A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of...

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

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

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).

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

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

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)

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)

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

Convolution Method

I h(z) = [z]+

I η(z) =

C exp( 1z2−1) if − 1 < z < 1

∫∞−∞ η(z)dz = 1

I ηε(z) =

Cε exp( 1

(z/ε)2−1) if − ε < z < ε

Convolution Method

I h(z) = [z]+

I η(z) =

C exp( 1z2−1) if − 1 < z < 1

∫∞−∞ η(z)dz = 1

I ηε(z) =

Cε exp( 1

(z/ε)2−1) if − ε < z < ε

Convolution Method

I h(z) = [z]+

I η(z) =

C exp( 1z2−1) if − 1 < z < 1

∫∞−∞ η(z)dz = 1

I ηε(z) =

Cε exp( 1

(z/ε)2−1) if − ε < z < ε

Convolution Method

I h(z) = [z]+

I η(z) =

C exp( 1z2−1) if − 1 < z < 1

∫∞−∞ η(z)dz = 1

I ηε(z) =

Cε exp( 1

(z/ε)2−1) if − ε < z < ε

Convolution Method

I h(z) = [z]+

I η(z) =

C exp( 1z2−1) if − 1 < z < 1

∫∞−∞ η(z)dz = 1

I ηε(z) =

Cε exp( 1

(z/ε)2−1) if − ε < z < ε

Convolution Method

φεβ(x,α)∈X×R

(x,α) = α+1

1 − β

(ηε ∗ h)(−xT y − α)p(y)dy

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

−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)

Convolution Method

φεβ(x,α)∈X×R

(x,α) = α+1

1 − β

(ηε ∗ h)(−xT y − α)p(y)dy

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

−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)

Piecewise Quadratic Approximation

I h(z) = [z]+

I ρε(z) =

0 if z 6 −ε

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].

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)

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)

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

11 − β

minx∈X

maxq∈Q

−qT Yx

Q = q ∈ RN : 1T q = 1, 0 6 q 61

1 − βp

problemminx∈X

maxQ∈QEQ[Yx]

Q = Q : 0 6∂Q

11 − β

minx∈X

maxq∈Q

−qT Yx

Q = q ∈ RN : 1T q = 1, 0 6 q 61

1 − βp

problemminx∈X

maxQ∈QEQ[Yx]

Q = Q : 0 6∂Q

11 − β

minx∈X

maxq∈Q

−qT Yx

Q = q ∈ RN : 1T q = 1, 0 6 q 61

1 − βp

problemminx∈X

maxQ∈QEQ[Yx]

Q = Q : 0 6∂Q

11 − β

minx∈X

maxq∈Q

−qT Yx

Q = q ∈ RN : 1T q = 1, 0 6 q 61

1 − βp

problemminx∈X

maxQ∈QEQ[Yx]

Q = Q : 0 6∂Q

11 − β

minx∈X

maxq∈Q

−qT Yx

Q = q ∈ RN : 1T q = 1, 0 6 q 61

1 − βp

problemminx∈X

maxQ∈QEQ[Yx]

Q = Q : 0 6∂Q

11 − β

minx∈X

maxq∈Q

−qT Yx

Q = q ∈ RN : 1T q = 1, 0 6 q 61

1 − βp

problemminx∈X

maxQ∈QEQ[Yx]

Q = Q : 0 6∂Q

11 − β

minx∈X

maxq∈Q

−qT Yx

Q = q ∈ RN : 1T q = 1, 0 6 q 61

1 − βp

I Nesterov Procedure

D1 ←12(1 +

D2 ←−1

1 − β(β lnβ+ (1 − β) ln (1 − β))

σ2 ←1β

, Ω← maxi‖Yi‖22

K ← 1ε

√ΩD1D2

σ2, µ← ε

x(0) ← 1q

1, d2(q)←N∑

qi ln qi+(pi

1 − β−qi) ln (

1 − β− qi)

D1 ←12(1 +

D2 ←−1

1 − β(β lnβ+ (1 − β) ln (1 − β))

