On The Portfolio Selection Problem - EAFIT...On The Portfolio Selection Problem Outline 1....

On The Portfolio Selection Problem

On The Portfolio Selection ProblemHenry Laniado

[email protected]

Seminario de DoctoradoDoctorado en Ingenierıa Matem atica

Medellın, Abril 2015


Outline

1. Introduction a Fast Review

2. Solution Under Optimization, Mean-Variance

3. Solution Under Stochastic Order, Utility Function

4. Solution Under Simulation, Extremality

5. Conclusions and open problems

6. References


Introduction a Fast Review

Indice


Solution Under Optimization

Solution and Comparison of Portfolios under Stochastic Orders

Solution under Simulation

Conclusions

Futurer Research Lines

References



What is the Problem...?




◮ Consider an investor who has the possibility of investing inn different risky assets

X = (X1, X2, . . . , Xn)





X = (X1, X2, . . . , Xn)

◮ The investor has to allocate his budget C to the differentrisks. Without loss of generality C = 1





X = (X1, X2, . . . , Xn)

◮ The investor has to allocate his budget C to the differentrisks. Without loss of generality C = 1

◮ The investor has many alternatives to invest given by

w = (ω1, ω2, . . . , ωn),

n∑

i=1

ωi = 1, ωi ≥ 0, i = 1, . . . , n,

where ωi is the weight ( budget proportion ) assigned to therisk Xi.




◮ The portfolio is the random variable

Pw =

n∑

i=1

ωiXi





Pw =

n∑

i=1

ωiXi

◮ GivenX = (X1, X2, . . . , Xn)

How does the investor find the best portfolio...?





Pw =

n∑

i=1

ωiXi

◮ GivenX = (X1, X2, . . . , Xn)

How does the investor find the best portfolio...?

◮ Some answers will be given in this talk



What is the Problem...?◮ Assume that an investor cares only about the mean and

variance of portfolio.◮ A simple case of two risks

X = (X1, X2) such that E(X) = (µ1, µ2) and Σ =

(

σ21 σ12

σ12 σ22

)






(

σ21 σ12

σ12 σ22

)

◮ Let w = (ω1, ω2) be the vector of portfolio weights. Clearlythe portfolio is

Pw = ω1X1 + ω2X2 = ωX1 + (1− ω)X2, 0 ≤ ω ≤ 1.






(

σ21 σ12

σ12 σ22

)

◮ Let w = (ω1, ω2) be the vector of portfolio weights. Clearlythe portfolio is

Pw = ω1X1 + ω2X2 = ωX1 + (1− ω)X2, 0 ≤ ω ≤ 1.

◮ E(Pw) = ωµ1 + (1− ω)µ2

V AR(Pw) = ω2σ2

1+ (1− ω)2σ2

2+ 2ω(1− ω)σ12




Let X = (X1, X2) such thatµ1 = 0.5, µ2 = 0.3, σ2

1 = 4, σ22 = 1, σ12 = 1. If ω = 1 , then

Pw = ωX1 + (1− ω)X2 = X1

E(Pw) = ωµ1 + (1− ω)µ2 = 0.5

V AR(Pw) = ω2σ21 + (1 − ω)2σ2

2 + 2ω(1− ω)σ12 = 4

E(Pw)

V AR(Pw)





1 = 4, σ22 = 1, σ12 = 1. If ω = 1 , then

Pw = ωX1 + (1− ω)X2 = X1

E(Pw) = ωµ1 + (1− ω)µ2 = 0.5

V AR(Pw) = ω2σ21 + (1 − ω)2σ2

2 + 2ω(1− ω)σ12 = 4

E(Pw)

V AR(Pw)

⋆P1





1 = 4, σ22 = 1, σ12 = 1. If ω = 0 , then

Pw = ωX1 + (1− ω)X2 = X2

E(Pw) = ωµ1 + (1− ω)µ2 = 0.3

V AR(Pw) = ω2σ21 + (1 − ω)2σ2

2 + 2ω(1− ω)σ12 = 1

E(Pw)

V AR(Pw)

⋆P1⋆P0





1 = 4, σ22 = 1, σ12 = 1. If ω = 0.5 , then

Pw = ωX1 + (1− ω)X2 = 0.5X1 + 0.5X2

E(Pw) = ωµ1 + (1− ω)µ2 = 0.75

V AR(Pw) = ω2σ21 + (1 − ω)2σ2

2 + 2ω(1− ω)σ12 = 0.87

E(Pw)

V AR(Pw)

⋆P1⋆P0

⋆P0.5




0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Portf

olio M

ean

Portfolio Variance

Figure: Mean-Variance for Different ω Values.




