var vs cvar carisma conference 2010

8/20/2019 VaR vs CVaR CARISMA Conference 2010

1/75

Stan Uryasev

Joint presentation with

Sergey Sarykalin, Gaia Serraino and Konstantin Kalinchenko

Risk Management and Financial Engineering Lab, University of Floridaand

American Optimal Decisions

VaR vs CVaR in

Risk Management and Optimization

1


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Agenda

Compare Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)

definitions of VaR and CVaR

basic properties of VaR and CVaR

axiomatic definition of Risk and Deviation Measures

reasons affecting the choice between VaR and CVaR

risk management/optimization case studies conducted with

Portfolio Safeguard package by AORDA.com

2


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Risk Management

Risk Management is a procedure for shaping a loss distribution

Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are

popular function for measuring risk The choice between VaR and CVaR is affected by:

differences in mathematical properties,

stability of statistical estimation, simplicity of optimization procedures,

acceptance by regulators

Conclusions from these properties are contradictive

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Risk Management

Key observations:

CVaR has superior mathematical properties versus VaR

Risk management with CVaR functions can be done very efficiently

VaR does not control scenarios exceeding VaR

CVaR accounts for losses exceeding VaR

Deviation and Risk are different risk management concepts

CVaR Deviation is a strong competitor to the Standard Deviation

4


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VaR and CVaR Representation

5


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x

Risk

VaR

CVaR

CVaR+

CVaR-

VaR, CVaR, CVaR+ and CVaR-

6


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Value-at-Risk

is non convex and discontinuous function of the

confidence level α for discrete distributions

is non-sub-additive

difficult to control/optimize for non-normal distributions:VaR has many extremums for discrete distributions

7

[1,0]∈α })(|min{)( α α ≥= zF z X VaR X for

)( X VaRα

X a loss random variable

)( X VaRα


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Conditional Value-at-Risk

Rockafellar and Uryasev, “Optimization of Conditional Value-at-Risk”,

Journal of Risk, 2000 introduced the term Conditional Value-at-

Risk

For

where

8

∫+∞

∞−

= )()( z zdF X CVaR X α

α [1,0]∈α

⎪⎩

⎪

⎨

⎧

≥−

−

<

= )( when1

)(

)(nwhe 0

)( X VaR z zF

X VaR z

zF X X α

α α

α α


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CVaR+

(Upper CVaR): expected value of X strictly exceeding VaR(also called Mean Excess Loss and Expected Shortfall)

CVaR- (Lower CVaR): expected value of X weakly exceeding VaR(also called Tail VaR)

Property: is weighted average of and

zero for continuous distributions!!! 9

)](|[)( X VaR X X E X CVaR α α >=+

)( X C V a Rα ( )C V aR X α +

( )VaR X α

( ) ( ) (1 ( )) ( ) if ( ( )) 1( )

( ) if ( ( )) 1 X

X

X VaR X X CVaR X F VaR X CVaR X

VaR X F VaR X α α α α α α

α α

λ λ +⎧ + −


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Definition on previous page is a major innovation

and for general loss distributions are

discontinuous functions

CVaR is continuous with respect to α

CVaR is convex in X

VaR, CVaR- ,CVaR+ may be non-convex

VaR ≤ CVaR- ≤ CVaR ≤ CVaR+

10

)( X CVaR+

α )( X VaRα


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x

Risk

VaR

CVaR

CVaR+

CVaR-

VaR, CVaR, CVaR+ and CVaR-

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CVaR: Discrete Distributions

α does not “split” atoms: VaR < CVaR- < CVaR = CVaR+,λ = (Ψ- α)/(1- α) = 0

12

1 2 41 2 6 6 3 6

1 15 62 2

Six scenarios, ,

CVaR CVaR =

p p p

f f

α = = = = = =

= +

L

+

Probability CVaR

16

16

16

16

16

16

1 f

2 f

3 f 4 f 5 f 6 f

VaR --CVaR

+CVaR

Loss


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• α “splits” the atom: VaR < CVaR- < CVaR < CVaR+,λ = (Ψ- α)/(1- α) > 0

