stochastic optimization with risk measures · stochastic optimization with risk measures ima new...

Stochastic Optimization with Risk Measures

IMA New Directions Short Course on Mathematical OptimizationJim Luedtke

Department of Industrial and Systems EngineeringUniversity of Wisconsin-Madison

August 8, 2016

Jim Luedtke (UW-Madison) Risk Measures Lecture Notes 1 / 25

Motivating Application: Portfolio Optimization

Let’s make some money!

Given a set of assets I, what fraction of my portfolio should I investeach asset to maximize return?

Return on asset i ∈ I is random: Ri ≥ 0

Let xi = fraction of portfolio invested in asset i

Constraints: Invest all, and no short-selling∑i∈I xi = 1, x ≥ 0

What is optimal solution if we want to maximize expected value?

max E[∑

i∈I Rixi

]=∑

i∈I E[Ri]xi

Invest everything in one asset with highest expected return!

Anybody see a problem with that solution?


Mean-Variance Model

Markowitz key observation:

Variance of return is also important

For given expected return, want portfolio with smallest variance (orvice versa)

How to deal with two objectives?

Constrain expected return above L, minimize variance (vary L)

Constraint variance below L, maximize expected return (vary L)

X Maximize expected return less λ*variance (vary λ)


Mean-Variance Model

Random returns Ri, i ∈ I:

Expected return ri, i ∈ IVariance Σii, Covariance Σij

What is variance of return if xi is invested in asset i?

Recall V(Z) = E[(Z − µZ)2

], Cov(Z, Y ) = E[(Z − µZ)(Y − µY )]

E[(∑

i

Rixi −∑i

rixi)2]

=E[∑

i

∑j

(Ri − ri)(Rj − rj)xixjRiRj]

=∑i,j

Cov(Ri, Rj)xixj = x>Σx

where Σ is the variance-covariance matrix


Mean-Variance Model

Mean-variance model:

maxx>r︸︷︷︸

E[f(x,ξ)]

−λ x>Σx︸︷︷︸V[f(x,ξ)]

: e>x = 1, x ≥ 0

Σ 0⇒ Convex optimization model

This is one example of a mean-risk optimization model

Possible drawback: Variance is symmetric ⇒ Penalizes high returns aswell as low returns

If returns are joint normally distributed, portfolio return is normal ⇒FineBut is normal a reasonable distribution of return?


Value at Risk

Another popular quantification of risk used in finance

α Value at Risk (VaR):

VaRα(Z) = inft : P[Z ≤ t] ≥ α= inft : FZ(t) ≥ α

Constraint on Value at Risk is equivalent to a chance constraint:

VaRα(f(x, ξ)) ≤ 0⇔ P[f(x, ξ) ≤ 0] ≥ α.

NB: In portfolio problem, evaluate VaR of loss: f(x, ξ) = −R>x


Risk Measures More Generally

Classic stochastic programming model:

minE[f(x, ξ)] : x ∈ X

Replace expected value with a risk measure:

minρ[f(x, ξ)] : x ∈ X

where ρ is a risk measure:

Maps a random variable to a number

E.g., ρ(Z) = E[Z], ρ(Z) = E[Z] + λV(Z), ρ(Z) = E[Z] + λVaR(Z),ρ(Z) = E[u(Z)], where u is a “disutility function”


Average Value at Risk

Another popular risk measure, often called “Conditional Value at Risk”

Minimizing, so wish to measure/limit risk of large values

Definition

α Average value at risk (AVaR)

AVaRα(Z)def= min

y∈R

y +

1

1− αE[Z − y]+

Whaa?!?Theorem (Intuitive Definition)

If, P(Z = VaRα(Z)) = 0, then

AVaRα(Z) = E[Z | Z ≥ VaRα(Z)]


Proving Equivalent Definition

Lemma

Assume P(Z = VaRα(Z)) = 0. Then

E[Z | Z ≥ VaRα(Z)] = VaRα(Z) +1

1− αE[Z − VaRα(Z)]+.

AVaRα(Z)def= min

y∈R

y +

1

1− αE[Z − y]+

(*)

Using Lemma,AVaRα(Z) = E[Z | Z ≥ VaRα(Z)]

follows if we can show v := VaRα(Z) is a minimizer of (*).


