A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of...
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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
![Page 2: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/2.jpg)
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
![Page 3: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/3.jpg)
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
![Page 4: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/4.jpg)
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
![Page 5: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/5.jpg)
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
![Page 6: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/6.jpg)
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
![Page 7: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/7.jpg)
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
![Page 8: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/8.jpg)
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
![Page 9: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/9.jpg)
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
![Page 10: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/10.jpg)
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
![Page 11: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/11.jpg)
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
![Page 12: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/12.jpg)
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
![Page 13: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/13.jpg)
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
![Page 14: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/14.jpg)
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
![Page 15: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/15.jpg)
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
![Page 16: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/16.jpg)
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
![Page 17: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/17.jpg)
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
![Page 18: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/18.jpg)
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
![Page 19: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/19.jpg)
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
![Page 20: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/20.jpg)
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
![Page 21: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/21.jpg)
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
![Page 22: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/22.jpg)
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
![Page 23: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/23.jpg)
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
![Page 24: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/24.jpg)
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
![Page 25: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/25.jpg)
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
![Page 26: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/26.jpg)
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
![Page 27: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/27.jpg)
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
![Page 28: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/28.jpg)
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
![Page 29: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/29.jpg)
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
![Page 30: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/30.jpg)
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
![Page 31: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/31.jpg)
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
![Page 32: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/32.jpg)
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
![Page 33: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/33.jpg)
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
![Page 34: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/34.jpg)
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
![Page 35: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/35.jpg)
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
![Page 36: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/36.jpg)
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
![Page 37: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/37.jpg)
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
![Page 38: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/38.jpg)
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
![Page 39: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/39.jpg)
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
![Page 40: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/40.jpg)
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
![Page 41: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/41.jpg)
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
![Page 42: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/42.jpg)
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
![Page 43: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/43.jpg)
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
![Page 44: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/44.jpg)
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
![Page 45: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/45.jpg)
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
![Page 46: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/46.jpg)
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
![Page 47: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/47.jpg)
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
![Page 48: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/48.jpg)
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
![Page 49: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/49.jpg)
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
![Page 50: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/50.jpg)
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
![Page 51: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/51.jpg)
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
![Page 52: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/52.jpg)
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
![Page 53: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/53.jpg)
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
![Page 54: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/54.jpg)
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
![Page 55: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/55.jpg)
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
![Page 56: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/56.jpg)
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
![Page 57: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/57.jpg)
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
![Page 58: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/58.jpg)
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
![Page 59: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/59.jpg)
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
![Page 60: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/60.jpg)
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
![Page 61: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/61.jpg)
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
![Page 62: A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of Efficiency in CVaR Portfolio OptimizationMembers: Mark Glad, Chen Zhang, Bowen Yu, Yiran](https://reader034.fdocuments.in/reader034/viewer/2022051919/600b24703f41d377bc20394b/html5/thumbnails/62.jpg)
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