chapter 5 risk and rates of return © 2005 thomson/south-western
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Chapter 5
Risk and Rates of Return
© 2005 Thomson/South-Western
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Defining and Measuring Risk
Risk is the chance that an unexpected outcome will occur
A probability distribution is a listing of all possible outcomes with a probability assigned to each;
must sum to 1.0 (100%).
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Probability Distributions
It either will rain, or it will not.Only two possible outcomes.
Outcome (1) Probability (2)
Rain 0.40 = 40%
No Rain 0.60 = 60%
1.00 100%
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Probability Distributions
Martin Products and U. S. Electric
Martin Products U.S. Electric
Boom 0.2 110% 20%Normal 0.5 22% 16%Recession 0.3 -60% 10%
1.0
Probability of This State Occurring
State of the Economy
Rate of Return on Stock if This State Occurs
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Expected Rate of Return
Rate of return expected to be realized from an investment during its life
Mean value of the probability distribution of possible returns
Weighted average of the outcomes, where the weights are the probabilities
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(1) (2) (3) = (4) (5) = (6)Boom 0.2 110% 22% 20% 4%Normal 0.5 22% 11% 16% 8%Recession 0.3 -60% -18% 10% 3%
1.0 km = 15% km = 15%
State of the Economy
Martin Products U. S. Electric
Return if This State Occurs (ki)
Product: (2) x (5)
Probability of This State
Occurring (Pr i)
Return if This State
Occurs (ki)Product: (2) x (3)
^ ^
Expected Rate of Return
7
n
1ii
n21
k
kkkk
i
n21
Pr
PrPrPr ˆ
Expected Rate of Return
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Continuous versus Discrete Probability Distributions
Discrete Probability Distribution:number of possible outcomes is limited, or finite.
Discrete Probability Discrete Probability DistributionsDistributions
a. Martin ProductsProbability of Occurrence
b. U. S. ElectricProbability of Occurrence
-60 -45 -30 -15 0 15 22 30 45 60 75 90 110Rate of
Return (%)Expected Rate of Return (15%)
0.5 -
0.4 -
0.3 -
0.2 -
0.1 -
-10 -5 0 5 10 16 20 25Rate of
Return (%)Expected Rate
of Return (15%)
0.5 -
0.4 -
0.3 -
0.2 -
0.1 -
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Continuous versus Discrete Probability Distributions
Continuous Probability Distribution:number of possible outcomes is unlimited, or infinite.
Probability Density
-60 0 15 110Rate of Return
(%)Expected Rate of Return
Martin ProductsMartin Products
U. S. ElectricU. S. Electric
Continuous Probability Distributions
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Measuring Risk: The Standard Deviation
Calculating Martin Products’ Standard Deviation
(1) (2) (1) - (2) = (3) (4) (5) (4) x (5) = (6)110% 15% 95 9,025 0.2 1,805.0 22% 15% 7 49 0.5 24.5 -60% 15% -75 5,625 0.3 1,687.5
Payoff
ki(ki - k)
2Pr iProbability
Expected Return
kki - k (ki - k)
2^^
^ ^
%3.59 517,3 Deviation Standard
0.517,3 Variance
2mm
2
^^^^
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n
1iiikk̂ return of rate Expected Pr
Variance 2 k i - ˆ k 2Prii 1
n
Standard deviation 2 k i - ˆ k 2Prii 1
n
Measuring Risk: The Standard Deviation
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Standardized measure of risk per unit of return
Calculated as the standard deviation divided by the expected return
Useful where investments differ in risk and expected returns
k̂Return
Risk CV Coefficient of variation
Measuring Risk: Coefficient of Variation
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Risk Aversion
Risk-averse investors require higher rates of return to invest in higher-risk securities
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Risk Aversion and Required Returns
Risk Premium (RP):
The portion of the expected return that can be attributed to an investment’s risk beyond a riskless investment
The difference between the expected rate of return on a given risky asset and that on a less risky asset
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Portfolio Risk and theCapital Asset Pricing Model
CAPM:A model based on the proposition that any
stock’s required rate of return is equal to the risk-free rate of return plus a risk premium, where risk is based on diversification.
PortfolioA collection of investment securities
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ˆ k p w1ˆ k 1 w 2
ˆ k 2 w Nˆ k N
w jˆ k j
j1
N
Portfolio ReturnsExpected return on a portfolio,pk̂
The weighted average expected return on the stocks held in the portfolio
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Portfolio Returns
Realized rate of return, kThe return that is actually earnedActual return usually different from
expected return
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Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM:
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15
0
-10
Stock W
-10
0
15
25
Portfolio WM
-10
0
15
25
Stock M
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Stock M
0
15
25
-10
0
15
25
-10
Stock M’
0
15
25
-10
Stock MM’
Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM’:
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Portfolio RiskCorrelation Coefficient, r
Measures the degree of relationship between two variables.
