A Study of Efficiency in CVaR PortfolioOptimization
Team OneMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang,
Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher Bemis
January 15, 2011
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Outline
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
VaRβ and CVaRβ
Taken from Conditional Value-at-Risk (CVaR): Algorithms and Applications by Stanislav Uryasevwww-iam.mathematik.hu-berlin.de/∼romisch/SP01/Uryasev.pdf
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
CVaR Details
Recall:I β-CVaR is the average loss given the condition that a
(large) loss in excess of β-VaR has occurred.I CVaR incorporates tail behavior beyond the VaR value.
Specifically,
CVaRβ(x) =1
1 − β
∫−xT y>VaRβ(x)
f(x, y)p(y)dy
where x ∈ X = x ∈ Rm : x > 0,−µT x 6 −R, 1T x = 1 We candiscretize this in a natural way by sampling our scenariosdiscretely according to p(y).
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
CVaR Details
I Such an approach leads to another optimization problem
CVaRβ = minx∈X
CVaRβ(x) (1)
I An equivalent optimization problem is
min(x,α)∈X×R
Fβ(x,α) : =1
1 − β
∫−xT y>α
−xT yp(y)dy
= α+1
1 − β
∫[−xT y − α]+p(y)dy
(2)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
CVaR Details
I Such an approach leads to another optimization problem
CVaRβ = minx∈X
CVaRβ(x) (1)
I An equivalent optimization problem is
min(x,α)∈X×R
Fβ(x,α) : =1
1 − β
∫−xT y>α
−xT yp(y)dy
= α+1
1 − β
∫[−xT y − α]+p(y)dy
(2)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
CVaR Objective function
I The objective function Fβ(x,α) can be discretized as
Fβ(x,α) = α+1
q(1 − β)
q∑j=1
[−xT yj − α]+ (3)
where y1, . . . , yq are sampled from probability distributionp(y)
I The optimization problem becomes
min(x,α)∈X×R
Fβ(x,α) (4)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
CVaR Objective function
I The objective function Fβ(x,α) can be discretized as
Fβ(x,α) = α+1
q(1 − β)
q∑j=1
[−xT yj − α]+ (3)
where y1, . . . , yq are sampled from probability distributionp(y)
I The optimization problem becomes
min(x,α)∈X×R
Fβ(x,α) (4)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Convolution Method
I h(z) = [z]+
I η(z) =
C exp( 1z2−1) if − 1 < z < 1
0 otherwisewhere C is chosen so that
∫∞−∞ η(z)dz = 1
I ηε(z) =
Cε exp( 1
(z/ε)2−1) if − ε < z < ε
0 otherwiseI gε(z) = ηε ∗ h(z) =
∫R h(z − s)ηε(s)ds
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Convolution Method
I h(z) = [z]+
I η(z) =
C exp( 1z2−1) if − 1 < z < 1
0 otherwisewhere C is chosen so that
∫∞−∞ η(z)dz = 1
I ηε(z) =
Cε exp( 1
(z/ε)2−1) if − ε < z < ε
0 otherwiseI gε(z) = ηε ∗ h(z) =
