On The Portfolio Selection Problem
On The Portfolio Selection ProblemHenry Laniado
Seminario de DoctoradoDoctorado en Ingenierıa Matem atica
Medellın, Abril 2015
On The Portfolio Selection Problem
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
On The Portfolio Selection Problem
Introduction a Fast Review
Indice
Introduction a Fast Review
Solution Under Optimization
Solution and Comparison of Portfolios under Stochastic Orders
Solution under Simulation
Conclusions
Futurer Research Lines
References
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
◮ Consider an investor who has the possibility of investing inn different risky assets
X = (X1, X2, . . . , Xn)
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
◮ Consider an investor who has the possibility of investing inn different risky assets
X = (X1, X2, . . . , Xn)
◮ The investor has to allocate his budget C to the differentrisks. Without loss of generality C = 1
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
◮ Consider an investor who has the possibility of investing inn different risky assets
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.
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
◮ The portfolio is the random variable
Pw =
n∑
i=1
ωiXi
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
◮ The portfolio is the random variable
Pw =
n∑
i=1
ωiXi
◮ GivenX = (X1, X2, . . . , Xn)
How does the investor find the best portfolio...?
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
◮ The portfolio is the random variable
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
On The Portfolio Selection Problem
Introduction a Fast Review
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
)
On The Portfolio Selection Problem
Introduction a Fast Review
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
)
◮ Let w = (ω1, ω2) be the vector of portfolio weights. Clearlythe portfolio is
Pw = ω1X1 + ω2X2 = ωX1 + (1− ω)X2, 0 ≤ ω ≤ 1.
On The Portfolio Selection Problem
Introduction a Fast Review
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
)
◮ 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
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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)
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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)
⋆P1
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
Let X = (X1, X2) such thatµ1 = 0.5, µ2 = 0.3, σ2
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
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
Let X = (X1, X2) such thatµ1 = 0.5, µ2 = 0.3, σ2
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
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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.
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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
Figure: Mean-Variance for Different ω Values.
S =E(Pω)
V AR(Pω)
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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
Figure: Mean-Variance for Different ω Values.
S =E(Pω)
V AR(Pω)
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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
Figure: Mean-Variance for Different ω Values.
S =E(Pω)
V AR(Pω)
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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
Figure: Mean-Variance for Different ω Values.
S =E(Pω)
V AR(Pω)
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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
Figure: Mean-Variance for Different ω Values.
S =E(Pω)
V AR(Pω)
On The Portfolio Selection Problem
Introduction a Fast Review
What is the Problem...?
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
Figure: Mean-Variance for Different ω Values.
S =E(Pω)
V AR(Pω)
On The Portfolio Selection Problem
Introduction a Fast Review
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)
On The Portfolio Selection Problem
Introduction a Fast Review
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)
On The Portfolio Selection Problem
Introduction a Fast Review
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)
On The Portfolio Selection Problem
Introduction a Fast Review
Portfolio Problem
◮ Consider the random vector X = (X1, X2, . . . , Xn) and thePortfolio Random Variable
Pw =
n∑
i=1
ωiXi
On The Portfolio Selection Problem
Introduction a Fast Review
Portfolio Problem
◮ Consider the random vector X = (X1, X2, . . . , Xn) and thePortfolio Random Variable
Pw =
n∑
i=1
ωiXi
◮ Let U be his/her subjective utility function. Assume thatU′ ≥ 0 and U′′ ≤ 0. Increasing and Concave
On The Portfolio Selection Problem
Introduction a Fast Review
Portfolio Problem
◮ Consider the random vector X = (X1, X2, . . . , Xn) and thePortfolio Random Variable
Pw =
n∑
i=1
ωiXi
◮ Let U be his/her subjective utility function. Assume thatU′ ≥ 0 and U′′ ≤ 0. Increasing and Concave
◮ The portfolio problem in this case is given by
maxw
EU (Pw) s.t.n∑
i=1
ωi = 1.
On The Portfolio Selection Problem
Solution Under Optimization
Indice
Introduction a Fast Review
Solution Under Optimization
Solution and Comparison of Portfolios under Stochastic Orders
Solution under Simulation
Conclusions
Futurer Research Lines
References
On The Portfolio Selection Problem
Solution Under Optimization
Efficient Frontier
Figure: Best Portfolio
On The Portfolio Selection Problem
Solution Under Optimization
Efficient Frontier
Figure: Best Portfolio
On The Portfolio Selection Problem
Solution Under Optimization
Minimum- Variance Portfolio
Markowitz (1952)An investor who cares only about the mean and variance shouldhold a portfolio on the efficient frontier .
On The Portfolio Selection Problem
Solution Under Optimization
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
On The Portfolio Selection Problem
Solution Under Optimization
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
If you have data you can use estimators for Σ and E(Xi).
