Chapter

McGraw-Hill/Irwin Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.

13•Return, Risk, and the Security Market Line•Return, Risk, and the Security Market Line

13-2

Expected Returns

• Expected Rate of Return

All possible outcomes and probabilities known

• Kμ = E(R) = P1K1 + P2K2 + P3K3

Sample taken of past returns

• _

• K = ΣK/n

13-3

Example: Expected Returns

• Suppose you have predicted the following returns for stocks C and T in three possible states of nature. What are the expected returns?• State Probability C T• Boom 0.3 15

25• Normal 0.5 10 20• Recession ??? 2 1

• RC = .3(15) + .5(10) + .2(2) = 9.99%• RT = .3(25) + .5(20) + .2(1) = 17.7%

Sample of Observations

Returns:

2008 8.6%

2007 14.2%

2006 -4.6%

2007 8.8%

E(R) = Mean = (8.6+14.2-4.6+8.8)/4 = 6.75

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13-5

Variance and Standard Deviation

Variance: σ2 = P1(K1 - Kμ)2 + P2(K2 - Kμ)2 + P3(K3 - Kμ)2

_ _ _

Variance: S2 = [(K1 - K)2 + (K2 - K)2 + (K3 - K)2]/n-1

Standard Deviation = Square Root of Variance

Sample of Observations

Returns:

2008 8.6%

2007 14.2%

2006 -4.6%

2007 8.8%

E(R) = Mean = (8.6+14.2-4.6+8.8)/4 = 6/75

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13-7

Example: Variance andStandard Deviation

• Consider the previous example. What are the variance and standard deviation for each stock?

• Stock C2 = .3(15-9.9)2 + .5(10-9.9)2 + .2(2-9.9)2 = 20.29 = 4.5

• Stock T2 = .3(25-17.7)2 + .5(20-17.7)2 + .2(1-17.7)2 =

74.41 = 8.63

Sample Variance

Variance

= [(8.6 – 6.75)^2 + (14.2 – 6.75)^2 + (-4.6 – 6.75)^2 + (8.8 – 6.75)^2] /3

= [3.4224 + 55,5025 + 128.8225 + 4.2025]/3 = 63.98333

Standard Deviation

= 63.98333^(1/2) = 7.998

Average Distance Around Mean

[(8.6 – 6.75) + (14.2 – 6.75) + (-4.6 – 6.75) + (8.8 – 6.75)] /4 = 5.675

13-8

13-9

Portfolios vs. Individual Stocks

• A portfolio is a collection of assets

• An asset’s risk and return are important in how they affect the risk and return of the portfolio

• The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets

13-10

Individual Stocks’ Mean Return

θ P K W

1 .2 -10 402 .2 40 -103 .2 -5 354 .2 35 -55 .2 15 15

Kμ = .2(-10) + .2(40) + .2(-5) + .2(35) + .2(15) = 15

Wμ = .2(40) + .2(-10) + .2(35) + .2(-5) + .2(15) = 15

13-11

Individual Stocks’ Varianceand Standard Deviation

θ P K W (K-Kμ)2 (W-Wμ)2

1 .2 -10 40 625 625

2 .2 40 -10 625 625

3 .2 -5 35 400 400

4 .2 35 -5 400 400

5 .2 15 15 0 0

σK = [.2(625) + .2(625) + .2(400) +.2(400) + .2(0)]½ = 20.25

σW = [.2(625) + .2(625) + .2(400) +.2(400) + .2(0)]½ =20.25

13-12

Equally WeightedPortfolio Returns

θ P K W (K-Kμ)2 (W-Wμ)2 .5K+.5W

1 .2 -10 40 625 625 15

2 .2 40 -10 625 625 15

3 .2 -5 35 400 400 15

4 .2 35 -5 400 400 15

5 .2 15 15 0 0 15

Ex: (1) .5 x -10 + .5 x 40 = 15

Ex: (2) .5 x 40 + .5 x -10 = 15

13-13

Unequally Weighted Portfolio Variance

θ P K W (K-Kμ)2 (W-Kμ)2 .75K+.25W

1 .2 -10 40 625 625 2.5

2 .2 40 -10 625 625 27.5

3 .2 -5 35 400 400 5.0

4 .2 35 -5 400 400 25.0

5 .2 15 15 0 0 15.0

Ex: (1) .75 x -10 + .25 x 40 = 2.5

Ex: (2) .75 x 40 + .25 x -10 = 27.5

13-14

Correlation of Security Returns

Perfect Positive = +1

Perfect Negative = -1

Uncorrelated = 0

13-15

Diversification

Total Risk = Nondiversifiable Risk + Diversifiable Risk

Total Risk = Systematic Risk + Unsystematic Risk

Total Risk = Market Risk + Firm Risk

13-16

Systematic Risk

• Risk factors that affect a large number of assets

• Also known as non-diversifiable risk or market risk

• Includes such things as changes in GDP, inflation, interest rates, etc.

