk:biwibi x wi bi x wi x sMCategory
Labels for chart.Stock 10.625.0%0.1503.0%b1w1sMStock
21.225.0%0.3006.0%b2w2sMStock
31.225.0%0.3006.0%b3w3sMStock
41.425.0%0.3507.0%b4w4sM1.10022.0%b5w5sMEstimating
BetaWe can use the data shown previously for MicroDrive and
SnailDrive to estimate their betas.Calculating
BetaMarketMicroDriveSnailDriveStandard deviation
(annual):19.89%51.75%34.17%Correlation with the
market:0.5110.264bi = riM(si/sM)1.3300.454Beta can also be
calculated as the slope of a regression of the stock (on the y-
axis) and the market (on the x-axis). This can be done using the
SLOPE function or by plotting the returns and specifying that
the chart show the TRENDLINE.Calculating Beta as the Slope
of a Regression Using Excel Functions (See Excel explanations
to right)MicroDriveSnailDrivebi =

riM(si/sM)1.3300.454=SLOPE(F232:F279,$D$232:$D$279)Inte
rcept-0.0020.004=INTERCEPT(F232:F279,$D$232:$D$279)R
squared0.2610.070=RSQ(F232:F279,$D$232:$D$279)Calculatin
g Confidence Intervals using Excel FunctionsSee the comment
in this cell for instructions for how to use LINEST and estimate
confidence intervals.
Mike Ehrhardt: Show below are the outputs from the array
function LINEST. The yellow area shows what statistics are
output by LINEST. To calculate the statistics for MicroDrive,
highlight the gray area. Enter the formula
=LINEST(E209:E256,D209:D256,TRUE,TRUE), then hit Ctrl-
Shift-Enter. Use a similar process to calculate the regression
statistics for SnailDrive in the blue region.
Recall from statistics that if you take an estimated coefficient
from a simple regression, subtract a target value (which is often
zero), and then divide that difference by the standard error of
the estimated the coefficient, the result will be a t-statistic,
which conform to the Student's t-distribution with n-k-1 degrees
of freedom, where n is the number of observations and k is the
number of explanatory variables. This approach is used to
determine whether the estimated coefficient is statistically
significantly different from zero. To find the confidence
interval around an estimated regression coefficient, you use this
relationship, but you pick a target probability rather than a
target value. Using the target probability and the degrees of
freedom, the TINV function will provide the corresponding
value (i.e., t-stat) for a two-tailed t-test. Therefore, the
confidence interval corresponding to the target probability has a
range equal to 2(t-stat)(standard error of coefficient). The lower
end of the range is equal to the estimated coefficient minus (t-
stat)(standard error of coefficient); the upper end of the ra nge is
equal to the estimated coefficient plus (t-stat)(standard error of
coefficient).Input desired probability for confidence
interval95%95%Excel function LINEST output in an
arrayLINEST Output for MicroDriveLINEST Output for

SnailDriveLower boundary of confidence interval for beta0.666-
0.037See explanation to right.SlopeIntercept1.330-
0.0020.4540.004Upper boundary of confidence interval for
beta1.9940.946See explanation to right.Standard error
slopeStandard error intercept0.3300.0190.2440.014Lower
boundary of confidence interval for intercept-0.040-0.025See
explanation to right.R squaredStandard
error0.2610.1300.0700.096Upper boundary of confidence
interval for intercept0.0360.032See explanation to
right.FDegrees of freedom1646346SS RegressionSS
Residual0.2740.7750.0320.425Figure 6-8Stock Returns of
MicroDrive and the Market: Estimating BetaShown below is the
output from Excel's regression tool. From the menu bar, select
Data, then Data Analysis, the Regression.MicroDriveSnail
DriveSUMMARY OUTPUTSUMMARY OUTPUTRegression
StatisticsRegression StatisticsMultiple R0.5108093315Multiple
R0.2625490104R Square0.2609261731R
Square0.0689319829Adjusted R Square0.2445023103Adjusted
R Square0.0482415825Standard Error0.1312392274Standard
Error0.0950833686Observations47Observations47ANOVAANO
VAdfSSMSFSignificance FdfSSMSFSignificance
FRegression10.27363375230.273633752315.88701610530.0002
437609Regression10.03012041810.03012041813.33159250750.
0746025035Residual450.77506806630.0172237348Residual450.
40683811420.009040847Total461.0487018186Total460.436958
5323CoefficientsStandard Errort StatP-valueLower 95%Upper
95%Lower 95.0%Upper 95.0%CoefficientsStandard Errort
StatP-valueLower 95%Upper 95%Lower 95.0%Upper
95.0%Intercept-0.00197719060.0193616376-
0.10211897630.9191159351-0.04097353060.0370191493-
0.04097353060.0370191493Intercept0.00089492940.014027587
40.06379781280.9494137787-0.02735808180.0291479406-
0.02735808180.02914794060.02370456321.33013536730.33371
418953.98585199240.00024376090.65800048722.00227024740.
65800048722.00227024740.02370456320.44130770090.241777
32471.82526505130.0746025035-0.04565682810.9282722298-

0.04565682810.9282722298EXAMPLE: CALCULATING BETA
COEFFICIENTS FOR AN ACTUAL COMPANYNow we show
how to calculate beta for an actual company, General Electric.
Step 1. Retrieve DataWe downloaded stock prices and
dividends from http://finance.yahoo.com for General Electric,
using its ticker symbol GE, and for the S&P 500 Total Retun
Index ( symbol ^SP500TR), which is an index incorporating the
prices and dividends of the 500 companies listed in the S&P
500, which tracks 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 first 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 xlsx by opening a new Excel worksheet,
copying the date and adjusted index price data to it, and saving
as an xlsx 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
500 Total Return Index, as we show below.Step 2. Calculate
ReturnsNext, 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.
Yahoo actually adjusts the stock prices to reflect any stock
splits or dividend payments. For example, suppose the stock
price is $100 in July, the company has a 2-for-1 split, and the
actual price in August is $60. The reported adjusted price for
August would be $60, but the reported price for July would be
$50, which reflects the stock split. This gives an accurate stock
return of 20%: ($60-$50)/$50 = 20%, the same as if there had
not been a split, in which case the return would have been
($120-$100)/$100 = 20%.

Or suppose the actual price in September is $50, the
company pays a $10 dividend, and the actual price in October is
$60. Shareholders have had a return of ($60+$10-$50)/$50 =
40%. Yahoo reports an adjusted price of $60 for October, and
an adjusted price of $42.857 for September, which gives a
return of ($60-$42.857)/$42.857 = 40%.
In other words, the percent change in the adjusted price
accurately reflects the actual return. At this point, we are ready
to calculate some statistics and to find GE's beta coefficient.
This is shown below the data.Not in Textbook: Stock Return
Data for GE and the S&P 500 Total Return IndexMonthMarket
Level
(S&P 500 Total Return Index) at Month EndMarket's
ReturnGE Adjusted Stock Price at Month End
Michael C. Ehrhardt: Yahoo actually adjusts the stock prices to
reflect any stock splits or dividend payments. For example,
suppose the stock price is $100 in July, the company has a 2-
for-1 split, and the actual price in August is $60. The reported
adjusted price for August would be $60, but the reported price
for July would be $50, which reflects the stock split. This gives
an accurate stock return of 20%: ($60-$50)/$50 = 20%, the same
as if there had not been a split, in which case the return would
have been ($120-$100)/$100 = 20%.
Or suppose the actual price in September is $50,the company
pays a $10 dividend, and the actual price in October is $60.
Shareholders have had a return of ($60+$10-$50)/$50 = 40%.
Yahoo reports an adjusted price of $60 for October, and an
adjusted price of $42.857 for September, which gives a return
of ($60-$42.857)/$42.857 = 40%.
accurately reflects the actual return.
GE's
ReturnNovember 20175,155.443.1%$18.29-9.3%October

20175,002.032.3%$20.16-15.8%September
20174,887.972.1%$23.94-1.5%August
20174,789.180.3%$24.31-4.1%July 20174,774.562.1%$25.36-
4.4%June 20174,678.360.6%$26.52-1.4%May
20174,649.341.4%$26.88-5.6%April 20174,584.821.0%$28.46-
2.7%March 20174,538.210.1%$29.260.8%February
20174,532.934.0%$29.040.4%January
20174,359.811.9%$28.93-5.3%December
20164,278.662.0%$30.552.7%November
20164,195.733.7%$29.745.7%October 20164,045.89-
1.8%$28.13-1.0%September 20164,121.060.0%$28.41-
5.2%August 20164,120.290.1%$29.970.3%July
20164,114.513.7%$29.87-0.3%June
20163,968.210.3%$29.974.1%May 20163,957.951.8%$28.78-
1.7%April 20163,888.130.4%$29.28-3.3%March
20163,873.116.8%$30.2710.0%February 20163,627.06-
0.1%$27.520.1%January 20163,631.96-5.0%$27.49-
5.9%December 20153,821.60-1.6%$29.204.0%November
20153,882.840.3%$28.073.5%October
20153,871.338.4%$27.1115.7%September 20153,570.17-
2.5%$23.430.8%August 20153,660.75-6.0%$23.24-4.2%July
20153,895.802.1%$24.25-0.9%June 20153,815.85-1.9%$24.48-
2.6%May 20153,891.181.3%$25.130.7%April
20153,841.781.0%$24.959.1%March 20153,805.27-1.6%$22.86-
3.7%February 20153,866.425.7%$23.738.8%January
20153,656.28-3.0%$21.81-4.6%December 20143,769.44-
0.3%$22.86-4.6%November
20143,778.962.7%$23.962.6%October
20143,679.992.4%$23.341.6%September 20143,592.25-
1.4%$22.98-1.4%August 20143,643.334.0%$23.303.3%July
20143,503.19-1.4%$22.56-3.5%June 20143,552.182.1%$23.38-
1.9%May 20143,480.292.3%$23.83-0.4%April
20143,400.460.7%$23.923.9%March
20143,375.510.8%$23.032.5%February
20143,347.384.6%$22.461.4%January 20143,200.95-
3.5%$22.16-9.6%December

