Microéconomie de la finance 7e édition
Christophe Boucher
[email protected]
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Chapitre 4 7e édition
La théorie Moyenne-Variance du choix de portefeuille
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Part 4. Mean-Variance Portfolio Theory
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
Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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Preamble: Mean/Variance Analysis Assumptions
Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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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.
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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
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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.
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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.
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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!
+
+⋯
•
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)].
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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) Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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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.
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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 )
Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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 13
4.1 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) = Microéconomie de la finance – Christophe BOUCHER – 2014/2015
2
] = E(R
2
) − ( E ( R ))
2
V ( R) 15
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=
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2
2
1
2
] − E(R )E(R ) 1
2
Cov( R1 , R2 ) σ ( R1 )σ ( R2 )
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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 )
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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
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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
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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 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
< σ ( RB ) < σ ( RA )
Diversification effect 20
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
Microéconomie de la finance – Christophe BOUCHER – 2014/2015
ρRA,RB = −1
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4.2 Asset Allocation with 2 Risky Assets
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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 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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Portfolio characteristics and correlations (cnt’d) • Special cases: Corr ( R1 , R2 ) = 1
⇒
p
Corr ( R1 , R2 ) = −1
⇒
Corr ( R1 , R2 ) = 0
⇒
Microéconomie de la finance – Christophe BOUCHER – 2014/2015
V ( R ) = [ xσ ( R ) + (1 − x ) σ ( R ) ]
2
1
2
V ( R ) = [ xσ ( R ) − (1 − x ) σ ( R ) ]
2
p
1
2
V ( R p ) = x V ( R1 ) + (1 − x ) V ( R2 ) 2
2
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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
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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
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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
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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
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4.3 Introducing a Risk Free Asset and the Tobin’s Separation Theorem
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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.
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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 )
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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 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
E ( R1 ) − R f σ ( R1 )
σ ( RP )
Slope 32
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
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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 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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The Capital Market Line
Expected return
CML
Rf
M
Risk Premium E(RM) – Rf
Standard error Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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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
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Optimal portfolio choice IB’’ IB’ IB
IA’’
CML
IA’ IA
Expected return
B M
A
Rf
Standard error
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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 )
Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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 ] 38
The Security Market Line (SML)
Expected return
SML
M
Risk Premium E(RM) – Rf)
Rf
0.5 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
1
2
β 39
CML
Security 1
M
β=1 Security 2
Rf
β=0,5 β=0
σ
Microéconomie de la finance – Christophe BOUCHER – 2014/2015
SML
Expected return
Expected return
CML vs SML
Security 1
M Security 2
Rf
0.5
1
2
β 40
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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4.4 Asset Allocation with N risky Assets
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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 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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Some notations
E(R ) 1 E ( R2 ) R= ⋮ E ( RN )
X
x 1 x = 2 ⋮ xN
σ ⋯σ 1N 11 Ω = ⋮ (σij ) ⋮ σ N1⋯σ NN
E ( RP ) = RT X = X T R
V ( RP ) = X T ΩX
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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
∑∑ xi x jσij
= X T ( R − Rf ) X T ΩX
−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 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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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
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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 ⋮ 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 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
E (RP ) = R f +
σ ( RM )
E (R1) − Rf σ ( RP ) σ (R1) 47
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)
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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
A = 1T Ω−1R = RT Ω−11 B = RT Ω−1R C = 1T Ω−11 D = BC − A2
It follows
Microéconomie de la finance – Christophe BOUCHER – 2014/2015
λ=
CE ( RP ) − A D
γ=
B − AE ( RP ) D
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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 Microéconomie de la finance – Christophe BOUCHER – 2014/2015
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)
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3.5 Portfolio Diversification
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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
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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)
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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’.
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Standard error
Random selection of stocks
Diversifiable / idiosyncratic risk
Market / non-diversifiable risk 0
1
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2 ...
20
40
No. of shares in portfolio
56
Portfolio risk as a function of number of securities
Source: BKM (2007) from Statman (1987)
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Thank you for your attention…
See you next week
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