principes de choix de portefeuille -...
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® 2009 Pearson Education France
transparents traduits par Vincent Dropsy
Principes de choix
de portefeuille7e édition
Christophe Boucher
1
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy
Chapitre 47e édition
La théorie Moyenne-Variance du choix de portefeuille
2
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Part 4. Mean-Variance Portfolio Theory
3
4.1 Measuring Risk and Return
4.2 Asset Allocation with 2 Risky Assets
4.3 Introducing a Risk Free Asset and the Tobin’s Separation Theorem
4.4 Asset Allocation with N risky Assets
4.5 Portfolio Diversification
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Preamble:
Mean/Variance Analysis Assumptions
4
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Mean-variance is simpler
5
• Early researchers in finance, such as Markowitz and Sharpe, used just the mean and the variance of the return rate of an asset to describe it.
• Characterizing the prospects of a gamble with its mean and variance is often easier than using an NM utility function, so it is popular.
• But is it compatible with VNM theory?
• The answer is yes … approximately … under some conditions.
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Mean-variance assumptions
6
• Investors maximise expected utility of end-of-perio d wealth
• Can be shown that above implies maximise a function of expected portfolio returns and portfolio variance p roviding
- Either utility is quadratic, or
- Portfolio returns are normally distributed (and utility is concave),
- Consider only small risks (then approximately true)
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Mean-variance: quadratic utility
7
Suppose utility is quadratic, U(y) = ay–by2,
with some conditions on a and b
Expected utility is then
2
2
[ ( )] ( ) ( )
( ) ( ) V ar( ) .
E U y aE y bE y
aE y b E y y
= −
= − +
Thus, expected utility is a function of:
the mean: E(y),
and the variance: Var(y)
BUT not intuitive Utility function : increasing ARA
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Mean-variance: joint normals
8
• Suppose all lotteries in the domain have normally distributed prized. (They need not be independent of each other).
• Any combination of such lotteries will also be normally distributed.
• The normal distribution is completely described by its first two moments.
• Therefore, the distribution of any combination of lotteries is also completely described by just the mean and the variance.
• As a result, expected utility can be expressed as a function of just these two numbers as well.
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Mean-variance: small risks
9
• The most relevant justification for mean-variance is probably the case of small risks.
• If we consider only small risks, we may use a second order Taylor approximation of the NM utility function.
• A second order Taylor approximation of a concave function is a quadratic function with a negative coefficient on the quadratic term.
• In other words, any risk-averse NM utility function can locally be approximated with a quadratic function.
• But the expectation of a quadratic utility function can be evaluated with the mean and variance. Thus, to evaluate small risks, mean and variance are enough.
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Mean-variance: small risks
10
• Let f : R R be a smooth function. The Taylor approximation is
• So f(x) can approximately be evaluated by looking at the value of fat another point x0, and making a correction involving the first nderivatives.
• We will use this idea to evaluate E[U(y)].
1 2
0 00 0 0
3
00
( ) ( )( ) ( ) '( ) ''( )
1! 2!
( )'''( )
3!
x x x xf x f x f x f x
x xf x
− −≈ + + +
−+⋯
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Mean-variance: small risks
11
• Consider first an additive risk, i.e. y = w+x where x is a zero mean random variable.
• For small variance of x, E[U(y)] is close to U(w).
• Consider the second order Taylor approximation,
• Let c be the certainty equivalent, U(c)=E[U(w+x)].
• For small variance of x, c is close to w, but let us look at the first order Taylor approximation.
[ ]2( )
( ) ( ) '( ) ( ) ''( )2
Var( )( ) ''( ) .
2
E xE U w x U w U w E x U w
xU w U w
+ ≈ + +
= +
( ) ( ) '( )( )U c U w U w c w≈ + −
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Mean-variance: small risks
12
• Since E[U(w+x)] = U(c), this simplifies to
• w – c is the risk premium.
• The risk premium is approximately a linear function of the variance of the additive risk, with the slope of the effect equal to half the coefficient of absolute risk.