13

711 2 6 6 12

1 4 1 2 24 5 65 5 5 5 5

Six scenarios, ,

CVaR VaR CVaR =

α = = = = =

= + + +

L p p p

f f f +

Probability CVaR

16

16

16

16

112

16

16

1 f 2 f 3 f 4 f 5 f 6 f

VaR

--

CVaR

+1 1

5 62 2

CVaR + = f f Loss


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• α “splits” the last atom: VaR = CVaR- = CVaR ,CVaR+ is not defined, λ = (Ψ - α)/(1- α) > 0

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CVaR: Equivalent Definitions

Pflug defines CVaR via an optimization problem, as in Rockafellar

and Uryasev (2000)

Acerbi showed that CVaR is equivalent to Expected Shortfalldefined by

15

Pflug, G.C., “Some Remarks on the Value-at-Risk and on the Conditional Value-at-Risk”, Probabilistic

Constrained Optimization: Methodology and Applications, (Uryasev ed), Kluwer, 2000

Acerbi, C., “Spectral Measures of Risk: a coherent representation of subjective risk aversion”, JBF, 2002


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RISK MANAGEMENT: INSURANCE

Accident lost

Payment

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∞ ∞

Premium

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TWO CONCEPTS OF RISK

Risk as a possible loss

Minimum amount of cash to be added to make a portfolio (orproject) sufficiently safe

Example 1. MaxLoss

-Three equally probable outcomes, { -4, 2, 5 };

MaxLoss = -4; Risk = 4

-Three equally probable outcomes, { 0, 6, 9 };

MaxLoss = 0; Risk = 0

Risk as an uncertainty in outcomes

Some measure of deviation in outcomes

Example 2. Standard Deviation

-Three equally probable outcomes, { 0, 6, 9 }; StandardDeviation > 0


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Risk Measures: axiomatic definition

A functional is a coherent risk measure in theextended sense if:

R1: for all constant C R2: for (convexity)

R3: when (monotonicity)

R4: when with (closedness)

A functional is a coherent risk measure in the basic

sense if it satisfies axioms R1, R2, R3, R4 and R5:

R5: for (positive homogeneity)

18

]0,1[λ ∈' X X ≤

0>λ


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A functional is an averse risk measure in theextended sense if it satisfies axioms R1, R2, R4 and R6:

R6: for all nonconstant X (aversity)

A functional is an averse risk measure in the basic

sense if it satisfies axioms R1, R2, R4, R6 and R5

Aversity has the interpretation that the risk of loss in a

nonconstant random variable X cannot be acceptable unless EX


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Examples of coherent risk measures: A

Z

Examples of risk measures not coherent:

, λ >0, violates R3 (monotonicity)

violates subadditivity

is a coherent measure of risk

in the basic sense and it is an averse measure of risk !!! Averse measure of risk might not be coherent, a coherent

measure might not be averse

20

Risk Measures: axiomatic definition


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A functional is called a deviation measure in the extendedsense if it satisfies:

D1: for constant C , but for nonconstant X

D2: for (convexity)

D3: when with (closedness)

A functional is called a deviation measure in the basic

sense if it satisfies axioms D1,D2, D3 and D4:

D4: (positive homogeneity)

A deviation measure in extended or basic sense is also coherent if itadditionally satisfies D5:

D5: (upper range boundedness)

21

Deviation Measures: axiomatic definition

[1,0]∈λ


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Examples of deviation measures in the basic sense: Standard Deviation

Standard Semideviations

Mean Absolute Deviation α-Value-at-Risk Deviation measure:

α-VaR Dev does not satisfy convexity axiom D2 it is not a

deviation measure

α-Conditional Value-at-Risk Deviation measure:

22


Coherent deviation measure in basic sense !!!


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23


CVaR Deviation Measure is a coherent deviation measure in the

basic sense


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Rockafellar et al. (2006) showed the existence of a one-to-one

correspondence between deviation measures in the extended sense

and averse risk measures in the extended sense:

24

Risk vs Deviation Measures

Rockafellar, R.T., Uryasev, S., Zabarankin, M., “Optimality conditions in portfolio analysis

with general deviation measures”, Mathematical Programming, 2006


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25

Risk vs Deviation Measures

Deviation

Measure

Counterpart Risk

Measure

where


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26

Chance and VaR Constraints

Let be some random loss function. By definition:

Then the following holds:

