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CHAPTER 6
Risk, Return, & the Capital Asset Pricing Model
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Topics in Chapter
Basic return concepts Basic risk concepts Stand-alone risk Portfolio (market) risk Risk and return: CAPM/SML
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Value = + + +FCF1 FCF2 FCF∞
(1 + WACC)1 (1 + WACC)∞
(1 + WACC)2
Free cash flow(FCF)
Market interest rates
Firm’s business riskMarket risk aversion
Firm’s debt/equity mixCost of debt
Cost of equity
Cost of debt
Cost of equity
Weighted averagecost of capital
(WACC)
Net operatingprofit after taxes
Required investmentsin operating capital−
=
Determinants of Intrinsic Value:
The Cost of Equity
...
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> Risk, > Return, (both + & -)
Stand – Alone Risk Risk in Portfolio Context b. Market Risk
Quantified by Beta & used in CAPM: Capital Asset Pricing
Model Relationship b/w market risk
& required return as depicted in SML
Req’d return = Risk-free return + Mrkt risk
Prem(Beta) SML: ri = rRF + (RM - rRF )bi
a. Diversifiable
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What are investment returns?
Investment returns measure financial results of an investment.
Returns may be historical or prospective (anticipated).
Returns can be expressed in: ($) dollar terms. (%) percentage terms.
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An investment costs $1,000 and is sold after 1 year for $1,100.
Dollar return:
Percentage return:
$ Received - $ Invested $1,100 - $1,000 = $100
$ Return/$ Invested $100/$1,000 = 0.10 = 10%
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What is investment risk?
Typically, investment returns are not known with certainty.
Investment risk pertains to the probability of earning a return less than expected.
Greater the chance of a return far below the expected return, greater the risk.
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Risk & Return
Student Sue Student Bob
Exam 1 70%
Exam 2 80%
X weightX 50%
X wt.X 50%-----------
Final grade = 75 %
Exam 1 x weight 50% x .50
Exam 2 x wt 100% x .50 -------
Final grade = 75 %
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Probability Distribution: Which stock is riskier? Why?
-30 -15 0 15 30 45 60
Returns (% )
Stock AStock B
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WedTech Co
Normal 40% Return 20% = .08 Bad 30% Return 5% = .015 Good 30% Return 35% = .105 =Expected ave return = 20%
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WedTech Co
Standard Deviation: Measure of stand-alone risk
Return-Exp Ret = Diff2 x Prob =
Variance: SD:
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Standard Deviation and Normal Distributions
1 SD = 68.26% likelihood 2 SD = 95.46% 3 SD = 99.74%
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WedTech Co vs. IBM
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Stand-Alone Risk
Standard deviation measures the stand-alone risk of an investment.
The larger the standard deviation, the higher the probability that returns will be far below the expected return.
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WedTech Co & IBM in 2 stock Portfolio
Ave Portfolio Return
Portfolio Standard Deviation
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WedTech Co & IBM & adding other stocks to Portfolio
IBM WedTech
Coke Microsoft
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Historical Risk vs. Return Return: Hi – Lo Small Co stock Large Co Stock LT Corp Bonds LT Treasuries ST T-Bills
Risk: Hi - Lo
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Reward-to-Variabilty Ratio (Sharpe’s)
Portfolio’s average return in excess of risk-free rate divided by standard deviation
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Comparing Different Stocks
Coefficient of Variation: = S.D. / Return; or Risk / Return
WalMart vs. Philip Morris 12% Return 12% S.D. = C.V. =
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Expected Return versus Coefficient of Variation
Security
Expected
ReturnRisk:
Risk:CV
Alta Inds 17.4% 20.0% 1.1
Market 15.0 15.3 1.0Am. Foam 13.8 18.8 1.4T-bills 8.0 0.0 0.0Repo Men 1.7 13.4 7.9
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Comparing Different Stocks
Correlation coefficient = r (rho): Measures tendency of 2 variables to
move together. Rho (r) = 1 = perfect + correlation & variables move together in unison.