20133,315.592.5%$24.525.1%November
20133,233.72NA$23.32NADescription of DataAverage return
(annual):12.2%-4.3%Standard deviation
(annual):9.6%18.7%Minimum monthly return:-6.0%-
15.8%Maximum monthly return:8.4%15.7%Correlation between
GE and the market:0.51Beta: bGE = rGE,M (sGE / sM)0.99Beta
(using the SLOPE function):0.99Intercept (using the
INTERCEPT function):-0.01R2 (using the RSQ
function):0.26Step 3. Examine the Data and Calculate
BetaUsing 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 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 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
beta.6-7 The Relationship between Risk and Return in the
Capital Asset Pricing ModelThe SML shows the relationship
between the stock's beta and its required return, as predicted by
the CAPM.rRF6%<< Varies over time, but is constant for all
firms at a given time.rM11%<< Varies over time, but is
constant for all firms at a given time.bi0.5<< Varies over time,
and varies from firm to firm.The SML predicts stock i's required
return to be:RPM = rM - rRF ri = rRF + bi(RPM)RPM = rM -
rRF =5%ri = rRF + bi(RPM)8.5%With the above data, we can
generate a Security Market Line that is flexible enough to allow
for changes inany of the input factors. We generate a table of
values for beta and expected returns, and then plot thegraph as a
scatter diagram.BetaSecurity Market Line: riRisk-Free
Rate0.006.0%6%0.508.5%6%1.0011.0%6%1.5013.5%6%2.0016.
0%6%Figure 6-9The Security Market LineThe 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
(rM – rRF), 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.Portfolio
ReturnsThe same relationship holds for required returns: The
required return on a portfolio is simply a weighted average of
the required returns of the individual assets in the portfolio. The
weights are the percentage of total portfolio funds invested in
each asset. The required return on a portfolio is also equal to:rp
= rRF + bp(RPM)The expected return on a portfolio is simply a
weighted average of the expected returns of the individual
assets in the portfolio. The weights are the percentage of total
portfolio funds invested in each asset. Consider the following
portfolio and the hypothetical illustrative returns
data.StockAmount of InvestmentPortfolio
WeightExpected
ReturnWeighted Expected
ReturnSouthwest
Airlines$300,0000.315.0%4.5%Starbucks$100,0000.112.0%1.2
%FedEx$200,0000.210.0%2.0%Dell$400,0000.49.0%3.6%Total
investment =$1,000,0001.0Por tfolio's Expected Return
=11.3%6-8 The Efficient Markets HypothesisThe Efficient
Markets Hypothesis (EMH) asserts that (1) stocks are always in
equilibrium and (2) it is impossible for an investor to “beat the
market” and consistently earn a higher rate of return than is
justified by the stock’s risk. 6-9 The Fama-French Three-Factor
ModelThe Fama-French 3-Factor model shows the actual stock
return given the risk-free rate, the return on the market, the
return on the SMB portfolio, and the return on the HML
portfolio:Suppose a company announces that it is going to
include more outsiders on its board of directors and that the
company’s stock falls by 2% on the day of the announcement.
Does that mean that investors don’t want outsiders on the

board? Actual return on announcement day =-2%Suppose you
estimate the following coefficients of the Fama-French model
using historical actual data prior to the announcement date:ai
=0.0bi =0.9ci =0.2di =0.3These are the returns on the
announcement day:rRF ≈0.0%rM =-3.0%rSMB =-1.0%rHML =-
2.0%The predicted return on the announcement day:Predicted
return =rRF,t + ai + bi(rM,t - rRF,t) + ci(rSMB,t) +
di(rHML,t)Predicted return =-3.5%Unexplained return =Actual
return - predicted returnUnexplained return =1.5%
Security Market Line: ri
0 0.5 1 1.5 2 0.06 8.4999999999999992E-2 0.11
0.13500000000000001 0.16 Risk-Free Rate
Risk-Free Rate
0 0.5 1 1.5 2 0.06 0.06 0.06 0.06 0.06
Beta
Required Return
0.37 0.11 -0.15 0.3 0.4 0.3
Outcomes: Market Returns for 3 Scenarios
Probability of Scenario
-0.66 -0.55000000000000004 -0.44000000000000006 -
0.33000000000000007 -0.22000000000000008 -
0.11000000000000008 0 0.11 0.22 0.33 0.44
0.55000000000000004 0.66 0.77 0.88
1.7511394806058752E-4 1.0680516885827041E-3
5.4186431553394105E-3 2.0452870329666931E-2
5.7451043229686055E-2 0.12012007299591695
0.18697232380635256 0.21668376169278963
0.18697232380635254 0.12012007299591698
5.7451043229685972E-2 2.0452870329666917E-2
5.4186431553394643E-3 1.0680516885827052E-3
1.7511394806057901E-4
Outcomes: Market Returns

Probability
Normal Distribution
-0.66 -0.55000000000000004 -0.44000000000000006 -
0.33000000000000007 -0.22000000000000008 -
0.11000000000000008 0 0.11 0.22 0.33 0.44
0.55000000000000004 0.66 0.77 0.88
1.7511394806058752E-4 1.0680516885827041E-3
5.4186431553394105E-3 2.0452870329666931E-2
5.7451043229686055E-2 0.12012007299591695
0.18697232380635256 0.21668376169278963
0.18697232380635254 0.12012007299591698
5.7451043229685972E-2 2.0452870329666917E-2
5.4186431553394643E-3 1.0680516885827052E-3
1.7511394806057901E-4
Return
Probability
MicroDrive 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22
23 24 25 26 27 28 29 30 31 32 33 34
35 36 37 38 39 40 41 42 43 44 45 46
47 48 2.8123990924714889E-2 0.14788012262092151
7.8502963553737266E-3 8.7129468545996486E-2
8.3412755288645168E-3 -0.32424777774866015
0.22562272254962037 -0.2421188972496903
8.0033999253312074E-2 -1.2880104827403847E-2
4.4340333222723204E-2 -6.3572272894571916E-2
0.11793465207595152 0.2132415493195787 -
0.10275248323559877 3.9638703647158331E-2
0.34980897959761476 2.5618545344283039E-2 -
0.10798804364762635 -6.6991550094965782E-2 -
2.309098107106064E-2 8.2631500961910756E-2
5.0805676359279997E-3 -0.14613977701929601 -
4.5612755984871939E-2 -0.12083292328306293
0.31684468589336262 4.4284410066492273E-2

3.1607939009120284E-3 4.5883106630769353E-2
2.07552456031646E-2 -0.15485866662189754
0.32391586170454112 0.15150100919688878 -
3.7185495271657959E-2 -0.1539751666991345 -
0.12667133730723446 -0.10429634808460107 -
7.1363934717645017E-2 -4.8514454737545759E-2
8.146235222622937E-2 -0.14717242637944389
0.32446717204181608 -0.2833663222356419 -
5.5520125185090474E-2 5.8097374046303288E-2
4.0169352047872364E-2 4.3819809396092653E-2
SnailDrive 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33
34 35 36 37 38 39 40 41 42 43 44 45
46 47 48 0.1492690115098104 -
3.2625626037028344E-2 -0.10573003868698551 -
0.10761017744603606 9.7091665313481013E-2
6.4019696075159227E-2 2.551740058812201E-3
5.1823111248978371E-3 -8.6675542060738534E-2
0.21619937238442177 3.8682354577748018E-2
4.9976714369936108E-2 -0.1231552558765608 -
2.4334023349728399E-2 4.8709991986939309E-2 -
0.17848886713987205 -0.10894557506251189 -
0.16677693278427547 9.02224897298773E-2
0.22602742043638951 0.14253323854601735 -
7.6768659825705354E-2 -2.5191647655802694E-2
9.3197313516893962E-2 -1.3762620727642588E-2
0.13919465840323408 3.9062205051343148E-2
8.9076694586209232E-2 -3.7587548782611876E-2 -
3.951356902881889E-2 -1.4254586969160453E-2 -
6.3757773671727844E-2 0.11996362520663377 -
2.003461329144321E-2 -0.12506 156308683511 -
4.944118628225227E-3 9.9145429710516583E-2 -
8.206824068176545E-2 -0.11269202815269945 -
0.10323090464415442 9.1937173710245873E-2 -
4.2822325102026349E-3 9.8823607828717221E-3

3.955891709074854E-2 -8.5047861043818507E-2
0.16096750035334298 8.8623069016098177E-2
1.3046620641216391E-2
Month of Return
Monthly Rate of Return
MicroDrive
y = 1.33x - 0.002
R² = 0.2612
2.3704563216878447E-2 0.12678838148916216 -
1.1271722117620619E-2 0.10934853098288892 -
1.5001243046439301E-4 -3.3094084606626398E-2
0.11894009781525601 -3.9637113175926407E-2 -
4.900848341 6118451E-2 7.1005972903079154E-2
2.9372997557913449E-2 -6.5174257007283729E-2
3.7154046682836443E-2 4.7403568255148722E-2 -
8.2053210024488593E-2 -5.1497959731137855E-2
3.9150545149045569E-2 1.0802516698457023E-2 -
2.48410615005232E-2 3.9150545149045569E-2
3.129112880648411E-2 1.6802075480611047E-3
5.1725910558059016E-2 2.5581205766142971E-2 -
5.4058128270569515E-2 -2.0870420462762541E-2
1.0802516698457023E-2 0.10467065567288585 -
3.7416426909536671E-2 2.9372997557913449E-2 -
9.5039047334366744E-2 -3.0986002707321395E-2
7.9483740655806087E-2 0.10934853098288892 -
1.6981645157961755E-2 -3.9637113175926407E-2
5.1725910558059016E-2 -7.5293925805213309E-3 -
9.0361172024363487E-2 -9.5039047334366744E-2
4.7403568255148722E-2 -3.8255645271969832E-3
4.3218973200413506E-2 -1.8916714871863298E-2 -
3.9637113175926407E-2 6.5807443379660199E-2 -
1.316071492587351E-2 4.7403568255148722E-2
2.8123990924714889E-2 0.14788012262092151
7.8502963553737266E-3 8.7129468545996486E-2