∫R h(z − s)ηε(s)ds
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Convolution Method
I h(z) = [z]+
I η(z) =
C exp( 1z2−1) if − 1 < z < 1
0 otherwisewhere C is chosen so that
∫∞−∞ η(z)dz = 1
I ηε(z) =
Cε exp( 1
(z/ε)2−1) if − ε < z < ε
0 otherwiseI gε(z) = ηε ∗ h(z) =
∫R h(z − s)ηε(s)ds
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Convolution Method
I h(z) = [z]+
I η(z) =
C exp( 1z2−1) if − 1 < z < 1
0 otherwisewhere C is chosen so that
∫∞−∞ η(z)dz = 1
I ηε(z) =
Cε exp( 1
(z/ε)2−1) if − ε < z < ε
0 otherwiseI gε(z) = ηε ∗ h(z) =
∫R h(z − s)ηε(s)ds
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Convolution Method
I h(z) = [z]+
I η(z) =
C exp( 1z2−1) if − 1 < z < 1
0 otherwisewhere C is chosen so that
∫∞−∞ η(z)dz = 1
I ηε(z) =
Cε exp( 1
(z/ε)2−1) if − ε < z < ε
0 otherwiseI gε(z) = ηε ∗ h(z) =
∫R h(z − s)ηε(s)ds
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Convolution Method
I h(z) = [z]+
I η(z) =
C exp( 1z2−1) if − 1 < z < 1
0 otherwisewhere C is chosen so that
∫∞−∞ η(z)dz = 1
I ηε(z) =
Cε exp( 1
(z/ε)2−1) if − ε < z < ε
0 otherwiseI gε(z) = ηε ∗ h(z) =
∫R h(z − s)ηε(s)ds
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Convolution Method
I
φεβ(x,α)∈X×R
(x,α) = α+1
1 − β
∫Rm
(ηε ∗ h)(−xT y − α)p(y)dy
(5)I
φεβ(x,α)∈X×R
(x,α) = α+1
q(1 − β)
q∑j=1
ηε ∗ h(−xT yj − α)
= α+1
q(1 − β)
q∑j=1
∫1
−1h(−xT yj − α− εs)η(s)ds
= α+1
q(1 − β)
q∑j=1
N∑n=1
ωnη(zn)h(−xT yj − α− εzn) (6)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Convolution Method
I
φεβ(x,α)∈X×R
(x,α) = α+1
1 − β
∫Rm
(ηε ∗ h)(−xT y − α)p(y)dy
(5)I
φεβ(x,α)∈X×R
(x,α) = α+1
q(1 − β)
q∑j=1
ηε ∗ h(−xT yj − α)
= α+1
q(1 − β)
q∑j=1
∫1
−1h(−xT yj − α− εs)η(s)ds
= α+1
q(1 − β)
q∑j=1
N∑n=1
ωnη(zn)h(−xT yj − α− εzn) (6)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Piecewise Quadratic Approximation
I h(z) = [z]+
I ρε(z) =
0 if z 6 −ε
z2
4ε +z2 + ε
4 if − ε < z < εz if z > ε
Figure: ρε(z) with ε = 0.1 and ε = 0.2 and z ∈ [−0.6, 0.6].
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Piecewise Quadratic Approximation
I The optimization of β-CVaR becomes
min(x,α)∈X×R
Fβ(x,α) := α+1
1 − β
∫Rmρε(−xT y − α)p(y)dy
I Discretization of the quadratic approximation is
min(x,α)∈X×R
Fβ(x,α) := α+1
q(1 − β)
q∑j=1
ρε(−xT y − α)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Piecewise Quadratic Approximation
I The optimization of β-CVaR becomes
min(x,α)∈X×R
Fβ(x,α) := α+1
1 − β
∫Rmρε(−xT y − α)p(y)dy
I Discretization of the quadratic approximation is
min(x,α)∈X×R
Fβ(x,α) := α+1
q(1 − β)
q∑j=1
ρε(−xT y − α)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Main Idea:I Converting the scenario-based mean-CVaR problem to the