0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Portfo

lio Me

an

Portfolio Variance


S =E(Pω)

V AR(Pω)




0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Portfo

lio Me

an

Portfolio Variance


S =E(Pω)

V AR(Pω)




0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Portfo

lio Me

an

Portfolio Variance

Best Portfolio


S =E(Pω)

V AR(Pω)



Efficient Frontier

0 0.5 1 1.5

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−3

Portfo

lio Me

an

Portfolio Variance

Figure: Feasible Portfolios

The set of couples risk-return that cannot be improved at the sametime is called Efficient Frontier. Markowitz (1952)



Efficient Frontier

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−3

Portfo

lio Me

an

Portfolio Variance

EfficientFrontier

Figure: Efficient Portfolios

The set of couples risk-return that cannot be improved at the sametime is called Efficient Frontier. Markowitz (1952)



Efficient Frontier

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

−3

Portfo

lio Me

an

Portfolio Variance

BestPortfolio

Figure: Best Portfolio

S =E(Pω)

V AR(Pω)



Portfolio Problem

◮ Consider the random vector X = (X1, X2, . . . , Xn) and thePortfolio Random Variable

Pw =

n∑

i=1

ωiXi



Portfolio Problem


Pw =

n∑

i=1

ωiXi

◮ Let U be his/her subjective utility function. Assume thatU′ ≥ 0 and U′′ ≤ 0. Increasing and Concave



Portfolio Problem


Pw =

n∑

i=1

ωiXi


◮ The portfolio problem in this case is given by

maxw

EU (Pw) s.t.n∑

i=1

ωi = 1.



Indice





Conclusions


References



Efficient Frontier




Minimum- Variance Portfolio

Markowitz (1952)An investor who cares only about the mean and variance shouldhold a portfolio on the efficient frontier .





Given the mean-value the best portfolio is the solution to th eoptimization problem.

minw

w′Σw

s.t. E(Pw) = µ

n∑

i=1

ωi = 1





Given the mean-value the best portfolio is the solution to th eoptimization problem.

minw

w′Σw

s.t. E(Pw) = µ

n∑

i=1

ωi = 1

If you have data you can use estimators for Σ and E(Xi).



Efficient Frontier




Maximum-Mean Portfolio

Following Markowitz Model (1952) this portfolio also will b e onthe efficient frontier . Therefore, given the variance, the bestportfolio is the solution to the optimization problem



Maximum-Mean Portfolio

Following Markowitz Model (1952) this portfolio also will b e onthe efficient frontier . Therefore, given the variance, the bestportfolio is the solution to the optimization problem

maxw

E(Pw)

s.t. w′Σw = σ

n∑

i=1

ωi = 1

If you have data you can use estimators for Σ and E(Xi).



Mean-Variance Portfolio

0 0.5 1 1.5

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−3

Portfo

lio Me

an

Portfolio Variance


maxw

w′Σw− 1

αE(Pw)

s.t.n∑

ωi = 1. α is the risk-aversion parameter





maxw

w′Σw− 1

αE(Pw)

s.t.n∑

ωi = 1. α = 1 is the risk-aversion parameter





maxw

w′Σw− 1

αE(Pw)

s.t.n∑

ωi = 1. α > 1 is the risk-aversion parameter





maxw

w′Σw − 1

αE(Pw)

s.t.n∑

ωi = 1. α −→ ∞ is the risk-aversion parameter



Risk Inverse Weighting Analysis PIR

Puerta and Laniado (2010)Let X = (X1, . . . , Xn) be a risky assets vector.

Pw =

n∑

i=1

ωiXi, ωi =

1ρ(Xi)

∑ni=1

1ρ(Xi)

ρ is a univariate positive risk measure.



Risk Inverse Weighting Analysis PIR

Puerta and Laniado (2010)Let X = (X1, . . . , Xn) be a risky assets vector.

Pw =

n∑

i=1

ωiXi, ωi =

1ρ(Xi)

∑ni=1

1ρ(Xi)

ρ is a univariate positive risk measure.