8.3412755288645168E-3 -0.32424777774866015
0.22562272254962037 -0.2421188972496903
8.0033999253312074E-2 -1.2880104827403847E-2
4.4340333222723204E-2 -6.3572272894571916E-2
0.11793465207595152 0.2132415493195787 -
0.10275248323559877 3.9638703647158331E-2
0.34980897959761476 2.5618545344283039E-2 -
0.10798804364762635 -6.6991550094965782E-2 -
2.309098107106064E-2 8.2631500961910756E-2
5.0805676359279997E-3 -0.14613977701929601 -
4.5612755984871939E-2 -0.12083292328306293
0.31684468589336262 4.4284410066492273E-2
3.1607939009120284E-3 4.5883106630769353E-2
2.07552456031646E-2 -0.15485866662189754
0.32391586170454112 0.151501009196888 78 -
3.7185495271657959E-2 -0.1539751666991345 -
0.12667133730723446 -0.10429634808460107 -
7.1363934717645017E-2 -4.8514454737545759E-2
8.146235222622937E-2 -0.14717242637944389
0.32446717204181608 -0.2833663222356419 -
5.5520125185090474E-2 5.8097374046303288E-2
4.0169352047872364E-2 4.3819809396092653E-2 Market
vs. Market
Market vs. Market
2.3704563216878447E-2 0.12678838148916216 -
1.1271722117620619E-2 0.10934853098288892 -
1.5001243046439301E-4 -3.3094084606626398E-2
0.11894009781525601 -3.9637113175926407E-2 -
4.9008483416118451E-2 7.1005972903079154E-2
2.9372997557913449E-2 -6.5174257007283729E-2
3.7154046682836443E-2 4.7403568255148722E-2 -
8.2053210024488593E-2 -5.1497959731137855E-2
3.9150545149045569E-2 1.0802516698457023E-2 -
2.48410615005232E-2 3.9150545149045569E-2
3.129112880648411E-2 1.6802075480611047E-3
5.1725910558059016E-2 2.5581205766142971E-2 -

5.4058128270569515E-2 -2.0870420462762541E-2
1.0802516698457023E-2 0.10467065567288585 -
3.7416426909536671E-2 2.9372997557913449E-2 -
9.5039047334366744E-2 -3.0986002707321395E-2
7.9483740655806087E-2 0.10934853098288892 -
1.6981645157961755E-2 -3.9637113175926407E-2
5.1725910558059016E-2 -7.5293925805213309E-3 -
9.0361172024363487E-2 -9.5039047334366744E-2
4.7403568255148722E-2 -3.8255645271969832E-3
4.3218973200413506E-2 -1.8916714871863298E-2 -
3.9637113175926407E-2 6.5807443379660199E-2 -
1.316071492587351E-2 4.7403568255148722E-2
2.3704563216878447E-2 0.12678838148916216 -
1.1271722117620619E-2 0.10934853098288892 -
1.50012430464 39301E-4 -3.3094084606626398E-2
0.11894009781525601 -3.9637113175926407E-2 -
4.9008483416118451E-2 7.1005972903079154E-2
2.9372997557913449E-2 -6.5174257007283729E-2
3.7154046682836443E-2 4.7403568255148722E-2 -
8.2053210024488593E-2 -5.1497959731137855E-2
3.9150545149045569E-2 1.0802516698457023E-2 -
2.48410615005232E-2 3.9150545149045569E-2
3.129112880648411E-2 1.6802075480611047E-3
5.1725910558059016E-2 2.5581205766142971E-2 -
5.4058128270569515E-2 -2.0870420462762541E-2
1.0802516698457023E-2 0.10467065567288585 -
3.7416426909536671E-2 2.9372997557913449E-2 -
9.5039047334366744E-2 -3.0986002707321395E-2
7.9483740655806087E-2 0.10934853098288892 -
1.6981645157961755E-2 -3.9637113175926407E-2
5.1725910558059016E-2 -7.5293925805213309E-3 -
9.0361172024363487E-2 -9.5039047334366744E-2
4.7403568255148722E-2 -3.8255645271969832E-3 4.321
8973200413506E-2 -1.8916714871863298E-2 -
3.9637113175926407E-2 6.5807443379660199E-2 -
1.316071492587351E-2 4.7403568255148722E-2

x-axis: Historical
Market Returns
y-axis: Historical
MicroDrive Returns
Portfolio standard deviation = 22%
b1w1sM = 3.0%
b2w2sM = 6.0%
b3w3sM = 6.0%
b4w4sM = 7.0%
b1w1sM b2w2sM b3w3sM b4w4sM 0.03 0.06 0.06
6.9999999999999993E-2 -0.66 -
0.55000000000000004 -0.44000000000000006 -
0.33000000000000007 -0.22000000000000008 -
0.11000000000000008 0 0.11 0.22 0.33 0.44
0.55000000000000004 0.66 0.77 0.88
1.977148949688342E-2 3.0692439939570888E-2
4.5197958685507564E-2 6.2478934368428322E-2
8.0632887903820935E-2 9.6879145814807499E-2
0.10820778739354575 0.11227871279487092
0.10820778739354575 9.6879145814807499E-2
8.0632887903820935E-2 6.2478934368428322E-2
4.5197958685507564E-2 3.0692439939570888E-2
1.977148949688342E-2
Outcomes: Stock Returns
Probability
35%
6-1SECTION 6-1SOLUTIONS TO SELF-TESTSuppose you pay
$500 for an investment that returns $600 in one year. What is
the annual rate of return?Amount invested$500Amount received
in one year$600Dollar return$100Rate of return20%

6-2SECTION 6-2SOLUTIONS TO SELF-TESTAn investment
has a 20% chance of producing a 25% return, a 60% chance of
producing a 10% return, and a 20% chance of producing a -15%
return. What is its expected return? What is its standard
deviation? ProbabilityReturnProb x
Ret.20%25%5.0%60%10%6.0%20%-15%-3.0%Expected return
=8.0%Alternatively, use the Excel SUMPRODUCT function,
which will multiply each value in the first array be the
corresponding value in the next array, and the sum them. This is
exactly the calculation shown in the step-by-step manner
above:Expected return =8.0%ProbabilityReturnDeviation from
expected returnDeviation2Prob x
Dev.220%25%17.0%2.890%0.578%60%10%2.0%0.040%0.024%
20%-15%-23.0%5.290%1.058%Variance =0.01660Standard
deviation =12.9%Alternatively, use the Excel SUMPRODUCT
function, which will multiply each value in the first array be the
corresponding value in the next array, and the sum them. If the
first array has probabilities and the second array subtracts the
mean from the array of outcomes and is squared, then this is
exactly the calculation shown in the step-by-step manner above
to find variance:Variance =0.01660Standard deviation =12.9%
6-4SECTION 6-4SOLUTIONS TO SELF-TESTA stock’s returns
for the past three years are 10%, -15%, and 35%. What is the
historical average return? What is the historical sample
standard deviation?RealizedYearreturn110%2-
15%335%Average =10.0%Standard deviation = 25.0%
6-5SECTION 6-5SOLUTIONS TO SELF-TESTStock A's returns
the past five years have been 10%, −15%, 35%, 10%, and −20%.
Stock B's returns have been −5%, 1%, −4%, 40%, and 30%.
What is the correlation coefficient for returns between Stock A
and Stock B?Realized ReturnsYearStock AStock B110%-5%2-
15%1%335%-4%410%40%5-20%30%Average
=4.0%12.4%Standard deviation = 22.2%21.1%Correlation
between Stock A and Stock B:-0.35
6-6SECTION 6-6SOLUTIONS TO SELF-TESTAn investor has a
3-stock portfolio with $25,000 invested in Apple, $50,000

invested in Ford, and $25,000 invested in Walmart. Dell’s beta
is estimated to be 1.20, Ford’s beta is estimated to be 0.80, and
Walmart's beta is estimated to be 1.0. What is the estimated
beta of the investor’s portfolio?
StockInvestmentBetaWeightBeta x
WeightApple$25,0001.20.250.30Ford$50,0000.80.500.40Walma
rt$25,0001.00.250.25Total$100,000Portfolio beta =0.95
6-7SECTION 6-7SOLUTIONS TO SELF-TESTA stock has a
beta of 0.8. Assume that the risk-free rate is 5.5% and that the
market risk premium is 6%. What is the stock’s required rate of
return? Beta0.8Risk-free rate5.5%Market risk
premium6.0%Required rate of return10.30%
6-9SECTION 6-9SOLUTIONS TO SELF-TEST An analyst has
modeled the stock of a company using a Fama-French three-
factor model and has estimate that ai = 0, bi = 0.7, ci = 1.2, and
di = 0.7. Suppose the daily risk-free rate is approximately equal
to zero, the market return is 11%, the return on the SMB
portfolio is 3.2%, and the return on the HML portfolio is 4.8%
on a particular day. The stock had an actual return of 16.9% on
that day. What is the stock's predicted return for that day?
What is the stock’s unexplained return for the
day?ai0.0%bi0.70ci1.20di0.70Actual stock return16.9%Risk-
free rate0.0%Market return11.0%SMB return3.2%HML
return4.8%Predicted return14.90%Unexplained return2.00%
ChapterTool KitChapter 611/20/18Risk and Return6-1
Investment Returns and RiskAmount invested$1,000Amount
received in one year$1,100Dollar return (Profit)$100Rate of
return = Profit/Investment =10%6-2 Measuring Risk for
Discrete DistributionsThe 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.A listing of possible outcomes and their probabilities is
called a probability distribution, as shown
below.ScenarioProbability of ScenarioRate of Return in
ScenarioBest Case0.3037%Most Likely0.4011%Worst
Case0.30−15%1.00Figure 6-1Discrete Probability Distribution
for Three ScenariosGiven the probabilities and the outcomes for
possible returns, it is possible to calculate the expected return
and standard deviation.Figure 6-2Calculating Expected Returns
and Standard Deviations: Discrete
ProbabilitiesINPUTS:Expected ReturnStandard
DeviationScenarioProbability of Scenario
(1)Market Rate of Return
(2)Product of Probability and Return
(3) = (1) × (2) Deviation from Expected Return
(4) = (2) − D66Squared Deviation
(5) = (4)2 Prob. × Sq. Dev.
(6) = (1) × (5)Best
Case0.3037%11.1%0.26000.06760.0203Excel functions for
finding expected return and standard deviation of discrete
eventsMost Likely0.4011%4.4%0.00000.00000.0000Use
SUMPRODUCT to find expected return by putting probabilities
in first argument array and rates of return in the second
argument array.11%Worst Case0.30−15%−4.5%-
0.26000.06760.0203=SUMPRODUCT(B63:B65,C63:C65)1.00E
xp. ret. = Sum = 11.0% Sum = Variance =0.0406Use
SUMPRODUCT to find variance. If the first array has
probabilities and the second array subtracts the mean from the
array of outcomes and is squared, then this is exactly the
calculation shown in the step-by-step manner above to find
variance:0.0406Std. Dev. = Square
root of variance =20.1%=SUMPRODUCT(B63:B65,(C63:C65-
D66)^2)Note: Calculations are not rounded in intermediate
steps.6-3 Risk in a Continuous DistributionIt is possible to add
more scenarios.ScenarioPanel A: Probability of Market Return

