risk and rates of return chapter 11. defining and measuring risk urisk is the chance that an outcome...
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Risk and Rates of ReturnRisk and Rates of Return
Chapter 11
Defining and Defining and Measuring RiskMeasuring Risk
Risk is the chance that an outcome other than expected will occur
Probability distribution is a listing of all possible outcomes with a probability assigned to each must sum to 1.0 (100%)
Probability DistributionsProbability 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%
Probability DistributionsProbability 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
Expected Rate of ReturnExpected Rate of Return
The rate of return expected to be realized from an investment
The mean value of the probability distribution of possible returns
The weighted average of the outcomes, where the weights are the probabilities
^ ^
Expected Rate of ReturnExpected Rate of Return
(1) (2) (3) = (4)Boom 0.2 110% 22%Normal 0.5 22% 11%Recession 0.3 -60% -18%
1.0 km = 15%
State of the Economy
Martin ProductsProbability of This State
Occurring (Pr i)
Return if This State
Occurs (ki)Product: (2) x (3)
^ ^
Expected Rate of ReturnExpected Rate of Return
(1) (2) (3) = (4)Boom 0.2 20% 4%Normal 0.5 16% 8%Recession 0.3 10% 3%
1.0 km= 15%
State of the Economy
U. S. Electric
Return if This
State Occurs (ki)Product: (2) x (3)
Probability of This State
Occurring (Pr i)
Expected Rate of ReturnExpected Rate of Return
n
iii
nn2211
k
kkkk̂
1
Pr
PrPrPr
Continuous versus Discrete Continuous versus Discrete Probability DistributionsProbability Distributions
Discrete probability distribution the number of possible outcomes is limited,
or finite
-10 -5 0 5 10 16 20 25 Rate of Return (%)
Expected Rate of Return (15%)
b. U. S. ElectricProbability of Occurrence
0.5 -
0.4 -
0.3 -
0.2 -
0.1 -
Discrete Probability Discrete Probability DistributionsDistributions
-60 -45 -30 -15 0 15 22 30 45 60 75 90 110Rate of
Return (%)Expected Rate
of Return (15%)
a. Martin ProductsProbability of Occurrence
0.5 -
0.4 -
0.3 -
0.2 -
0.1 -
Continuous versus Discrete Continuous versus Discrete Probability DistributionsProbability Distributions
Continuous probability distribution the number of possible outcomes is
unlimited, or infinite
-60 0 15 110Rate of Return
(%)Expected Rate of
Return
Martin Products
Probability Density
U. S. Electric
Continuous Probability Continuous Probability DistributionsDistributions
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^
^^ ^
59.3% 3,517σσ Deviation Standard
3,517.0σ Variance
2MM
2
Measuring Risk: Measuring Risk: The Standard DeviationThe Standard Deviation
n
1iii
2 k̂-k Variance Pr2
n
1iii
2 k̂-k deviation Standard Pr2
Measuring Risk: Measuring Risk: The Standard DeviationThe Standard Deviation
Measuring Risk: Measuring Risk: Coefficient of VariationCoefficient of Variation
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 variation oft Coefficien
Risk-averse investors require higher rates of return to invest in higher-risk securities
Risk AversionRisk Aversion
Risk Aversion and Risk Aversion and Required ReturnsRequired Returns
Risk premium (RP) the portion of the expected return that can
be attributed to the additional risk of an investment
the difference between the expected rate of return on a given risky asset and that on a less risky asset
Portfolio Risk and thePortfolio Risk and the Capital Asset Pricing Model Capital 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 reflects diversification
Portfolio a collection of investment securities
Expected return on a portfolio, kp
the weighted average expected return on the stocks held in the portfolio
N
1jjj
NN2211p
k̂w
k̂wk̂wk̂wk̂
Portfolio ReturnsPortfolio Returns
Realized rate of return, k the return that is actually earned actual return is generally different from
the expected return
Portfolio ReturnsPortfolio Returns
Portfolio RiskPortfolio Risk
Correlation coefficient, r a measure of the degree of relationship
between two variables positively correlated stocks rates of return
move together in the same direction negatively correlated stocks have rates of
return than move in opposite directions
Portfolio RiskPortfolio Risk
Risk reduction combining stocks that are not perfectly
positively correlated will reduce the portfolio risk by 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
