portfolio choice via quantilesorfe.princeton.edu/op5/slides/he.pdf · f = {f(·) : r → [0,1] |...
TRANSCRIPT
![Page 1: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/1.jpg)
Portfolio Choice via Quantiles
Xuedong He
Oxford
Princeton University/March 28, 2009
Based on the joint work with Prof Xunyu Zhou
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 1 / 16
![Page 2: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/2.jpg)
Models of Risk Choice
Study on continuous-time portfolio choice has predominantly centredaround expected utility maximisation
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 2 / 16
![Page 3: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/3.jpg)
Models of Risk Choice
Study on continuous-time portfolio choice has predominantly centredaround expected utility maximisation
However, some of the basic tenets of expected utility aresystematically violated in practice
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 2 / 16
![Page 4: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/4.jpg)
Models of Risk Choice
Study on continuous-time portfolio choice has predominantly centredaround expected utility maximisation
However, some of the basic tenets of expected utility aresystematically violated in practice
Many alternative preferences have been put forth, especially inbehavioural finance
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 2 / 16
![Page 5: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/5.jpg)
Models of Risk Choice
Study on continuous-time portfolio choice has predominantly centredaround expected utility maximisation
However, some of the basic tenets of expected utility aresystematically violated in practice
Many alternative preferences have been put forth, especially inbehavioural finance
Yaari’s “dual theory of choice” [Yaari (1987)]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 2 / 16
![Page 6: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/6.jpg)
Models of Risk Choice
Study on continuous-time portfolio choice has predominantly centredaround expected utility maximisation
However, some of the basic tenets of expected utility aresystematically violated in practice
Many alternative preferences have been put forth, especially inbehavioural finance
Yaari’s “dual theory of choice” [Yaari (1987)]Kahneman and Tversky’s prospect theory [Kahneman and Tversky(1979), Tversky and Kahneman (1992)]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 2 / 16
![Page 7: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/7.jpg)
Models of Risk Choice
Study on continuous-time portfolio choice has predominantly centredaround expected utility maximisation
However, some of the basic tenets of expected utility aresystematically violated in practice
Many alternative preferences have been put forth, especially inbehavioural finance
Yaari’s “dual theory of choice” [Yaari (1987)]Kahneman and Tversky’s prospect theory [Kahneman and Tversky(1979), Tversky and Kahneman (1992)]Lopes’ SP/A theory [Lopes (1987) and Lopes and Oden (1999)]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 2 / 16
![Page 8: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/8.jpg)
Alternative Models
Another large set of portfolio choice problems involve probability andVaR/CVaR, instead of expectation, in objectives and/or constraints
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 3 / 16
![Page 9: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/9.jpg)
Alternative Models
Another large set of portfolio choice problems involve probability andVaR/CVaR, instead of expectation, in objectives and/or constraints
Goal achieving problem [Kulldorff (1993), Heath (1993) and Browne(1999)]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 3 / 16
![Page 10: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/10.jpg)
Alternative Models
Another large set of portfolio choice problems involve probability andVaR/CVaR, instead of expectation, in objectives and/or constraints
Goal achieving problem [Kulldorff (1993), Heath (1993) and Browne(1999)]
VaR/CVaR [Rockafellar and Uryasev (2000)]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 3 / 16
![Page 11: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/11.jpg)
Alternative Models
Another large set of portfolio choice problems involve probability andVaR/CVaR, instead of expectation, in objectives and/or constraints
Goal achieving problem [Kulldorff (1993), Heath (1993) and Browne(1999)]
VaR/CVaR [Rockafellar and Uryasev (2000)]
Law-invariant coherent risk measure [Artzner, Delbaen, Eber and Heath(1999) and Kusuoka (2001)]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 3 / 16
![Page 12: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/12.jpg)
Portfolio Choice with New Preferences
Some of them have been studied case by case, and many of them arecompletely unexplored
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 4 / 16
![Page 13: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/13.jpg)
Portfolio Choice with New Preferences
Some of them have been studied case by case, and many of them arecompletely unexplored
