Financial Markets & Portfolio Choice 2011/2012 Session 3

Benjamin HAMIDI Christophe BOUCHER [email protected]

Part 3. Mean-Variance Portfolio Theory

3.1 Measuring Risk and Return 3.2 Asset Allocation with 2 Risky Assets 3.3 Introducing a Risk Free Asset and the Tobin’s Separation Theorem 3.4 Asset Allocation with N risky Assets 3.5 Portfolio Diversification

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Preamble: Mean/Variance Analysis Assumptions

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Mean-variance is simpler • 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.

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Mean-variance assumptions

• Investors maximise expected utility of end-of-period wealth • Can be shown that above implies maximise a function of expected portfolio returns and portfolio variance providing - Either utility is quadratic, or - Portfolio returns are normally distributed (and utility is concave), - Consider only small risks (then approximately true)

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Mean-variance: quadratic utility Suppose utility is quadratic, U(y) = ay–by2, with some conditions on a and b

E[U ( y )] = aE ( y ) − bE ( y 2 ) Expected utility is then

= aE ( y ) − b  E ( y ) 2 + V ar( y )  .

Thus, expected utility is a function of:

the mean: E(y), and the variance: Var(y) BUT not intuitive Utility function : increasing ARA

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Mean-variance: joint normals • 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.

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Mean-variance: small risks • 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.

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Mean-variance: small risks • Let f : R
R be a smooth function. The Taylor approximation is

f ( x) ≈ f ( x0 ) + f '( x0 ) f '''( x0 )

( x − x0 )1

( x − x0 )3 3!

1!

+ f ''( x0 )

( x − x0 ) 2 2!

+

+L

•

So f(x) can approximately be evaluated by looking at the value of f at another point x0, and making a correction involving the first n derivatives.

•

We will use this idea to evaluate E[U(y)].

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Mean-variance: small risks • 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, E( x2 ) E [U ( w + x) ] ≈ U ( w) + U '( w) E ( x) + U ''( w) 2 V ar( x) = U ( w) + U ''( w) . 2 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.

U (c ) ≈ U ( w) + U '( w)(c − w) Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Mean-variance: small risks • Since E[U(w+x)] = U(c), this simplifies to

V ar( x) w − c ≈ A( w) 2 • 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.

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Example: Simple mean-variance utility function • Consider the simple Mean-Variance Utility function: U = E (RP ) − 1 A⋅σ 2 (RP ) 2 E (RP ) = xE (RM ) + (1− x)Rf     1 1 MaxU = Max  E ( R ) − A ⋅σ 2 ( R ) = Max  Rf + x  E ( R ) − Rf  − A ⋅ x2σ 2 ( R ) M x x  x    2 P P  M  2    

⇒  E ( R ) − Rf  M  

⇒

x=

− 1 2 A⋅ xσ 2 (RM ) = 0 2

E (RM ) − Rf

Aσ 2 (RM )

Optimal position in the risky asset is inversely proportional to the level of risk aversion and the level of risk and directly proportional to the risk premium

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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3.1 Measuring Risk and Return

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Risk and return of a 2-asset portfolio Returns of assets 1 and 2: Weight of asset1:

R1 , R2

x

Return of the portfolio:

Rp

E ( RP ) = E ( xR1 + (1 − x) R2 ) = xE ( R1 ) + (1 − x) E ( R2 )

[

V ( R ) = E ( R − E ( R )) σ ( R) = Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

2

] = E(R

2

) − ( E ( R ))

2

V ( R) 14

Risk and return of a 2-asset portfolio V ( R p ) = E ( RP ) − ( E ( RP )) = x V ( R1 ) + (1 − x ) V ( R2 ) + 2 x (1 − x )Cov ( R1 , R2 ) 2

2

2

2

V ( R p ) = x V ( R1 ) + (1 − x ) V ( R2 ) + 2 x (1 − x ) σ ( R1 ) σ ( R2 )Corr ( R1 , R2 ) 2

2

Cov ( R , R ) = E [ ( R − E ( R )( R − E ( R ) ] = E [ R × R 1

2

1

1

ρ=

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

2

2

1

2

] − E(R )E(R ) 1

2

Cov( R1 , R2 ) σ ( R1 )σ ( R2 )

