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Risk, Return, and CAPM
Finance 221Summer 2006
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Review
ãTradeoff between risk and returnãWhat is the difference between systematicand unsytematicrisk?ãWhy are investors compensated only forsystematic risk?ãWhat is a risk premium?
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Average Return and St. Dev. forIndividual Securities, 19942003
05101520250 10 20 30 40 50 60
Average Return (%)Standard Deviation (%)WalMart AnheuserBusch Archer Daniels Midland American Airlines
Standard Deviation captures total risk
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Review: Diversification
ãResearch suggests that investors are not asdiversified as they should beãCan also diversify across asset class: a portfoliocomposed of 80% stocks and 20% bonds gets97% of S&P 500 return with 85% of the risk.ãInternational diversification; Correlation betweenUS and Brazil Markets –0.09 in late 1970s, 0.68in early 1990s. Home bias (Japan –98%).ãCampbell, Lettau, Malkieland Xu (2001): marketvolatility relatively constant over 19621997,firmspecific volatility doubled. Correlationsdeclined from 0.28 to 0.08 in the sample period.
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Today: CAPM (Capital Asset Pricing Model)
Nobel Prize in Economics, 1990, Awarded to Harry Markowitz, Merton Miller, and William Sharpe for “their pioneering work in the theory of financial economics”.
⇒
CAPM
A stock’s
beta
(
β
)
,
a measure of how much astock’s returns vary in response to changes inthe overall market, is an importantdeterminant of the stock’s
expected return
: aforecast of the return that an asset will earnover some period of time.
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Expected Returns
ãExpected returns reflect our estimate of how an investment will perform in thefuture. Think about the expected returnsof stocks vs. bonds.ãExpected returns are
not observable
. Thismeans we need to develop methods toestimate expected returns.
2
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Expected Returns
Decisions must be based on expected returnsMethods used to estimate expected return
Historical approachProbabilistic approach
Riskbased approach
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Historical Approach forEstimating Expected Returns
Assumption: the distribution of expected returns willbe similar to historical distribution of returns.
Using 19002003 annual returns, the average risk premium for U.S. stocks relative to Treasury bills is7.6%. Treasury bills currently offer a 2% yield tomaturityExpected return on U.S. stocks: 7.6% + 2% = 9.6%
Can historical approach be used to estimate the expected return of an individual stock?
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Historical Approach forEstimating Expected Returns
Assume General Motors longrun average return is17.0%. Treasury bills average return over sameperiod was 4.1%GM historical risk premium: 17.0% 4.1% = 12.9%GM expected return = Current Tbill rate + GMhistorical risk premium = 2% + 12.9% = 14.9%Limitationsof historicalapproach forindividualstocksMay reflect GM’s past more thanits future
Many stocks do not have a longhistory to forecast expected return
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Probabilistic Approach forEstimating Expected Returns
Identify all possible outcomes of returns and assigna probability to each possible outcome:GM Expected Return = 0.20(30%) + 0.70(15%)+0.10(55%) = 10%For example, assign probabilities for possible statesof economy: boom, expansion, recession and projectthe returns of GM stock for the three states
55%10%Boom15%70%Expansion30%20%Recession
GM ReturnProbabilityOutcome
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RiskBased Approach forEstimating Expected Returns
1. Measure the risk of the asset2. Use the risk measure to estimate the expectedreturn
How can we capture the systematic risk component of a stock’s volatility?
1. Measure the risk of the asset
ã
Systematic risks simultaneously affect manydifferent assets
ã
Investors can diversify away the unsystematic risk
ã
Market rewards only the systematic risk: onlysystematic risk should be related to the expectedreturn
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ã
Collect data on a stock’s returns and returns on amarket index
ã
Plot the points on a scatter plot graph
–
Y–axis measures stock’s return
–
Xaxis measures market’s return
ã
Plot a line (using linear regression) throughthepoints
RiskBased Approach forEstimating Expected Returns
Slope of line equals beta, the sensitivity of a stock’sreturns relative to changes in overall market returns
Beta is a measure of systematic risk for a particularsecurity.