Less Weight to Higher Risk



Risk Inverse Weighting Analysis

DeMiguel et al. (2009) and Muller and Stoyan (2002) showed th eadvantages of using 1

n-rule (Naive Portfolio) .






Pw =

n∑

i=1

ωiXi =

n∑

i=1

1

nXi






Pw =

n∑

i=1

ωiXi =

n∑

i=1

1

nXi

◮ if X = (X1, . . . , Xn) is exhangeable, then PIR ≡ 1n

-rule






Pw =

n∑

i=1

ωiXi =

n∑

i=1

1

nXi


-rule◮ if X = (X1, . . . , Xn) is comonotonic and ρ is comonotonic

risk measure, then the risk of PIR is smaller than the risk of1n

-rule.






Pw =

n∑

i=1

ωiXi =

n∑

i=1

1

nXi


-rule◮ if X = (X1, . . . , Xn) is comonotonic and ρ is comonotonic

risk measure, then the risk of PIR is smaller than the risk of1n

-rule.◮ if X = (X1, X2), the variance of PIR is smaller than the

variance of 1n

-rule.


Solution and Comparison of Portfolios under Stochastic Ord ers

Indice





Conclusions


References



Portfolio Problem


Pw =n∑

i=1

ωiXi


◮ The portfolio problem in this case is given by

maxw

EU (Pw) s.t.n∑

i=1

ωi = 1.



Portfolio Problem

maxw

EU (Pw) s.t.n∑

i=1

ωi = 1. (1)

Hadar and Russel (1971)Investigated the problem (1) for iid random variables in thebivariate case. They showed that the solution to the problem (1)is the 1

n-rule

P∗

w= P 1

2



Portfolio Problem

maxw

EU (Pw) s.t.n∑

i=1

ωi = 1. (1)

Hadar and Russel (1971)Investigated the problem (1) for iid random variables in thebivariate case. They showed that the solution to the problem (1)is the 1

n-rule

P∗

w= P 1

2

Ma (2000)Showed that if (X1, X2, . . . , Xn) are exchangeable . Then thesolution of (1) is the 1

n-rule .

P∗

w= P 1

n



Portfolio Problem

maxw

EU (Pw) s.t.n∑

i=1

ωi = 1. (2)

Pellerey and Semeraro (2005)They considered X = (X1, X2), S = X1 +X2 and D = X2 −X1.They showed that if (S,D) is PQD and E(X2) ≤ E(X1), then

EU [(1− α)X1 + αX2]

is decreasing in α ∈ [ 12 , 1].The solution to the problem (2) is the 1

n-rule

P∗

w = P 12



Portfolio Problem

Laniado et al. (2012)Consider X = (X1, X2) and assume that there is a vectoru = (u1, u2) with ‖ u ‖= 1. If(

u1 u2

−u2 u1

)(

X1

X2

)

is PQD and u1E(X2)− u2E(X1) ≤ 0.

E

[

U

(√2

2(u1 + u2 − 2u2α)X1 +

√2

2(2u1α− u1 + u2)X2

)]

is decreasing in α ∈ [ 12 , 1].



Portfolio Problem

Laniado et al. (2012)Consider X = (X1, X2) and assume that there is a vectoru = (u1, u2) with ‖ u ‖= 1. If(

u1 u2

−u2 u1

)(

X1

X2

)

is PQD and u1E(X2)− u2E(X1) ≤ 0.

E

[

U

(√2

2(u1 + u2 − 2u2α)X1 +

√2

2(2u1α− u1 + u2)X2

)]

is decreasing in α ∈ [ 12 , 1].

P∗

w=

√2

2u1X1 +

√2

2u2X2



Elliptical Distributions

DefinitionThe random vector X = (X1, . . . , Xn)

′ is said to have an ellipticaldistribution with parameters µ and Σ if its characteristic functioncan be expressed as

E[exp(it′X)] = exp(it′µ)φ (t′Σt) , t = (t1, . . . , tn)′, (3)

for some function φ, and if Σ is such that Σ = AA′ for some

matrix A(n×m).



Property

Laniado et al. (2012)Let X = (X1, X2) be a random vector elliptically distributed withparameters µ = 0 and ΣX. Then there exists a rotation matrixsuch that RX is exchangeable.



Property

Laniado et al. (2012)Let X = (X1, X2) be a random vector elliptically distributed withparameters µ = 0 and ΣX. Then there exists a rotation matrixsuch that RX is exchangeable.