σ2 ←1β

K ← 1ε

√ΩD1D2

σ2, µ← ε

x(0) ← 1q

1, d2(q)←N∑

qi ln qi+(pi

1 − β−qi) ln (

1 − β− qi)

D1 ←12(1 +

D2 ←−1

1 − β(β lnβ+ (1 − β) ln (1 − β))

σ2 ←1β

K ← 1ε

√ΩD1D2

σ2, µ← ε

x(0) ← 1q

1, d2(q)←N∑

qi ln qi+(pi

1 − β−qi) ln (

1 − β− qi)

D1 ←12(1 +

D2 ←−1

1 − β(β lnβ+ (1 − β) ln (1 − β))

σ2 ←1β

K ← 1ε

√ΩD1D2

σ2, µ← ε

x(0) ← 1q

1, d2(q)←N∑

qi ln qi+(pi

1 − β−qi) ln (

1 − β− qi)

D1 ←12(1 +

D2 ←−1

1 − β(β lnβ+ (1 − β) ln (1 − β))

σ2 ←1β

K ← 1ε

√ΩD1D2

σ2, µ← ε

x(0) ← 1q

1, d2(q)←N∑

qi ln qi+(pi

1 − β−qi) ln (

1 − β− qi)

D1 ←12(1 +

D2 ←−1

1 − β(β lnβ+ (1 − β) ln (1 − β))

σ2 ←1β

K ← 1ε

√ΩD1D2

σ2, µ← ε

x(0) ← 1q

1, d2(q)←N∑

qi ln qi+(pi

1 − β−qi) ln (

1 − β− qi)

for k ← 0 to K do

q(k) ← arg maxq∈Q

qT Yx(k) − µd2(q)

y(k+1) ← arg min

−(q(k))T Yy(k) +

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)

for k ← 0 to K do

y(k+1) ← arg min

−(q(k))T Yy(k) +

2µσ2‖y − x(k)‖22

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)

for k ← 0 to K do

y(k+1) ← arg min

−(q(k))T Yy(k) +

2µσ2‖y − x(k)‖22

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)

for k ← 0 to K do

y(k+1) ← arg min

−(q(k))T Yy(k) +

2µσ2‖y − x(k)‖22

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)

for k ← 0 to K do

y(k+1) ← arg min

−(q(k))T Yy(k) +

2µσ2‖y − x(k)‖22

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)

for k ← 0 to K do

y(k+1) ← arg min

−(q(k))T Yy(k) +

2µσ2‖y − x(k)‖22

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)

I Testing Results

I M: infinityI ε: error toleranceI final result: sensitive to εI Iyengar’s model works well, β = 0.99

I Testing Results

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.

I Main Idea:

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]+

Fβ(x,α) = α+ (1 − β)−1∫

min(x,α)∈X×R

Fβ(x,α)

Fβ(x,α) = α+1

q(1 − β)

q∑k=1

[f(x, yk ) − α]+

Fβ(x,α) = α+ (1 − β)−1∫

min(x,α)∈X×R

Fβ(x,α)

Fβ(x,α) = α+1

q(1 − β)

q∑k=1

[f(x, yk ) − α]+

Fβ(x,α) = α+ (1 − β)−1∫

min(x,α)∈X×R

Fβ(x,α)

Fβ(x,α) = α+1

q(1 − β)

q∑k=1

[f(x, yk ) − α]+

minimize α+1

q(1 − β)

q∑k=1

subject to xT y + α+ uk > 0uk > 0x > 01T x = 1µ(x) 6 −R.

minimize α+1

q(1 − β)

q∑k=1

subject to xT y + α+ uk > 0uk > 0x > 01T x = 1µ(x) 6 −R.

Comparison

Figure: Scenarios vs. CVaR Difference

Comparison

Figure: Scenarios vs. Runtime

Comparison

Figure: Assets vs. Runtime

Out of Sample Performance

I Reference: S&P 500, Google Finance

Thank you!

A Study of Efficiency in CVaR Portfolio Optimization(x, )2X R F~ (x, ) (4) Team One A Study of...

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