Firm-Specific Risk versus Firm-Specific Risk versus Market RiskMarket 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 versus Firm-Specific Risk versus Market RiskMarket Risk
Firm-specific risk that part of a security’s risk associated with
random outcomes generated by events, or behaviors, specific to the firm
it can be eliminated by proper diversification
Firm-Specific Risk versus Firm-Specific Risk versus Market RiskMarket Risk
Market risk that part of a security’s risk that cannot be
eliminated by diversification because it is associated with economic, or market factors that systematically affect most firms
Firm-Specific Risk versus Firm-Specific Risk versus Market RiskMarket Risk
Relevant risk the risk of a security that cannot be
diversified away--its market risk this reflects a security’s contribution to the
risk of a portfolio
The Concept of BetaThe Concept of Beta
Beta 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 is of average risk = 2.0: stock is twice as risky as the
average stock
The beta of any set of securities is the weighted average of the individual securities’ betas
N
1jjj
NN2211p
βw
βwβwβwβ
Portfolio Beta CoefficientsPortfolio Beta Coefficients
stock j the on return of rate k̂ thj expected
The Relationship between The Relationship between Risk and Rates of ReturnRisk and Rates of Return
The Relationship between The Relationship between Risk and Rates of ReturnRisk and Rates of Return
stock j the on return of rate k
stock j the on return of rate k̂
thj
thj
required
expected
The Relationship between The Relationship between Risk and Rates of ReturnRisk and Rates of Return
return of rate k
stock j the on return of rate k
stock j the on return of rate k̂
RF
thj
thj
freerisk
required
expected
The Relationship between The Relationship between Risk and Rates of ReturnRisk and Rates of Return
premium riskmarket k-k RP
return of rate k
stock j the on return of rate k
stock j the on return of rate k̂
RFMM
RF
thj
thj
freerisk
required
expected
The Relationship between The Relationship between Risk and Rates of ReturnRisk and Rates of Return
stock j the on premium risk βk-k RP
premium riskmarket k-k RP
return of rate k
stock j the on return of rate k
stock j the on return of rate k̂
thjRFMj
RFMM
RF
thj
thj
freerisk
required
expected
Market Risk PremiumMarket 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% Thus, the market risk premium is 8%:
RPM = kM - kRF = 14% - 6% = 8%
Risk Premium for a StockRisk Premium for a Stock
Risk premium for stock j
= RPj = RPM j
j stockfor return of rate k j required
jRFMRF
jMRFj
β kk k
β RP k k
The Required Rate of Return The Required Rate of Return for a Stockfor a Stock
The Required Rate of Return The Required Rate of Return for a Stockfor 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
Required Rate of
Return (%)
Risk-Free Rate: 6%
0 0.5 1.0 2.0 Risk, j
khigh = 22
kM = kA = 14
kLow = 10
kRF = 6
Safe Stock: Risk Premium: 4%
Market (Average Stock): Risk Premium: 8%
jRFMRFj β kk k k:SML
Relatively Risky Stock: Risk Premium = 16%
Security Market LineSecurity Market Line
The Impact of InflationThe 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, k* an inflation premium (IP)
An increase in expected inflation would increase the risk-free rate, kRF
Changes in Risk AversionChanges 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, which increases the slope
Changes in a Stock’s Beta Changes in a Stock’s Beta CoefficientCoefficient
The risk of a stock is affected by composition of its assets use of debt financing increased competition expiration of patents
Any change in the required return (from change in or in expected inflation) affects the stock price
Stock Market EquilibriumStock 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
jj k k̂
Changes in Equilibrium Changes in Equilibrium Stock PricesStock 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
Changes in expected dividends, D0(1+g)
Riskiness of a physical asset is only relevant in terms of its effect on the stock’s risk
Physical AssetsPhysical Assetsversus Securitiesversus Securities
Word of CautionWord of Caution
CAPM based on expected conditions only have historical data as conditions change, future volatility may
differ from past volatility estimates are subject to error
End of Chapter 11End of Chapter 11
Risk and Rates of Return