ScenarioPanel B: Probability of Stock Return Scenario Rate of
Return in Scenario10.00020.0198-66%20.00110.0307-
55%30.00540.0452-44%40.02050.0625-33%50.05750.0806-
22%60.12010.0969-
11%70.18700.10820%80.21670.112311%90.18700.108222%100
.12010.096933%110.05750.080644%120.02050.062555%130.00
540.045266%140.00110.030777%150.00020.019888%1.00001.0
000Average =11.0%11.0%Std. dev. =20.2%36.2%Figure 6-
3Discrete Probability Distributions for 15 ScenariosPanel A:
Market Return for 15 Scenarios: Standard Devation =
20.2%Panel B: Single Company's Stock Return for 15
Scenarios: Standard Devation = 36.2%At some point, it becomes
impractical to keep adding scenarios. Many analysts use the
normal distribution to estimate stock returns.Here is an example
of a normal distribution with a similar mean and standard
deviation as the discrete distribution shown above.6-4 Using
Historical Data to Estimate RiskInvestors often use historical
data to estimate risk. This is quite easy in Excel by using the
AVERAGE and STDEV functions.Standar d Deviation Based On
a Sample of Historical
DataInputs:RealizedYearreturn201715.0%2018−5.0%201920.0%
Calculations:=AVERAGE(E183:E185)10.0%=STDEV(E183:E18
5)13.2%Use STDEV, the function for a sample.Measuring the
Standard Deviation of MicroDriveThe monthly stock returns for
MicroDrive and one of its competitors, SnailDrive, during the
past 48 months are shown in the figure below. The actual data
are below the figure.Figure 6-5Historical Monthly Stock
Returns for MicroDrive and
SnailDriveMicroDriveSnailDriveAverage Return
(annualized)12.0%9.3%Standard Deviation
(annualized)51.8%34.2%Weights are specified in Section 6-
5.Portfolio
weightsSnailDrive:75%MicroDrive:25%PeriodMarketMicroDriv
eSnailDrivePortfolio12.37%2.81%14.93%11.90%212.68%14.79
%−3.26%1.25%3−1.13%0.79%−10.57%−7.73%410.93%8.71%−
10.76%−5.89%5−0.02%0.83%9.71%7.49%6−3.31%−32.42%6.4

0%−3.30%711.89%22.56%0.26%5.83%8−3.96%−24.21%0.52%
−5.66%9−4.90%8.00%−8.67%−4.50%107.10%−1.29%21.62%15
.89%112.94%4.43%3.87%4.01%12−6.52%−6.36%5.00%2.16%1
33.72%11.79%−12.32%−6.29%144.74%21.32%−2.43%3.51%15
−8.21%−10.28%4.87%1.08%16−5.15%3.96%−17.85%−12.40%1
73.92%34.98%−10.89%0.57%181.08%2.56%−16.68%−11.87%1
9−2.48%−10.80%9.02%4.07%203.92%−6.70%22.60%15.28%21
3.13%−2.31%14.25%10.11%220.17%8.26%−7.68%−3.69%235.1
7%0.51%−2.52%−1.76%242.56%−14.61%9.32%3.34%25−5.41
%−4.56%−1.38%−2.17%26−2.09%−12.08%13.92%7.42%271.08
%31.68%3.91%10.85%2810.47%4.43%8.91%7.79%29−3.74%0.
32%−3.76%−2.74%302.94%4.59%−3.95%−1.82%31−9.50%2.08
%−1.43%−0.55%32−3.10%−15.49%−6.38%−8.65%337.95%32.3
9%12.00%17.10%3410.93%15.15%−2.00%2.28%35−1.70%−3.7
2%−12.51%−10.31%36−3.96%−15.40%−0.49%−4.22%375.17%
−12.67%9.91%4.27%38−0.75%−10.43%−8.21%−8.76%39−9.04
%−7.14%−11.27%−10.24%40−9.50%−4.85%−10.32%−8.96%41
4.74%8.15%9.19%8.93%42−0.38%−14.72%−0.43%−4.00%434.
32%32.45%0.99%8.85%44−1.89%−28.34%3.96%−4.12%45−3.9
6%−5.55%−8.50%−7.77%466.58%5.81%16.10%13.52%47−1.32
%4.02%8.86%7.65%484.74%4.38%1.30%2.07%Full 48
(annual):19.9%51.8%34.2%27.1%Maximum of monthly
returns:12.7%34.98%22.60%17.1%Minimum of monthly
returns:-9.5%-32.42%-17.85%-12.4%Past 12
MonthsMonthMarketMicroDriveSnailDrivePortfolio375.2%-
12.7%9.9%4.3%38-0.8%-10.4%-8.2%-8.8%39-9.0%-7.1%-
11.3%-10.2%40-9.5%-4.9%-10.3%-
9.0%414.7%8.1%9.2%8.9%42-0.4%-14.7%-0.4%-
4.0%434.3%32.4%1.0%8.9%44-1.9%-28.3%4.0%-4.1%45-4.0%-
5.6%-8.5%-7.8%466.6%5.8%16.1%13.5%47-
1.3%4.0%8.9%7.7%484.7%4.4%1.3%2.1%Past 12

return:-0.11%-2.41%0.97%0.12%Average return (annual):-
1.3%-28.9%11.6%1.5%Standard deviation
(annual):18.9%52.4%31.4%29.1%Total compound return:-2.9%-
34.3%7.3%-2.4%The total compound return is the total return
on $1 invested at the end of month 36. The Excel function
=FVSCHEDULE calculates the ending value given an initial
amount and a series of returns.6-5 Risk in a Portfolio
ContextNow we are going to analyze the risk of a portfolio
instead of the stand-alone risk of individual assets.Creating a
PortfolioLook at the data for MicroDrive and SnailDrive shown
above. The last column shows a portfolio with the weights
shown below. Here are the results for the two companies and for
the portfolio. Notice that the portfolio has a higher return than
SnailDrive and less risk than either of the two stocks.Portfolio
weightsSnailDrive:75%MicroDrive:25%Full 48
(annual):19.9%51.8%34.2%27.1%CorrelationLoosely speaking,
correlation measures the tendency of two variables to move
together.Correlation between MicroDrive and SnailDrive:r =-
0.133=CORREL(E232:E279,F232:F279)6-6 The Relevant Risk
of a Stock: The Capital Asset Pricing Model (CAPM)The
Capital Asset Pricing Model (CAPM) provides a measure of
risk.Contribution to Market Risk: BetaThe relevant risk of an
individual stock as defined by its beta. Beta measures how much
risk a stock contributes to a well-diversified portfolio.Beta for
Stock i = bi = riM(si/sM)A portfolio's beta is the weighted
average of the stock's individual betas. Consider the following
example.Stock Beta:Weight in Portfolio:Contribution of Stock
to Portfolio Beta:biwibi x wi x sMStock 10.625.0%0.150Stock
21.225.0%0.300Stock 31.225.0%0.300Stock
41.425.0%0.350Portfolio beta = 1.100The standard deviation of
a well-diversified portfolio is:Std. Dev. of portfolio = sp = bp
(sM)Note: if the bp is negative, then σp = |bp| (σM).If the

example portfolio had more than 4 stocks and was well -
diversified, then its standard deviation would be:Beta of
portfolio = bp =1.1Std. Dev. of market = sM =20%Std. Dev. of
portfolio = sp =22%Figure 6-7The Contribution of Individual
Stocks to Portfolio Risk: The Effect of BetaMarket standard
deviation = sM =20.0%Stock Beta:Weight in
Portfolio:Contribution of Stock to Portfolio Beta:Contribution
of Stock to Portfolio Risk:biwibi x wi bi x wi x sMCategory
Labels for chart.Stock 10.625.0%0.1503.0%b1w1sMStock
21.225.0%0.3006.0%b2w2sMStock
31.225.0%0.3006.0%b3w3sMStock
41.425.0%0.3507.0%b4w4sM1.10022.0%b5w5sMEstimating
BetaWe can use the data shown previously for MicroDrive and
SnailDrive to estimate their betas.Calculating
BetaMarketMicroDriveSnailDriveStandard deviation
(annual):19.89%51.75%34.17%Correlation with the
market:0.5110.264bi = riM(si/sM)1.3300.454Beta can also be
calculated as the slope of a regression of the stock (on the y-
axis) and the market (on the x-axis). This can be done using the
SLOPE function or by plotting the returns and specifying that
the chart show the TRENDLINE.Calculating Beta as the Slope
of a Regression Using Excel Functions (See Excel explanations
to right)MicroDriveSnailDrivebi =
riM(si/sM)1.3300.454=SLOPE(F232:F279,$D$232:$D$279)Inte
rcept-0.0020.004=INTERCEPT(F232:F279,$D$232:$D$279)R
squared0.2610.070=RSQ(F232:F279,$D$232:$D$279)Calculatin
g Confidence Intervals using Excel FunctionsSee the comment
in this cell for instructions for how to use LINEST and estimate
confidence intervals.
Mike Ehrhardt: Show below are the outputs from the array
function LINEST. The yellow area shows what statistics are
output by LINEST. To calculate the statistics for MicroDrive,
highlight the gray area. Enter the formula
=LINEST(E209:E256,D209:D256,TRUE,TRUE), then hit Ctrl-
Shift-Enter. Use a similar process to calculate the regression