Does not help with diversification See text figures 6-9 thru 6-11
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Two-Stock Portfolios Two stocks can be combined to form
a riskless portfolio if = -1.0. Risk is not reduced at all if the two
stocks have = +1.0. In general, stocks have ≈ 0.35, so
risk is lowered but not eliminated. Investors typically hold many stocks. What happens when = 0?
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Adding Stocks to a Portfolio
What would happen to the risk of an average 1-stock portfolio as more randomly selected stocks were added?
p would decrease because the added stocks would not be perfectly correlated, but the expected portfolio return would remain relatively constant.
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stock ≈ 35%Many stocks ≈ 20%
-75 -60 -45 -30 -15 0 15 30 45 60 75 90 105
Returns (% )
1 stock2 stocksMany stocks
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2510 20 30 40 2,000 stocks
Company Specific (Diversifiable) Risk
Market Risk
20%
0
Stand-Alone Risk, p
p
35%
Risk vs. Number of Stock in Portfolio
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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.
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Conclusions As more stocks are added, each new
stock has a smaller risk-reducing impact on the portfolio.
p falls very slowly after about 40 stocks are included. The lower limit for p is about 20% = M .
By forming well-diversified portfolios, investors can eliminate about half the risk of owning a single stock.
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Can an investor holding one stock earn a return commensurate with its risk?
No. Rational investors will minimize risk by holding portfolios.
They bear only market risk, so prices and returns reflect this lower risk.
The one-stock investor bears higher (stand-alone) risk, so the return is less than that required by the risk.
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How is market risk measured for individual securities?
Market risk, which is relevant for stocks held in well-diversified portfolios, is defined as the contribution of a security to the overall riskiness of the portfolio.
It is measured by a stock’s beta coefficient. For stock i, its beta is:
bi = (i,M i) / M
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How are betas calculated?
In addition to measuring a stock’s contribution of risk to a portfolio, beta also measures the stock’s volatility relative to the market.
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Using a Regression to Estimate Beta
Run a regression with returns on the stock in question plotted on the Y axis and returns on the market portfolio plotted on the X axis.
The slope of the regression line, which measures relative volatility, is defined as the stock’s beta coefficient, or b.
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Use the historical stock returns to calculate the beta for PQU.
Year Market PQU
1 25.7%
40.0%
2 8.0%
-15.0%
3 -11.0%
-15.0%
4 15.0%
35.0%
5 32.5%
10.0%
6 13.7%
30.0%
7 40.0%
42.0%
8 10.0%
-10.0%
9 -10.8%
-25.0%
10 -13.1%
25.0%
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Calculating Beta for PQU
rPQU = 0.8308 rM + 0.0256
R2 = 0.3546-30%-20%-10%
0%10%20%30%40%50%
-30% -20% -10% 0% 10% 20% 30% 40% 50%
Market Return
PQ
U R
etu
rn
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Beta & PQU Co.
Beta reflects slope of line via regression y = mx + b m=slope + b= y intercept Rpqu = 0.8308 rM + 0.0256 So, PQU’s beta is .8308 & y-intercept @
2.56%
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Beta & PQU Co. & R2
R2 measures degree of dispersion about regression line (ie – measures % of variance explained by regression equation)
PQU’s R2 of .3546 means about 35% of PQU’s returns are explained by the market returns (32% for a typical stock)
R2 of .95 on portfolio of 40 randomly selected stocks would reflect a regression line with points tightly clustered to it.
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Two-Stock Portfolios Two stocks can be combined to form
a riskless portfolio if = -1.0. Risk is not reduced at all if the two
stocks have = +1.0. In general, stocks have ≈ 0.35, so
risk is lowered but not eliminated. Investors typically hold many stocks. What happens when = 0?