saddle-point problemI Using Nesterov Procedure to solve the saddle-point
problemminx∈X
CVaRβ(Yx) = minx∈X
maxQ∈QEQ[Yx]
Q = Q : 0 6∂Q
∂P6
11 − β
minx∈X
maxq∈Q
−qT Yx
Q = q ∈ RN : 1T q = 1, 0 6 q 61
1 − βp
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Main Idea:I Converting the scenario-based mean-CVaR problem to the
saddle-point problemI Using Nesterov Procedure to solve the saddle-point
problemminx∈X
CVaRβ(Yx) = minx∈X
maxQ∈QEQ[Yx]
Q = Q : 0 6∂Q
∂P6
11 − β
minx∈X
maxq∈Q
−qT Yx
Q = q ∈ RN : 1T q = 1, 0 6 q 61
1 − βp
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Main Idea:I Converting the scenario-based mean-CVaR problem to the
saddle-point problemI Using Nesterov Procedure to solve the saddle-point
problemminx∈X
CVaRβ(Yx) = minx∈X
maxQ∈QEQ[Yx]
Q = Q : 0 6∂Q
∂P6
11 − β
minx∈X
maxq∈Q
−qT Yx
Q = q ∈ RN : 1T q = 1, 0 6 q 61
1 − βp
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Main Idea:I Converting the scenario-based mean-CVaR problem to the
saddle-point problemI Using Nesterov Procedure to solve the saddle-point
problemminx∈X
CVaRβ(Yx) = minx∈X
maxQ∈QEQ[Yx]
Q = Q : 0 6∂Q
∂P6
11 − β
minx∈X
maxq∈Q
−qT Yx
Q = q ∈ RN : 1T q = 1, 0 6 q 61
1 − βp
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Main Idea:I Converting the scenario-based mean-CVaR problem to the
saddle-point problemI Using Nesterov Procedure to solve the saddle-point
problemminx∈X
CVaRβ(Yx) = minx∈X
maxQ∈QEQ[Yx]
Q = Q : 0 6∂Q
∂P6
11 − β
minx∈X
maxq∈Q
−qT Yx
Q = q ∈ RN : 1T q = 1, 0 6 q 61
1 − βp
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Main Idea:I Converting the scenario-based mean-CVaR problem to the
saddle-point problemI Using Nesterov Procedure to solve the saddle-point
problemminx∈X
CVaRβ(Yx) = minx∈X
maxQ∈QEQ[Yx]
Q = Q : 0 6∂Q
∂P6
11 − β
minx∈X
maxq∈Q
−qT Yx
Q = q ∈ RN : 1T q = 1, 0 6 q 61
1 − βp
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Main Idea:I Converting the scenario-based mean-CVaR problem to the
saddle-point problemI Using Nesterov Procedure to solve the saddle-point
problemminx∈X
CVaRβ(Yx) = minx∈X
maxQ∈QEQ[Yx]
Q = Q : 0 6∂Q
∂P6
11 − β
minx∈X
maxq∈Q
−qT Yx
Q = q ∈ RN : 1T q = 1, 0 6 q 61
1 − βp
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Nesterov Procedure
D1 ←12(1 +
2M)2
D2 ←−1
1 − β(β lnβ+ (1 − β) ln (1 − β))
σ2 ←1β
, Ω← maxi‖Yi‖22
K ← 1ε
√ΩD1D2
σ2, µ← ε
2D2
x(0) ← 1q
1, d2(q)←N∑
i=1
qi ln qi+(pi
1 − β−qi) ln (
pi
1 − β− qi)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Nesterov Procedure
D1 ←12(1 +
2M)2
D2 ←−1
1 − β(β lnβ+ (1 − β) ln (1 − β))
σ2 ←1β
, Ω← maxi‖Yi‖22
K ← 1ε
√ΩD1D2
σ2, µ← ε
2D2
x(0) ← 1q
1, d2(q)←N∑