In general is nonconvex w.r.t. x, (e.g., discretedistributions)

may be nonconvex constraints

mi x f i ,..1 ),,( =ω }}),(Pr{:min{)( α ε ω ε α ≥≤= x f xVaR

ε α ε ω α ≤↔≥≤ )(VaR }),(Pr{ X x f

)( xVaRα

}),(Pr{and)(VaR α ε ω ε α ≥≤≤ x f X


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27

VaR vs CVaR in optimization

VaR is difficult to optimize numerically when losses are not

normally distributed

PSG package allows VaR optimization

In optimization modeling, CVaR is superior to VaR:

For elliptical distribution minimizing VaR, CVaR or Variance is

equivalent CVaR can be expressed as a minimization formula (Rockafellar

and Uryasev, 2000)

CVaR preserve convexity


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CVaR optimization

Theorem 1:

1. is convex w.r.t. α

2. α -VaR is a minimizer of F with respect to ζ :

3. α - CVaR equals minimal value (w.r.t. ζ ) of function F :

),( ζ α xF

( ) ( )VaR ( , ) ( , ) arg min ( , ) f x f x F xα α α ζ

ξ ζ ξ ζ = =

( )CVaR ( , ) min ( , ) f x F xα α ζ

ξ ζ =

}]),({[1

1),( +−

−+= ζ ξ

α ζ ζ α x f E xF


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29

CVaR optimization

Preservation of convexity: if f(x,ξ ) is convex in x thenis convex in x

If f(x,ξ ) is convex in x then is convex in x and ζ

If f(x*,ζ *) minimizes over then

is equivalent to

)( X CVaRα

),( ζ α xF

ℜ× X


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30

CVaR optimization

In the case of discrete distributions:

The constraint can be replaced by a system ofinequalities introducing additional variables η k :

ω ζ α ≤),( xF

N1,..,k ,0),( ,0 =≤−−≥ k k k y x f η ζ η

ω η α

ζ ≤−

+ ∑=

N

k

k k p11

1

1

1

( , ) (1 ) [ ( , ) ]

max { , 0}

N k

k

k

F x p f x

z z

α ζ ζ α ξ ζ − +

=+

= + − −

=

∑


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31

Generalized Regression Problem

Approximate random variable by random variables

Error measure satisfies axioms

Y 1 2, ,..., .n X X X

Rockafellar, R. T., Uryasev, S. and M. Zabarankin:

“Risk Tuning with Generalized Linear Regression”, accepted for publication in Mathematics of

Operations Research, 2008


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Error, Deviation, Statistic

32

For an error measure :

the corresponding deviation measure is

the corresponding statistic

is


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Theorem: Separation Principle

General regression problem

is equivalent to

33


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Percentile Regression and CVaR Deviation

34

Koenker, R., Bassett, G.

Regression quantiles.

Econometrica 46, 33–50(1978)

( )1min [ ] ( 1) [ ] ( )C R

E X C E X C CVaR X EX α α −

+ −∈− + − − = −


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35

Stability of Estimation

VaR and CVaR with same confidence level measure “differentparts” of the distribution

For a specific distribution the confidence levels α1 and α2 for

comparison of VaR and CVaR should be found from the equation

Yamai and Yoshiba (2002), for the same confidence level:

VaR estimators are more stable than CVaR estimators

The difference is more prominent for fat-tailed distributions

Larger sample sizes increase accuracy of CVaR estimation

More research needed to compare stability of estimators for the

same part of the distribution.

)()(21

X CVaR X VaR α α =

Decomposition According to Risk Factors


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36

For continuous distributions the following decompositions ofVaR and CVaR hold:

When a distribution is modeled by scenarios it is easier to

estimate than

Estimators of are more stable than estimators

of

iii

n

i i

z X VaR X X E z z

X VaR X VaR )](|[

)()(

1 α

α

α

==∂

∂=

∑=

iii

n

i i

z X VaR X X E z z

X CVaR X CVaR )](|[

)()(

1

α α

α ≥=∂∂

= ∑=

)](|[ X VaR X X E i α ≥ )](|[ X VaR X X E i α =

i z X CVaR

∂∂)(α

i z

X VaR

∂∂ )(α

Decomposition According to Risk Factors

Contributions


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Generalized Master Fund Theorem and CAPM

Assumptions:• Several groups of investors each with utility function

• Utility functions are concave w.r.t. mean and deviation

increasing w.r.t. mean

decreasing w.r.t. deviation

• Investors maximize utility functions s.t. budget constraint.

