finance 9. optimal portfolio choice professor andré farber solvay business school université libre...
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![Page 1: FINANCE 9. Optimal Portfolio Choice Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2007](https://reader031.vdocument.in/reader031/viewer/2022032201/56649d3f5503460f94a18981/html5/thumbnails/1.jpg)
FINANCE9. Optimal Portfolio Choice
Professor André Farber
Solvay Business SchoolUniversité Libre de BruxellesFall 2007
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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
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MBA 2007 Portfolio choice |3April 18, 2023
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
),(
),(
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MBA 2007 Portfolio choice |4April 18, 2023
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
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MBA 2007 Portfolio choice |5April 18, 2023
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%
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MBA 2007 Portfolio choice |6April 18, 2023
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
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MBA 2007 Portfolio choice |7April 18, 2023
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
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MBA 2007 Portfolio choice |8April 18, 2023
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
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MBA 2007 Portfolio choice |9April 18, 2023
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
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MBA 2007 Portfolio choice |10April 18, 2023
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
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MBA 2007 Portfolio choice |11April 18, 2023
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
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MBA 2007 Portfolio choice |12April 18, 2023
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
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MBA 2007 Portfolio choice |13April 18, 2023
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
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MBA 2007 Portfolio choice |14April 18, 2023
Some intuition
Var Cov Cov Cov CovCov Var Cov Cov CovCov Cov Var Cov CovCov Cov Cov Var CovCov Cov Cov Cov Var
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MBA 2007 Portfolio choice |15April 18, 2023
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
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MBA 2007 Portfolio choice |16April 18, 2023
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
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MBA 2007 Portfolio choice |17April 18, 2023
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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MBA 2007 Portfolio choice |18April 18, 2023
Optimal portofolio with borrowing and lending
Optimal portfolio: maximize Sharpe ratio
M
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Efficient markets
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MBA 2007 Portfolio choice |20April 18, 2023
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
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MBA 2007 Portfolio choice |21April 18, 2023
Efficient markets: intuition
Expectation
Time
Price
Realization
Price change is unexpected
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MBA 2007 Portfolio choice |22April 18, 2023
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)
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MBA 2007 Portfolio choice |23April 18, 2023
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
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MBA 2007 Portfolio choice |24April 18, 2023
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
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MBA 2007 Portfolio choice |25April 18, 2023
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)