Thé i Fi ièThéorie Financière Ri k d d (2)Risk and ... 2009 08 Risk and return... · Thé i Fi...

Thé i Fi ièThéorie Financière

Ri k d d (2)Risk and expected returns (2)

P f A d é F bProfesseur André Farber

Risk and return

• Objectives for this session:Object ves o t s sess o :• 1. Review: 2 risky assets• 2. Many risky assets• 3. Beta• 4. Optimal portfolio• 5 Eq ilibri m: CAPM• 5. Equilibrium: CAPM

November 14, 2009 Tfin 08 Risk and return (2) |2

Review: The efficient set for two assets

30.00

25.00

A

BOptimal risky portfolio

20.00

etur

n

Optimal asset allocation10.00

15.00

Expe

cted

r

Riskless rate

Optimal asset allocation

5.00

10.00

0.000.00 10.00 20.00 30.00 40.00 50.00 60.00

( )


Risk ( standard deviation)

Formulas

),(~ PPP RNR σReturns: normal distribution

21 21 RXRXRP +=Expected return:

PP

122122

22

21

21

2 2 σσσσ XXXXP ++=Variance:

211212 σσρσ =

[ ] ⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

2

12221

1221

212

XX

XXP σσσσ

σ

XXP Ω= '2σ


Choosing portfolios from many stocksg p y

• Porfolio composition :o o o co pos t o :• (X1, X2, ... , Xi, ... , XN)• X1 + X2 + ... + Xi + ... + XN = 1

• E pected ret rn: NNP RXRXRXR +++= ...2211• Expected return:

• Risk:∑∑ ∑∑∑

≠

=+=i ij i j

ijjiijjijj

jP XXXXX σσσσ 222

• Note:• N terms for variances• N(N-1) terms for covariances• Covariances dominate


• Covariances dominate

Using matricesg

⎤⎡ X ⎤⎡ R ⎤⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

X

XX ...

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=R

RR ...

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=Ω

NNN

N

σσ

σσ

............

...

1

111

⎥⎦⎢⎣ NX ⎥⎦

⎢⎣ NR ⎥⎦⎢⎣ NNN σσ ...1

XXRXRP

Ω

=

''

2 XXP Ω= '2σ


Some intuition

Var Cov Cov Cov CovCov Var Cov Cov CovCov Cov Var Cov CovCov Cov Cov Var CovC C C C VCov Cov Cov Cov Var


Examplep

• Consider the risk of an equally weighted portfolio of N "identical« stocks:Co s de t e s o a equa y we g ted po t o o o N de t ca « stoc s:

• Equally weighted:N

X j1

=

cov),(,, === jijj RRCovRR σσ

• Variance of portfolio:

Nj

cov)11(1 22

NNP −+= σσ

• If we increase the number of securities ?:

NN

• Variance of portfolio:

∞→→

NP cov2σ


Diversification

Risk Reduction of Equally Weighted Portfoliosq y g

30 00%

35.00%

25.00%

30.00%

atio

n Unique risk

15.00%

20.00%

olio

sta

ndar

d de

vi

5 00%

10.00%

Portf

o

Market risk

0.00%

5.00%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49


# stocks in portfolio

Conclusion

• 1. Diversification pays - adding securities to the portfolio decreases risk. . ve s cat o pays add g secu t es to t e po t o o dec eases s .This is because securities are not perfectly positively correlated

• 2. There is a limit to the benefit of diversification : the risk of the portfolio 't b l th th i ( ) b t th t kcan't be less than the average covariance (cov) between the stocks

• The variance of a security's return can be broken down in the following way:

Total risk of individual security

Portfolio risk

Unsystematic or di ifi bl i k

• The proper definition of the risk of an individual security in a portfolio M is the covariance of the security with the portfolio:

security diversifiable risk

y p


The efficient set for many securitiesy

• Portfolio choice: choose an efficient portfolioo t o o c o ce: c oose a e c e t po t o o• 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)Expected ReturnReturn

i kNovember 14, 2009 Tfin 08 Risk and return (2) |11

Risk

Optimal portofolio with borrowing and lendingp p g g

M

O i l f liOptimal portfolio: maximize Sharpe ratio


Capital asset pricing model (CAPM)p p g ( )

• Sharpe (1964) Lintner (1965)S a pe ( 96 ) t e ( 965)• Assumptions

• Perfect capital markets• Homogeneous expectations

• Main conclusions: Everyone picks the same optimal portfolio• Main implications:• 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 portfolioreturn 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


Market equilibrium: illustrationq

Wealth Risk free Market Firm 1 Firm 2 Firm 3asset Portfolio

Optimal portfolio 100% 20% 50% 30%

Alan 10 -10 20 4 10 6

Ben 20 -5 25 5 12.5 7.5

Clara 30 15 15 3 7.5 4.5

Market 60 0 60 12 30 18Market 60 0 60 12 30 18


Capital Asset Pricing Modelp g

Expected return jFMFj RRRR β×−+= )(

R

j

Rj

RM

Risk free interest rate

Beta1βj


BBeta

Prof. André FarberSOLVAY BUSINESS SCHOOLUNIVERSITÉ LIBRE DE BRUXELLES

Measuring the risk of an individual assetg

• The measure of risk of an individual asset in a portfolio has to incorporate e easu e o s o a d v dua asset a po t o o as to co po atethe impact of diversification.