On The Portfolio Selection Problem
Solution Under Optimization
Efficient Frontier
Figure: Best Portfolio
On The Portfolio Selection Problem
Solution Under Optimization
Efficient Frontier
Figure: Best Portfolio
On The Portfolio Selection Problem
Solution Under Optimization
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
On The Portfolio Selection Problem
Solution Under Optimization
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).
On The Portfolio Selection Problem
Solution Under Optimization
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
Figure: Best Portfolio
maxw
w′Σw− 1
αE(Pw)
s.t.n∑
ωi = 1. α is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw− 1
αE(Pw)
s.t.n∑
ωi = 1. α = 1 is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw− 1
αE(Pw)
s.t.n∑
ωi = 1. α = 1 is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw− 1
αE(Pw)
s.t.n∑
ωi = 1. α = 1 is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw− 1
αE(Pw)
s.t.n∑
ωi = 1. α = 1 is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw− 1
αE(Pw)
s.t.n∑
ωi = 1. α > 1 is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw− 1
αE(Pw)
s.t.n∑
ωi = 1. α > 1 is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw− 1
αE(Pw)
s.t.n∑
ωi = 1. α > 1 is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw − 1
αE(Pw)
s.t.n∑
ωi = 1. α −→ ∞ is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw − 1
αE(Pw)
s.t.n∑
ωi = 1. α −→ ∞ is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
Mean-Variance Portfolio
Figure: Best Portfolio
maxw
w′Σw − 1
αE(Pw)
s.t.n∑
ωi = 1. α −→ ∞ is the risk-aversion parameter
On The Portfolio Selection Problem
Solution Under Optimization
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.
On The Portfolio Selection Problem
Solution Under Optimization
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
On The Portfolio Selection Problem
Solution Under Optimization
Risk Inverse Weighting Analysis
DeMiguel et al. (2009) and Muller and Stoyan (2002) showed th eadvantages of using 1
n-rule (Naive Portfolio) .
On The Portfolio Selection Problem
Solution Under Optimization
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
On The Portfolio Selection Problem
Solution Under Optimization
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
◮ if X = (X1, . . . , Xn) is exhangeable, then PIR ≡ 1n
-rule
On The Portfolio Selection Problem
Solution Under Optimization
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
◮ if X = (X1, . . . , Xn) is exhangeable, then PIR ≡ 1n
-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.
On The Portfolio Selection Problem
Solution Under Optimization
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
◮ if X = (X1, . . . , Xn) is exhangeable, then PIR ≡ 1n
-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.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Indice
Introduction a Fast Review
Solution Under Optimization
Solution and Comparison of Portfolios under Stochastic Orders
Solution under Simulation
Conclusions
Futurer Research Lines
References
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Portfolio Problem
◮ Consider the random vector X = (X1, X2, . . . , Xn) and thePortfolio Random Variable
Pw =n∑
i=1
ωiXi
◮ Let U be his/her subjective utility function. Assume thatU′ ≥ 0 and U′′ ≤ 0. Increasing and Concave
◮ The portfolio problem in this case is given by
maxw
EU (Pw) s.t.n∑
i=1
ωi = 1.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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].
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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).
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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)
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Elliptical Distribution
X
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Rotated Elliptical Distribution
RX
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
For any concave function f
Ef (a′RX) ≥ Ef (b′RX) .
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
For any concave function f
EU (a′RX) ≥ EU (b′RX) .
In particular for an utility function U
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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
For any concave function f
EU (a′RX) ≥ EU (b′RX) .
In particular for an utility function U
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
Therefore, given the portfolios Pω1 and Pω2 such that
Pω1 ≤st Pω2 ,
then an investor with increasing utility function prefers Pω2 .
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Portafolio Comparison
Shaked and Shanthikumar (2007)
X ≤icx Y ⇐⇒ E[φ(X)] ≤ E[φ(Y )],
for all increasing concave function φ for which the expectationexist.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Portafolio Comparison
Shaked and Shanthikumar (2007)
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 .
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Portfolio Problem
max~ω
EU (Pω) s.t.n∑
i=1
ωi = 1. (4)
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
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.
Shaked and Shanthikumar (2007)
X ≤lr Y ⇐⇒ fY (t)
fX(t)↑t
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Portfolio Problem
max~ω
EU (Pω) s.t.n∑
i=1
ωi = 1. (5)
Muller and Stoyan (2002)If X1, . . . , Xn are independent with
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.
On The Portfolio Selection Problem
Solution and Comparison of Portfolios under Stochastic Ord ers
Portfolio Problem
max~ω
EU (Pω) s.t.n∑
i=1
ωi = 1. (5)
Muller and Stoyan (2002)If X1, . . . , Xn are independent with
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.