statistics for SnailDrive in the blue region.
Recall from statistics that if you take an estimated coefficient
from a simple regression, subtract a target value (which is often
zero), and then divide that difference by the standard error of
the estimated the coefficient, the result will be a t-statistic,
which conform to the Student's t-distribution with n-k-1 degrees
of freedom, where n is the number of observations and k is the
number of explanatory variables. This approach is used to
determine whether the estimated coefficient is statistically
significantly different from zero. To find the confidence
interval around an estimated regression coefficient, you use this
relationship, but you pick a target probability rather than a
target value. Using the target probability and the degrees of
freedom, the TINV function will provide the corresponding
value (i.e., t-stat) for a two-tailed t-test. Therefore, the
confidence interval corresponding to the target probability has a
range equal to 2(t-stat)(standard error of coefficient). The lower
end of the range is equal to the estimated coefficient minus (t-
stat)(standard error of coefficient); the upper end of the range is
equal to the estimated coefficient plus (t-stat)(standard error of
coefficient).Input desired probability for confidence
interval95%95%Excel function LINEST output in an
arrayLINEST Output for MicroDriveLINEST Output for
SnailDriveLower boundary of confidence interval for beta0.666-
0.037See explanation to right.SlopeIntercept1.330-
0.0020.4540.004Upper boundary of confidence interval for
beta1.9940.946See explanation to right.Standard error
slopeStandard error intercept0.3300.0190.2440.014Lower
boundary of confidence interval for intercept-0.040-0.025See
explanation to right.R squaredStandard
error0.2610.1300.0700.096Upper boundary of confidence
interval for intercept0.0360.032See explanation to
right.FDegrees of freedom1646346SS RegressionSS
Residual0.2740.7750.0320.425Figure 6-8Stock Returns of
MicroDrive and the Market: Estimating BetaShown below is the
output from Excel's regression tool. From the menu bar, select

Data, then Data Analysis, the Regression.MicroDriveSnail
DriveSUMMARY OUTPUTSUMMARY OUTPUTRegression
StatisticsRegression StatisticsMultiple R0.5108093315Multiple
R0.2625490104R Square0.2609261731R
Square0.0689319829Adjusted R Square0.2445023103Adjusted
R Square0.0482415825Standard Error0.1312392274Standard
Error0.0950833686Observations47Observations47ANOVAANO
VAdfSSMSFSignificance FdfSSMSFSignificance
FRegression10.27363375230.273633752315.88701610530.0002
437609Regression10.03012041810.03012041813.33159250750.
0746025035Residual450.77506806630.0172237348Residual450.
40683811420.009040847Total461.0487018186Total460.436958
5323CoefficientsStandard Errort StatP-valueLower 95%Upper
95%Lower 95.0%Upper 95.0%CoefficientsStandard Errort
StatP-valueLower 95%Upper 95%Lower 95.0%Upper
95.0%Intercept-0.00197719060.0193616376-
0.10211897630.9191159351-0.04097353060.0370191493-
0.04097353060.0370191493Intercept0.00089492940.014027587
40.06379781280.9494137787-0.02735808180.0291479406-
0.02735808180.02914794060.02370456321.33013536730.33371
418953.98585199240.00024376090.65800048722.00227024740.
65800048722.00227024740.02370456320.44130770090.241777
32471.82526505130.0746025035-0.04565682810.9282722298-
0.04565682810.9282722298EXAMPLE: CALCULATING BETA
COEFFICIENTS FOR AN ACTUAL COMPANYNow we show
how to calculate beta for an actual company, General Electric.
Step 1. Retrieve DataWe downloaded stock prices and
dividends from http://finance.yahoo.com for General Electric,
using its ticker symbol GE, and for the S&P 500 Total Retun
Index ( symbol ^SP500TR), which is an index incorporating the
prices and dividends of the 500 companies listed in the S&P
500, which tracks 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 first 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 xlsx by opening a new Excel worksheet,
copying the date and adjusted index price data to it, and saving
as an xlsx 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
500 Total Return Index, as we show below.Step 2. Calculate
ReturnsNext, 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.
Yahoo actually adjusts the stock prices to reflect any stock
splits or dividend payments. For example, suppose the stock
price is $100 in July, the company has a 2-for-1 split, and the
actual price in August is $60. The reported adjusted price for
August would be $60, but the reported price for July would be
$50, which reflects the stock split. This gives an accurate stock
return of 20%: ($60-$50)/$50 = 20%, the same as if there had
not been a split, in which case the return would have been
($120-$100)/$100 = 20%.
Or suppose the actual price in September is $50, the
company pays a $10 dividend, and the actual price in October is
$60. Shareholders have had a return of ($60+$10-$50)/$50 =
40%. Yahoo reports an adjusted price of $60 for October, and
an adjusted price of $42.857 for September, which gives a
return of ($60-$42.857)/$42.857 = 40%.
accurately reflects the actual return. At this point, we are ready
to calculate some statistics and to find GE's beta coefficient.
This is shown below the data.Not in Textbook: Stock Return
Data for GE and the S&P 500 Total Return IndexMonthMarket
Level
(S&P 500 Total Return Index) at Month EndMarket's

ReturnGE Adjusted Stock Price at Month End
Michael C. Ehrhardt: Yahoo actually adjusts the stock prices to
reflect any stock splits or dividend payments. For example,
suppose the stock price is $100 in July, the company has a 2-
for-1 split, and the actual price in August is $60. The reported
adjusted price for August would be $60, but the reported price
for July would be $50, which reflects the stock split. This gives
an accurate stock return of 20%: ($60-$50)/$50 = 20%, the same
as if there had not been a split, in which case the return would
have been ($120-$100)/$100 = 20%.
Or suppose the actual price in September is $50,the company
pays a $10 dividend, and the actual price in October is $60.
Shareholders have had a return of ($60+$10-$50)/$50 = 40%.
Yahoo reports an adjusted price of $60 for October, and an
adjusted price of $42.857 for September, which gives a return
of ($60-$42.857)/$42.857 = 40%.
accurately reflects the actual return.
GE's
ReturnNovember 20175,155.443.1%$18.29-9.3%October
20175,002.032.3%$20.16-15.8%September
20174,887.972.1%$23.94-1.5%August
20174,789.180.3%$24.31-4.1%July 20174,774.562.1%$25.36-
4.4%June 20174,678.360.6%$26.52-1.4%May
20174,649.341.4%$26.88-5.6%April 20174,584.821.0%$28.46-
2.7%March 20174,538.210.1%$29.260.8%February
20174,532.934.0%$29.040.4%January
20174,359.811.9%$28.93-5.3%December
20164,278.662.0%$30.552.7%November
20164,195.733.7%$29.745.7%October 20164,045.89-
1.8%$28.13-1.0%September 20164,121.060.0%$28.41-
5.2%August 20164,120.290.1%$29.970.3%July
20164,114.513.7%$29.87-0.3%June

20163,968.210.3%$29.974.1%May 20163,957.951.8%$28.78-
1.7%April 20163,888.130.4%$29.28-3.3%March
20163,873.116.8%$30.2710.0%February 20163,627.06-
0.1%$27.520.1%January 20163,631.96-5.0%$27.49-
5.9%December 20153,821.60-1.6%$29.204.0%November
20153,882.840.3%$28.073.5%October
20153,871.338.4%$27.1115.7%September 20153,570.17-
2.5%$23.430.8%August 20153,660.75-6.0%$23.24-4.2%July
20153,895.802.1%$24.25-0.9%June 20153,815.85-1.9%$24.48-
2.6%May 20153,891.181.3%$25.130.7%April
20153,841.781.0%$24.959.1%March 20153,805.27-1.6%$22.86-
3.7%February 20153,866.425.7%$23.738.8%January
20153,656.28-3.0%$21.81-4.6%December 20143,769.44-
0.3%$22.86-4.6%November
20143,778.962.7%$23.962.6%October
20143,679.992.4%$23.341.6%September 20143,592.25-
1.4%$22.98-1.4%August 20143,643.334.0%$23.303.3%July
20143,503.19-1.4%$22.56-3.5%June 20143,552.182.1%$23.38-
1.9%May 20143,480.292.3%$23.83-0.4%April
20143,400.460.7%$23.923.9%March
20143,375.510.8%$23.032.5%February
20143,347.384.6%$22.461.4%January 20143,200.95-
3.5%$22.16-9.6%December
20133,315.592.5%$24.525.1%November
20133,233.72NA$23.32NADescription of DataAverage return
(annual):12.2%-4.3%Standard deviation
(annual):9.6%18.7%Minimum monthly return:-6.0%-
15.8%Maximum monthly return:8.4%15.7%Correlation between
GE and the market:0.51Beta: bGE = rGE,M (sGE / sM)0.99Beta
(using the SLOPE function):0.99Intercept (using the
INTERCEPT function):-0.01R2 (using the RSQ
function):0.26Step 3. Examine the Data and Calculate
BetaUsing 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 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 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
beta.6-7 The Relationship between Risk and Return in the
Capital Asset Pricing ModelThe SML shows the relationship
between the stock's beta and its required return, as predicted by
the CAPM.rRF6%<< Varies over time, but is constant for all
firms at a given time.rM11%<< Varies over time, but is
constant for all firms at a given time.bi0.5<< Varies over time,
and varies from firm to firm.The SML predicts stock i's required
return to be:RPM = rM - rRF ri = rRF + bi(RPM)RPM = rM -
rRF =5%ri = rRF + bi(RPM)8.5%With the above data, we can
generate a Security Market Line that is flexible enough to allow
for changes inany of the input factors. We generate a table of
values for beta and expected returns, and then plot thegraph as a
scatter diagram.BetaSecurity Market Line: riRisk-Free
Rate0.006.0%6%0.508.5%6%1.0011.0%6%1.5013.5%6%2.0016.
0%6%Figure 6-9The Security Market LineThe 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
(rM – rRF), 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.Portfolio
ReturnsThe same relationship holds for required returns: The
required return on a portfolio is simply a weighted average of
the required returns of the individual assets in the portfolio. The
weights are the percentage of total portfolio funds invested in
each asset. The required return on a portfolio is also equal to:rp
= rRF + bp(RPM)The expected return on a portfolio is simply a

