Risk Management Using Conditional Value-at-Risk (CVaR) (Mathematical Science of Optimization)

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Title

Risk Management Using Conditional Value-at-Risk (CVaR)

(Mathematical Science of Optimization)

Author(s)

Uryasev, Stanislav

Citation

数理解析研究所講究録 (2000), 1174: 159-160

Issue Date

2000-10

URL

http://hdl.handle.net/2433/64463

Right

Type

Departmental Bulletin Paper

Textversion

publisher

(2)

Risk

uanagement

Using

Conditional

$\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}-\mathrm{a}\mathrm{t}$

-Risk

$(\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R})$

Stanislav

Uryasev

University

$0\overline{\mathrm{r}}$

Florida,

USA

$\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{v}\emptyset \mathrm{i}\mathrm{s}\mathrm{e}.\mathrm{u}\mathrm{f}\mathrm{l}$

.

edu

Value-at-Risk

$(\mathrm{V}\mathrm{a}\mathrm{R})$

,

a

widely

used

performance

measure,

answers

the

question:

what

is

the

maximum loss

with

a

specified

confidence level? Although

$\mathrm{V}\mathrm{a}\mathrm{R}$

is

a

very

popular

measure

of

risk,

it

has undesirable

properties

such

as

lack of sub-additivity, i.e.,

$\mathrm{V}\mathrm{a}\mathrm{R}$

of

a

portfolio with

two

instruments

may

be

greater than

the

sum

of individual

$\mathrm{V}\mathrm{a}\mathrm{R}\mathrm{s}$

ofthese

two

instruments.

Also,

$\mathrm{V}\mathrm{a}\mathrm{R}$

is

difficult

to

optimize

when

calculated using scenarios. In

this case,

$\mathrm{V}\mathrm{a}\mathrm{R}$

is

non-convex,

non-smooth

as a

ffinction

ofpositions, and it has multiple

local

extrema.

An

alternative

measure

oflosses,

with

more

attractive properties, is Conditional

Value-at-Risk

$(\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R})$

,

which

is

also called Mean Excess

Loss,

Mean

Shortfall,

or

Tail

$\mathrm{V}\mathrm{a}\mathrm{R}$

.

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

is

a

more

consistent

measure

of risk

since it

is

sub-additive and

convex.

Moreover,

as

it

was

shown

recently

$[3,4]$

,

it

can

be

optimized using linear programming

$(\mathrm{L}\mathrm{P})$

and

nonsmooth

optimization

algorithms, which allow handling portfolios with

$\mathrm{v}\mathrm{e}\iota\gamma$

large

numbers of instruments and

scenarios.

Numerical

experiments

indicate that the

minimization of

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

also leads

to

near

optimal solutions in

$\mathrm{V}\mathrm{a}\mathrm{R}$

terms

because

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

is

always greater

than

or

equal

to

$\mathrm{V}\mathrm{a}\mathrm{R}$

.

Moreover,

when

the

return-loss distribution is

normal,

these

two

measures are

equivalent

[3],

i.e.,

they

provide the

same

optimal

portfolio.

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

can

be

used in conjunction

with

$\mathrm{V}\mathrm{a}\mathrm{R}$

and

is applicable

to

the

estimation

of risks

with

non-symmetric return-loss distributions.

Although

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

has

not

become

a

standard

in

the finance industry,

it

is

likely

to

play

a

major role

as

it

currently does

in

the

insurance

industry. Similar

to

the

Markowitz

mean-variance

approach,

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

can

be

used

in

return-risk analyses. For

instance,

we can

calculate

a

portfolio with

a

specified

return

and

minimal

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

.

Alternatively,

we

can

constrain

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

and find

a

portfolio with

maximal

return,

see

[2].

Also,

rather than

constraining

the

variance,

we can

$\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}6^{r}$

several

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

constraints

simultaneously

with

various

confidence levels

(thereby

shaping the loss

distribution),

which provides

a

flexible

and

powerffil risk

management

tool.

Several

case

studies

showed that risk

optimization

with the

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

performance

ffinction

and

constraints

can

be done for large

portfolios

and

a

large number of

scenarios

with

relatively

small

computational

resources.

For instance,

a

problem with 1,000 instruments

and 20,000

scenarios

can

be optimized

on a 300

MHz PC

in

less than

one

minute

using

the CPLEX LP solver.

A

case

study

on

the hedging of

a

portfolio of

options

using the

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

minimization

technique

is included in

[3].