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30%20%10%0%10%20%30%30% 20% 10% 0% 10% 20% 30%
Slope =Beta = 1.44
Scatter Plot for Returns on SharperImage and S&P 500
S&P 500 weekly returns
SharperImageweeklyreturns
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15%10%5%0%5%10%15%15% 10% 5% 0% 5% 10% 15%
Beta = 0.11
Scatter Plot for Returns on ConAgraand S&P 500
S&P 500 weekly returns
ConAgraweeklyreturns
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Beta
Beta is a standardized measure of the systematic risk of an individual asset
Sharper Image –
β
= 1.44
⇒
On average, if themarket’s return moves by 1%, the return for SharperImage moves in the same direction by 1.44%
⇒
Verysensitive to overall market movements.ConAgra –b = 0.11
⇒
On average, if the market’sreturn moves by 1%, the return for ConAgra moves inthe same direction by 0.11%
⇒
Less sensitive tomarket movements.Which stock has the highest systematic risk?
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RiskBased Approach forEstimating Expected Returns
Beta measures systematic risk and links the risk and expected return of an asset.
2. Use the risk measure to estimate the expectedreturn:
ã
Plot beta against expected return for two assets:

A riskfree asset that pays 4% with certainty,with zero systematic risk and

An “average stock”, with beta equal to 1, withan expected return of 10%.
ã
Draw a straight line connecting the two points.
ã
Investors holding a stock with beta of 0.5 or 1.5,for example, can find the expected return on theline.
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Risk and Expected Returns
Expected returns
ãã
10%1
Riskfree asset
ã ã ã ã
0.20.40.60.821.21.41.61.8
ã ã ã ã ã
Beta
ã
4%
ã
18%
ã
14%
“average” stock
What is the expected return for stock with beta = 1.5?
ß= 1.5
ãã
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Portfolio Expected Returns
The portfolio expected return equals the weightedaverage of the portfolio assets’expected returns
E(R
p
) = w
1
E(R
1
)+ w
2
E(R
2
)+…+w
n
E(R
n
)
ã
w
1
, w
2
, …, w
n
: portfolio weights
ã
E(R
1
), E(R
2
), …, E(R
N
): expected returns of securitiesExpected return of a portfolio with N securities
How does the expected return of a portfolio relate to the expected returns of the securities in the portfolio?
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Portfolio Expected Returns
$10,000$2,500$5,000$2,500
$ Invested
0.1258%Sears0.514%Pfizer0.2512%GE0.12510%IBM
WeightsE(R)Portfolio
E(R
p
) = (0.125)(10%) + (0.25)(12%) +(0.125)(8%) + (0.5)(14%) = 12.25%
E(R
p
) = w
1
E(R
1
)+ w
2
E(R
2
)+…+w
n
E(R
n
)
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Short Selling
ãPortfolio weights must sum up to oneãNot all weights have to be positive. A negativeweight
⇒
Short SellingRocket.comand BricksNMortarInc. both sell for $10per share. You expect returns on Rocket.comto be25%, but returns on BricksNMortarInc. to be only5%. You have $1,000 to invest.Borrow 50 shares of BricksNMortarsellimmediately. Buy 150 shares of Rocket.com.
%35%)25()5.1(%)5()5.0()(
=×+×−=
p
R E
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Short Selling
$1,350Net Cash Earned($1,350$1,000)/$1,000 = 0.35 = 35%Rate of Return =(price = $10.50)$525Return BorrowedShares(price = $12.50)$1875Sell Rocket Shares
End of Year
(150 shares)$1500Rocket Shares(50 shares)$500Borrowed$1000Initial Investment
Beginning of Year
*This is called leveraging
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Portfolio Risk
Portfolio risk is the weighted average of systematicrisk (beta) of the portfolio constituent securities.
$10,000$2,500$5,000$2,500
$ Invested
0.1250.67Sears0.51.67Pfizer0.251.33GE0.1251.00IBM
WeightsBetaPortfolio
ß
P
= (0.125)(1.00) + (0.25)(1.33) + (0.125)(0.67)+ (0.50)(1.67) = 1.38
But portfolio volatility is not the same as the weighted average of all portfolio security volatilities
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Security Market Line
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Beta
E(R
m
)Market portfolio
R
f
Riskfree asset
E(R)Portfolio
Portfolio composed of the following two assets:
ã
An asset that pays a riskfree return R
f,
, and
ã
A market portfolio that contains some of everyrisky asset in the market.Security market line: the line connecting the riskfree asset and the market portfolio
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The Security Market Line
β
i
E(R
P
)R
F
SML
Slope = E(R
m
) R
F
=MarketRisk Premium
ãã
ã
R
M
β
=1.0