V ar( )( )
2
xw c A w− ≈
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Example: Simple mean-variance utility function
13
• Consider the simple Mean-Variance Utility function:
21( ) ( )2P PU E R A Rσ= − ⋅
2 2 2)(1 1( ) ( ) ( )2 2Mx x xP P MRf x E R RfMaxU Max E R A R Max A x Rσ σ
=
+ −= − ⋅ − ⋅
( ) ( ) (1 )P ME R xE R x Rf= + −
2)( 21 ( ) 02M ME R Rf A x Rσ ⇒
− − ⋅ =
2
)(( )
M
M
E R Rfx
A Rσ⇒−=
Optimal position in the risky asset is inverselyproportional to the level of risk aversion and the level ofrisk and directly proportional to the risk premium
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
4.1 Measuring Risk and Return
14
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Returns of assets 1 and 2:
Weight of asset1:
Return of the portfolio:
Risk and return of a 2-asset portfolio
15
1 2, R R
x
pR
1 2 1 2( ) ( (1 ) ) ( ) (1 ) ( )PE R E xR x R xE R x E R= + − = + −
[ ]2 2 2( ) ( ( )) ( ) ( ( ))V R E R E R E R E R= − = −
( ) ( )R V Rσ =
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Risk and return of a 2-asset portfolio
16
2 2 2 2
1 2 1 2( ) ( ) ( ( )) ( ) (1 ) ( ) 2 (1 ) ( , )
p P PV R E R E R x V R x V R x x Cov R R= − = + − + −
2 2
1 2 1 2 1 2( ) ( ) (1 ) ( ) 2 (1 ) ( ) ( ) ( , )
pV R x V R x V R x x R R Corr R Rσ σ= + − + −
[ ] [ ]1 2 1 1 2 2 1 2 1 2
( , ) ( ( )( ( ) ( ) ( )Cov R R E R E R R E R E R R E R E R= − − = × −
1 2
1 2
( , )
( ) ( )
Cov R R
R R=ρσ σ
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Minimum -variance portfolio
17
V( )0
dPd R
x=
1 2 2 1 2 1 2 1 2 1 2( ) 2 ( ) 2 ( ) 2 ( ) ( ) ( , ) 4 ( ) ( ) ( , ) 02 V R V R V R R R Corr R R R R Corr R Rx x xσ σ σ σ+ − + − =⇔
2 1 2 1 2
1 2 1 2 1 2
( ) ( ) ( ) ( , )
( ) ( ) 2 ( ) ( ) ( , )
V R R R Corr R R
V R V R R R Corr R Rx
σ σ
σ σ+
−−
=
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Example
18
Consider 2 stocks, A and B:
State Probabilities A B
1234
¼¼¼¼
0%5%
15%20%
30%20%0%
-10%
1( ) (0% 5% 15% 20%) 10% ( )4 BAE R E R= + + + = =
2 2 2 2 21( ) (0%) (5%) (15%) (20%) 0,1 0.00634AV R
= + + + − =
2 2 2 2 21( ) (30%) (20%) (0%) ( 10%) 0.1 0.0254BV R
= + + + − − =
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Example (cnt’d)
19
( ) 0,079AR =σ
( ) 0,158BR =σ
21( , ) (0.00)(0.3) (0.05)(0.2) (0.15)(0.0) (0.2)( 0.1) 0.1 0.01254BACOV R R
= + + + − − = −
0.0125( , ) 10.079x0.158BACorr R R −= = −
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Example (cnt’d)
20
Risk and Return of the equally-weighted portfolio: 0,5x=
State Probabilities 1/N Portfolio
1234
¼¼¼¼
0,5 x 0% + 0,5 x 30% = 15%12,5%7,5%5%
( ) 0.1PE R =
2 2 2 2 21( ) (15%) (12.5%) (7.5%) (5%) 0.1 0.00164PV R
= + + + − =
or
2 2( ) (0.5 )(0.0063) (0.5 )(0.025) 2x0.5x0;5x(-1)(0.079)(0.158) 0.0016PV R = + + =
( ) 0.067PR =σ ( ) ( )B AR R< <σ σ Diversification effect
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Example (cnt’d)
21
The minimum Variance portfolio:
2 1 2 1 2
1 2 1 2 1 2
( ) ( ) ( ) ( , )( ) ( ) 2 ( ) ( ) ( , )
V R R R Corr R RV R V R R R Corr R R
x+
−−
= σ σ
σ σ
0.025 0.079x0.158x( 1) 230.0063 0.025 2x0.079x0.158x( 1)
x − −= =+ − −
2 2( ) (0.67 )(0.0063) (0.33 )(0.025) 2x(0.67)x(0.33)x(-1)(0.079)(0.158) 0PV R = + + =
since , 1BAR R =−ρ
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
4.2 Asset Allocation with 2 Risky Assets
22
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Portfolio characteristics and correlations
23
• Correlation (covariance) between 2 assets ⇒ Diversification gain
• General case:
4
6
8
10
12
14
16
18
2,00 3,00 4,00 5,00 6,00 7,00 8,00 9,00
A
B
(x>100%)
(x < 0)
4
6
8
10
12
14
16
18
2,00 3,00 4,00 5,00 6,00 7,00 8,00 9,004
6
8
10
12
14
16
18
2,00 3,00 4,00 5,00 6,00 7,00 8,00 9,00
A
B
(x>100%)
(x < 0)
Exp
ecte
d re
turn
Standard error
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Portfolio characteristics and correlations (cnt’d)
24
• Special cases:
1 2( , ) 1Corr R R =
1 2( , ) 1Corr R R = −
1 2( , ) 0Corr R R =
[ ]2
1 2( ) ( ) (1 ) ( )
pV R x R x R= + −σ σ⇒
[ ]2
1 2( ) ( ) (1 ) ( )
pV R x R x Rσ σ= − −⇒
2 2
1 2( ) ( ) (1 ) ( )
pV R x V R x V R= + −⇒
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
No diversification gain
25
1 2( , ) 1Corr R R =
4
6
8
10
12
14
16
18
3,0 3,5 4,0 4,5 5,0 5,5 6,0 6,5 7,0
Ren
dem
ent e
spér
é du
por
tefe
uille
(%
)
A
B
x = 1
x = 0
Exp
ecte
d re
turn
Standard error
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Full diversification gain
26
1 2( , ) 1Corr R R = −E
xpec
ted
retu
rn
Standard error
8
9
10
11
12
13
14
15
16
0,00 1,00 2,00 3,00 4,00 5,00 6,00 7,00Ecart type du portefeuille (%)
Ren
dem
ent e
spér
é du
por
tefe
uille
(%
)
A
B
C
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Partial diversification gain
27
1 2( , ) 0Corr R R =E
xpec
ted
retu
rn
Standard error
8
9
10
11
12
13
14
15
16
3,00 3,50 4,00 4,50 5,00 5,50 6,00 6,50 7,00
Ren
dem
ent e
spér
é du
por
tefe
uille
(%
)
A
B
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Efficient portfolios of risky assets
28
Exp
ecte
d re
turn
Standard error
8
9
10
11
12
13
14
15
16
3,00 3,50 4,00 4,50 5,00 5,50 6,00 6,50 7,00
Ren
dem
ent e
spér
é du
por
tefe
uille
(%
)
A
B
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
4.3 Introducing a Risk Free Asset and the Tobin’s Separation Theorem
29
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Introducing borrowing and lending:Risk free asset
30
• You are now allowed to borrow and lend at the risk free rate Rf while still investing in any SINGLE ‘risky bundle’ on the efficient frontier.
• For each SINGLE risky bundle, this gives a new set of risk return combination known as the ‘transformation line’.
• The risk-return combination is a straight line (for each single risky bundle) - transformation line.
• You can be anywhere you like on this line.