R =

√2

2

(

q11 + q21 q21 − q11q11 − q21 q11 + q21

)

.

ΣX = QDQ′ and Q = (qij)



Elliptical Distribution

X



Rotated Elliptical Distribution

RX



Theorem 3.A.35. Shaked and Shanthikumar (2007)Let X1, . . . , Xn be exchangeable random variables. Leta = (a1, . . . , an)

′ and b = (b1, . . . , bn)′ such that a ≺ b, then

n∑

i=1

aiXi ≥cv

n∑

i=1

biXi

Laniado et al. (2012)Let X = (X1, X2)

′ be elliptically distributed such that EX = 0 andlet a = (a1, a2)

′ and b = (b1, b2)′ be two vectors of constants. If

a ≺ b, thena′RX ≥cv b

′RX.





n∑

i=1

aiXi ≥cv

n∑

i=1

biXi





′RX.

For any concave function f

Ef (a′RX) ≥ Ef (b′RX) .





n∑

i=1

aiXi ≥cv

n∑

i=1

biXi





′RX.


EU (a′RX) ≥ EU (b′RX) .

In particular for an utility function U



Elliptical Distribution, n > 2

PropertyLet X = (X1, . . . , Xn)

′ be a random vector elliptically distributedwith parameters µX = 0 and ΣX is such that it has at least n− 1equal eigenvalues given by λ1 ≥ λ2 = · · · = λn = λ > 0. Thenthere exists a rotation matrix R such that RX has exchangeablecomponents.



Elliptical Distribution, n > 2

PropertyLet X = (X1, . . . , Xn)

′ be a random vector elliptically distributedwith parameters µX = 0 and ΣX is such that it has at least n− 1equal eigenvalues given by λ1 ≥ λ2 = · · · = λn = λ > 0. Thenthere exists a rotation matrix R such that RX has exchangeablecomponents.

If a = (a1, . . . , an)′ is majorized by b = (b1, . . . , bn)

′, then

a′RX ≥cv b

′RX


EU (a′RX) ≥ EU (b′RX) .

In particular for an utility function U



Portafolio Comparison

Shaked and Shanthikumar (2007)

X ≤st Y ⇐⇒ E[φ(X)] ≤ E[φ(Y )],

for all increasing function φ for which the expectation exist.





X ≤st Y ⇐⇒ E[φ(X)] ≤ E[φ(Y )],

for all increasing function φ for which the expectation exist.

Therefore, given the portfolios Pω1 and Pω2 such that

Pω1 ≤st Pω2 ,

then an investor with increasing utility function prefers Pω2 .





X ≤icx Y ⇐⇒ E[φ(X)] ≤ E[φ(Y )],

for all increasing concave function φ for which the expectationexist.





X ≤icx Y ⇐⇒ E[φ(X)] ≤ E[φ(Y )],

for all increasing concave function φ for which the expectationexist.

Therefore, given the portfolios Pω1 and Pω2 such that

Pω1 ≤icv Pω2 ,

then an investor with increasing and concave utility functi onprefers Pω2 .



Portfolio Problem

max~ω

EU (Pω) s.t.n∑

i=1

ωi = 1. (4)



Portfolio Problem

max~ω

EU (Pω) s.t.n∑

i=1

ωi = 1. (4)

Muller and Stoyan (2002)If X1, . . . , Xn are independent with

X1 ≥lr X2 ≥lr · · · ≥lr Xn,

and U is increasing. Then the optimization problem (4) has anoptimal solution with ω1 ≥ ω2 ≥ · · · ≥ ωn.



Portfolio Problem

max~ω

EU (Pω) s.t.n∑

i=1

ωi = 1. (4)


X1 ≥lr X2 ≥lr · · · ≥lr Xn,

and U is increasing. Then the optimization problem (4) has anoptimal solution with ω1 ≥ ω2 ≥ · · · ≥ ωn.


X ≤lr Y ⇐⇒ fY (t)

fX(t)↑t



Portfolio Problem

max~ω

EU (Pω) s.t.n∑

i=1

ωi = 1. (5)


X1 ≥rh X2 ≥rh · · · ≥rh Xn,

and U is increasing and concave. Then the optimization problem(5) has an optimal solution with. ω1 ≥ ω2 ≥ · · · ≥ ωn.