15

Minimum-variance portfolio

dV( RP ) =0 dx

⇔ 2 xV ( R1 ) + 2 xV ( R2 ) − 2V ( R2 ) + 2σ ( R1 )σ ( R2 )Corr ( R1 , R2 ) − 4 xσ ( R1 )σ ( R2 )Corr ( R1 , R2 ) = 0

x=

V ( R2 ) − σ ( R1 )σ ( R2 )Corr ( R1 , R2 ) V ( R1 ) + V ( R2 ) − 2σ ( R1 )σ ( R2 )Corr ( R1 , R2 )

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Example Consider 2 stocks, A and B: State

Probabilities

A

B

1 2 3 4

¼ ¼ ¼ ¼

0% 5% 15% 20%

30% 20% 0% -10%

E ( RA ) = 1 (0% + 5% +15% + 20%) = 10% = E ( RB ) 4 V ( RA ) = 1 (0%)2 + (5%)2 + (15%)2 + (20%)2  − 0,12 = 0.0063  4

V ( RB ) = 1 (30%)2 + (20%)2 + (0%)2 + (−10%)2  − 0.12 = 0.025  4

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Example (cnt’d) σ ( RA ) = 0,079 σ ( RB ) = 0,158 COV (RA , RB ) = 1 (0.00)(0.3) + (0.05)(0.2) + (0.15)(0.0) + (0.2)(−0.1) − 0.12 = −0.0125 4

Corr (RA, RB ) =

−0.0125 = −1 0.079x0.158

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Example (cnt’d) Risk and Return of the equally-weighted portfolio: x = 0,5 State

Probabilities

1/N Portfolio

1 2 3 4

¼ ¼ ¼ ¼

0,5 x 0% + 0,5 x 30% = 15% 12,5% 7,5% 5%

E ( RP ) = 0.1 V ( RP ) = 1 (15%)2 + (12.5%)2 + (7.5%)2 + (5%)2  − 0.12 = 0.0016  4

or V ( RP ) = (0.52 )(0.0063) + (0.52 )(0.025) + 2x0.5x0;5x(-1)(0.079)(0.158) = 0.0016

σ ( RP ) = 0.067

< σ ( RB ) < σ ( RA )

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

Diversification effect 19

Example (cnt’d) The minimum Variance portfolio: x=

V ( R2 ) − σ ( R1 )σ ( R2 )Corr ( R1 , R2 ) V ( R1 ) + V ( R2 ) − 2σ ( R1 )σ ( R2 )Corr ( R1 , R2 )

x=

0.025 − 0.079x0.158x(−1) =2 0.0063 + 0.025 − 2x0.079x0.158x(−1) 3

V ( RP ) = (0.672 )(0.0063) + (0.332 )(0.025) + 2x(0.67)x(0.33)x(-1)(0.079)(0.158) = 0

since

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

ρRA,RB = −1

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3.2 Asset Allocation with 2 Risky Assets

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Portfolio characteristics and correlations • Correlation (covariance) between 2 assets ⇒ Diversification gain

• General case: 18

(x < 0)

16

B Expected return

14

12

A

10

8

(x>100%)

6

4 2,00

3,00

4,00

5,00

6,00

7,00

8,00

9,00

Standard error Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Portfolio characteristics and correlations (cnt’d) • Special cases: Corr ( R1 , R2 ) = 1

⇒

Corr ( R1 , R2 ) = −1

⇒

Corr ( R1 , R2 ) = 0

⇒

V ( R ) = [ xσ ( R ) + (1 − x ) σ ( R ) ]

2

p

1

V ( R ) = [ xσ ( R ) − (1 − x ) σ ( R ) ]

2

p

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

2

1

2

V ( R p ) = x V ( R1 ) + (1 − x ) V ( R2 ) 2

2

23

No diversification gain

Corr ( R1 , R2 ) = 1

Rendement espéré du portefeuille (%)

18 16

Expected return

14

B x=0

12 10

A x=1

8 6 4 3,0

3,5

4,0

4,5

5,0

5,5

6,0

6,5

7,0

Standard error

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Full diversification gain Corr ( R1 , R2 ) = −1

