Risk and Rates of Return Chapter 11. Defining and Measuring Risk uRisk is the chance that an outcome...

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

^ ^


(1) (2) (3) = (4)Boom 0.2 110% 22%Normal 0.5 22% 11%Recession 0.3 -60% -18%

1.0 km = 15%


Martin ProductsProbability of This State

Occurring (Pr i)

Return if This State

Occurs (ki)Product: (2) x (3)

^ ^


(1) (2) (3) = (4)Boom 0.2 20% 4%Normal 0.5 16% 8%Recession 0.3 10% 3%

1.0 km= 15%


U. S. Electric

Return if This

State Occurs (ki)Product: (2) x (3)

Probability of This State

Occurring (Pr i)


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 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


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


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


stock j the on return of rate k

stock j the on return of rate k̂

thj

thj

required

expected


return of rate k



RF

thj

thj

freerisk

required

expected


premium riskmarket k-k RP

return of rate k



RFMM

RF

thj

thj

freerisk

required

expected


stock j the on premium risk βk-k RP

premium riskmarket k-k RP

return of rate k



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

Risk and Rates of Return Chapter 11. Defining and Measuring Risk uRisk is the chance that an outcome...

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