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Adding Stocks to a Portfolio
What would happen to the risk of an average 1-stock portfolio as more randomly selected stocks were added?
p would decrease because the added stocks would not be perfectly correlated, but the expected portfolio return would remain relatively constant.
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stock ≈ 35%Many stocks ≈ 20%
-75 -60 -45 -30 -15 0 15 30 45 60 75 90 105
Returns (% )
1 stock2 stocksMany stocks
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3910 20 30 40 2,000 stocks
Company Specific (Diversifiable) Risk
Market Risk
20%
0
Stand-Alone Risk, p
p
35%
Risk vs. Number of Stock in Portfolio
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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.
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Conclusions As more stocks are added, each new
stock has a smaller risk-reducing impact on the portfolio.
p falls very slowly after about 40 stocks are included. The lower limit for p is about 20% = M .
By forming well-diversified portfolios, investors can eliminate about half the risk of owning a single stock.
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Can an investor holding one stock earn a return commensurate with its risk?
No. Rational investors will minimize risk by holding portfolios.
They bear only market risk, so prices and returns reflect this lower risk.
The one-stock investor bears higher (stand-alone) risk, so the return is less than that required by the risk.
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How is market risk measured for individual securities?
Market risk, which is relevant for stocks held in well-diversified portfolios, is defined as the contribution of a security to the overall riskiness of the portfolio.
It is measured by a stock’s beta coefficient. For stock i, its beta is:
bi = (i,M i) / M
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How are betas calculated?
In addition to measuring a stock’s contribution of risk to a portfolio, beta also measures the stock’s volatility relative to the market.
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Using a Regression to Estimate Beta
Run a regression with returns on the stock in question plotted on the Y axis and returns on the market portfolio plotted on the X axis.
The slope of the regression line, which measures relative volatility, is defined as the stock’s beta coefficient, or b.
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Use the historical stock returns to calculate the beta for PQU.
Year Market PQU
1 25.7%
40.0%
2 8.0%
-15.0%
3 -11.0%
-15.0%
4 15.0%
35.0%
5 32.5%
10.0%
6 13.7%
30.0%
7 40.0%
42.0%
8 10.0%
-10.0%
9 -10.8%
-25.0%
10 -13.1%
25.0%
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Calculating Beta for PQU
rPQU = 0.8308 rM + 0.0256
R2 = 0.3546-30%-20%-10%
0%10%20%30%40%50%
-30% -20% -10% 0% 10% 20% 30% 40% 50%
Market Return
PQ
U R
etu
rn
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Expected Return versus Market Risk: Which investment is best?
Security
ExpectedReturn
(%) Risk, bAlta 17.4 1.29Market 15.0 1.00Am. Foam 13.8 0.68T-bills 8.0 0.00Repo Men
1.7 -0.86
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Capital Asset Pricing Model
The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM).
Return = Risk Free + Beta (RetMrkt –Rf)
SML: ri = rRF + (RPM)bi . Assume rRF = 8%; rM = rM = 15%. RPM = (rM - rRF) = 15% - 8% = 7%.
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Use the SML to calculate eachalternative’s required return.
The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM).
SML: ri = rRF + (RPM)bi . Assume rRF = 8%; rM = rM = 15%. RPM = (rM - rRF) = 15% - 8% = 7%.
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Required Rates of Return
rAlta = 8.0% + (7%)(1.29) = 17%. rM = 8.0% + (7%)(1.00) = 15.0%. rAm. F. = 8.0% + (7%)(0.68) =
12.8%. rT-bill = 8.0% + (7%)(0.00) = 8.0%. rRepo = 8.0% + (7%)(-0.86) = 2.0%.
