Chapter 13

Return, Risk, and the Security Market Line

Copyright © 2012 by McGraw-Hill Education. All rights reserved.

Key Concepts and Skills

• Know how to calculate expected returns• Understand the impact of diversification• Understand the systematic risk principle• Understand the security market line• Understand the risk-return trade-off• Be able to use the Capital Asset Pricing

Model

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Chapter Outline

• Expected Returns and Variances• Portfolios• Announcements, Surprises, and Expected

Returns• Risk: Systematic and Unsystematic• Diversification and Portfolio Risk• Systematic Risk and Beta• The Security Market Line• The SML and the Cost of Capital: A Preview

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Expected Returns

• “ The return on a risky asset expected in the future.”

• Risk premium is defined as “ the difference between the return on a risky investment and that on a risk-free investment.”

• Risk premium = Expected return – Risk free rate

Expected Returns

• Example: If the risk free investments are currently offering 8%. Stock Z offers rate of return 20%. What is stock Z risk premium?

• Risk premium = 20% - 8% = 12%

Expected Returns• Expected returns are based on the

probabilities of possible outcomes• In this context, “expected” means average

if the process is repeated many times• The “expected” return does not even have

to be a possible return

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Example: Expected Returns• Suppose you have predicted the following returns

for stocks C and T in three possible states of the economy. What are the expected returns?

• State Probability C TBoom 0.3 15% 25%Normal 0.5 10 20Recession ??? 2 1

• RC = .3(15) + .5(10) + .2(2) = 9.9%• RT = .3(25) + .5(20) + .2(1) = 17.7%

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Variance and Standard Deviation

• Variance and standard deviation measure the volatility of returns

• Using unequal probabilities for the entire range of possibilities

• Weighted average of squared deviations

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Example: Variance and Standard Deviation

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

• Stock C 2 = .3(15 - 9.9)2 + .5(10 - 9.9)2 + .2(2 - 9.9)2 = 20.29 = 4.50%

• Stock T 2 = .3(25 - 17.7)2 + .5(20 - 17.7)2 + .2(1 - 17.7)2 = 74.41 = 8.63%

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Another Example

• Consider the following information:State Probability ABC, Inc. (%)

Boom .25 15

Normal .50 8

Slowdown .15 4

Recession .10 -3

• What is the expected return?• What is the variance?• What is the standard deviation?

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Solution

1) The expected return = (0.25×15) + (0.5×8) + (0.15×4) + (0.1×-3) = 8.05

2) The variance = 0.25(15-8.05)2 + 0.5(8-8.05)2

+ 0.15(4-8.05)2 + 0.1(-3-8.05)2 = 26.75

3) The standard deviation = 5.17

Portfolios

• 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

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Portfolio Weights

• Portfolio Weight: “The percentage of a portfolio’s total value that is in a particular asset.”

• Notice that: the weights have to add up to 1 because all our money is invested in it.

Example: Portfolio Weights

• Suppose you have $15,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security?

– $2000 of DCLK– $3000 of KO– $4000 of INTC– $6000 of KEI

•DCLK: 2/15 = .133

•KO: 3/15 = .2

•INTC: 4/15 = .267

•KEI: 6/15 = .4

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Portfolio Expected Returns

• The expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio

• You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities

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Example: Expected Portfolio Returns

• Consider the portfolio weights computed previously. If the individual stocks have the following expected returns, what is the expected return for the portfolio?– DCLK: 19.69%

– KO: 5.25%

– INTC: 16.65%

– KEI: 18.24%

• E(RP) = .133(19.69) + .2(5.25) + .267(16.65) + .4(18.24) = 15.41%

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Portfolio Weights

• N.B.: This says that the expected return on a

portfolio is a straightforward combination of the expected returns on the assets in that portfolio.

Portfolio Variance

• Compute the portfolio return for each state:RP = w1R1 + w2R2 + … + wmRm

• Compute the expected portfolio return using the same formula as for an individual asset

• Compute the portfolio variance and standard deviation using the same formulas as for an individual asset

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Example 13.3 & 13.4

• Suppose we have the following projections for three stocks:

State of the economy

Probability of the state of economy

Returns if state occurs on stock A

Returns if state occurs on stock B

Returns if state occurs on stock C

Boom 0.4 10% 15% 20%

Bust 0.6 8 4 0

Example 13.3 & 13.4

• What would be the expected return if half of the portfolio were in A, with the remainder equally divided between B and C?

• What are the standard deviation of the portfolio?

Example 13.3 & 13.4

• First calculate the expected returns on the individual stocks as follows:

• E(RA) = (0.4×10) + (0.6×8) = 8.8%

• E(RB) = (0.4×15) + (0.6×4) = 8.4%

• E(RC) = (0.4×20) + (0.6×0) = 8.0%

Then,

• E(RP) = (0.5×8.8) + (0.25×8.4) + (0.25×8)

= 8.5%

Example 13.3 & 13.4

• First, calculate the portfolio returns in the two states.

(1) The portfolio return when the economy booms is calculated as:

• E(RP) = (0.5×10) + (0.25×15) + (0.25×20)

= 13.75%

Example 13.3 & 13.4

(2) The portfolio return when the economy busts is calculated as:

•E(RP) = (0.5×8) + (0.25×4) + (0.25×0)

= 5%

•E(RP) = (0.4×13.75) + (0.6×5) = 8.5%

Example 13.3 & 13.4

• The variance is:

σ2P = 0.4 × (0.1375 – 0.085)2 + 0.6 ×

(0.05 – 0.085) 2 = 0.0018375

• The standard deviation is:

σ = 4.3%

End of Chapter

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