• The standard deviation is not an correct measure for the risk of an i di id l it i tf liindividual 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.

h d fi i i f b i• The definition of beta is:

)( RRC22 )(

),( iM

M

Mii R

RRCovσσ

σβ ==


)(MM

Beta

• Several interpretations of beta are possible:Seve a te p etat o s o beta a e poss b e:

• (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 portfolioportfolio

• (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 slopep

30

15, 25

20, 27.5

20

25

15, 15

10

15

20

5

10

Ret

urn

on a

sset

Slope = Beta = 1.5

-5, -5 -5

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

R

-5, -15

-10, -17.5

-15

-10


-20

Return on market

A measure of systematic risk : betay

• Consider the following linear modelCo s de t e o ow g ea ode

• Rt Realized return on a security during period t

tMtt uRR +×+= βα

• α A constant : a return that the stock will realize in any period• RMt Realized return on the market as a whole during period t• β A meas re of the response of the ret rn on the sec rit to the ret rn• β 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:• Partition of yearly return into:– Market related part ß RMt

– Company specific part α + ut


p y p p t

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[R |R ] 2% + 1 2 10% 14%• E[Rt |RMt] = 2% + 1.2 x 10% = 14%

• If Rt = 17%, ut = 17%-14% = 3%t , t


Measuring Betag

• Data: past returns for the security and for the marketata: past etu s o t e secu ty a d o t e a et• Do linear regression : slope of regression = estimated beta

1A B C D E F G H I

Beta Calculation - monthly data23456

Market A BMean 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. =D5^2

789

1011

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

Intercept 0 -1.32% 1.64% D8. =INTERCEPT(D12:D23,$B$12:$B$23)

DataDate Rm RA RB

1213141516

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%

% % %171819202122

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%


2223

11 13.08% 9.71% 19.22%12 0.62% -1.67% 3.77%

Decomposing of the variance of a portfoliop g p

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

• The variance of the portfolio with 2 risky assets

• can be written as22222 2 BBABBAAAP XXXX σσσσ ++=

BBABABABBAAA

BBABBAABBAAAP

XXXXXX

XXXXXX

σσσσ

σσσσσ

+++=

+++=

)()(

)()(22

22222

• The variance of the portfolio is the weighted average of the covariances of h i di id l i h h f li

BPBAPA

BBABABABBAAA

XXXXXXXX

σσσσσσ

+=

+++ )()(

each individual asset with the portfolio.


Examplep

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

Prop Variance covarianceProp. 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. R 17.50


Beta and the decomposition of the variance

• The variance of the market portfolio can be expressed as:e va a ce o t e a et po t o o ca be e p essed as:

nMniMiMMM XXXX σσσσσ +++++= ......22112

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

1...... 2222

221

1 =+++++M

nMn

M

iMi

M

M

M

M XXXXσσ

σσ

σσ

σσ

1......2211 =+++++ nMniMiMM

MMMM

XXXXor

ββββ

σσσσ

1......2211 +++++ nMniMiMM XXXX ββββ


Marginal contribution to risk: some mathg

• Consider portfolio M. What happens if the fraction invested in stock ICo s de po t o o . W at appe s t e act o vested stocchanges?

• Consider a fraction X invested in stock i

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

22222 )1(2)1( iiMMP XXXX σσσσ +−+−=

2dσ

• Risk of portfolio increase if and only if:

)(2 2

0MiM

X

P

dXd σσσ

−==

Th i l ib i f k t th i k i

2MiM σσ >

• The marginal contribution of stock i to the risk is iMσ


Marginal contribution to risk: illustrationg

35.00

30.00

20.00

25.00

tfolio

15.00

Ris

k of

por

t

5.00

10.00

0.000 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


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

Beta and marginal contribution to riskg

• Increase (sightly) the weight of i:c ease (s g t y) t e we g t o i:

• The risk of the portfolio increases if: 122 >=⇔> iM

iMMiMσ

βσσ

• The risk of the portfolio is nchanged if:

2M

iMMiM σβ

• The risk of the portfolio is unchanged if:12

2 ==⇔=M

iMiMMiM σ

σβσσ

• The risk of the portfolio decreases if:12

2 <=⇔< iMiMMiM

σβσσ 2

Mσ


Inside beta

• Remember the relationship between the correlation coefficient and the e e be t e e at o s p betwee t e co e at o coe c e t a d t ecovariance:

iMiM

σρ =

• Beta can be written as:Mi

iM σσρ

M

iiM

M

iMiM σ

σρ

σσ

β == 2

• Two determinants of beta– the correlation of the security return with the market

h l ili f h i l i h l ili f h k– the volatility of the security relative to the volatility of the market