Shaked and Shanthikumar (2007)
X ≤rh Y ⇐⇒ FY (t)
FX(t)↑t
On The Portfolio Selection Problem
Solution under Simulation
Indice
Introduction a Fast Review
Solution Under Optimization
Solution and Comparison of Portfolios under Stochastic Orders
Solution under Simulation
Conclusions
Futurer Research Lines
References
On The Portfolio Selection Problem
Solution under Simulation
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.
On The Portfolio Selection Problem
Solution under Simulation
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
On The Portfolio Selection Problem
Solution under Simulation
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
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.
On The Portfolio Selection Problem
Solution under Simulation
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
PortfoiosEfficient FrontierBest Portfolio
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
On The Portfolio Selection Problem
Solution under Simulation
Alternative efficient frontier
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
PortfoiosEfficient FrontierBest Portfolio
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
PortfoiosEfficient FrontierBest Portfolio
Figure: u = 1√2[−1,−1]′ q u = 1√
2[1,−1]′ p
Criterion 1 Returns -1Criterion 2 Variance 1Criterion 3 Sharpe ratio -1Criterion 4 Entropy -1
On The Portfolio Selection Problem
Solution under Simulation
Alternative efficient frontier
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
PortfoiosEfficient FrontierBest Portfolio
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
PortfoiosEfficient FrontierBest Portfolio
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
On The Portfolio Selection Problem
Solution under Simulation
Alternative efficient frontier
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
PortfoiosEfficient FrontierBest Portfolio
Figure: u = 1√3[−1, 1,−1]′
Criterion 1 Returns -1Criterion 2 Variance 1Criterion 3 Sharpe ratio -1Criterion 4 Entropy -1
On The Portfolio Selection Problem
Solution under Simulation
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
On The Portfolio Selection Problem
Solution under Simulation
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
On The Portfolio Selection Problem
Solution under Simulation
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)
On The Portfolio Selection Problem
Solution under Simulation
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)
On The Portfolio Selection Problem
Solution under Simulation
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)
On The Portfolio Selection Problem
Conclusions
Indice
Introduction a Fast Review
Solution Under Optimization
Solution and Comparison of Portfolios under Stochastic Orders
Solution under Simulation
Conclusions
Futurer Research Lines
References
On The Portfolio Selection Problem
Conclusions
Conclusions◮ A fast review of different approaches to face the portfolio
selection problem
On The Portfolio Selection Problem
Conclusions
Conclusions◮ A fast review of different approaches to face the portfolio
selection problem◮ The strategy PIR was introduced as a novel methodology
and easy of implementing which it has advantages on the 1n.
On The Portfolio Selection Problem
Conclusions
Conclusions◮ A fast review of different approaches to face the portfolio
selection problem◮ The strategy PIR was introduced as a novel methodology
and easy of implementing which it has advantages on the 1n.
◮ 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.
On The Portfolio Selection Problem
Conclusions
Conclusions◮ A fast review of different approaches to face the portfolio
selection problem◮ The strategy PIR was introduced as a novel methodology
and easy of implementing which it has advantages on the 1n.
◮ 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.
◮ 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.
On The Portfolio Selection Problem
Conclusions
Conclusions◮ A fast review of different approaches to face the portfolio
selection problem◮ The strategy PIR was introduced as a novel methodology
and easy of implementing which it has advantages on the 1n.
◮ 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.
◮ 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
On The Portfolio Selection Problem
Futurer Research Lines
Indice
Introduction a Fast Review
Solution Under Optimization
Solution and Comparison of Portfolios under Stochastic Orders
Solution under Simulation
Conclusions
Futurer Research Lines
References
On The Portfolio Selection Problem
Futurer Research Lines
Conclusions
◮ Find good estimation for the variance and covariance matrix
On The Portfolio Selection Problem
Futurer Research Lines
Conclusions
◮ Find good estimation for the variance and covariance matrix◮ Consider the the result of the PIR strategy for high
dimensions
On The Portfolio Selection Problem
Futurer Research Lines
Conclusions
◮ Find good estimation for the variance and covariance matrix◮ Consider the the result of the PIR strategy for high
dimensions◮ Study conditions under which some other distributions can
be exchangeable through rotations.
On The Portfolio Selection Problem
Futurer Research Lines
Conclusions
◮ Find good estimation for the variance and covariance matrix◮ Consider the the result of the PIR strategy for high
dimensions◮ Study conditions under which some other distributions can
be exchangeable through rotations.◮ To face the portfolio problem considering other interestin g
measure risk.
On The Portfolio Selection Problem
References
Indice
Introduction a Fast Review
Solution Under Optimization
Solution and Comparison of Portfolios under Stochastic Orders
Solution under Simulation
Conclusions
Futurer Research Lines
References
On The Portfolio Selection Problem
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.
On The Portfolio Selection Problem
References
thanks for your attention