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
The risk free asset
31
• Consider a risk-free asset (e.g. T-bill rate)
fR
0fRσ =
1( , ) 0fCorr R R =
• Consider a portfolio with a risky asset 1 and a risk-free asset:
1
1
( ) ( ) (1 )
( ) ² ( )P f
P
E R xE R x R
V R x V R
= + −
=
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
The Capital Market Line
32
• From the precedent mean and variance expressions, we obtain:
1( ) ( )P f fE R R x E R R = + −
1
( )
( )PR
xR
σσ
=
• The Capital Market Line (CML):
1
1
( )( ) ( ) ( )
( )f
P f P
E R RE R E R R
R
−= + σ
σ
Slopeintercept
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
The Capital Market Line with one risky asset
33
0
2
4
6
8
10
12
14
16
18
0 1 2 3 4 5 6 7
Ren
dem
ent e
spér
é du
por
tefe
uille
(%
)
R1
Rf
x > 1
all wealth in risky asset
Exp
ecte
d re
turn
Standard error
all wealth in risk-free asset
Lending
Borrowing
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
The Capital Market Line with N risky assets- Best transformation line
34
Exp
ecte
d re
turn
Standard error
Transformation line 1
Transformation line 2
Transformation line 3 – best possible one
Rf A
M
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
The Capital Market Line
35
Exp
ecte
d re
turn
Standard error
CML
Rf
M
Risk PremiumE(RM) – Rf
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
The CML and the separation theorem
36
• The CML dominates all other possible portfolios
• An agent invests along the CML (where ? ⇒ risk preferences)
• James Tobin’s separation theorem:
- Invest in a risky portfolio (optimal combination of risky securities)
- Borrow-lend at the risk-free rate
• Depending on your attitude toward risk: how much lend or borrow
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Optimal portfolio choice
37
Exp
ecte
d re
turn
CML
Rf
M
Standard error
A
B
IA’IA’’ IB
IA
IB’’ IB’
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
From the CML to the SML
38
( )( ) ( ) ( )
( )M
P f PM
E R RfE R E R R
R
−= + σσ
CML:
[ ]( )( ) ( ) ( )
( )P
P f MM
RE R E R E R Rf
R= + −σ
σ
Note that:,
( , )
( ) ( )M p
M pR R
M p
Cov R R
R R=ρσ σ
and since: , 1M pR R =ρ
[ ]2
( ) ( )( ) ( ) ( )
( )P M
P f MM
R RE R E R E R Rf
R= + −σ σ
σ
[ ]2
( , )( ) ( ) ( )
( )M p
P f MM
Cov R RE R E R E R Rf
R= + −
σ
Security Market Line: [ ]( ) ( )P f ME R R E R Rf= + −β
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
The Security Market Line (SML)
39
Exp
ecte
d re
turn
SML
Rf
M
β10.5 2
Risk PremiumE(RM) – Rf)
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
CML vs SML
40
βE
xpec
ted
retu
rn SML
Rf
M
10.5 2
Security 1
Security 2
σ
Exp
ecte
d re
turn CML
Rf
M
Security 2
Security 1
β=0
β=0,5
β=1
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
CML vs SML
41
• The CML: combination the market portfolio and the riskless asset
• Portfolios on the CML are efficient portfolios
• Any portfolio on the CML has a correlation of 1 with the market portfolio
• CML is applicable only to an investor’s efficient portfolio
• SML is applicable to any security, asset or portfolio (CAPM World)
• In the CML, risk is measured by σ
• In the SML, risk is measured by β
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
4.4 Asset Allocation with N risky Assets
42
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Efficient portfolios
43
• Efficient portfolios are those that maximize the ex pected return for a given level of expected risk
• or minimize the risk for a given level of expected return
• Two kinds of efficient portfolios:
- only risky assets
- both risky assets and a riskless-asset (separation theorem)
• Suppose stable expected-returns and VCV matrix
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Some notations
44
1
2
( )
( )
( )N
E R
E RR
E R
=⋮
1
2
N
xx
X
x
=⋮
11 1
1
( )N
ij
NNN
Ω =⋯
⋮ ⋮
⋯
σ σ
σ
σ σ
( ) T TPE R R X X R= =
( ) TPV R X X= Ω
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Efficient frontier with a riskless asset
45
( )Max
( )M
M
E R RfR
−Θ =σ
• Recall the transformation lines and CML: Maximise the slope
11
N
ii
x=
=∑u.c
1f f
N
ii
R x R=
=∑set
1/21
1 1
( )Max ( )
N
i i fT Ti
N N
i j iji j
x R RX R Rf X X
x x
−=
= =
−Θ = = − Ω
∑
∑∑ σ
1/2 3/21( ) ( ) ( )2 02
T T Tf f
d R R X X X R R X X XdX
− −Θ = − Ω + − − Ω Ω =
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Efficient frontier with a riskless asset (cnt’d)
46
multiplying by1/2
TX X
Ω
11( ) ( ) ( )2 02
T Tf fR R X R R X X X
−− + − − Ω Ω =
1( )T T
fX R R X X
−= − ΩλDefine (SR)
( ) 0fR R X− − Ω =λ
X Z=λ
( )fR R Z− = Ω
Define Z such that:
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Efficient frontier with a riskless asset (cnt’d)
47
1 1 11 2 12 1
2 1 21 2 22 2
1 1 2 2
...