This problem

was

first studied

at

Algorithmics,

Inc.

with the

minimum expected

regret

approach.

Also,

the

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

minimization

approach

was applied

to

credit risk

management

of

a portfolio

of bonds

[1].

This

portfolio

was

put together by

several

banks

to test

various

credit

risk

modeling

techniques.

Earlier,

the

minimum

expected

regret

optimization

technique

was

applied

to

the

same

portfolio

at

Algorithmics,

Inc.;

we

have

used the

same

set

of

scenarios

to

test

the

minimum

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

technique.

A

case

study

on

optimization

of

a

portfolio of stocks with

数理解析研究所講究録

(3)

$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}$

constraints

is

included

in

[2].

REFERENCES

1.

Andersson, F., Mausser,

H., Rosen,

D.,

and

S. Uryasev

(2000):

Credit Risk

Optimization

With

Conditional Value-At-Risk

Criterion. Mathematical Programming

to

apper.

(Drafl

can

be

downloaded:

$\mathrm{w}\mathrm{w}\mathrm{w}.\mathrm{i}\mathrm{s}\mathrm{e}.\mathrm{u}\mathrm{f}\mathrm{l}.\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{u}\mathrm{r}\mathrm{y}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{v}/\mathrm{a}\mathrm{n}\mathrm{d}_{-}\mathrm{m}\mathrm{p}.\mathrm{p}\mathrm{d}\mathrm{f}\cdot$

,

relevant

Report

99-9

ofthe

Center for

Applied

Optimization, University

of

Florida,

can

be

downloaded:

$\backslash \mathrm{w}\mathrm{w}.\mathrm{i}\mathrm{s}\mathrm{e}$

.

ufl.

$\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{u}\mathrm{r}\mathrm{y}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{v}/\mathrm{p}\mathrm{u}\mathrm{b}\mathrm{s}$

.

html#t)

2. Palmquist,

J.,

Uryasev,

S.,

and P.

Krokhmal

(1999):

Portfolio Optimization with

Conditional Value-At-Risk Objective and Constraints. Research

$\mathrm{R}\mathrm{e}\mathrm{p}\mathrm{o}\iota \mathrm{t}$

99-14, Center for

Applied Optimization, University of Florida.

(Submitted

to

The Joumal

of

Risk,

Can be

downloaded:

$\mathrm{w}\mathrm{w}\mathrm{w}.\mathrm{i}\mathrm{s}\mathrm{e}.\mathrm{u}\mathrm{f}\mathrm{l}.\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{u}\mathrm{r}\mathrm{y}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{v}/\mathrm{p}\mathrm{a}\mathrm{l}.\mathrm{p}\mathrm{d}\mathit{0}$

3.

Rockafellar R.T. and

S. Uryasev

(2000):

Optimization of Conditional Value-at-Risk.

The

Joumal

of

Risk,

Vol. 2,

$\#$

3.

(Can

be downloaded:

$\mathrm{w}\mathrm{w}\mathrm{w}.\mathrm{i}\mathrm{s}\mathrm{e}.\mathrm{u}\mathrm{f}\mathrm{l}.\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{u}\mathrm{r}\mathrm{y}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{v}/\mathrm{c}\mathrm{v}\mathrm{a}\mathrm{r}.\mathrm{p}\mathrm{d}\mathrm{f}\cdot$

,

relevant Report

99-4

of the Center

for Applied

Optimization, University of

Florida,

can

be

downloaded:

$\mathrm{w}\mathrm{w}\mathrm{w}.\mathrm{i}\mathrm{s}\mathrm{e}.\mathrm{u}\mathrm{f}\mathrm{l}.\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{u}\mathrm{r}\mathrm{y}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{v}/\mathrm{p}\mathrm{u}\mathrm{b}\mathrm{s}.\mathrm{h}\mathrm{t}\mathrm{m}\mathrm{l}\#\mathrm{t})$

4.

Uryasev,

S.

Conditional

Value-at-Risk: Optimization Algorithms and Applications.

Financial

Engineering

News,

No.

14,

$\mathrm{F}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{u}\mathrm{a}\iota\gamma$

,

2000

(can

be

downloaded:

$\mathrm{r}\mathrm{w}.\mathrm{i}\mathrm{s}\mathrm{e}.\mathrm{u}\mathrm{f}\mathrm{l}.\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{u}\mathrm{r}\mathrm{y}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{v}/\mathrm{p}\mathrm{u}\mathrm{b}\mathrm{s}$

.

html#t).

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