Main difficulties include the non-concavity and time-inconsistency
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 4 / 16
![Page 14: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/14.jpg)
Portfolio Choice with New Preferences
Some of them have been studied case by case, and many of them arecompletely unexplored
Main difficulties include the non-concavity and time-inconsistency
In this work, we propose a new framework to accommodate most ofthe aforementioned preferences and develop a new technique to solvethe model
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 4 / 16
![Page 15: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/15.jpg)
Portfolio Selection in Continuous Time
Continuous time market
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 5 / 16
![Page 16: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/16.jpg)
Portfolio Selection in Continuous Time
Continuous time market
Tame portfolios
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 5 / 16
![Page 17: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/17.jpg)
Portfolio Selection in Continuous Time
Continuous time market
Tame portfolios
Arbitrage-free and complete market
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 5 / 16
![Page 18: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/18.jpg)
Portfolio Selection in Continuous Time
Continuous time market
Tame portfolios
Arbitrage-free and complete market
Dynamic portfolio selection can be translated into a static problem ofchoosing the optimal terminal payoff
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 5 / 16
![Page 19: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/19.jpg)
Portfolio Selection in Continuous Time
Continuous time market
Tame portfolios
Arbitrage-free and complete market
Dynamic portfolio selection can be translated into a static problem ofchoosing the optimal terminal payoff
“The more money the better”
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 5 / 16
![Page 20: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/20.jpg)
A Non-Expected Utility Maximisation Model
We consider the following portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 6 / 16
![Page 21: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/21.jpg)
A Non-Expected Utility Maximisation Model
We consider the following portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
u(·): utility function; T (·): distortion function
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 6 / 16
![Page 22: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/22.jpg)
A Non-Expected Utility Maximisation Model
We consider the following portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
u(·): utility function; T (·): distortion function
F is the set of distribution functions consistent with tame portfolios
F = {F (·) : R → [0, 1] | F (·) is increasing, cadlag and F (c) = 0 for some c ∈ R}
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 6 / 16
![Page 23: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/23.jpg)
A Non-Expected Utility Maximisation Model
We consider the following portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
u(·): utility function; T (·): distortion function
F is the set of distribution functions consistent with tame portfolios
F = {F (·) : R → [0, 1] | F (·) is increasing, cadlag and F (c) = 0 for some c ∈ R}
D is a subset of F, specifying the constraints imposed on the terminalpayoff
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 6 / 16
![Page 24: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/24.jpg)
A Non-Expected Utility Maximisation Model
We consider the following portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
u(·): utility function; T (·): distortion function
F is the set of distribution functions consistent with tame portfolios
F = {F (·) : R → [0, 1] | F (·) is increasing, cadlag and F (c) = 0 for some c ∈ R}
D is a subset of F, specifying the constraints imposed on the terminalpayoff
Both preference and constraints (other than the initial budgetconstraint) are law-invariant
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 6 / 16
![Page 25: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/25.jpg)
Examples
Expected Utility:∫ ∞
0
u(x)dFX(x)
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 7 / 16
![Page 26: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/26.jpg)
Examples
Expected Utility:∫ ∞
0
u(x)dFX(x)
Goal Achieving:∫ ∞
0
1{x≥b}dFX(x)
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 7 / 16
![Page 27: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/27.jpg)
Examples
Expected Utility:∫ ∞
0
u(x)dFX(x)
Goal Achieving:∫ ∞
0
1{x≥b}dFX(x)
Yaari:∫ ∞
0
xd [−T (1 − FX(x))]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 7 / 16
![Page 28: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/28.jpg)
Examples
Expected Utility:∫ ∞
0
u(x)dFX(x)
Goal Achieving:∫ ∞
0
1{x≥b}dFX(x)
Yaari:∫ ∞
0
xd [−T (1 − FX(x))]
SP/A:∫ ∞
0
xd [−T (1 − FX(x))]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 7 / 16
![Page 29: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/29.jpg)
Examples
Expected Utility:∫ ∞
0
u(x)dFX(x)
