Chapter 4

Risk and Rates of Return

© 2005 Thomson/South-Western

2

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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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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Expected Rate of Return

State of the economy

Prob. Martin ProductRETURNS

US ElectricRETURNS

(1) (2) (3) (4)(2 x 3)

(5) (6)(2 x 5)

Boom 0.2 110% 20%Normal 0.5 22% 16%Recession 0.3 -60% 10%

E(K)E(K)

5

n

1ii

n21

k

kkkk

i

n21

Pr

PrPrPr ˆ

Expected Rate of Return

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Continuous versus Discrete Probability Distributions

Continuous Probability Distribution:number of possible outcomes is unlimited, or infinite.

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Measuring Risk: The Standard Deviation

Martin ProductK E(K) K – E(K) [K –

E(K)]2

Pr. [K – E(K)]2 x Pr

(1) (2) (3)(1 – 2)

(4) (5) (6)(4 x 5)

110%

0.2

22% 0.5

-60% 0.3

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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 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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Portfolio Risk and Return

The goal of finance manager is to createAN EFFICIENT PORTFOLIO: “Maximizes returnfor a given level of risk or minimizes risk for agiven level of return”.

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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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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 size and risk

Inc size of a portfolio risk dec Risk dec to a certain point .. (Co. risk

= 0) If we take all sec in the stock mkt as

one portfolio (max size) still some risk exist (Market Risk) = relevant risk = the contribution of a sec’s risk to a portfolio.

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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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NO GOOD…

.. of thinking how risky a security is if helf in isolation – you need to measure its market risk … measure how sensitive it is to market … this sensitivitysensitivity is called BETABETA

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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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Steps in deriving Beta

Plot mkt ret (X) and asset ret (Y) at various point in time.

Regression: the slope = betaThe higher the beta the higher the riskBeta for the market = 1, all other betas

are viewed in relation to this value.Beta may be +ve or –v, +ve is the normMajority of betas fall between .5 and 2.

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N

1jjj

nn2211

w

www

p

Portfolio Beta Coefficients

The beta of any set of securities is the weighted average of the individual securities’ betas

IF mkt ret inc by 10%, a port with a beta of .75 will experience a 7.5% inc in its return (.75 x 10)

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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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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 - CAPM

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.