weighted average of the expected returns of the individual
assets in the portfolio. The weights are the percentage of total
portfolio funds invested in each asset. Consider the following
portfolio and the hypothetical illustrative returns
data.StockAmount of InvestmentPortfolio
WeightExpected
ReturnWeighted Expected
ReturnSouthwest
Airlines$300,0000.315.0%4.5%Starbucks$100,0000.112.0%1.2
%FedEx$200,0000.210.0%2.0%Dell$400,0000.49.0%3.6%Total
investment =$1,000,0001.0Portfolio's Expected Return
=11.3%6-8 The Efficient Markets HypothesisThe Efficient
Markets Hypothesis (EMH) asserts that (1) stocks are always in
equilibrium and (2) it is impossible for an investor to “beat the
market” and consistently earn a higher rate of return than is
justified by the stock’s risk. 6-9 The Fama-French Three-Factor
ModelThe Fama-French 3-Factor model shows the actual stock
return given the risk-free rate, the return on the market, the
return on the SMB portfolio, and the return on the HML
portfolio:Suppose a company announces that it is going to
include more outsiders on its board of directors and that the
company’s stock falls by 2% on the day of the announcement.
Does that mean that investors don’t want outsiders on the
board? Actual return on announcement day =-2%Suppose you
estimate the following coefficients of the Fama-French model
using historical actual data prior to the announcement date:ai
=0.0bi =0.9ci =0.2di =0.3These are the returns on the
announcement day:rRF ≈0.0%rM =-3.0%rSMB =-1.0%rHML =-
2.0%The predicted return on the announcement day:Predicted
return =rRF,t + ai + bi(rM,t - rRF,t) + ci(rSMB,t) +
di(rHML,t)Predicted return =-3.5%Unexplained return =Actual
return - predicted returnUnexplained return =1.5%
0 0.5 1 1.5 2 0.06 8.4999999999999992E-2 0.11
0.13500000000000001 0.16 Risk-Free Rate

Risk-Free Rate
0 0.5 1 1.5 2 0.06 0.06 0.06 0.06 0.06
Beta
Required Return
0.37 0.11 -0.15 0.3 0.4 0.3
Outcomes: Market Returns for 3 Scenarios
Probability of Scenario
-0.66 -0.55000000000000004 -0.44000000000000006 -
0.33000000000000007 -0.22000000000000008 -
0.11000000000000008 0 0.11 0.22 0.33 0.44
0.55000000000000004 0.66 0.77 0.88
1.7511394806058752E-4 1.0680516885827041E-3
5.4186431553394105E-3 2.0452870329666931E-2
5.7451043229686055E-2 0.12012007299591695
0.18697232380635256 0.21668376169278963
0.18697232380635254 0.12012007299591698
5.7451043229685972E-2 2.0452870329666917E-2
5.4186431553394643E-3 1.0680516885827052E-3
1.7511394806057901E-4
Outcomes: Market Returns
Probability
Normal Distribution
-0.66 -0.55000000000000004 -0.44000000000000006 -
0.33000000000000007 -0.22000000000000008 -
0.11000000000000008 0 0.11 0.22 0.33 0.44
0.55000000000000004 0.66 0.77 0.88
1.7511394806058752E-4 1.0680516885827041E-3
5.4186431553394105E-3 2.0452870329666931E-2
5.7451043229686055E-2 0.12012007299591695
0.18697232380635256 0.21668376169278963
0.18697232380635254 0.12012007299591698

5.7451043229685972E-2 2.0452870329666917E-2
5.4186431553394643E-3 1.0680516885827052E-3
1.7511394806057901E-4
Return
Probability
MicroDrive 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22
23 24 25 26 27 28 29 30 31 32 33 34
35 36 37 38 39 40 41 42 43 44 45 46
47 48 2.8123990924714889E-2 0.14788012262092151
7.8502963553737266E-3 8.7129468545996486E-2
8.3412755288645168E-3 -0.32424777774866015
0.22562272254962037 -0.2421188972496903
8.0033999253312074E-2 -1.2880104827403847E-2
4.4340333222723204E-2 -6.3572272894571916E-2
0.11793465207595152 0.2132415493195787 -
0.10275248323559877 3.9638703647158331E-2
0.34980897959761476 2.5618545344283039E-2 -
0.10798804364762635 -6.6991550094965782E-2 -
2.309098107106064E-2 8.2631500961910756E-2
5.0805676359279997E-3 -0.14613977701929601 -
4.5612755984871939E-2 -0.12083292328306293
0.31684468589336262 4.4284410066492273E-2
3.1607939009120284E-3 4.5883106630769353E-2
2.07552456031646E-2 -0.15485866662189754
0.32391586170454112 0.15150100919688878 -
3.7185495271657959E-2 -0.1539751666991345 -
0.12667133730723446 -0.10429634808460107 -
7.1363934717645017E-2 -4.8514454737545759E-2
8.146235222622937E-2 -0.14717242637944389
0.32446717204181608 -0.2833663222356419 -
5.5520125185090474E-2 5.8097374046303288E-2
4.0169352047872364E-2 4.3819809396092653E-2
SnailDrive 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33

34 35 36 37 38 39 40 41 42 43 44 45
46 47 48 0.1492690115098104 -
3.2625626037028344E-2 -0.10573003868698551 -
0.10761017744603606 9.7091665313481013E-2
6.4019696075159227E-2 2.551740058812201E-3
5.1823111248978371E-3 -8.6675542060738534E-2
0.21619937238442177 3.8682354577748018E-2
4.9976714369936108E-2 -0.1231552558765608 -
2.4334023349728399E-2 4.8709991986939309E-2 -
0.17848886713987205 -0.10894557506251189 -
0.16677693278427547 9.02224897298773E-2
0.22602742043638951 0.14253323854601735 -
7.6768659825705354E-2 -2.5191647655802694E-2
9.3197313516893962E-2 -1.3762620727642588E-2
0.13919465840323408 3.9062205051343148E-2
8.9076694586209232E-2 -3.7587548782611876E-2 -
3.951356902881889E-2 -1.4254586969160453E-2 -
6.3757773671727844E-2 0.11996362520663377 -
2.003461329144321E-2 -0.12506 156308683511 -
4.944118628225227E-3 9.9145429710516583E-2 -
8.206824068176545E-2 -0.11269202815269945 -
0.10323090464415442 9.1937173710245873E-2 -
4.2822325102026349E-3 9.8823607828717221E-3
3.955891709074854E-2 -8.5047861043818507E-2
0.16096750035334298 8.8623069016098177E-2
1.3046620641216391E-2
Month of Return
Monthly Rate of Return
MicroDrive
y = 1.33x - 0.002
R² = 0.2612
2.3704563216878447E-2 0.12678838148916216 -
1.1271722117620619E-2 0.10934853098288892 -
1.5001243046439301E-4 -3.3094084606626398E-2

0.11894009781525601 -3.9637113175926407E-2 -
4.900848341 6118451E-2 7.1005972903079154E-2
2.9372997557913449E-2 -6.5174257007283729E-2
3.7154046682836443E-2 4.7403568255148722E-2 -
8.2053210024488593E-2 -5.1497959731137855E-2
3.9150545149045569E-2 1.0802516698457023E-2 -
2.48410615005232E-2 3.9150545149045569E-2
3.129112880648411E-2 1.6802075480611047E-3
5.1725910558059016E-2 2.5581205766142971E-2 -
5.4058128270569515E-2 -2.0870420462762541E-2
1.0802516698457023E-2 0.10467065567288585 -
3.7416426909536671E-2 2.9372997557913449E-2 -
9.5039047334366744E-2 -3.0986002707321395E-2
7.9483740655806087E-2 0.10934853098288892 -
1.6981645157961755E-2 -3.9637113175926407E-2
5.1725910558059016E-2 -7.5293925805213309E-3 -
9.0361172024363487E-2 -9.5039047334366744E-2
4.7403568255148722E-2 -3.8255645271969832E-3
4.3218973200413506E-2 -1.8916714871863298E-2 -
3.9637113175926407E-2 6.5807443379660199E-2 -
1.316071492587351E-2 4.7403568255148722E-2
2.8123990924714889E-2 0.14788012262092151
7.8502963553737266E-3 8.7129468545996486E-2
8.3412755288645168E-3 -0.32424777774866015
0.22562272254962037 -0.2421188972496903
8.0033999253312074E-2 -1.2880104827403847E-2
4.4340333222723204E-2 -6.3572272894571916E-2
0.11793465207595152 0.2132415493195787 -
0.10275248323559877 3.9638703647158331E-2
0.34980897959761476 2.5618545344283039E-2 -
0.10798804364762635 -6.6991550094965782E-2 -
2.309098107106064E-2 8.2631500961910756E-2
5.0805676359279997E-3 -0.14613977701929601 -
4.5612755984871939E-2 -0.12083292328306293
0.31684468589336262 4.4284410066492273E-2
3.1607939009120284E-3 4.5883106630769353E-2

2.07552456031646E-2 -0.15485866662189754
0.32391586170454112 0.151501009196888 78 -
3.7185495271657959E-2 -0.1539751666991345 -
0.12667133730723446 -0.10429634808460107 -
7.1363934717645017E-2 -4.8514454737545759E-2
8.146235222622937E-2 -0.14717242637944389
0.32446717204181608 -0.2833663222356419 -
5.5520125185090474E-2 5.8097374046303288E-2
4.0169352047872364E-2 4.3819809396092653E-2 Market
vs. Market
Market vs. Market
2.3704563216878447E-2 0.12678838148916216 -
1.1271722117620619E-2 0.10934853098288892 -
1.5001243046439301E-4 -3.3094084606626398E-2
0.11894009781525601 -3.9637113175926407E-2 -
4.9008483416118451E-2 7.1005972903079154E-2
2.9372997557913449E-2 -6.5174257007283729E-2
3.7154046682836443E-2 4.7403568255148722E-2 -
8.2053210024488593E-2 -5.1497959731137855E-2
3.9150545149045569E-2 1.0802516698457023E-2 -
2.48410615005232E-2 3.9150545149045569E-2
3.129112880648411E-2 1.6802075480611047E-3
5.1725910558059016E-2 2.5581205766142971E-2 -
5.4058128270569515E-2 -2.0870420462762541E-2
1.0802516698457023E-2 0.10467065567288585 -
3.7416426909536671E-2 2.9372997557913449E-2 -
9.5039047334366744E-2 -3.0986002707321395E-2
7.9483740655806087E-2 0.10934853098288892 -
1.6981645157961755E-2 -3.9637113175926407E-2
5.1725910558059016E-2 -7.5293925805213309E-3 -
9.0361172024363487E-2 -9.5039047334366744E-2
4.7403568255148722E-2 -3.8255645271969832E-3
4.3218973200413506E-2 -1.8916714871863298E-2 -
3.9637113175926407E-2 6.5807443379660199E-2 -
1.316071492587351E-2 4.7403568255148722E-2
2.3704563216878447E-2 0.12678838148916216 -