37

))(( , j j j j R ERU D

Rockafellar, R.T., Uryasev, S., Zabarankin, M. “Master Funds in Portfolio Analysis with

General Deviation Measures”, JBF, 2005

Rockafellar, R.T., Uryasev, S., Zabarankin, M. “Equilibrium with Investors using a Diversity of

Deviation Measures”, JBF, 2007


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Equilibrium exists w.r.t. Each investor has its own master fund and invests in its own master

fund and in the risk-free asset

Generalized CAPM holds:

is expected return of asset i in group jis risk-free rate

is expected return of market fund for investor group j

is the risk identifier for the market fund j

40

)])([()r ,Covar(G ij j ijij j j Er r EGG E −−=


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41

When then

When then

When then


Cl i l CAPM Di ti F l


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Classical CAPM => Discounting Formula

( )

( )2

0

0

00

,cov

)(1

1

)(1

M

M ii

M iii

M i

ii

r r

r r E r

r r r

E

σ

β

β ζ π

β

ζ π

=

−−+=

−++=

All investors have the same risk preferences:

standard deviation

Discounting by risk-free rate with adjustment foruncertainties (derived in PhD dissertation of

Sarykalin)

G li d CAPM


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Generalized CAPM

There are different groups of investors k=1,…,K

Risk attitude of each group of investors can be expressedthrough its deviation measure Dk

Consequently:

Each group of investors invests its own Master Fund M

G li d CAPM


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Generalized CAPM

( )( )

( ))(1

1

)(1

,cov

0

0

00

r r E

r

r r r E

r D

Qr

M iii

M i

ii

M

D

M ii

−−

+

=

−++=

−=

β ζ π

β ζ π

β

= deviation measure

Q

M

= risk identifier for corresponding to M

Investors Buying Out-of-the-money


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Investors Buying Out-of-the-money

S&P500 Put Options

Group of investors buys S&P500 options

Risk preferences are described by mixed CVaR deviation

Assume that S&P500 is their Master Fund

Out-of-the-money put option is an investments in low tail of

price distribution. CVaR deviations can capture the tail.

( )∑=

Δ=n

i

i X CVaR X D i1

)( α λ

Mixed CVaR Deviation Risk Envelope


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Mixed CVaR Deviation Risk Envelope

EX XQ X DQ

−=∈Q

sup)(

( )

( ){ } X VaR X Q

X CVaR X D

X α

α

ω ω ≥=

= Δ

)()(

)(

1

( )

( ){ }∑∑

=

=

Δ

≥=

=n

i

i X

n

ii

X VaR X Q

X CVaR X D

i

i

1

1

)()(

)(

α

α

ω λ ω

λ

1

)(ω X Q

)(ω X

( ) X VaRα

)(ω X Q

)(ω X ( ) X VaR

1α

( ) X VaR2α

( ) X VaR3α

α 1

1

1

α λ

2

2

1

1

α

λ

α

λ +

3

3

2

2

1

1

α λ

α λ

α λ ++

Data


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Data

Put options prices on Oct,20 2009 maturing on Nov, 20 2009

S&P500 daily prices for 2000-2009

2490 monthly returns (overlapping daily):

every trading day t: r t=lnSt-lnSt-21

Mean return adjusted to 6.4% annually.

2490 scenarios of S&P500 option payoffs on Nov, 20 2009

CVaR Deviation


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CVaR Deviation

Risk preferences of Put Option buyers:

We want to estimate values for coefficients

( )

%50%75%85%95%99

)(

54321

5

1

=====

=

∑=Δ

α α α α α

λ α j

j X CVaR X D j

jλ

Prices Implied Volatilities


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Prices Implied Volatilities

Option prices with different strike prices vary very significantly

Black-Scholes formula: prices implied volatilities

Graphs: ( )XCV RXD Δ)(


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Graphs: ( ) X CVaR X D = %50)(

0%

5%

10%

15%

20%

25%

30%

35%

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Market IV CAPM price IV No Risk IV Generalized CAPM IV alpha=50%