Optimizing Average Value at Risk

Assume finite support distribution: P(ξ = ξs) = ps, s = 1, . . . , S

minxAVaRα(f(x, ξ)) : x ∈ X

= minx,y

y +

1

1− αE[f(x, ξ)− y]+ : x ∈ X, y ∈ R

= min

x,yy +

1

1− α

S∑s=1

psws

s.t. ws ≥ f(x, ξs)− y, s = 1, . . . , S

w ≥ 0, y ∈ R, x ∈ X

Convex program when X is convex and f(x, ξs) is convex for each s


Properties of Average Value at Risk

Assume (only for simplicity) probability space is finite: |Ω| = S.

Rndom variable Z ⇔ vector z ∈ RS .

AVaRα: Function from RS → R.

Theorem

ρ(·) = AVaRα(·) satisfies the following properties:

(A1) Convexity. β ∈ (0, 1), Z1 and Z2 rvs ⇒

ρ(βZ1 + (1− β)Z2) ≤ βρ(Z1) + (1− β)ρ(Z2)

(A2) Monotonicity. Z1 ≥ Z2 a.s. ⇒ ρ(Z1) ≥ ρ(Z2)

(A3) Translation invariance. If Z is a r.v. and a ∈ R, thenρ(Z + a) = ρ(Z) + a.

(A4) Positive homogeneity. ρ(tZ) = tρ(Z) ∀t > 0.

Reference: Rockafellar and Uryasev [2000]


Coherent Risk Measures

AVaRα is an example of a coherent risk measure

Definition

A risk measure ρ(·) is called coherent if it satisfies the following proper-ties:

(A1) Convexity. β ∈ (0, 1), Z1 and Z2 rvs ⇒

ρ(βZ1 + (1− β)Z2) ≤ βρ(Z1) + (1− β)ρ(Z2)

(A2) Monotonicity. Z1 ≥ Z2 a.s. ⇒ ρ(Z1) ≥ ρ(Z2)

(A3) Translation invariance. If Z is a r.v. and a ∈ R, thenρ(Z + a) = ρ(Z) + a.

(A4) Positive homogeneity. ρ(tZ) = tρ(Z) ∀t > 0.

References: Artzner et al. [1999], Ruszczynski and Shapiro [2006]


Distributional Robustness

Recall assumption: |Ω| = S

Distribution P⇔ vector µ in RS+ with∑

s µs = 1

Random variable Z ⇔ vector z ∈ RS

In practice, distribution µ of random variable is not known with certainty

Assume instead µ ∈ A, where A is a collection of distributions

Find minimum cost over worst case distribution

Distributionally Robust Stochastic Optimization

minx∈X

maxµ∈A

Eµ[f(x, ξ)]


Distributional Robustness ⇒ Coherent Risk Measures

Suppose we are given a family of distributions ALet ρA(Z) := maxµ∈A Eµ[Z] (= maxµ∈A

∑s µszs)

Then DRO problem is equivalent to:

minx∈X

ρA[f(x, ξ)]

Distributional robustness provides a means to define a risk measure

NB: Evaluating ρA(Z) is tractable ⇔ A is a tractable convex set

Theorem

Suppose e>µ = 1 and µ ≥ 0 for all µ ∈ A. Then ρA is a coherent riskmeasure.


Coherent Risk Measures ⇒ Distributional Robustness

Theorem

Suppose ρ is a coherent risk measure. Then there exists a convex set Asuch that for any random variable Z

ρ[Z] = maxµ∈A

Eµ[Z]

where for each µ ∈ A,∑

s µs = 1 and µ ≥ 0.

Example: Consider random variable with P(Z = zs) = ps, where p ∈ RS+and

∑s ps = 1

AVaRα(Z) = maxµ∈Aα

Eµ[Z]

where Aα = µ ∈ RS+ :∑

s µs = 1, µs ≤ ps/(1− α)


Proof Background

Given a function ρ : RS → R, its conjugate ρ∗ : RS → R is:

ρ∗(µ) = maxz∈RSµ>z − ρ(z)

Let A := µ ∈ RS : ρ∗(µ) < +∞ρ∗ is a convex function and A is a convex set

Theorem

If ρ is a convex function, then

ρ(z) = maxµ∈Aµ>z − ρ∗(µ).