i=1
qi ln qi+(pi
1 − β−qi) ln (
pi
1 − β− qi)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Nesterov Procedure
D1 ←12(1 +
2M)2
D2 ←−1
1 − β(β lnβ+ (1 − β) ln (1 − β))
σ2 ←1β
, Ω← maxi‖Yi‖22
K ← 1ε
√ΩD1D2
σ2, µ← ε
2D2
x(0) ← 1q
1, d2(q)←N∑
i=1
qi ln qi+(pi
1 − β−qi) ln (
pi
1 − β− qi)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Nesterov Procedure
D1 ←12(1 +
2M)2
D2 ←−1
1 − β(β lnβ+ (1 − β) ln (1 − β))
σ2 ←1β
, Ω← maxi‖Yi‖22
K ← 1ε
√ΩD1D2
σ2, µ← ε
2D2
x(0) ← 1q
1, d2(q)←N∑
i=1
qi ln qi+(pi
1 − β−qi) ln (
pi
1 − β− qi)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Nesterov Procedure
D1 ←12(1 +
2M)2
D2 ←−1
1 − β(β lnβ+ (1 − β) ln (1 − β))
σ2 ←1β
, Ω← maxi‖Yi‖22
K ← 1ε
√ΩD1D2
σ2, µ← ε
2D2
x(0) ← 1q
1, d2(q)←N∑
i=1
qi ln qi+(pi
1 − β−qi) ln (
pi
1 − β− qi)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Nesterov Procedure
D1 ←12(1 +
2M)2
D2 ←−1
1 − β(β lnβ+ (1 − β) ln (1 − β))
σ2 ←1β
, Ω← maxi‖Yi‖22
K ← 1ε
√ΩD1D2
σ2, µ← ε
2D2
x(0) ← 1q
1, d2(q)←N∑
i=1
qi ln qi+(pi
1 − β−qi) ln (
pi
1 − β− qi)
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
for k ← 0 to K do
q(k) ← arg maxq∈Q
qT Yx(k) − µd2(q)
y(k+1) ← arg min
y∈X
−(q(k))T Yy(k) +
Ω
2µσ2‖y − x(k)‖22
z(k+1) ← arg minz∈X
−
k∑t=0
t + 12
q(t)Yz +Ω
2µσ2‖z‖22
x(k+1) ← 2k + 1
z(k+1) +k + 1k + 3
y(k+1)
return x = y(K) q =
K∑k=0
2(k + 1)(K + 1)(K + 2)
q(k).
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
for k ← 0 to K do
q(k) ← arg maxq∈Q
qT Yx(k) − µd2(q)
y(k+1) ← arg min
y∈X
−(q(k))T Yy(k) +
Ω
2µσ2‖y − x(k)‖22
z(k+1) ← arg minz∈X
−
k∑t=0
t + 12
q(t)Yz +Ω
2µσ2‖z‖22
x(k+1) ← 2k + 1
z(k+1) +k + 1k + 3
y(k+1)
return x = y(K) q =
K∑k=0
2(k + 1)(K + 1)(K + 2)
q(k).
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
for k ← 0 to K do
q(k) ← arg maxq∈Q
qT Yx(k) − µd2(q)
y(k+1) ← arg min
y∈X
−(q(k))T Yy(k) +
Ω
2µσ2‖y − x(k)‖22
z(k+1) ← arg minz∈X
−
k∑t=0
t + 12
q(t)Yz +Ω
2µσ2‖z‖22
x(k+1) ← 2k + 1
z(k+1) +k + 1k + 3
y(k+1)
return x = y(K) q =
K∑k=0
2(k + 1)(K + 1)(K + 2)
q(k).
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
for k ← 0 to K do
q(k) ← arg maxq∈Q
qT Yx(k) − µd2(q)
y(k+1) ← arg min
y∈X
−(q(k))T Yy(k) +
Ω
2µσ2‖y − x(k)‖22
z(k+1) ← arg minz∈X
−
k∑t=0
t + 12
q(t)Yz +Ω
2µσ2‖z‖22
x(k+1) ← 2k + 1
z(k+1) +k + 1k + 3
y(k+1)
return x = y(K) q =
K∑k=0
2(k + 1)(K + 1)(K + 2)
q(k).
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
for k ← 0 to K do
q(k) ← arg maxq∈Q
qT Yx(k) − µd2(q)
y(k+1) ← arg min
y∈X
−(q(k))T Yy(k) +
Ω
2µσ2‖y − x(k)‖22
z(k+1) ← arg minz∈X
−
k∑t=0
t + 12
q(t)Yz +Ω
2µσ2‖z‖22
x(k+1) ← 2k + 1
z(k+1) +k + 1k + 3
y(k+1)
return x = y(K) q =
K∑k=0
2(k + 1)(K + 1)(K + 2)
q(k).