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Expected versus Required Returns (%)
Exp. Req. r rAlta 17.4 17.0
Undervalued Market 15.0 15.0 Fairly
valuedAm. Foam
13.8 12.8 Undervalued
T-bills 8.0 8.0 Fairly valued
Repo 1.7 2.0 Overvalued
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SML: ri = rRF + (RPM) bi
ri = 8% + (7%) bi
..Repo
.Alta
T-bills
.Am. Foam
rM = 15
rRF = 8
-1 0 1 2
.
ri (%)
Risk, bi
Market
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Calculate beta for a portfolio with 50% Alta and 50% Repo
bp = Weighted average= 0.5(bAlta) + 0.5(bRepo)= 0.5(1.29) + 0.5(-0.86)= 0.22.
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Required Return on the Alta/Repo Portfolio?
rp = Weighted average r = 0.5(17%) + 0.5(2%) = 9.5%.
Or use SML:
rp = rRF + (RPM) bp
= 8.0% + 7%(0.22) = 9.5%.
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SML1
Original situation
r (%)
SML2
0 0.5 1.0 1.5 Risk, bi
1815
11 8
New SML I = 3%
Impact of Inflation Change on SML
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SML1
Original situation
r (%)SML2
After change
Risk, bi
18
15
8
1.0
RPM = 3%
Impact of Risk Aversion Change
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Has the CAPM been completely confirmed or refuted?
No. The statistical tests have problems that make empirical verification or rejection virtually impossible. Investors’ required returns are based
on future risk, but betas are calculated with historical data.
Investors may be concerned about both stand-alone and market risk.
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Below are per book mini-case
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Consider the FollowingInvestment Alternatives
Econ. Prob.
T-Bill Alta Repo Am F. MP
Bust
0.10 8.0%-22.0%
28.0%
10.0%
-13.0%
Below avg.
0.20
8.0 -2.0 14.7 -10.0 1.0
Avg.
0.408.0 20.0 0.0 7.0 15.0
Above avg.
0.20
8.0 35.0 -10.0 45.0 29.0
Boom
0.10 8.0 50.0 -20.0 30.0 43.0
1.00
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What is unique about T-bill returns?
T-bill returns 8% regardless of the state of the economy.
Is T-bill riskless? Explain.
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Alta Inds. and Repo Men vs. Economy
Alta moves with economy, so it is positively correlated with economy. This is typical
Repo Men moves counter to economy. Such negative correlation is unusual.
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Calculate the expected rate of return on each alternative.
r = expected rate of return(think wtd average)
rAlta = 0.10(-22%) + 0.20(-2%) + 0.40(20%) + 0.20(35%) + 0.10(50%) = 17.4%.
^
^ n∑r =
^
i=1riPi.
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Alta has the highest rate of return. Does that make it best?
Expected return
Alta 17.4%Market 15.0Am. Foam 13.8T-bill 8.0Repo Men 1.7
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What is the standard deviationof returns for each alternative?
σ = Standard deviation
σ = √ Variance = √ σ2
n∑i=1
= √ (ri – r)2 Pi.^
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= [(-22 - 17.4)20.10 + (-2 - 17.4)20.20 + (20 - 17.4)20.40 + (35 - 17.4)20.20 + (50 - 17.4)20.10]1/2 = 20.0%.
Standard Deviation of Alta Industries
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T-bills = 0.0%.
Alta = 20.0%.
Repo = 13.4%.
Am Foam = 18.8%.
Market = 15.3%.
Standard Deviation of Alternatives
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Expected Return versus Risk
SecurityExpected
Return Risk, Alta Inds. 17.4% 20.0%Market 15.0 15.3Am. Foam 13.8 18.8T-bills 8.0 0.0Repo Men
1.7 13.4
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Coefficient of Variation (CV)
CV = Standard deviation / Expected return
CVT-BILLS = 0.0% / 8.0% = 0.0. CVAlta Inds = 20.0% / 17.4% = 1.1. CVRepo Men = 13.4% / 1.7% = 7.9. CVAm. Foam = 18.8% / 13.8% = 1.4. CVM = 15.3% / 15.0% = 1.0.