Portfolio Problem

max~ω

EU (Pω) s.t.n∑

i=1

ωi = 1. (5)


X1 ≥rh X2 ≥rh · · · ≥rh Xn,

and U is increasing and concave. Then the optimization problem(5) has an optimal solution with. ω1 ≥ ω2 ≥ · · · ≥ ωn.


X ≤rh Y ⇐⇒ FY (t)

FX(t)↑t



Indice





Conclusions


References



Alternative efficient frontiersLet Θ be a set of k criteria for evaluating the performance of theportfolio.In the classical Markowitz model k = 2 and corresponds to mean andvariance of the portfolio. Consider any criterion ci ∈ Θ, i = 1, . . . , k anddenote.




θci =

{

1 if the investor wants a portfolio with a low value of the crite rion ci

−1 if the investor wants a portfolio with a high value of the crit erion ci




θci =

{

1 if the investor wants a portfolio with a low value of the crite rion ci

−1 if the investor wants a portfolio with a high value of the crit erion ci

For example, if

Θ = {return , risk , Sharpe-ratio , entropy } = {c1, c2, c3, c4},

then

θreturn = θc1 = −1, θrisk = θc2 = 1, θSr = θc3 = −1, θentropy = θc4 = −1.



Alternative efficient frontier

0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.0540.0092

0.0094

0.0096

0.0098

0.01

0.0102

0.0104

0.0106

0.0108

0.011

0.0112

Criterion 2

Crit

erio

n 1

PortfoiosEfficient FrontierBest Portfolio

0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.250.0094

0.0096

0.0098

0.01

0.0102

0.0104

0.0106

0.0108

0.011

0.0112

Criterion 3C

riter

ion

1


Figure: u = 1√2[1,−1]′ p u = 1√

2[−1,−1]′ q

Criterion 1 Returns -1Criterion 2 Variance 1Criterion 3 Sharpe ratio -1Criterion 4 Entropy -1




0.8 1 1.2 1.4 1.6 1.8 20.0094

0.0096

0.0098

0.01

0.0102

0.0104

0.0106

0.0108

0.011

0.0112

Criterion 4

Crit

erio

n 1


0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.0540.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

Criterion 2

Crit

erio

n 3


Figure: u = 1√2[−1,−1]′ q u = 1√

2[1,−1]′ p





0.04 0.042 0.044 0.046 0.048 0.05 0.0520.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Criterion 2

Crit

erio

n 4


0.19 0.2 0.21 0.22 0.23 0.24 0.250.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Criterion 3

Crit

erio

n 4


Figure: u = 1√2[1,−1]′ p u = 1√

2[−1,−1]′ q





0.009 0.01 0.011 0.012 0.04

0.05

0.060.8

1

1.2

1.4

1.6

1.8

2

Criterion 1

Criterion 2

Crite

rion 4


Figure: u = 1√3[−1, 1,−1]′




Portfolio selection under extremality

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10−4

2

4

6

8

10

12

14x 10

−4

Risk

Re

turn

s AC

B

Efficient Frontier

30% of thebest portfolios

MinimumExtremalityPortfolio

Figure: Feasible Portfolios



Application to real data

Table: Portfolios notation in this work

Criteria returns and variance returns and Sharpe ratioPortfolio notation P12 P13

Criteria returns and entropy variance and Sharpe ratioPortfolio notation P14 P23

Criteria variance and entropy Sharpe ratio and entropyPortfolio notation P24 P34

Table: Portfolios notation for comparisons

1n

Equally-weighted PortfolioMEAN Mean-variance portfolio with shortsales constrained

MEANU Mean-Variance portfolio with shortsales unconstrainedMIN Minimum-Variance portfolio with shortsales constrained

MINU Minimum-Variance portfolio with shortsales unconstrained



ResultsTest proposed by Memmel (2003). 1

n-rule is a good benchmark DeMiguel et al. (2009b)

Table: Portfolio Sharpe ratios

Strategy 5Spain 6Spain 10Spain 25Spain 40Spain 48Ind 8Indexes

in this workP12 0.7218

(0.6948)0.5333(0.1315)

0.5498(0.0418)

0.5006(0.0314)

0.3700(0.0956)

0.2929(0.0965)

0.1070(0.3158)

P13 0.7478(0.6084)

0.5279(0.1399)

0.5989(0.0378)

0.5056(0.0854)

0.4044(0.0179)

0.2789(0.5170)

0.1003(0.4829)

P14 0.7196(0.6466)