Goal Achieving:∫ ∞
0
1{x≥b}dFX(x)
Yaari:∫ ∞
0
xd [−T (1 − FX(x))]
SP/A:∫ ∞
0
xd [−T (1 − FX(x))]
Prospect Theory:∫ ∞
B
u+(x − B)d [−T+ (1 − FX(x))] −
∫ B
−∞
u−(B − x)d [T− (FX(x))]
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 7 / 16
![Page 30: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/30.jpg)
Change the Decision Variable
The portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 8 / 16
![Page 31: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/31.jpg)
Change the Decision Variable
The portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
The major difficulty comes from the distortion function
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 8 / 16
![Page 32: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/32.jpg)
Change the Decision Variable
The portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
The major difficulty comes from the distortion function
Change of variable z = FX(x)
∫
∞
−∞
u(x)d [−T (1 − FX(x))] =
∫ 1
0u
(
F−1X (z)
)
T ′(1 − z)dz
= E[
u(F−1X (Z))T ′(1 − Z)
]
where Z ∼ U(0, 1)
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 8 / 16
![Page 33: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/33.jpg)
Change the Decision Variable
The portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
The major difficulty comes from the distortion function
Change of variable z = FX(x)
∫
∞
−∞
u(x)d [−T (1 − FX(x))] =
∫ 1
0u
(
F−1X (z)
)
T ′(1 − z)dz
= E[
u(F−1X (Z))T ′(1 − Z)
]
where Z ∼ U(0, 1)
If we regard F−1X (·) as the variable, the distortion function is
separated and we restore the concavity if u(·) is concave
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 8 / 16
![Page 34: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/34.jpg)
Change the Decision Variable (Cont’d)
This suggests that it is better to regard the quantile function F−1X (·)
as the decision variable. It works in the objective function
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 9 / 16
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Change the Decision Variable (Cont’d)
This suggests that it is better to regard the quantile function F−1X (·)
as the decision variable. It works in the objective function
It also works in the constraints
FX(·) ∈ F ∩ D ⇔ F−1X (·) ∈ G ∩ M
where
G := {G(·) : (0, 1) → R | G(·) is increasing, Caglad and G(0+) > −∞}
and M is a subset of G
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Change the Decision Variable (Cont’d)
This suggests that it is better to regard the quantile function F−1X (·)
as the decision variable. It works in the objective function
It also works in the constraints
FX(·) ∈ F ∩ D ⇔ F−1X (·) ∈ G ∩ M
where
G := {G(·) : (0, 1) → R | G(·) is increasing, Caglad and G(0+) > −∞}
and M is a subset of G
It, however, does not work in the budget constraint
E [ρX] ≤ x0
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 9 / 16
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Dual Argument
A dual argument is applied to dealing with the budget constraint
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Dual Argument
A dual argument is applied to dealing with the budget constraint
Primal: to maximise the performance of the investment
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 10 / 16
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Dual Argument
A dual argument is applied to dealing with the budget constraint
Primal: to maximise the performance of the investment
Dual: to minimise the cost (budget) while keeping the performanceat some level
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 10 / 16
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Dual Argument
A dual argument is applied to dealing with the budget constraint
Primal: to maximise the performance of the investment
Dual: to minimise the cost (budget) while keeping the performanceat some level
The performance only depends on the distribution of the terminalpayoff, thus the dual problem is to minimise the cost of replicatingthe terminal payoffs following a given distribution
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 10 / 16
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Dual Argument
A dual argument is applied to dealing with the budget constraint
Primal: to maximise the performance of the investment
Dual: to minimise the cost (budget) while keeping the performanceat some level
The performance only depends on the distribution of the terminalpayoff, thus the dual problem is to minimise the cost of replicatingthe terminal payoffs following a given distribution
Given a distribution function F (·), formulate the following dualproblem
MinX
E [ρX]
Subject to X is F (·) distributed
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 10 / 16
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Terminal Payoff with Minimal Cost
Lemma (Jin and Zhou 2008)
If ρ has no atom, then Z := 1 − Fρ(ρ) is uniformly distributed and
E[
ρF−1(Z)]
≤ E [ρX] for any F (·) distributed r.v. X. Moreover, if
E[
ρF−1(Z)]
< ∞, then the inequality is equality iff X = F−1(Z)
X = F−1(Z) = F−1(1 − Fρ(ρ)), where Z ∼ U(0, 1), uniquely solvesthe dual problem