Graphs: ( )XCVaRXD Δ)(


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0%

5%

10%

15%

20%

25%

30%

35%

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110


Graphs: ( )XCVaRXD Δ=)(


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0%

5%

10%

15%

20%

25%

30%

35%

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110


Graphs: ( )XCVaRXD Δ=)(


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Graphs: ( )∑ Δ=5

)( XCVaRXD λ


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0%

5%

10%

15%

20%

25%

30%

35%

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Market IV CAPM price IV No Risk IV Ceneralized CAPM Mixed

Graphs:

111.0219.0227.0239.0204.0

%50%75%85%95%99

54321

54321

==========

λ λ λ λ λ

α α α α α

( )∑=

=1

)( j

j X CVaR X D jα λ


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VaR: Pros

1. VaR is a relatively simple risk management concept and has a clearinterpretation

2. Specifying VaR for all confidence levels completely defines thedistribution (superior to σ)

3. VaR focuses on the part of the distribution specified by theconfidence level

4. Estimation procedures are stable

5. VaR can be estimated with parametric models

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VaR: Cons

1. VaR does not account for properties of the distribution beyond

the confidence level

2. Risk control using VaR may lead to undesirable results for skewed

distributions

3. VaR is a non-convex and discontinuous function for discrete

distributions

56


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CVaR: Pros

1. VaR has a clear engineering interpretation

2. Specifying CVaR for all confidence levels completely defines the

distribution (superior to σ)3. CVaR is a coherent risk measure

4. CVaR is continuous w.r.t. α

5. is a convex function w.r.t.

6. CVaR optimization can be reduced to convex programming and insome cases to linear programming

57

)...( 11 nn X w X wCVaR ++α ),...,( 1 nww


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CVaR: Cons

1. CVaR is more sensitive than VaR to estimation errors

2. CVaR accuracy is heavily affected by accuracy of tail modeling

58


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VaR or CVaR in financial applications?

VaR is not restrictive as CVaR with the same confidence levela trader may prefer VaR

A company owner may prefer CVaR; a board of director may

prefer reporting VaR to shareholders VaR may be better for portfolio optimization when good models

for the tails are not available

CVaR should be used when good models for the tails are available CVaR has superior mathematical properties

CVaR can be easily handled in optimization and statistics

Avoid comparison of VaR and CVaR for the same level α

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Adequate accounting for risks, classical and downside risk measures:

-Value-at-Risk (VaR)

- Conditional Value-at-Risk (CVaR)

- Drawdown

- Maximum Loss- Lower Partial Moment

- Probability (e.g. default probability)

- Variance

- St.Dev.- and many others

Various data inputs for risk functions: scenarios and covariances

- Historical observations of returns/prices- Monte-Carlo based simulations, e.g. from RiskMetrics or S&P CDO

evaluator

- Covariance matrices, e.g. from Barra factor models

PSG: Portfolio Safeguard

60

PSG P tf li S f d


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Powerful and robust optimization tools:

four environments:

Shell (Windows-dialog)

MATLAB

C++

Run-file

simultaneous constraints on many functions at various times

(e.g., multiple constraints on standard deviations obtained by

resampling in combination with drawdown constraints)

PSG: Portfolio Safeguard

61


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PSG: Risk Functions

orRisk function

(e.g., St.Dev, VaR, CVaR, Mean,…)covariance

matrix

matrix of

scenarios

matrix with

one rowLinear function

62

PSG E l


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PSG: Example

63

PSG O ti ith F ti


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PSG: Operations with Functions

Risk functions

Optimization Sensitivities Graphics

64

C St d Ri k C t l i V R


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Case Study: Risk Control using VaR

Risk control using VaR may lead to paradoxical results for skeweddistributions

Undesirable feature of VaR optimization: VaR minimization mayincrease the extreme losses

Case Study main result: minimization of 99%-VaR Deviation leadsto 13% increase in 99%-CVaR compared to 99%-CVaR of optimal

99%-CVaR Deviation portfolio

Consistency with theoretical results: CVaR is coherent, VaR is not

coherent

65

Larsen, N., Mausser, H., Uryasev, S., “ Algorithms for Optimization of Value-at-Risk”,