Perfectly correlated stocks have rates of return that move in the same direction.
Negatively correlated stocks have rates of return that move in opposite directions.
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Portfolio RiskRisk Reduction
Combining stocks that are not perfectly correlated will reduce the portfolio risk through diversification.
The riskiness of a portfolio is reduced as the number of stocks in the portfolio increases.
The smaller the positive correlation, the lower the risk.
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Firm-Specific Risk versus Market Risk
Firm-Specific Risk:That part of a security’s risk
associated with random outcomes generated by events, or behaviors, specific to the firm.
Firm-specific risk can be eliminated through proper diversification.
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Firm-Specific Risk versus Market Risk
Market Risk:That part of a security’s risk that
cannot be eliminated through diversification because it is associated with economic, or market factors that systematically affect all firms.
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Firm-Specific Risk versus Market Risk
Relevant Risk:The risk of a security that cannot be
diversified away, or its market risk.This reflects a security’s
contribution to a portfolio’s total risk.
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The Concept of BetaBeta Coefficient,
A measure of the extent to which the returns on a given stock move with the stock market.
= 0.5: Stock is only half as volatile, or risky, as the average stock.
= 1.0: Stock has the same risk as the average risk.
= 2.0: Stock is twice as risky as the average stock.
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w
www
p
Portfolio Beta CoefficientsThe beta of any set of securities is the
weighted average of the individual securities’ betas
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stock j on thereturn of rate k̂ thj expected
The Relationship Between Risk and Rates of Return
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stock j on thereturn of rate k
stock j on thereturn of rate k̂th
j
thj
required
expected
The Relationship Between Risk and Rates of Return
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return of rate k
stock j on thereturn of rate k
stock j on thereturn of rate k̂
RF
thj
thj
freerisk
required
expected
The Relationship Between Risk and Rates of Return
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premiumrisk market k-k RP
return of rate freerisk k
stock j on thereturn of rate required k
stock j on thereturn of rate expected k̂
RFMM
RF
thj
thj
The Relationship Between Risk and Rates of Return
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stock j on the premiumrisk k-k RP
premiumrisk market k-k RP
return of rate k
stock j on thereturn of rate k
stock j on thereturn of rate k̂
thjRFMj
RFMM
RF
thj
thj
freerisk
required
expected
The Relationship Between Risk and Rates of Return
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Market Risk Premium
RPM is the additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.
Assuming: Treasury bonds yield = 6%, Average stock required return = 14%, Then the market risk premium is 8 percent:
RPM = kM - kRF = 14% - 6% = 8%.
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Risk Premium for a Stock
Risk Premium for Stock j = RPj = RPM xj
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jstock for return of rate k j required
jRFMRF
jMRFj
kk k
RP k k
The Required Rate of Return for a Stock
Security Market Line (SML):The line that shows the relationship
between risk as measured by beta and the required rate of return for individual securities.
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Security Market Line jRFMRFj kk k k:SML
khigh = 22
kM = kA = 14
kLOW = 10
kRF = 6
Risk, j0 0.5 1.0 1.5 2.0
Required Rate of Return (%)
Risk-Free Rate: 6%
Safe Stock Risk Premium: 4%
Market (Average Stock) Risk Premium: 8%
Relatively Risky Stock’s Risk Premium: 16%
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The Impact of Inflation
kRF is the price of money to a riskless borrower.
The nominal rate consists of: a real (inflation-free) rate of return, and an inflation premium (IP).
An increase in expected inflation would increase the risk-free rate.
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Changes in Risk Aversion
The slope of the SML reflects the extent to which investors are averse to risk.
An increase in risk aversion increases the risk premium and increases the slope.
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Changes in a Stock’s Beta Coefficient
The Beta risk of a stock is affected by:composition of its assets,use of debt financing, increased competition, andexpiration of patents.
Any change in the required return (from change in beta or in expected inflation) affects the stock price.
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Stock Market Equilibrium
The condition under which the expected return on a security is just equal to its required return
Actual market price equals its intrinsic value as estimated by the marginal investor, leading to price stability
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Changes in Equilibrium Stock Prices
Stock prices are not constant due to changes in:
Risk-free rate, kRF,
Market risk premium, kM – kRF,
Stock X’s beta coefficient, x,
Stock X’s expected growth rate, gX, and
Changes in expected dividends, D0.
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Physical AssetsVersus Securities
Riskiness of corporate assets is only relevant in terms of its effect on the stock’s risk.
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Word of Caution
CAPMBased on expected conditionsOnly have historical dataAs conditions change, future volatility
may differ from past volatilityEstimates are subject to error
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End of Chapter 4
Risk and Rates of Return