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 11 / 16
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Terminal Payoff with Minimal Cost
Lemma (Jin and Zhou 2008)
If ρ has no atom, then Z := 1 − Fρ(ρ) is uniformly distributed and
E[
ρF−1(Z)]
≤ E [ρX] for any F (·) distributed r.v. X. Moreover, if
E[
ρF−1(Z)]
< ∞, then the inequality is equality iff X = F−1(Z)
X = F−1(Z) = F−1(1 − Fρ(ρ)), where Z ∼ U(0, 1), uniquely solvesthe dual problem
The assumption ρ is atomless is crucial and we will assume it in thefollowing context
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 11 / 16
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Terminal Payoff with Minimal Cost
Lemma (Jin and Zhou 2008)
If ρ has no atom, then Z := 1 − Fρ(ρ) is uniformly distributed and
E[
ρF−1(Z)]
≤ E [ρX] for any F (·) distributed r.v. X. Moreover, if
E[
ρF−1(Z)]
< ∞, then the inequality is equality iff X = F−1(Z)
X = F−1(Z) = F−1(1 − Fρ(ρ)), where Z ∼ U(0, 1), uniquely solvesthe dual problem
The assumption ρ is atomless is crucial and we will assume it in thefollowing context
This dual problem dates back to Dybvig (1988) and is revived in Jinand Zhou (2008)
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 11 / 16
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Change the Budget Constraint
We only need to consider the terminal payoff in the form of F−1(Z)where Z = 1 − Fρ(ρ)
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 12 / 16
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Change the Budget Constraint
We only need to consider the terminal payoff in the form of F−1(Z)where Z = 1 − Fρ(ρ)
Rewrite the budget constraint
x ≥ E[
ρF−1(Z)]
= E[
F−1ρ (1 − Z)F−1(Z)
]
(F−1ρ (Fρ(ρ)) = ρ, a.s.)
=
∫ 1
0F−1
ρ (1 − z)F−1(z)dz
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 12 / 16
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Quantile Formulation
Recall the portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
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Quantile Formulation
Recall the portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
Let G(·) := F−1(·)
MaxG(·)
U(G(·)) :=∫ 10 u(G(z))T ′(1 − z)dz
Subject to G(·) ∈ G ∩ M,∫ 10 F−1
ρ (1 − z)G(z)dz ≤ x0
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 13 / 16
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Quantile Formulation
Recall the portfolio selection problem
MaxX
V (X) :=∫
∞
−∞u(x)d [−T (1 − FX(x))]
Subject to FX(·) ∈ F ∩ D,
E [ρX] ≤ x0
Let G(·) := F−1(·)
MaxG(·)
U(G(·)) :=∫ 10 u(G(z))T ′(1 − z)dz
Subject to G(·) ∈ G ∩ M,∫ 10 F−1
ρ (1 − z)G(z)dz ≤ x0
We call it quantile formulation
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 13 / 16
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Quantile formulation (Cont’d)
If X∗ is optimal to the portfolio selection problem, then F−1X∗ (·) is
optimal to the quantile formulation
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Quantile formulation (Cont’d)
If X∗ is optimal to the portfolio selection problem, then F−1X∗ (·) is
optimal to the quantile formulation
If G∗(·) is optimal to the quantile formulation, thenG∗(Z) = G∗(1 − Fρ(ρ)) is optimal to the portfolio selection problem
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Quantile formulation (Cont’d)
If X∗ is optimal to the portfolio selection problem, then F−1X∗ (·) is
optimal to the quantile formulation
If G∗(·) is optimal to the quantile formulation, thenG∗(Z) = G∗(1 − Fρ(ρ)) is optimal to the portfolio selection problem
The optimal solution to the portfolio selection problem must beanti-comonotonic w.r.t the pricing kernel ρ
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Quantile formulation (Cont’d)
If X∗ is optimal to the portfolio selection problem, then F−1X∗ (·) is
optimal to the quantile formulation
If G∗(·) is optimal to the quantile formulation, thenG∗(Z) = G∗(1 − Fρ(ρ)) is optimal to the portfolio selection problem
The optimal solution to the portfolio selection problem must beanti-comonotonic w.r.t the pricing kernel ρ
Solvable by Lagrange
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 14 / 16
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Quantile formulation (Cont’d)
If X∗ is optimal to the portfolio selection problem, then F−1X∗ (·) is
optimal to the quantile formulation
If G∗(·) is optimal to the quantile formulation, thenG∗(Z) = G∗(1 − Fρ(ρ)) is optimal to the portfolio selection problem
The optimal solution to the portfolio selection problem must beanti-comonotonic w.r.t the pricing kernel ρ
Solvable by Lagrange
All the aforementioned examples can be solved explicitly
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 14 / 16
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Conclusions
Motivated by behavioural finance, we formulate a generalnon-expected utility maximisation problem that covers many models(rational or irrational)
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Conclusions
Motivated by behavioural finance, we formulate a generalnon-expected utility maximisation problem that covers many models(rational or irrational)
