FINANCE 10. Risk and expected returns Professor André Farber Solvay Business School Université...

FINANCE10. Risk and expected returns

Professor André Farber

Solvay Business SchoolUniversité Libre de BruxellesFall 2006

MBA 2006 Risk and return (2) |2

Measuring the risk of an individual asset

• La mesure du risque d’un titre dans un portefeuille doit tenir compte de l’impact de la diversification.

• L’écart type n’est donc pas la bonne mesure.

• Le risque se mesure par la contribution du titre au risque du portefeuille.

• 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


Beta

• Plusieurs interprétations du beta:

• (1) Beta mesure la sensibilité de Ri par rapport au marché

• (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


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


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


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%


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


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


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


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


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


CAPM - Illustration

Expected Return

Beta1

MR

FR


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


Pratical implications

• Efficient market hypothesis + CAPM: passive investment

• Buy index fund

• Choose asset allocation

Arbitrage Pricing Model

Professeur André Farber


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


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


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


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:


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 )(

FINANCE 10. Risk and expected returns Professor André Farber Solvay Business School Université...

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