Rendement espéré du portefeuille (%)

Expected return

16 15 14

B 13 12

C

11 10

A 9 8 0,00

1,00

2,00

3,00 4,00 5,00 Ecart type du portefeuille (%)

6,00

7,00

Standard error

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Partial diversification gain Corr ( R1 , R2 ) = 0

Rendement espéré du portefeuille (%)

Expected return

16 15 14

B 13 12 11 10

A 9 8 3,00

3,50

4,00

4,50

5,00

5,50

6,00

6,50

7,00

Standard error

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Efficient portfolios of risky assets

Rendement espéré du portefeuille (%)

Expected return

16 15 14

B 13 12 11 10

A 9 8 3,00

3,50

4,00

4,50

5,00

5,50

6,00

6,50

7,00

Standard error

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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3.3 Introducing a Risk Free Asset and the Tobin’s Separation Theorem

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Introducing borrowing and lending: Risk free asset • 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.

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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The risk free asset • Consider a risk-free asset (e.g. T-bill rate)

Rf σRf = 0 Corr ( R1 , R f ) = 0

• Consider a portfolio with a risky asset 1 and a risk-free asset:

E ( RP ) = xE ( R1 ) + (1 − x) R f V ( RP ) = x ²V ( R1 )

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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The Capital Market Line • From the precedent mean and variance expressions, we obtain:

E ( RP ) = R f + x  E ( R1 ) − R f  x=

σ ( RP ) σ ( R1 )

• The Capital Market Line (CML): E ( RP ) = E ( R f ) +

intercept

E ( R1 ) − R f σ ( R1 )

σ ( RP )

Slope

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The Capital Market Line with one risky asset

18

Borrowing

Rendement espéré du portefeuille (%)

16 14

Expected return

12

R1 all wealth in risk-free asset

10

x>1

8

all wealth in risky asset

6 4

Lending

Rf

2 0 0

1

2

3

4

5

6

7

Standard error

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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The Capital Market Line with N risky assets - Best transformation line

Expected return

Transformation line 3 – best possible one

Rf

M Transformation line 2 Transformation line 1

A

Standard error Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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The Capital Market Line

Expected return

CML

M

Rf

Risk Premium E(RM) – Rf)

Standard error Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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The CML and the separation theorem

• 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

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Optimal portfolio choice IB’’ IB’ IB

IA’’

CML

IA’ IA

Expected return

B M

A

Rf

Standard error

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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From the CML to the SML CML:

E ( RM ) − Rf E ( RP ) = E ( R f ) + σ ( RP ) σ ( RM ) σ ( RP ) E ( RP ) = E ( R f ) + [ E ( RM ) − Rf ] σ ( RM ) E ( RP ) = E ( R f ) +

Note that: ρR , R = M p

Cov( RM , R p )

and since: ρRM , R p = 1

σ ( RM )σ ( R p )

E ( RP ) = E ( R f ) +

Security Market Line:

σ ( RP )σ ( RM ) [ E ( RM ) − Rf ] 2 σ ( RM )

Cov ( RM , R p ) σ ( RM )

2

[ E ( RM ) − Rf ]

E ( RP ) = R f + β [ E ( RM ) − Rf ]

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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The Security Market Line (SML)

Expected return

SML

M

Risk Premium E(RM) – Rf)

Rf

0.5

1

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

2

β 38

CML

Security 1

M

β=1 Security 2

Rf

β=0,5 β=0

σ

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

SML

Expected return

Expected return

CML vs SML

Security 1

M Security 2

Rf

0.5

1

2

β 39

CML vs SML

• 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

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3.4 Asset Allocation with N risky Assets

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Efficient portfolios • Efficient portfolios are those that maximize the expected 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 Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Some notations

 E(R )   1     E ( R2 )  R=   M     E ( RN )  

X

x   1  x  =  2  M    xN   

 σ Lσ  1N   11 Ω = M (σij ) M    σ N1Lσ NN 

E ( RP ) = RT X = X T R

V ( RP ) = X T ΩX

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Efficient frontier with a riskless asset • Recall the transformation lines and CML: Maximise the slope