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Expected Return versus Coefficient of Variation
Security
Expected
ReturnRisk:
Risk:CV
Alta Inds 17.4% 20.0% 1.1
Market 15.0 15.3 1.0Am. Foam 13.8 18.8 1.4T-bills 8.0 0.0 0.0Repo Men 1.7 13.4 7.9
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Return vs. Risk (Std. Dev.): Which investment is best?
T-bills
Repo
MktAm. Foam
Alta
0.0%
5.0%
10.0%
15.0%
20.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Risk (Std. Dev.)
Ret
urn
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Portfolio Risk and Return
Assume a two-stock portfolio with $50,000 in Alta Inds. and $50,000 in Repo Men.
Calculate rp and p.^
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Portfolio Expected Return
rp = Σ wi ri
rp is a weighted average (wi is % ofportfolio in stock i):
rp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.^
^
^ ^n
i = 1
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Alternative Method: Find portfolio return in each economic state
Economy Prob. Alta Repo
Port.= 0.5(Alta)
+ 0.5(Rep
o)Bust 0.10 -22.0% 28.0% 3.0%Below avg.
0.20 -2.0 14.7 6.4
Average 0.40 20.0 0.0 10.0Above avg.
0.20 35.0 -10.0 12.5
Boom 0.10 50.0 -20.0 15.0
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Use portfolio outcomes to estimate risk and expected return
rp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40 + (12.5%)0.20 + (15.0%)0.10 = 9.6%
^
p = ((3.0 - 9.6)20.10 + (6.4 - 9.6)20.20
+(10.0 - 9.6)20.40 + (12.5 - 9.6)20.20
+ (15.0 - 9.6)20.10)1/2 = 3.3%
CVp = 3.3%/9.6% = .34
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Portfolio vs. Its Components Portfolio expected return (9.6%) is
between Alta (17.4%) and Repo (1.7%) returns.
Portfolio standard deviation is much lower than: either stock (20% and 13.4%). average of Alta and Repo (16.7%).
The reason is due to negative correlation () between Alta and Repo returns.
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Two-Stock Portfolios Two stocks can be combined to form
a riskless portfolio if = -1.0. Risk is not reduced at all if the two
stocks have = +1.0. In general, stocks have ≈ 0.35, so
risk is lowered but not eliminated. Investors typically hold many stocks. What happens when = 0?
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Adding Stocks to a Portfolio
What would happen to the risk of an average 1-stock portfolio as more randomly selected stocks were added?
p would decrease because the added stocks would not be perfectly correlated, but the expected portfolio return would remain relatively constant.
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stock ≈ 35%Many stocks ≈ 20%
-75 -60 -45 -30 -15 0 15 30 45 60 75 90 105
Returns (% )
1 stock2 stocksMany stocks
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8010 20 30 40 2,000 stocks
Company Specific (Diversifiable) Risk
Market Risk
20%
0
Stand-Alone Risk, p
p
35%
Risk vs. Number of Stock in Portfolio
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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.
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Conclusions As more stocks are added, each new
stock has a smaller risk-reducing impact on the portfolio.
p falls very slowly after about 40 stocks are included. The lower limit for p is about 20% = M .
By forming well-diversified portfolios, investors can eliminate about half the risk of owning a single stock.
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Can an investor holding one stock earn a return commensurate with its risk?
No. Rational investors will minimize risk by holding portfolios.
They bear only market risk, so prices and returns reflect this lower risk.
The one-stock investor bears higher (stand-alone) risk, so the return is less than that required by the risk.
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How is market risk measured for individual securities?
Market risk, which is relevant for stocks held in well-diversified portfolios, is defined as the contribution of a security to the overall riskiness of the portfolio.
It is measured by a stock’s beta coefficient. For stock i, its beta is:
bi = (i,M i) / M
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How are betas calculated?
In addition to measuring a stock’s contribution of risk to a portfolio, beta also measures the stock’s volatility relative to the market.
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Using a Regression to Estimate Beta
Run a regression with returns on the stock in question plotted on the Y axis and returns on the market portfolio plotted on the X axis.