Properties of betap

• Two importants properties of beta to rememberwo po ta ts p ope t es o beta to e e be

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

1......2211 =+++++ nMniMiMM XXXX ββββ

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

nMnPiMiPMPMPP XXXX βββββ +++++= ......2211 nMnPiMiPMPMPP βββββ 2211


Risk premium and betap

• 3. The expected return on a security is positively related to its beta3. e e pected etu o a secu ty s pos t ve y e ated to ts beta

• Capital-Asset Pricing Model (CAPM) :

• The e pected ret rn on a sec rit eq als:β×−+= )( FMF RRRR

• The expected return on a security equals:the risk-free rate

pluspthe excess market return (the market risk premium)

timesBeta of the security


CAPM - Illustration

Expected Return

MR

FR

Beta1


CAPM - Examplep

• Assume: Risk-free rate = 6% Market risk premium = 8.5%ssu e: s ee ate 6% a et s p e u 8.5%• Beta Expected Return (%)• American Express 1.5 18.75• BankAmerica 1.4 17.9• Chrysler 1.4 17.9• Digital Eq ipement 1 1 15 35• 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 40November 14, 2009 Tfin 08 Risk and return (2) |33

• Gold Bullion -0.07 5.40

CAPM – two formulations

Consider a future uncertain cash flow C to be received in 1 year.

)~(CE )cov( rr

PV calculation based on CAPM:

β)(1)(

fMf rrrCEV−++

=2

),cov(M

Mrrσ

β =

( )C V E C−% %

2

( )Here: cov( , )1 ( ) M

f M fM

C V E Cr VV C rr r r

Vσ

= ⇒ =+ + −

%

)~(Cfrr −)~()),cov(1( CE

VrCrV M

f =++⇒ λ2 :DefineM

fM rr

σλ =

rCCE − equivalentCertainty)~cov()~( λ

ff

M

rrrCCEV

+=

+=

1equivalentCertainty

1),cov()( λ

See Breale and M ers Chap 9


See Brealey and Myers Chap 9

Risk-adjusted expected cash flowj p

Using risk-adjusted discount rates is OK if you know beta.g j y

The adjusted risk-adjusted discount rate does not work for OPTIONS or projects with unknown betas.

To understand how to proceed in that case, we need to go deeper into valuation theory.


Example (see Introduction)p ( )

You observe the following data:

Value Up market (u)P b 0 75

Down market (d)P b 0 25

Expected return

g

Proba = 0.75 Proba = 0.25Bond 95.24 105 105 5%

Market Portfolio

100 1.2 0.80 10%

Wh i h l f h f ll i ? Wh i d ?

NewAsset ? 200 100 ?

What is the value of the following asset? What are its expected returns?


Valuation of project with CAPMp j

MrCCEV =−

=1

equivalentCertainty 1

),cov()( λ

ff rr ++ 11Step 1: calculate statistics for the market portfolio:

Up mkt Proba = .75

Down mkt Proba = .25

Return 20% -20%Return 20% -20%

Expected return: (0.75)(20%) (0.25)( 20%) 10%Mr = + − =

M k i k i 10% 5% 5%

Variance: 2 (0.75)(.20)² (0.25)( .20)² (.10)² 0.030Mσ = + − − =

Market risk premium: 10% 5% 5%M Fr r− = − =

Price of covariance: 05Price of covariance:2

.05 1.67

.30M F

M

r rλσ−

= = =


Valuation of project with CAPM (2)p j ( )

Step 2: Calculate statistics for the projectp p j

Expected cash flow: ( ) 0.75 200 0.25 100 175E C = × + × =%

Covariance with market portfolio:

cov( , ) (0.75)(200 .2) (0.25)(100 ( .2)) (175)(0.10)7.5

MC r = × + × − −=

% %

cov( , ) ( ) ( ) ( )x y E xy E x E y= −(Reminder: )

Step 3: Value the projectStep 3: Value the project

( ) cov( , ) 175 (1.67)(7.5)1 1 05

ME C C rVr

λ− −= =

+

% % %

1 1.05162.49 154.76

1 05

Fr+

= =


1.05

Valuation of project with CAPM (3)p j ( )

Once the value of the project is known, the beta can be calculated.

Value Up mkt Proba Down mkt = .75 Proba = .25

Cash flow 154.76 200 100Returns 29 23% 35 38%Returns 29.23% -35.38%

Expected return: (0 75)( 2923) (0 25)( 0 3538) 13 08%r = + − =Expected return: (0.75)(.2923) (0.25)( 0.3538) 13.08%r = + =

Beta:13.08% 5% 1.61

10% 5%Fr rβ − −

= = =10% 5%M Fr r

β− −


Thé i Fi ièThéorie Financière Ri k d d (2)Risk and ... 2009 08 Risk and return... · Thé i Fi...

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Transcript of Thé i Fi ièThéorie Financière Ri k d d (2)Risk and ... 2009 08 Risk and return... · Thé i Fi...