0.4391(0.0519)

0.4438(0.2303)

0.4558(0.0978)

0.3564(0.0819)

0.2793(0.3309)

0.0896(0.8759)

P23 0.7080(0.9093)

0.4962(0.2988)

0.5375(0.1723)

0.5406(0.0178)

0.3166(0.5215)

0.2801(0.4466)

0.0985(0.5582)

P24 0.6941(0.8454)

0.3446(0.3012)

0.3656(0.7308)

0.4735(0.0610)

0.3182(0.5137)

0.2836(0.1533)

0.0848(0.6856)

P34 0.7114(0.6893)

0.4308(0.1397)

0.4881(0.0025)

0.4514(0.0198)

0.3766(0.0204)

0.2731(0.8809)

0.0910(0.7383)

for comparison1/n 0.6997 0.3753 0.3815 0.3791 0.2955 0.2719 0.0883

MEAN 0.4132(0.0750)

0.0804(0.1902)

0.1075(0.1999)

0.2213(0.4145)

−0.1400(0.0024)

0.2296(0.4806)

0.0555(0.7131)

MEANU 0.6632(0.7598)

0.4750(0.3314)

0.5354(0.1060)

0.4201(0.8452)

0.1960(0.6209)

0.0921(0.0519)

−0.0267(0.4246)

MIN 0.6502(0.5314)

0.1373(0.2605)

0.2745(0.5303)

0.2881(0.5073)

0.3500(0.5276)

0.2293(0.4326)

0.0961(0.8968)

MINU 0.6199(0.4932)

0.0871(0.1989)

0.2577(0.4981)

−0.1271(0.0276)

0.0012(0.0948)

0.1123(0.0393)

−0.0426(0.0640)


Conclusions

Indice





Conclusions


References


Conclusions

Conclusions◮ A fast review of different approaches to face the portfolio

selection problem


Conclusions


selection problem◮ The strategy PIR was introduced as a novel methodology

and easy of implementing which it has advantages on the 1n.


Conclusions




◮ If the random variables represents risky assets, we look forrotations of the distribution such that, the rotateddistribution satisfies conditions already studied in theliterature allowing to find one portfolio that maximizes anutility function.


Conclusions





◮ For the case of random variables elliptically distributed w ithmean zero, in n = 2 we showed that always is possible tofind a rotation where the rotated distribution hasexchangeable components so we can find what linearcombinations of the random variables improve an utilityfunction.


Conclusions





◮ For the case of random variables elliptically distributed w ithmean zero, in n = 2 we showed that always is possible tofind a rotation where the rotated distribution hasexchangeable components so we can find what linearcombinations of the random variables improve an utilityfunction.

◮ New concept of efficient frontier was introduced, taking int oaccount different criteria considered in Markowitz Model



Indice





Conclusions


References



Conclusions

◮ Find good estimation for the variance and covariance matrix



Conclusions

◮ Find good estimation for the variance and covariance matrix◮ Consider the the result of the PIR strategy for high

dimensions



Conclusions


dimensions◮ Study conditions under which some other distributions can

be exchangeable through rotations.



Conclusions


dimensions◮ Study conditions under which some other distributions can

be exchangeable through rotations.◮ To face the portfolio problem considering other interestin g

measure risk.


References

Indice





Conclusions


References


References

Hadar, J., Russel, W.R., 1971. Stochastic dominance and diversification. Journal of Economic Theory 3, 288-305.

Ma, C., 2000. Convex order for linear combinations of random variables. Journal of Statistical Planning and Inference 84, 11-25.

Laniado, H., Puerta, M. 2010. Diseno de estrategias optimas para portfolios, un analisis de la ponderacion inversa al riesgo.

Lecturas de Economıa 73, 243-273.

Laniado, H., Lillo, R.E., Pellerey, F., Romo, J. 2012. Portfolio selection through an extremality stochastic order. Insurance:

Mathematics and Economics 51, 1-9.

Markowitz, H. M., 1952. Mean-variance analysis in portfolio choice and capital markets. Journal of Finance 7, 77-91.

Muller, A., Stoyan, D., 2002. Comparison methods for stochastic models and risk. Wiley, New York.

Pellerey, F., Semeraro, P., 2005. A note on the portfolio selection problem. Theory and Decision 59, 295-306.

Shaked, M., Shanthikumar, J.G., 2007. Stochastic orders. Springer, New York.


References

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