1.1271722117620619E-2 0.10934853098288892 -
1.50012430464 39301E-4 -3.3094084606626398E-2
0.11894009781525601 -3.9637113175926407E-2 -
4.9008483416118451E-2 7.1005972903079154E-2
2.9372997557913449E-2 -6.5174257007283729E-2
3.7154046682836443E-2 4.7403568255148722E-2 -
8.2053210024488593E-2 -5.1497959731137855E-2
3.9150545149045569E-2 1.0802516698457023E-2 -
2.48410615005232E-2 3.9150545149045569E-2
3.129112880648411E-2 1.6802075480611047E-3
5.1725910558059016E-2 2.5581205766142971E-2 -
5.4058128270569515E-2 -2.0870420462762541E-2
1.0802516698457023E-2 0.10467065567288585 -
3.7416426909536671E-2 2.9372997557913449E-2 -
9.5039047334366744E-2 -3.0986002707321395E-2
7.9483740655806087E-2 0.10934853098288892 -
1.6981645157961755E-2 -3.9637113175926407E-2
5.1725910558059016E-2 -7.5293925805213309E-3 -
9.0361172024363487E-2 -9.5039047334366744E-2
4.7403568255148722E-2 -3.8255645271969832E-3 4.321
8973200413506E-2 -1.8916714871863298E-2 -
3.9637113175926407E-2 6.5807443379660199E-2 -
1.316071492587351E-2 4.7403568255148722E-2
x-axis: Historical
Market Returns
y-axis: Historical
MicroDrive Returns
Portfolio standard deviation = 22%
b1w1sM = 3.0%
b2w2sM = 6.0%
b3w3sM = 6.0%
b4w4sM = 7.0%

b1w1sM b2w2sM b3w3sM b4w4sM 0.03 0.06 0.06
6.9999999999999993E-2 -0.66 -
0.55000000000000004 -0.44000000000000006 -
0.33000000000000007 -0.22000000000000008 -
0.11000000000000008 0 0.11 0.22 0.33 0.44
0.55000000000000004 0.66 0.77 0.88
1.977148949688342E-2 3.0692439939570888E-2
4.5197958685507564E-2 6.2478934368428322E-2
8.0632887903820935E-2 9.6879145814807499E-2
0.10820778739354575 0.11227871279487092
0.10820778739354575 9.6879145814807499E-2
8.0632887903820935E-2 6.2478934368428322E-2
4.5197958685507564E-2 3.0692439939570888E-2
1.977148949688342E-2
Outcomes: Stock Returns
Probability
35%
6-1SECTION 6-1SOLUTIONS TO SELF-TESTSuppose you pay
$500 for an investment that returns $600 in one year. What is
the annual rate of return?Amount invested$500Amount received
in one year$600Dollar return$100Rate of return20%
6-2SECTION 6-2SOLUTIONS TO SELF-TESTAn investment
has a 20% chance of producing a 25% return, a 60% chance of
producing a 10% return, and a 20% chance of producing a -15%
return. What is its expected return? What is its standard
deviation? ProbabilityReturnProb x
Ret.20%25%5.0%60%10%6.0%20%-15%-3.0%Expected return
=8.0%Alternatively, use the Excel SUMPRODUCT function,
which will multiply each value in the first array be the
corresponding value in the next array, and the sum them. This is
exactly the calculation shown in the step-by-step manner
above:Expected return =8.0%ProbabilityReturnDeviation from
expected returnDeviation2Prob x
Dev.220%25%17.0%2.890%0.578%60%10%2.0%0.040%0.024%

20%-15%-23.0%5.290%1.058%Variance =0.01660Standard
deviation =12.9%Alternatively, use the Excel SUMPRODUCT
function, which will multiply each value in the first array be the
corresponding value in the next array, and the sum them. If the
first array has probabilities and the second array subtracts the
mean from the array of outcomes and is squared, then this is
exactly the calculation shown in the step-by-step manner above
to find variance:Variance =0.01660Standard deviation =12.9%
6-4SECTION 6-4SOLUTIONS TO SELF-TESTA stock’s returns
for the past three years are 10%, -15%, and 35%. What is the
historical average return? What is the historical sample
standard deviation?RealizedYearreturn110%2-
15%335%Average =10.0%Standard deviation = 25.0%
6-5SECTION 6-5SOLUTIONS TO SELF-TESTStock A's returns
the past five years have been 10%, −15%, 35%, 10%, and −20%.
Stock B's returns have been −5%, 1%, −4%, 40%, and 30%.
What is the correlation coefficient for returns between Stock A
and Stock B?Realized ReturnsYearStock AStock B110%-5%2-
15%1%335%-4%410%40%5-20%30%Average
=4.0%12.4%Standard deviation = 22.2%21.1%Correlation
between Stock A and Stock B:-0.35
6-6SECTION 6-6SOLUTIONS TO SELF-TESTAn investor has a
3-stock portfolio with $25,000 invested in Apple, $50,000
invested in Ford, and $25,000 invested in Walmart. Dell’s beta
is estimated to be 1.20, Ford’s beta is estimated to be 0.80, and
Walmart's beta is estimated to be 1.0. What is the estimated
beta of the investor’s portfolio?
StockInvestmentBetaWeightBeta x
WeightApple$25,0001.20.250.30Ford$50,0000.80.500.40Walma
rt$25,0001.00.250.25Total$100,000Portfolio beta =0.95
6-7SECTION 6-7SOLUTIONS TO SELF-TESTA stock has a
beta of 0.8. Assume that the risk-free rate is 5.5% and that the
market risk premium is 6%. What is the stock’s required rate of
return? Beta0.8Risk-free rate5.5%Market risk
premium6.0%Required rate of return10.30%
6-9SECTION 6-9SOLUTIONS TO SELF-TEST An analyst has

modeled the stock of a company using a Fama-French three-
factor model and has estimate that ai = 0, bi = 0.7, ci = 1.2, and
di = 0.7. Suppose the daily risk-free rate is approximately equal
to zero, the market return is 11%, the return on the SMB
portfolio is 3.2%, and the return on the HML portfolio is 4.8%
on a particular day. The stock had an actual return of 16.9% on
that day. What is the stock's predicted return for that day?
What is the stock’s unexplained return for the
day?ai0.0%bi0.70ci1.20di0.70Actual stock return16.9%Risk-
free rate0.0%Market return11.0%SMB return3.2%HML
return4.8%Predicted return14.90%Unexplained return2.00%
Risk and Return
CHAPTER 6
© 2020 Cengage Learning. All Rights Reserved. May not be
copied, scanned, or duplicated, in whole or in part, except for
use as permitted in a license distributed with a certain product
or service or otherwise on a password-protected website for
classroom use.
Topics in Chapter
Basic return and risk concepts
Stand-alone risk
Portfolio (market) risk
Risk and return: CAPM/SML
Market equilibrium and market efficiency
classroom use.
Determinants of Intrinsic Value: The Cost of Equity

classroom use.
What are investment returns?
Investment returns measure the financial results of an
investment.
Returns may be historical or prospective (anticipated).
Returns can be expressed in:
Dollar terms.
Percentage terms.
classroom use.
An investment costs $1,000 and is sold after
1 year for $1,060.
Dollar return:
$ Received$ Invested
$1,060 $1,000$60.
Percentage return:
$ Return/$ Invested
$60/$1,0000.066%.

classroom use.
What is investment risk?
Investment risk is exposure to the chance of earning less than
expected.
The greater the chance of a return far below the expected return,
the greater the risk.
classroom use.
Scenarios and Returns for the 10-Year Zero Coupon T-bond
Over the Next YearScenarioProbabilityReturnWorst Case0.10
−14%Poor Case0.20 −4%Most Likely0.406%Good
Case0.2016%Best Case0.1026%1.00
classroom use.
Discrete Probability Distribution for Scenarios
classroom use.

Example of a Continuous Probability Distribution
classroom use.
Calculate the expected rate of return (r ̂ ) on the bond for the
next year.
= 0.10(-14%) + 0.20(-4%) + 0.40(6%)
+ 0.20(16%) + 0.10(26%)
= 6%
classroom use.
Use Excel to Calculate the Expected Value of a Discrete
Distribution
= SUMPRODUCT(Probabilities,Returns)
SUMPRODUCT:
Multiplies each value in the first array (the range of cells with
probabilities) by its corresponding value in the second array
(the range of cells with returns).
Sums the products.
This is identical to the formula on the previous slide.
See Ch06 Mini Case.xlsx

classroom use.
Consider these probability distributions for two investments.
Which riskier? Why?
classroom use.
Stand-Alone Risk: Standard Deviation
Stand-alone risk is the risk of each asset held by itself.
Standard deviation measures the dispersion of possible
outcomes.
For a single asset:
Stand-alone risk = Standard deviation
classroom use.
Variance (σ2) and Standard Deviation (σ) for Discrete
Probabilities

classroom use.
Standard Deviation of the Bond’s Return During the Next Year
σ2 = 0.10 (-0.14 – 0.06)2
+ 0.20 (-0.04 – 0.06)2
+ 0.40 (0.06 – 0.06)2
+ 0.20 (0.16 – 0.06)2
+ 0.10 (0.26 – 0.06)2
σ2 = 0.0120
σ =
σ = 0.1095 = 10.95%
classroom use.
Use Excel to Calculate the Variance and Standard Deviation of
a Discrete Distribution
= SUMPRODUCT(Probabilities,Returns− ,Returns−)
SUMPRODUCT:
Multiplies each value in the first array (the range of cells with
probabilities) by its corresponding value in the second array
(the range of cells with returns less the expected return) and by
the third array (which is identical to the second array).
Sums the products; the result is variance.
Take the square root of the variance to get the standard
deviation. See Ch06 Mini Case.xlsx
classroom use.