...
...
N Nf
N Nf
N N NNN Nf
R R Z Z Z
R R Z Z Z
R R Z Z Z
− = + + +− = + + +
− = + + +⋮
σ σ σ
σ σ σ
σ σ σ
1
ii N
ii
ZX
Z=
=∑
Since: 1 1 1
N N N
i i ii i i
X Z Z= = =
= ⇒ =∑ ∑ ∑λ λ
Then we can calculate: ( )ME R ( )MRσand
1
1
( )( ) ( )
( )P Pf
E R RfE R R R
R−= + σ
σand the CML
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Efficient frontier with no riskless asset
48
1Min2
TX XΩ( )
1 1
TP
T
X R E R
X
==u.c
1£ ( ) 1 12
T T TPX X E R X R X
= Ω + − + −λ γ
£ 1 0
£ ( ) 0
£ 1 1 0
TP
T
d X RdXd E R X Rdd Xd
=Ω − − =
= − =
= − =
λ γ
λ
γ
1 1X R
−= Ω +λ γ
Eq. 1
Eq. 2
Eq. 3
Eq. 4 (from Eq. 1)
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Efficient frontier with no riskless asset (Cnt’d)
49
1 1 ( )T TPR X R R E R
−= Ω + =λ γ
11 1 1 1T TX R
−= Ω + =λ γ
1 1
1
1
2
1 1
1 1
T T
T
T
A R RB R RCD BC A
− −
−
−
= Ω = Ω= Ω= Ω= −
We define
( )PCE R AD
−=λ
( )PB AE RD
−=γ
Eq. 5 (from Eq. 4 and 2)
Eq. 6 (from Eq. 4 and 3)
It follows
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Efficient frontier with no riskless asset (Cnt’d)
Recall: 1 1X R
−= Ω +λ γ
( )PX g hE R= + 1 1
1 1
1
1
B A RgD
C R AhD
− −
− −
Ω − Ω=
Ω − Ω=
with
Portfolio g represent the optimal weights of a portfolio with ( ) 0PE R =
X g h= + represents the optimal weights of a portfolio with ( ) 1PE R =
2( ) ( ) ( )P P PV R aE R bE R c= + +
It follows: Analytic solution
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Characterization of frontier portfolios
51
• The entire set of frontier portfolios can be generated by 2 efficient portfolios, e.g. g and g+h
• A combination of efficient portfolios would be efficient too
⇒To find a efficient portfolio with
⇒ Build a portfolio with of g and of g+h
⇒ The structure of the portfolio is:
1( )E R
11 ( )E R
−1( )E R
1 1 11 ( ) ( )( ) ( )E R g E R g h g hE R
− + + = +
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
3.5 Portfolio Diversification
52
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Diversification and Portfolio risk
53
• Market risk
- Systematic or Nondiversifiable (Business cycle, geopolitics, etc.)
• Firm -specific risk
- Diversifiable or nonsystematic (Firm specific factors: management, sector, loss of a patent , etc.)
• Total risk = Systematic risk + idiosyncratic risk
Can disappear with diversification
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Power of Diversification
54
• As the number of assets (N) in the portfolio increases, the SD (total riskiness) falls
• Assumptions:
- All assets have the same variance : σi2 = σ2
- All assets have the same covariance : σij = ρ σ 2
- Invest equally in each asset (i.e. 1/N)
® 2009 Pearson Education France
transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016
Power of Diversification
55
• General formula for calculating the portfolio variance
σ2p = Σ wi
2 σi2 + ΣΣ wiwj σij
• Formula with assumptions imposed
σ2p = (1/N) σ2 + ((N-1)/N) ρσ2
• If N is large, (1/N) is small and ((N-1)/N) is close to 1.
• Hence : σ2p ≈ ρ σ 2
• Portfolio risk is ‘covariance risk’.
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016 56
Sta
ndar
d er
ror
No. of shares in portfolio
Diversifiable / idiosyncratic risk
Market / non-diversifiable risk
20 400 1 2 ...
Random selection of stocks
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transparents traduits par Vincent Dropsy Principes de choix de portefeuille –Christophe BOUCHER – 2015/2016 57
Portfolio risk as a function of number of securities
Source: BKM (2007) from Statman (1987)