13-17

Unsystematic Risk

• Risk factors that affect a limited number of assets

• Also known as unique risk and asset-specific risk

• Includes such things as labor strikes, part shortages, etc.

13-18

Diversification

• Portfolio diversification is the investment in several different asset classes or sectors

• Diversification is not just holding a lot of assets

• For example, if you own 50 internet stocks, you are not diversified

• However, if you own 50 stocks that span 20 different industries, then you are diversified

13-19

Table 13.7

13-20

The Principle of Diversification

• Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns

• This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another

• However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion

13-21

Figure 13.1

13-22

Diversifiable Risk

• The risk that can be eliminated by combining assets into a portfolio

• Often considered the same as unsystematic, unique or asset-specific risk

• If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away

13-23

Total Risk

• Total risk = systematic risk + unsystematic risk

• The standard deviation of returns is a measure of total risk

• For well-diversified portfolios, unsystematic risk is very small

• Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk

13-24

Systematic Risk Principle

• There is a reward for bearing risk

• There is not a reward for bearing risk unnecessarily

• The expected return on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified away

13-25

Measuring Systematic Risk

• How do we measure systematic risk?• We use the beta coefficient to measure

systematic risk• What does beta tell us?

• A beta of 1 implies the asset has the same systematic risk as the overall market

• A beta < 1 implies the asset has less systematic risk than the overall market

• A beta > 1 implies the asset has more systematic risk than the overall market

13-26

Total versus Systematic Risk

• Consider the following information: Standard Deviation Beta• Security C 20% 1.25• Security K 30% 0.95

• Which security has more total risk?

• Which security has more systematic risk?

• Which security should have the higher expected return?

13-27

Examples of Betas

Edison Electricity .55

Coor’s Brewing .75

Sony .85

General Motors 1.25

Intel 1.40

Dell 1.55

13-28

Example: Portfolio Betas

• Consider the following four securities• Security Weight Beta

• DCLK .133 2.685

• KO .2 0.195

• INTC .267 2.161

• KEI .4 2.434

• What is the portfolio beta?• .133(2.685) + .2(.195) + .267(2.161) + .4(2.434)

= 1.9467

13-29

Beta and the Risk Premium

• Remember that the risk premium = expected return – risk-free rate

• The higher the beta, the greater the risk premium should be

• Can we define the relationship between the risk premium and beta so that we can estimate the expected return?• YES!

13-30

Example: Portfolio Expected Returns and Betas

0%

5%

10%

15%

20%

25%

30%

0 0.5 1 1.5 2 2.5 3

Beta

Exp

ecte

d R

etur

n

Rf

E(RA)

A

13-31

Reward-to-Risk Ratio: Definition and Example

• The reward-to-risk ratio is the slope of the line illustrated in the previous example• Slope = (E(RA) – Rf) / (A – 0)• Reward-to-risk ratio for previous example =

(20 – 8) / (1.6 – 0) = 7.5

• What if an asset has a reward-to-risk ratio of 8 (implying that the asset plots above the line)?

• What if an asset has a reward-to-risk ratio of 7 (implying that the asset plots below the line)?

13-32

Market Equilibrium

• In equilibrium, all assets and portfolios must have the same reward-to-risk ratio and they all must equal the reward-to-risk ratio for the market

M

fM

A

fA RRERRE

)()(

13-33

Security Market Line

• The security market line (SML) is the representation of market equilibrium

• The slope of the SML is the reward-to-risk ratio: (E(RM) – Rf) / M

• But since the beta for the market is ALWAYS equal to one, the slope can be rewritten

• Slope = E(RM) – Rf = market risk premium

13-34

The Capital Asset Pricing Model (CAPM)

• The capital asset pricing model defines the relationship between risk and return

• E(RA) = Rf + A(E(RM) – Rf)

• If we know an asset’s systematic risk, we can use the CAPM to determine its expected return

• This is true whether we are talking about financial assets or physical assets

13-35

Factors Affecting Expected Return

• Pure time value of money – measured by the risk-free rate

• Reward for bearing systematic risk – measured by the market risk premium

• Amount of systematic risk – measured by beta

13-36

Example - CAPM

• Consider the betas for each of the assets given earlier. If the risk-free rate is 2.13% and the market risk premium is 8.6%, what is the expected return for each?

Security Beta Expected Return

DCLK 2.685 2.13 + 2.685(8.6) = 25.22%

KO 0.195 2.13 + 0.195(8.6) = 3.81%

INTC 2.161 2.13 + 2.161(8.6) = 20.71%

KEI 2.434 2.13 + 2.434(8.6) = 23.06%

13-37

Figure 13.4

Chapter

McGraw-Hill/Irwin Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.

12

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