Understanding the Standard Deviation
If the returns are normally distributed:
Outcome will be more than 1 σ away from about 31.74% ≈ 32%
of the time:
16% of the time below −σ
16% of the time above +σ.
If = 6% and σ =10.95% ≈ 11%:
16% of the time return <−5% = 6% − 11%
16% of the time return > 17% = 6% + 11
classroom use.
Useful in Comparing Investments
Investments with bigger standard deviations have more risk.
High risk doesn’t mean you should reject the investment, but:
You should know the risk before investing
You should expect a higher return as compensation for bearing
the risk.
classroom use.
Using Historical Data to Estimate Risk
Analysts often use discrete outcomes to analyze risk for
projects; see Chapter 11.
But for investments, most analysts normally use historical data
rather than discrete forecasts to estimate an investment’s r isk
unless it is a very special situation.
Most analysts use:

48 to 60 months of monthly data, or
52 weeks of weekly data, or
Shorter period using daily data.
Use annual returns here for sake of simplicity.
classroom use.
Formulas for a Sample of T Historical Returns
Tedious to calculate by hand, easy in Excel. See next slide.
classroom use.
Excel Functions a Sample of T Historical Returns
Suppose “SampleData” is the cell range with the T historical
returns.
=AVERAGE(SampleData)
=STDEV(SampleData)
classroom use.

Historical Data for Stock
ReturnsYearMarketBlandyGourmange1 30% 26% 47%2 7
15 −543 18 −14 154 −22 −15 75 −14 2 −286 10
−18 407 26 42 178 −10 30 −239 −3 −32 −410 38
28 75
classroom use.
Average and Standard Deviations for
Stand-Alone Investments
Use formulas shown previously (tedious) or use Excel (easy)
What is Blandy’s stand-alone risk?
Note: analysts often use past risk as a predictor of future risk,
but past returns are not a good prediction of future
returns.MarketBlandyGourmangeAverage return 8.0% 6.4%
9.2%Standard deviation 20.1% 25.2% 38.6%
classroom use.
How risky is Blandy stock?
Assumptions:
Returns are normally distributed.
σ is 25.2%
Expected return is about 6.4%.
16% of the time (approximately), return will be:
< −18.8% (6.4%−25.2% = −18.8)
> 32.6% (6.4%+25.2% = 32.6)
Stocks are very risky!

classroom use.
Portfolio Returns
The percentage of a portfolio’s value that is invested in Stock i
is denoted by the “weight” wi. Notice that the sum of all the
weights must equal 1.
With n stocks in the portfolio, its return each year will be:
classroom use.
Example: 2-Stock Portfolio
Form a portfolio by selling 25% of the Blandy stock and
investing it in the higher-risk Gourmange stock.
The portfolio return each year will be:
classroom use.
Historical Data for Stocks and Portfolio
ReturnsYearBlandyGourmangePortfolio of Blandy

and Gourmange1 26% 47% 31.3%2 15 −54 −2.33
−14 15 −6.84 −15 7 −9.55 2 −28 −5.56
−18 40 −3.57 42 17 35.88 30 −23 16.89
−32 −4 −25.010 28 75 39.8
classroom use.
Portfolio Historical Average and
Standard Deviation
The portfolio’s average return is the weighted average of the
stocks’ average returns.
The portfolio’s standard deviation is less than either stock’s σ!
What explains this?BlandyGourmangePortfolioAverage return
6.4% 9.2% 7.1%Standard deviation 25.2% 38.6%
22.2%
classroom use.
How closely do the returns follow one another?
Notice that the returns don’t move in perfect lock-step:
Sometimes one is up and the other is down.
classroom use.

Correlation Coefficient (ρi,j)
Loosely speaking, the correlation (r) coefficient measures the
tendency of two variables to move together.
Estimating ri,j with historical data is tedious:
classroom use.
Excel Functions to Estimate
the Correlation Coefficient (ρi,j)
“Stocki” and “Stockj” are the cell ranges with historical returns
for Stocks i and j.
Est. ρi,j = Rij =Correl(Stocki,Stockj)
Correlation between Blandy (B) and Gourmange (G):
Est. ρB,G = 0.11
classroom use.
2-Stock Portfolios
r = −1
2 stocks can be combined to form a riskless portfolio: σp = 0.
r = +1
Risk is not “reduced”
σp is just the weighted average of the 2 stocks’ standard
deviations.

−1 < r < −1
Risk is reduced but not eliminated.
classroom use.
Adding Stocks to a Portfolio
What would happen to the risk of an average 1-stock portfolio
as more randomly selected stocks were added?
sp would decrease because the added stocks would not be
perfectly correlated.
classroom use.
Risk vs. Number of Stocks in Portfolio
classroom use.
Stand-alone risk = Market risk + Diversifiable risk
Market risk is that part of a security’s stand-alone risk that
cannot be eliminated by diversification.
Firm-specific, or diversifiable, risk is that part of a security’s
stand-alone risk that can be eliminated by diversification.

classroom use.
Conclusions
As more stocks are added, each new stock has a smaller risk-
reducing impact on the portfolio.
sp falls very slowly after about 40 stocks are included. The
lower limit for sp is about 20% = sM .
By forming well-diversified portfolios, investors can eliminate
about half the risk of owning a single stock.
classroom use.
Can an investor holding one stock earn a return commensurate
with its risk?
No. Rational investors will minimize risk by holding
portfolios.
Investors bear only market risk, so prices and returns reflect the
amount of market risk an individual stock brings to a portfolio,
not the stand-alone risk of individual stock.
classroom use.
Market Risk Due to an Individual Stock
How do you measure the amount of market risk that an

individual stock brings to a well-diversified portfolio?
William Sharpe developed the Capital Asset Pricing Model
(CAPM) to answer this question.
And the answer is….. See next slide.
classroom use.
Market Risk as Defined by the CAPM
Define:
wi is the percent of the portfolio invested in Stock i.
σM is the standard deviation of the market index.
ri,M is the correlation between Stock i and the market.
The contribution of Stock i to the standard deviation of a well-
diversified portfolio (σp) is:
classroom use.
Market Risk and Beta (1 of 2)
The beta of Stock i (bi) is defined:
The contribution of Stock i to the standard deviation of a well -
diversified portfolio (σp) is:
Given the standard deviation of the market and the percent of

the portfolio invested in Stock i, beta measures the impact of
Stock i on σp.
classroom use.
Market Risk and Beta (2 of 2)
So Stock i contributes more risk (has a higher beta) if
its correlation with the market, , is larger and/or
its stand-alone risk, , is larger
classroom use.
Required Return and Risk: General Concept
Investors require a return for time (for tying their funds up in
the investment).
rRF, the risk-free rate
Investors require a return for risk, which is the extra return
above the risk-free rate that investors require to induce them to
invest in Stock i.
RPi, the risk premium of Stock i.
classroom use.

Required Return and Risk: The CAPM
RPM is the market risk premium. It is the extra return above the
risk-free rate that that investors require to invest in the overall
stock market:
RPM = rM − rRF.
The CAPM defines the risk premium for Stock i as:
RPi = bi (RPM)
classroom use.
The Security Market Line: Relating Risk and Required Return
(1 of 2)
The Security Market Line (SML) puts the pieces together,
showing how to determine the return required for bearing a
stock’s risk:
SML: ri = rRF + (RPM)bi
classroom use.
The Security Market Line: Relating Risk and Required Return
Risk depends on beta:
The part of σp due to Stock i = wi σM bi
Required return depends on beta:
ri = rRF + (RPM) bi

classroom use.
Correlation Between Blandy and the Market
Using the formula for correlation or the Excel function
=CORREL, Blandy’s correlation with the market (ρB,M) is:
ρB,M = 0.481
classroom use.
Beta for Blandy
Use the previously calculated standard deviations for Blandy
and the market to estimate Blandy’s beta:
b = ρB,M (σB/σM)
b = 0.481(.252/.201) = 0.60
The average beta is equal to 1.0, so Blandy’s stock contributes
less risk to a well-diversified portfolio than does the average
stock.
classroom use.
Required Return for Blandy
Inputs:
rRF = 4% (given)
RPM = 5%(given)
b = 0.60 (estimated)
ri = rRF + bi (RPM)

ri = 4% + 0.60(5%) = 7%
classroom use.
Comparing Risk and Return for Different Stocks
The beta of an average stock is 1.0; Gourmange’s beta is 1.3.
How do their required returns compare with Blandy’s?
rAvg_company = 4%+ 1.0 (5%) = 9%
rG = 4%+ 1.3 (5%) = 10.5%
rB = 4%+ 0.6 (5%) = 7%
Blandy’s stock contributes less risk to a well-diversified
portfolio than do Gourmange or the average stock, so Blandy’s
investors require a lower rate of return.
classroom use.
Using a Regression to Estimate Beta
Run a regression with returns on the stock plotted on the Y-axis
and returns on the market portfolio plotted on the X-axis.
The slope of the regression line is equal to the stock’s beta
coefficient.
classroom use.

Excel: Plot Trendline Right on Chart
classroom use.
Estimated Beta from Regression
The trendline is plotted on the previous slide, including the
regression equation.
y = 0.6027x + 0.0158
b = Slope = 0.6027 (same as before)
Easier way—use the Excel SLOPE function.
b =SLOPE(y_values,x_values)
classroom use.
Web Sites for Beta
http://finance.yahoo.com
Enter the ticker symbol for a “Stock Quote”, such as IBM or
Dell, then click GO.
When the quote comes up, select Key Statistics from panel on
left.
www.valueline.com
Enter a ticker symbol at the top of the page (registration is
free).
Most stocks have betas in the range of 0.5 to 1.5.

classroom use.
Impact on SML of Increase in Risk-Free Rate
classroom use.
Impact on SML of Increase in Risk Aversion
classroom use.
Calculate the weights for a portfolio with $1.4 million in
Blandy and $0.6 million in Gourmange.
Find the weights:
wB = $1.4/($1.4+$0.6) = 70%
wG = $0.6/($1.4+$0.6) = 30%
The portfolio beta is the weighted average of the stocks’ betas:
classroom use.

ChapterTool KitChapter 6112018Risk and Return6-1 Investment Retu

ChapterTool KitChapter 6112018Risk and Return6-1 Investment Retu

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