FINANCE10. Risk and expected returns

Professor André Farber

Solvay Business SchoolUniversité Libre de BruxellesFall 2006

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Risk and return

• Objectives for this session:

• 1. Efficient set

• 2. Beta

• 3. Optimal portfolio

• 4. CAPM

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The efficient set for many securities

• Portfolio choice: choose an efficient portfolio

• Efficient portfolios maximise expected return for a given risk

• They are located on the upper boundary of the shaded region (each point in this region correspond to a given portfolio)

Risk

Expected Return

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Choosing between 2 risky assets

• Choose the asset with the highest ratio of excess expected return to risk:

• Example: RF = 6%

• Exp.Return Risk

• A 9% 10%

• B 15% 20%

• Asset Sharpe ratio

• A (9-6)/10 = 0.30

• B (15-6)/20 = 0.45 **

i

Fi RR

ratio Sharpe

A

B

A

Risk

Expected return

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Optimal portofolio with borrowing and lending

Optimal portfolio: maximize Sharpe ratio

M

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Capital asset pricing model (CAPM)

• Sharpe (1964) Lintner (1965)

• Assumptions

• Perfect capital markets

• Homogeneous expectations

• Main conclusions: Everyone picks the same optimal portfolio

• Main implications:

– 1. M is the market portfolio : a market value weighted portfolio of all stocks

– 2. The risk of a security is the beta of the security:

• Beta measures the sensitivity of the return of an individual security to the return of the market portfolio

• The average beta across all securities, weighted by the proportion of each security's market value to that of the market is 1

Beta

Prof. André FarberSOLVAY BUSINESS SCHOOLUNIVERSITÉ LIBRE DE BRUXELLES

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Measuring the risk of an individual asset

• The measure of risk of an individual asset in a portfolio has to incorporate the impact of diversification.

• The standard deviation is not an correct measure for the risk of an individual security in a portfolio.

• The risk of an individual is its systematic risk or market risk, the risk that can not be eliminated through diversification.

• Remember: the optimal portfolio is the market portfolio.

• The risk of an individual asset is measured by beta.

• The definition of beta is:

22 )(

),(

M

iM

M

Mii

R

RRCov

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Beta

• Several interpretations of beta are possible:

• (1) Beta is the responsiveness coefficient of Ri to the market

• (2) Beta is the relative contribution of stock i to the variance of the market portfolio

• (3) Beta indicates whether the risk of the portfolio will increase or decrease if the weight of i in the portfolio is slightly modified

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Beta as a slope

15, 25

15, 15

-5, -5

-5, -15

-10, -17.5

20, 27.5

-20

-15

-10

-5

0

5

10

15

20

25

30

-15 -10 -5 0 5 10 15 20 25

Return on market

Ret

urn

on

ass

et

Slope = Beta = 1.5

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A measure of systematic risk : beta

• Consider the following linear model

• Rt Realized return on a security during period t

A constant : a return that the stock will realize in any period

• RMt Realized return on the market as a whole during period t

A measure of the response of the return on the security to the return on the market

• ut A return specific to the security for period t (idosyncratic return or unsystematic return)- a random variable with mean 0

• Partition of yearly return into:

– Market related part ß RMt

– Company specific part + ut

tMtt uRR

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Beta - illustration

• Suppose Rt = 2% + 1.2 RMt + ut

• If RMt = 10%

• The expected return on the security given the return on the market

• E[Rt |RMt] = 2% + 1.2 x 10% = 14%

• If Rt = 17%, ut = 17%-14% = 3%

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Measuring Beta

• Data: past returns for the security and for the market

• Do linear regression : slope of regression = estimated beta

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

A B C D E F G H IBeta Calculation - monthly data

Market A B

Mean 2.08% 0.00% 4.55% D3. =AVERAGE(D12:D23)

StDev 5.36% 4.33% 10.46% D4. =STDEV(D12:D23)

Correl 78.19% 71.54% D5. =CORREL(D12:D23,$B$12:$B$23)

R² 61.13% 51.18% D6. =D5 2̂

Beta 1 0.63 1.40 D7. =SLOPE(D12:D23,$B$12:$B$23)

I ntercept 0 -1.32% 1.64% D8. =I NTERCEPT(D12:D23,$B$12:$B$23)

Data

Date Rm RA RB

1 5.68% 0.81% 20.43%

2 -4.07% -4.46% -7.03%

3 3.77% -1.85% -10.14%

4 5.22% -1.94% 6.91%

5 4.25% 3.49% 4.65%

6 0.98% 3.44% 7.64%

7 1.09% -4.27% 8.41%

8 -6.50% -2.70% -1.25%

9 -4.19% -4.29% -11.19%

10 5.07% 3.75% 13.18%

11 13.08% 9.71% 19.22%

12 0.62% -1.67% 3.77%

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Decomposing of the variance of a portfolio

• How much does each asset contribute to the risk of a portfolio?

