FINANCE 9. Optimal Portfolio Choice Professor André Farber Solvay Business School Université Libre...

FINANCE9. Optimal Portfolio Choice

Professor André Farber

Solvay Business SchoolUniversité Libre de BruxellesFall 2007

MBA 2007 Portfolio choice |2April 18, 2023

Introduction: random portfolios

A B

RF

Risky portfolio

C DOptimal asset

allocation

Optimal portfolio

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60

Risk (standard deviation)

Expe

cted

ret

urn


Covariance and correlation

• Statistical measures of the degree to which random variables move together

• Covariance

• Like variance figure, the covariance is in squared deviation units.• Not too friendly ...

• Correlation

• covariance divided by product of standard deviations• Covariance and correlation have the same sign

– Positive : variables are positively correlated– Zero : variables are independant– Negative : variables are negatively correlated

• The correlation is always between –1 and + 1

)])([(),cov( BBAABAAB RRRRERR

BA

BABAAB

RRCovRRCorr

),(

),(


Risk and expected returns for porfolios

• In order to better understand the driving force explaining the benefits from diversification, let us consider a portfolio of two stocks (A,B)

• Characteristics:

– Expected returns :

– Standard deviations :

– Covariance :

• Portfolio: defined by fractions invested in each stock XA , XB XA+ XB= 1

• Expected return on portfolio:

• Variance of the portfolio's return:

BA RR ,

BA ,

BAABAB

BBAAP RXRXR

22222 2 BBABBAAAP XXXX


Example

• Invest $ 100 m in two stocks:

• A $ 60 m XA = 0.6

• B $ 40 m XB = 0.4

• Characteristics (% per year) A B

• • Expected return 20% 15%

• • Standard deviation 30% 20%

• Correlation 0.5

• Expected return = 0.6 × 20% + 0.4 × 15% = 18%

• Variance = (0.6)²(.30)² + (0.4)²(.20)²+2(0.6)(0.4)(0.30)(0.20)(0.5)

²p = 0.0532 Standard deviation = 23.07 %

• Less than the average of individual standard deviations:

• 0.6 x0.30 + 0.4 x 0.20 = 26%


Example:

Exp.Return Sigma Variance

Riskless rate 5 0 0

A 20 30 900

B 15 20 400

Correlation 0.5

Prop. Variance-covariance

A 0.60 900 300

B 0.40 300 400

Cov(Ri,Rp) 660 340

Exp.Return 18.00

Variance 532

St.deviation 23.07

Beta 1.24 0.64


A

B

Riskless rate

Risky portfolio

Optimal asset allocation

0.00

5.00

10.00

15.00

20.00

25.00

30.00

0.00 10.00 20.00 30.00 40.00 50.00 60.00

Risk (standard deviation)

Expe

cted

ret

urn


Combining the Riskless Asset and a single Risky Asset

• Consider the following portfolio P:

• Fraction invested

– in the riskless asset 1-x (40%)

– in the risky asset x (60%)

• Expected return on portfolio P:

• Standard deviation of portfolio :

Riskless asset

Risky asset

Expected return

6% 12%

Standard deviation

0% 20%

SFP RxRxR )1(

%60.912.060.006.040.0 PR

SP x

%1220.060.0 P


Relationship between expected return and risk

• Combining the expressions obtained for :

• the expected return

• the standard deviation

• leads to

SFP RxRxR )1(

SP x

PS

FSFP

RRRR

SSPR 30.006.020.0

06.012.006.0

P

PR

S

SR

FR


Risk aversion

• Risk aversion :

• For a given risk, investor prefers more expected return

• For a given expected return, investor prefers less risk

Expected return

Risk

Indifference curve

P


Utility function

• Mathematical representation of preferences

• a: risk aversion coefficient

• u = certainty equivalent risk-free rate

• Example: a = 2

• A 6% 0 0.06

• B 10% 10% 0.08 = 0.10 - 2×(0.10)²

• C 15% 20% 0.07 = 0.15 - 2×(0.20)²

• B is preferred

2),( PPPP aRRU

PR P Utility


Optimal choice with a single risky asset

• Risk-free asset : RF Proportion = 1-x

• Risky portfolio S: Proportion = x

• Utility:

• Optimum:

• Solution:

• Example: a = 2

SSR ,22 ²])1[( SSFPP axRxRxaRu

02)( 2 SFS axRRdx

du

22

1

S

FS RR

ax

375.0)20.0(

06.012.0

22

1

2

122

S

FS RR

ax


Choosing portfolios from many stocks

• Porfolio composition :

• (X1, X2, ... , Xi, ... , XN)

• X1 + X2 + ... + Xi + ... + XN = 1

• Expected return:

• Risk:

• Note:

• N terms for variances

• N(N-1) terms for covariances

• Covariances dominate

NNP RXRXRXR ...2211

i ij i j

ijjiijjijj

jP XXXXX 222


Some intuition

Var Cov Cov Cov CovCov Var Cov Cov CovCov Cov Var Cov CovCov Cov Cov Var CovCov Cov Cov Cov Var


Example

• Consider the risk of an equally weighted portfolio of N "identical« stocks:

• Equally weighted:

• Variance of portfolio:

• If we increase the number of securities ?:

• Variance of portfolio:

NX j

1

cov)1

1(1 22

NNP

NP cov2

cov),(,, jijj RRCovRR


Diversification

Risk Reduction of Equally Weighted Portfolios

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

# stocks in portfolio

Po

rtfo

lio

sta

nd

ard

de

via

tio

n

Market risk

Unique risk


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


Optimal portofolio with borrowing and lending

Optimal portfolio: maximize Sharpe ratio

M

Efficient markets


Notions of Market Efficiency

• An Efficient market is one in which:

– Arbitrage is disallowed: rules out free lunches

– Purchase or sale of a security at the prevailing market price is never a positive NPV transaction.

– Prices reveal information

• Three forms of Market Efficiency

• (a) Weak Form Efficiency

• Prices reflect all information in the past record of stock prices

• (b) Semi-strong Form Efficiency

• Prices reflect all publicly available information

• (c) Strong-form Efficiency

• Price reflect all information


Efficient markets: intuition

Expectation

Time

Price

Realization

Price change is unexpected


Weak Form Efficiency

• Random-walk model:

– Pt -Pt-1 = Pt-1 * (Expected return) + Random error

– Expected value (Random error) = 0

– Random error of period t unrelated to random component of any past period

• Implication:

– Expected value (Pt) = Pt-1 * (1 + Expected return)

– Technical analysis: useless

• Empirical evidence: serial correlation

– Correlation coefficient between current return and some past return

– Serial correlation = Cor (Rt, Rt-s)


Random walk - illustration

Bourse de Bruxelles 1980-1999

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

-30.00 -25.00 -20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00

Rentabilité mois t

Re

nta

bili

té m

ois

t+

1


Semi-strong Form Efficiency

• Prices reflect all publicly available information

• Empirical evidence: Event studies

• Test whether the release of information influences returns and when this influence takes place.

• Abnormal return AR : ARt = Rt - Rmt

• Cumulative abnormal return:

• CARt = ARt0 + ARt0+1 + ARt0+2 +... + ARt0+1


Strong-form Efficiency

• How do professional portfolio managers perform?

• Jensen 1969: Mutual funds do not generate abnormal returns

• Rfund - Rf = + (RM - Rf)

• Insider trading

• Insiders do seem to generate abnormal returns

• (should cover their information acquisition activities)

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