We turn the problem into an equivalent optimisation problem —quantile formulation where quantiles serve as the decision variable
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 15 / 16
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Conclusions
Motivated by behavioural finance, we formulate a generalnon-expected utility maximisation problem that covers many models(rational or irrational)
We turn the problem into an equivalent optimisation problem —quantile formulation where quantiles serve as the decision variable
The key is a dual argument in which we minimise the cost whilekeeping the performance
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 15 / 16
![Page 58: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/58.jpg)
Conclusions
Motivated by behavioural finance, we formulate a generalnon-expected utility maximisation problem that covers many models(rational or irrational)
We turn the problem into an equivalent optimisation problem —quantile formulation where quantiles serve as the decision variable
The key is a dual argument in which we minimise the cost whilekeeping the performance
The problem can be solved by Lagrange
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 15 / 16
![Page 59: Portfolio Choice via Quantilesorfe.princeton.edu/op5/slides/he.pdf · F = {F(·) : R → [0,1] | F(·) is increasing, c`adl`ag and F(c) = 0 for some c ∈ R} D is a subset of F, specifying](https://reader033.vdocument.in/reader033/viewer/2022043012/5faaa890f621b114aa781686/html5/thumbnails/59.jpg)
Conclusions
Motivated by behavioural finance, we formulate a generalnon-expected utility maximisation problem that covers many models(rational or irrational)
We turn the problem into an equivalent optimisation problem —quantile formulation where quantiles serve as the decision variable
The key is a dual argument in which we minimise the cost whilekeeping the performance
The problem can be solved by Lagrange
Incomplete market case can also be dealt with and mutual fundtheorem is derived
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Conclusions
The idea of quantile formulation works far beyond our specificnon-expected utility model
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Conclusions
The idea of quantile formulation works far beyond our specificnon-expected utility model
It essentially only needs the following two assumptions
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Conclusions
The idea of quantile formulation works far beyond our specificnon-expected utility model
It essentially only needs the following two assumptions
Preference and constraints (other than budget constraint) of theportfolio selection problem only depend on the distribution of theterminal payoff
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 16 / 16
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Conclusions
The idea of quantile formulation works far beyond our specificnon-expected utility model
It essentially only needs the following two assumptions
Preference and constraints (other than budget constraint) of theportfolio selection problem only depend on the distribution of theterminal payoffThe more money one starts with, the higher performance one achieves
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 16 / 16
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Conclusions
The idea of quantile formulation works far beyond our specificnon-expected utility model
It essentially only needs the following two assumptions
Preference and constraints (other than budget constraint) of theportfolio selection problem only depend on the distribution of theterminal payoffThe more money one starts with, the higher performance one achieves
The quantile formulation can be applied to all the aforementionedmodels
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 16 / 16
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Conclusions
The idea of quantile formulation works far beyond our specificnon-expected utility model
It essentially only needs the following two assumptions
Preference and constraints (other than budget constraint) of theportfolio selection problem only depend on the distribution of theterminal payoffThe more money one starts with, the higher performance one achieves
The quantile formulation can be applied to all the aforementionedmodels
Prospect theory has been solved in Jin and Zhou (2008)
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 16 / 16
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Conclusions
The idea of quantile formulation works far beyond our specificnon-expected utility model
It essentially only needs the following two assumptions
Preference and constraints (other than budget constraint) of theportfolio selection problem only depend on the distribution of theterminal payoffThe more money one starts with, the higher performance one achieves
The quantile formulation can be applied to all the aforementionedmodels
Prospect theory has been solved in Jin and Zhou (2008)
SP/A model and model with law-invariant coherent risk measure havebeen solved by He and Zhou recently
Xuedong He (Oxford) Portfolio Choice via Quantiles March 28, 2009 16 / 16