• The variance of the portfolio with 2 risky assets

• can be written as

• The variance of the portfolio is the weighted average of the covariances of each individual asset with the portfolio.

22222 2 BBABBAAAP XXXX

BPBAPA

BBABABABBAAA

BBABBAABBAAAP

XX

XXXXXX

XXXXXX

)()(

)()(22

22222

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Example

Exp.Return Sigma VarianceRiskless rate 5 0 0A 15 20 400B 20 30 900Correlation 0

Prop. Variance-covarianceA 0.50 400 0B 0.50 0 900

Cov(Ri,Rp) 200.00 450.00X 0.50 0.50

Variance 325.00St.dev. 18.03Exp.Ret. Rp 17.50

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Beta and the decomposition of the variance

• The variance of the market portfolio can be expressed as:

• To calculate the contribution of each security to the overall risk, divide each term by the variance of the portfolio

nMniMiMMM XXXX ......22112

1......

1......

2211

2222

221

1

nMniMiMM

M

nMn

M

iMi

M

M

M

M

XXXX

or

XXXX

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Marginal contribution to risk: some math

• Consider portfolio M. What happens if the fraction invested in stock I changes?

• Consider a fraction X invested in stock i

• Take first derivative with respect to X for X = 0

• Risk of portfolio increase if and only if:

• The marginal contribution of stock i to the risk is

22222 )1(2)1( iiMMP XXXX

)(2 2

0

2

MiM

X

P

dX

d

2MiM

iM

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Marginal contribution to risk: illustration

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Fraction in B

Ris

k o

f p

ort

folio

Cor = 0 Cor = 0.25 Cor = 0.50 Cor = 0.75 Cor = 1.0

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Beta and marginal contribution to risk

• Increase (sightly) the weight of i:

• The risk of the portfolio increases if:

• The risk of the portfolio is unchanged if:

• The risk of the portfolio decreases if:

12

2 M

iMiMMiM

12

2 M

iMiMMiM

12

2 M

iMiMMiM

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Inside beta

• Remember the relationship between the correlation coefficient and the covariance:

• Beta can be written as:

• Two determinants of beta

– the correlation of the security return with the market

– the volatility of the security relative to the volatility of the market

Mi

iMiM

M

iiM

M

iMiM

2

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Properties of beta

• Two importants properties of beta to remember

• (1) The weighted average beta across all securities is 1

• (2) The beta of a portfolio is the weighted average beta of the securities

1......2211 nMniMiMM XXXX

nMnPiMiPMPMPP XXXX ......2211

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Risk premium and beta

• 3. The expected return on a security is positively related to its beta

• Capital-Asset Pricing Model (CAPM) :

• The expected return on a security equals:

the risk-free rate

plus

the excess market return (the market risk premium)

times

Beta of the security

)( FMF RRRR

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CAPM - Illustration

Expected Return

Beta1

MR

FR

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CAPM - Example

• Assume: Risk-free rate = 6% Market risk premium = 8.5%

• Beta Expected Return (%)

• American Express 1.5 18.75

• BankAmerica 1.4 17.9

• Chrysler 1.4 17.9

• Digital Equipement 1.1 15.35

• Walt Disney 0.9 13.65

• Du Pont 1.0 14.5

• AT&T 0.76 12.46

• General Mills 0.5 10.25

• Gillette 0.6 11.1

• Southern California Edison 0.5 10.25

• Gold Bullion -0.07 5.40

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Pratical implications

• Efficient market hypothesis + CAPM: passive investment

• Buy index fund

• Choose asset allocation

Arbitrage Pricing Model

Professeur André Farber

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Market Model

• Consider one factor model for stock returns:

• Rj = realized return on stock j

• = expected return on stock j

• F = factor – a random variable E(F) = 0

• εj = unexpected return on stock j – a random variable

• E(εj) = 0 Mean 0

• cov(εj ,F) = 0 Uncorrelated with common factor

• cov(εj ,εk) = 0 Not correlated with other stocks

jjjj FRR

jR

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Diversification

• Suppose there exist many stocks with the same βj.

• Build a diversified portfolio of such stocks.

• The only remaining source of risk is the common factor.

FRR jjj

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Created riskless portfolio

• Combine two diversified portfolio i and j.

• Weights: xi and xj with xi+xj =1

• Return:

• Eliminate the impact of common factor riskless portfolio

• Solution:

FxxRxRx

RxRxR

jjiijjii

jjiiP

)()(

0 jiii xx

ji

jix

ji

ijx

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Equilibrium

• No arbitrage condition:

• The expected return on a riskless portfolio is equal to the risk-free rate.

Fjji

ii

ji

j RRR

j

Fj

i

FiRRRR

At equilibrium:

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Risk – expected return relation

jFj RR

FM RR

Linear relation between expected return and beta

For market portfolio, β = 1

Back to CAPM formula:

jFMFj RRRR )(