The slope of the regression line, which measures relative volatility, is defined as the stock’s beta coefficient, or b.
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Use the historical stock returns to calculate the beta for PQU.
Year Market PQU
1 25.7%
40.0%
2 8.0%
-15.0%
3 -11.0%
-15.0%
4 15.0%
35.0%
5 32.5%
10.0%
6 13.7%
30.0%
7 40.0%
42.0%
8 10.0%
-10.0%
9 -10.8%
-25.0%
10 -13.1%
25.0%
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Calculating Beta for PQU
rPQU = 0.8308 rM + 0.0256
R2 = 0.3546-30%-20%-10%
0%10%20%30%40%50%
-30% -20% -10% 0% 10% 20% 30% 40% 50%
Market Return
PQ
U R
etu
rn
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What is beta for PQU?
The regression line, and hence beta, can be found using a calculator with a regression function or a spreadsheet program. In this example, b = 0.83.
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Calculating Beta in Practice
Many analysts use the S&P 500 to find the market return.
Analysts typically use four or five years’ of monthly returns to establish the regression line.
Some analysts use 52 weeks of weekly returns.
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How is beta interpreted?
If b = 1.0, stock has average risk. If b > 1.0, stock is riskier than
average. If b < 1.0, stock is less risky than
average. Most stocks have betas in the range
of 0.5 to 1.5. Can a stock have a negative beta?
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Other Web Sites for Beta
Go to 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.
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Expected Return versus Market Risk: Which investment is best?
Security
ExpectedReturn
(%) Risk, bAlta 17.4 1.29Market 15.0 1.00Am. Foam 13.8 0.68T-bills 8.0 0.00Repo Men
1.7 -0.86
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Use the SML to calculate eachalternative’s required return.
The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM).
SML: ri = rRF + (RPM)bi . Assume rRF = 8%; rM = rM = 15%. RPM = (rM - rRF) = 15% - 8% = 7%.
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Required Rates of Return
rAlta = 8.0% + (7%)(1.29) = 17%. rM = 8.0% + (7%)(1.00) = 15.0%. rAm. F. = 8.0% + (7%)(0.68) =
12.8%. rT-bill = 8.0% + (7%)(0.00) = 8.0%. rRepo = 8.0% + (7%)(-0.86) = 2.0%.
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Expected versus Required Returns (%)
Exp. Req. r rAlta 17.4 17.0
Undervalued Market 15.0 15.0 Fairly
valuedAm. Foam
13.8 12.8 Undervalued
T-bills 8.0 8.0 Fairly valued
Repo 1.7 2.0 Overvalued
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SML: ri = rRF + (RPM) bi
ri = 8% + (7%) bi
..Repo
.Alta
T-bills
.Am. Foam
rM = 15
rRF = 8
-1 0 1 2
.
ri (%)
Risk, bi
Market
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Calculate beta for a portfolio with 50% Alta and 50% Repo
bp = Weighted average= 0.5(bAlta) + 0.5(bRepo)= 0.5(1.29) + 0.5(-0.86)= 0.22.
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Required Return on the Alta/Repo Portfolio?
rp = Weighted average r = 0.5(17%) + 0.5(2%) = 9.5%.
Or use SML:
rp = rRF + (RPM) bp
= 8.0% + 7%(0.22) = 9.5%.
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SML1
Original situation
r (%)
SML2
0 0.5 1.0 1.5 Risk, bi
1815
11 8
New SML I = 3%
Impact of Inflation Change on SML
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SML1
Original situation
r (%)SML2
After change
Risk, bi
18
15
8
1.0
RPM = 3%
Impact of Risk Aversion Change
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Has the CAPM been completely confirmed or refuted?
No. The statistical tests have problems that make empirical verification or rejection virtually impossible. Investors’ required returns are based
on future risk, but betas are calculated with historical data.
Investors may be concerned about both stand-alone and market risk.