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
for k ← 0 to K do
q(k) ← arg maxq∈Q
qT Yx(k) − µd2(q)
y(k+1) ← arg min
y∈X
−(q(k))T Yy(k) +
Ω
2µσ2‖y − x(k)‖22
z(k+1) ← arg minz∈X
−
k∑t=0
t + 12
q(t)Yz +Ω
2µσ2‖z‖22
x(k+1) ← 2k + 1
z(k+1) +k + 1k + 3
y(k+1)
return x = y(K) q =
K∑k=0
2(k + 1)(K + 1)(K + 2)
q(k).
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Testing Results
I M: infinityI ε: error toleranceI final result: sensitive to εI Iyengar’s model works well, β = 0.99
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Testing Results
I M: infinityI ε: error toleranceI final result: sensitive to εI Iyengar’s model works well, β = 0.99
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Testing Results
I M: infinityI ε: error toleranceI final result: sensitive to εI Iyengar’s model works well, β = 0.99
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Testing Results
I M: infinityI ε: error toleranceI final result: sensitive to εI Iyengar’s model works well, β = 0.99
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Iterative Gradient Descent Methodology
I Testing Results
I M: infinityI ε: error toleranceI final result: sensitive to εI Iyengar’s model works well, β = 0.99
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
I Main Idea:
Rockafellar and Uryasev introduce a performance functionand auxiliary variables to transfer the original problem intoa linear program and minimize CVaR by sampling.It gets rid of the assumption of the distribution of the returnof assets.
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
I Main Idea:
Rockafellar and Uryasev introduce a performance functionand auxiliary variables to transfer the original problem intoa linear program and minimize CVaR by sampling.It gets rid of the assumption of the distribution of the returnof assets.
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
I Main Idea:
Rockafellar and Uryasev introduce a performance functionand auxiliary variables to transfer the original problem intoa linear program and minimize CVaR by sampling.It gets rid of the assumption of the distribution of the returnof assets.
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
Fβ(x,α) = α+ (1 − β)−1∫
f(x,y)>VaRβ(x)[f(x, y) − α]+p(y)dy
min(x,α)∈X×R
Fβ(x,α)
Fβ(x,α) = α+1
q(1 − β)
q∑k=1
[f(x, yk ) − α]+
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
Fβ(x,α) = α+ (1 − β)−1∫
f(x,y)>VaRβ(x)[f(x, y) − α]+p(y)dy
min(x,α)∈X×R
Fβ(x,α)
Fβ(x,α) = α+1
q(1 − β)
q∑k=1
[f(x, yk ) − α]+
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
Fβ(x,α) = α+ (1 − β)−1∫
f(x,y)>VaRβ(x)[f(x, y) − α]+p(y)dy
min(x,α)∈X×R
Fβ(x,α)
Fβ(x,α) = α+1
q(1 − β)
q∑k=1
[f(x, yk ) − α]+
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
Fβ(x,α) = α+ (1 − β)−1∫
f(x,y)>VaRβ(x)[f(x, y) − α]+p(y)dy
min(x,α)∈X×R
Fβ(x,α)
Fβ(x,α) = α+1
q(1 − β)
q∑k=1
[f(x, yk ) − α]+
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
minimize α+1
q(1 − β)
q∑k=1
uk ,
subject to xT y + α+ uk > 0uk > 0x > 01T x = 1µ(x) 6 −R.
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Linear Program Methodology
minimize α+1
q(1 − β)
q∑k=1
uk ,
subject to xT y + α+ uk > 0uk > 0x > 01T x = 1µ(x) 6 −R.
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Comparison
Figure: Scenarios vs. CVaR Difference
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Comparison
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Comparison
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Comparison
Figure: Scenarios vs. Runtime
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Comparison
Figure: Assets vs. Runtime
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Out of Sample Performance
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Out of Sample Performance
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Out of Sample Performance
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Out of Sample Performance
I Reference: S&P 500, Google Finance
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
Thank you!
Team One Members: Mark Glad, Chen Zhang, Bowen Yu, Yiran Zhang, Feiyu Pang, Haochen Kang, Liqiong ZhaoMentor: Christopher BemisA Study of Efficiency in CVaR Portfolio Optimization
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