E ( RM ) − Rf Max Θ = σ ( RM )

N

xi = 1 ∑ i =1

u.c

N

set

Rf = ∑ xi Rf i =1

N

Max Θ =

xi ( Ri − R f ) ∑ i=1 N N

= X T ( R − Rf )  X T ΩX 

∑∑ xi x jσij



−1/2



i=1 j =1

d Θ = ( R − R )  X T ΩX  −1/2 +  X T ( R − R ) (− 1 )2ΩX  X T ΩX  −3/2 = 0     f  f  2    dX  Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Efficient frontier with a riskless asset (cnt’d)

1/2

multiplying by

 T   X ΩX   

 (R − R f ) +  X T (R − R f 

Define (SR)

 ) (− 1 )2ΩX  X T ΩX  2   

λ =  X T ( R − R f )  X T ΩX  



−1

=0

−1



( R − R f ) − λΩX = 0

Define Z such that: λX = Z ( R − R f ) = ΩZ

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Efficient frontier with a riskless asset (cnt’d)  R − R = Z σ + Z σ + ... + Z σ  1 N 1N 1 11 2 12 f   R2 − R f = Z1σ21 + Z 2σ22 + ... + Z N σ2 N  M  R − R = Z σ + Z2σ N 2 + ... + Z N σ NN 1 N1 f  N

Xi =

Zi N

Zi ∑ i =1

Since:

N

N

N

i =1

i =1

i =1

∑ X i λ = ∑ Z i ⇒ λ = ∑ Zi

Then we can calculate: E (RM ) and and the CML

E (RP ) = R f +

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

σ ( RM )

E (R1) − Rf σ ( RP ) σ (R1) 46

Efficient frontier with no riskless asset

Min 1 X T ΩX 2

u.c

 T  X R = E ( RP )  T  X 1 = 1

£ = 1 X T ΩX + λ  E ( RP ) − X T R  + γ 1− X T 1 2       d £ = ΩX − λR − γ1 = 0  dX   d £ = E(R ) − X T R = 0  dλ P  d£ = 1− X T 1 = 0   dγ

X = Ω−1  λR + γ1 

Eq. 1 Eq. 2 Eq. 3

Eq. 4 (from Eq. 1)



Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Efficient frontier with no riskless asset (Cnt’d) RT X = RT Ω−1  λR + γ1 = E ( RP )

Eq. 5 (from Eq. 4 and 2)

1T X = 1T Ω−1  λR + γ1 = 1

Eq. 6 (from Eq. 4 and 3)





We define

It follows





A = 1T Ω−1R = RT Ω−11 B = RT Ω−1R C = 1T Ω−11 D = BC − A2 λ=

CE ( RP ) − A D

γ=

B − AE ( RP ) D

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Efficient frontier with no riskless asset (Cnt’d)

Recall: X = Ω−1  λR + γ1 

It follows:



X = g + hE (RP )

with

−1 −1 g = BΩ 1 − AΩ R D −1 −1 h = CΩ R − AΩ 1 D

Analytic solution

Portfolio g represent the optimal weights of a portfolio with E( RP ) = 0 X = g +h

represents the optimal weights of a portfolio with E( RP ) =1

V ( RP ) = aE ( RP ) 2 + bE ( RP ) + c Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

Characterization of frontier portfolios

• 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 E(R1) ⇒ Build a portfolio with 1− E(R1) of g and E(R1) of g+h ⇒ The structure of the portfolio is: 1− E(R1) g + E(R1)( g + h) = g + hE(R1)

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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3.5 Portfolio Diversification

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Diversification and Portfolio risk •

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

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Power of Diversification • 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)

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Power of Diversification

• General formula for calculating the portfolio variance σ2p = Σ wi2 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’.

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Standard error

Random selection of stocks

Diversifiable / idiosyncratic risk

Market / non-diversifiable risk 0

1

2 ...

20

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

40

No. of shares in portfolio

55

Portfolio risk as a function of number of securities

Source: BKM (2007) from Statman (1987)

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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Thank you for your attention…

See you next week

Financial Markets & Portfolio Choice – Christophe BOUCHER & Benjamin HAMIDI – 2011/2012

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