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Return, Risk, and the Security Market Line

Chapter 13

Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Chapter Outline

• Expected Returns and Variances of Individual Securities

• Expected Returns and Variances of Portfolios

• Systematic and Unsystematic Risk• Diversification• Systematic Risk and Beta• The Security Market Line & the Capital

Asset Pricing Model (CAPM)

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Expected Returns of Individual Securities

• The expected return is based on future returns and the probabilities of possible outcomes

where 1, 2, 3…i = states p = probability that a state occurs

R = future return

ii pRpRpRRE ...)( 2211

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Variance of Individual Securities

where 1, 2, 3…i = states

p = probability that a state occurs R = future return E(R)= expected return

ii pRER

pRERpRER2

22

212

12

))((

...))(())((σ

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Example: E(R) and VAR of Individual Security

Security A:State R p1 .30 .402 .15 .503 .05 .10

1.00E( RA ) = VARA =Security B:State R p 1 .25 .302 .17 .403 .10 .30

1.00E( RB ) = VARB =

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Expected Portfolio Returns

• The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio

where A, B, …j = securities

)(...)()()( jjBBAAP REwREwREwRE

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Example: Expected Portfolio Return

• You have a total of $15,000 to invest and have purchased $12,000 worth of security A and $3,000 worth of security B. Assume that E(RA)=.20 and E(RB) = .173 What are the portfolio weights and what is the expected portfolio return E(RP)?

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Variance of Portfolio Returns

• 1. Compute the future return of the portfolio for each state

• 2. Compute the expected portfolio return

• 3. Compute the portfolio variance

where 1, 2, 3…i = states p = probability that a state occurs R = future return E(R)= expected return

iPPi

PPPP

pRER

pRERpRER2

22

212

12P

))((

...))(())((σ

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Example: Variance of Portfolio Returns

Assuming that 30% of the portfolio is invested in Stock A and 70% in stock B, what is the portfolio variance?

State RA (wA=.3) RB(wB=.7) p Rp

1 .20 .25 40% 2 .18 .15 60%

E(Rp)=VARp=

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Systematic & Unsystematic Risk

• Realized returns are generally not equal to expected returns

• There is the expected component and the unexpected componentR = E(R)+U

U = systematic (non-diversifiable) portion + unsystematic (diversifiable) portion

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Diversification

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Diversification

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Systematic Risk and Beta• We use the beta coefficient to

measure systematic (non-diversifiable) 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

Sample Company Betas

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http://screener.finance.yahoo.com/stocks.html

Security Market Line & The Capital Asset Pricing

Model (CAPM)• 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

)()(

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Security Market Line & the CAPM

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

• The slope of the SML is the reward-to-risk ratio

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The CAPM

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

• E(Ri) = Rf + i(E(RM) – Rf)

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CAPM Example:

• Assume the risk-free rate is 3%, the return on the market is 9% and a particular stock has a beta of 1.5. What is the expected return of the stock?

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