Another Example: Mean α-Semi-deviation

Let α ≥ 1, and for λ ≥ 0 define:

ρλ[Z] = E[Z] + λE[(Z − E[Z])α+

]1/αFor discrete distribution with P(Z = zs) = ps, s = 1, . . . , S

ρλ[Z] = p>z + λ(∑

s

ps(zs −∑i

pizi)α+

)1/αTheorem

If λ ∈ [0, 1] then ρλ(·) is a coherent risk measure.

The dispersion part by itself is NOT a coherent risk measure (itviolates monotonicity).

Proof of monotonicity of ρλ(·) is nontrivial.


Properties of Other Risk Measures

Mean Variance: ρMV [Z] = E[Z] + λV[Z]

Convex? Yes

Monotone? No(!)

Translation invariant? Yes

Positively homogeneous? No

Mean Standard-deviation: ρMS [Z] = E[Z] + λV(Z)1/2

Convex? Yes

Monotone? No(!)


Positively homogeneous? Yes


Nonmonotonicity Example

Suppose there are two outcomes: P(1) = 0.95, P(2) = 0.05

ω Z1 Z2

1 10 102 10 0

E 10 9.5V 0 4.75

V1/2 0 2.18

Z1 ≥ Z2 with probability 1, but mean variance and meanstandard-deviation measures would prefer Z1 for modest values of λ


Properties of Risk Measures

VaRα(Z):

Convex? No

Monotone? Yes


Positively homogeneous? Yes, if P(Z ≥ 0) = 1


Nonconvexity of Value at Risk

Suppose there are three equally likely outcomes

ω Z1 Z2 12Z

1 + 12Z

2

1 300 0 1502 0 0 03 0 300 150

VaR0.6 0 0 150E 100 100 100

12VaR0.6(Z

1) + 12VaR0.6(Z

1) < VaR0.6(12Z

1 + 12Z

2)

Diversified “portfolio” looks worse, despite having same expectedvalue


Disutility Based Risk Measure

Let g : R→ R be a convex increasing function. Let

ρg[Z] = E[g(Z)]

Properties:

Convex? Yes

Montone? Yes

Translation invariant? No. (Unless g(z) ≡ z.)

Positively homogeneous? Yes, if g is positively homogeneous.


Modified Disutility Based Risk Measure

Let g : R→ R be a convex increasing function. Let

ρg[Z]def= inf

tE[Z + g(Z − t)]

Properties:

Convex? Yes

Monotone? Yes

Translation invariant? Yes!

Positively homgeneous? Yes, if g is positively homgeneous.


Optimizing Risk Measures

Now consider the problem:

minρ[f(x, ξ)] : x ∈ X

Theorem

Assume ρ is a convex and monotone risk function and f(x, ξs) is convexfor s = 1, . . . , S. Then φ(x) := ρ[f(x, ξ)] is a convex function of x.

Let h : Rn → RS , be defined by hs(z) = f(x, ξs).

h is a convex function. ρ is a convex and increasing function.

⇒ φ(x) = ρ(h(x)) is convex.


Optimizing Risk Measures

Now consider the problem:

minφ(x) = ρ[f(x, ξ)] : x ∈ X

Let ρ be a coherent risk measure with representation:

ρ(Z) = maxµ∈A

Eµ[Z]

Assume f(x, ξs) is convex for each s = 1, . . . , S.

Subgradients

Let x ∈ X and µ ∈ argmaxµ∈A∑

s µsf(x, ξs), and let ds be a sub-gradient of f(x, ξs) at x for s ∈ Ω. Then∑

s

µsds

is a subgradient of φ(x) at x.


Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath.Coherent measures of risk. Mathematical finance, 9(3):203–228, 1999.

R.T. Rockafellar and S. Uryasev. Optimization of conditional value-at-risk.Journal of Risk, 2:21–41, 2000.

A. Ruszczynski and A. Shapiro. Optimization of convex risk functions.Mathematics of operations research, 31(3):433–452, 2006.


stochastic optimization with risk measures · stochastic optimization with risk measures ima new...

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