Financial Engineering, E-commerce, Supply Chain, P. Pardalos , T.K. Tsitsiringos Ed.,

Kluwer, 2000

C St d Ri k C t l i V R


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Case Study: Risk Control using VaR

66

Columns 2, 3 report value of risk functions at optimal point of Problem 1 and 2;

Column “Ratio” reports ratio of Column 3 to Column 2

P.1 Pb.2

C St d Li R i H d i


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Case Study: Linear Regression-Hedging

67

Investigate performance of optimal hedging strategies based ondifferent deviation measures

Determining optimal hedging strategy is a linear regression problem

Benchmark portfolio value is the response variable, replicatingfinancial instruments values are predictors, portfolio weights arecoefficients of the predictors to be determined

Coefficients chosen to minimize a replication error

function depending upon the residual

I I x x θ θ θ ++= ...ˆ 11

θ θ ˆ0

− I x x ,...,1

C St d Li R i H d i


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68

Out-of-sample performance of hedging strategies significantly dependson the skewness of the distribution

Two-Tailed 90%-VaR has the best out-of-sample performance

Standard deviation has the worst out-of-sample performance

(1)

(2)

(3)

(4)

(5)

C St d Li R i H d i


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69

Each row reports value o different risk functions evaluated at optimal points of the 5

different hedging strategies (e.g., the first raw is for hedging strategy)

the best performance has TwoTailVar

Out-of-sample performance of different deviation measures evaluated at optimal points

of the 5 different hedging strategies (e.g., the first raw is for hedging strategy)0.9CVaRΔ

0.9CVaRΔ

0.9

Δ

Example:

h d i i l


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Chance and VaR constraints equivalence

70

We illustrate numerically the equivalence:

At optimality the two

problems selected

the same portfoliowith the same

objective function

value

Case Study: Portfolio Rebalancing Strategies,


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Risks and Deviations

71

We consider a portfolio rebalancing problem:

We used as risk functions VaR, CVaR, VaR Deviation, CVaRDeviation, Standard Deviation

We evaluated Sharpe ratio and mean value of each sequence ofportfolios

We found a good performance of VaR and VaR Deviationminimization

Standard Deviation minimization leads to inferior results

Case Study: Portfolio Rebalancing Strategies,

i k d i i ( d)


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Risks and Deviations (Cont'd)

72

Results depend on the scenario dataset and on k In the presence of mean reversion the tails of historical distribution

are not good predictors of the tail in the future

VaR disregards the unstable part of the distribution thus may leadto good out-of-sample performance

Sharpe ratio for the rebalancing strategy when different risk functions are

used in the objective for different values of the parameter k

Conclusions: key observations


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73

CVaR has superior mathematical properties: CVaR is coherent,CVaR of a portfolio is a continuous and convex function withrespect to optimization variables

CVaR can be optimized and constrained with convex and linear

programming methods; VaR is relatively difficult to optimize

VaR does not control scenarios exceeding VaR

VaR estimates are statistically more stable than CVaR estimates

VaR may lead to bearing high uncontrollable risk

CVaR is more sensitive than VaR to estimation errors

CVaR accuracy is heavily affected by accuracy of tail modeling

C l i k b i


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74

There is a one-to-one correspondence between Risk Measuresand Deviation Measures

CVaR Deviation is a strong competitor of Standard Deviation

Mixed CVaR Deviation should be used when tails are not

modeled correctly. Mixed CVaR Deviation gives different weight

to different parts of the distribution

Master Fund Theorem and CAPM can be generalized with the for

different deviation measure.

C l i C St di


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Conclusions: Case Studies

Case Study 1: risk control using VaR may lead to paradoxicalresults for skewed distribution. Minimization of VaR may lead to astretch of the tail of the distribution exceeding VaR

Case Study 2: determining optimal hedging strategy is a linear

regression problem. Out-of-sample performance based ondifferent Deviation Measures depends on the skewness of thedistribution, we found standard deviation have the worst

performance Case Study 3: chance constraints and percentiles of a distribution

are closely related, VaR and Chance constraints are equivalent

Case Study 4: the choice of the risk function to minimize in aportfolio rebalancing strategy depends on the scenario dataset. Inthe presence of mean reversion VaR neglecting tails may lead togood out-of-sample performance

var vs cvar carisma conference 2010

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