11

Translation invariant and

positive homogeneous risk measures

and portfolio management

Zinoviy LandsmanDepartment of Statistics,

University of Haifa,E-mail: landsman@stat.haifa.ac.il

IME 2007

22

-It is wide-spread opinionthe use of any positive homogeneous and translation invariant risk measure reduces to the same portfolio management as mean-variance risk measure does, which is equivalent to Markowitz portfolio

-We show that generally speaking

it is not true, because these measures

reduce to a different optimization problem.

33

-We provide the explicit closed form solution of this problem

-The results will be demonstrated with the data of stocks from NASDAQ/Computer.

44

Translation invariant and Translation invariant and positivepositive

homogeneous risk measures homogeneous risk measures

1) ( ) ( ) ,X X const

2) ( ) ( ), 0cX c X c const

55

Examples

( ) ( )

inf{ | ( ) }q

X

X VaR X

x F x q

( ) ( )

( ( ))

q

q

X ES X

E X X VaR X

Tail VaR, Tail conditional expectation (TCE),Conditional VaR (CVaR)

BASEL II

VaR

Expected Short Fall

66

Panjer (2002 (

Landsman and Valdez (2003, 2005(

H¨urlimann (2001)

77

Distorted risk measures

0( ) ( ( )) ,X g F x dx

Coherent risk measures

STD-premium

( ) ( ) ( ) X E X STD X

Denneberg (1994) and Wang (1996)

Artzner, Delbean, Eber, and Heath (1999)

88

Tail SDP

( ) ( | ( )) ( | ( ))q qX E X X VaR X STD X X VaR X

Furman and L (2006)

99

Portfolio management

1( ,..., ) ( , )Tn nX X N Σ X

1

,

n

j jj

P x X1

1

n

jj

x

( ) infP

1010

Translation invariant and positive homogeneous risk measure

reduces to minimization

of linear plus root of quadratic functional

( ) inf x x x xT Tf

11 xT

inf ( ) q qZ VaR PN(0,1)Z

( ) inf ( ) q qES Z ES P

( )Z Any translation invariant and positive homogeneous risk measure

1111

Classical mean - variance optimization

( ) ( ) ( ) xq E P Var P inf x x xT T

11 xTsubject to

1 11 1

1 1

1

2

* 1 1x = 1-

1 1 1 1

T

T T

Exact solution (see Boyle et.al (1998))

linear + quadratic functional

1212

Minimization of linear plus root of quadratic functional

We found the explicit closed form solution for minimization problem under system of equality constraints (Landsman(2007))

( ) inf

x x x x

x

T Tf

B c

, vectorB matrixm n -c

1313

The special case for only one constraintm=1

11

Tnn

( 1) ( 1) 11 n n

( ) inf

1

x x x x

1 x

T T

T

f

1414

( 1) ( 1) n nDefine matrix

111 1 1 11 111 T TTnnQ

0 0.QLemma

Theorem 1. Landsman (2007). If

1 1( ,..., ) Tn n n

1 TQ

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the minimization problem has the finite exact solution

11 1

11 2 1 1

1( )

( )( )

1x 1

1 1 1 1

T T TT T T

Q QQ

.Value-at-Risk and Expected Short Fall 3

( )

( ) ( )

1(1 ( ))

1q

q q

VaR Z

ES Z VaR Z

F x dxq

( )q qVaR Z Z

1616

Theorem 2

If VaR-optimal portfolio solution is finite ,the (ES, TSDP )-optimal portfolio solutions are automatically finite

Consider a portfolio of 10 stocks from NASDAQ/Computers

17

1818

The loss on the portfolio

1

n

j jj

L R x X

1( ) infT

T Tq qVaR L z

x 1=x x x

1( ) infT

T Tq qES L

x 1=x x x

From Theorem 1: Lower bound for solution

1 0.2142 TB Q

1919

( , )nN ΣX

VaR optimal portfolio finite 0.585q

ES optimal portfolio finite 0.112q 0.8q

2020

Companies

We

igh

ts

2 4 6 8 10

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

AdobeCompuware

NVDIA

StaplesVeriSign

Sandisk

Microsoft

Citrix

Intuit

Symantec

o o o - minVaR

______ - minES

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Including risk less asset

0

0Tnn

11

111 1 1 1 111 11 01T TT

nnQ

11 1

11 1 111 2 1 111

1( )

( )( )

T T TT T T

1x 1

1 1 1 1

111

TB

2222

Elliptical family

n nE g X

11

2Tn

n

cf g X x x x

density generator

( ) exp( ) ( , )n ng u u N

2323

Property

~ ( , )Tm mB E B ,B B gΣ X c c

guarantees

( )T T TX + Σ x xx x

1 1( ), ~ (0,1, )Z Z E g

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1 , , )p

n n p ng u u k t p

Multivariate Generalized t-distribution (GST)

( ) inf

1

x x x x

1 x

T T

T

f

qT VaR optimal

( )qES T ES optimal

1(0,1, )1T t p

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p

q-b

ou

nd

ary

1.5 2.0 2.5 3.0 3.5 4.0

0.2

0.4

0.6

0.8

q-bound.ES.norm

q-bound.VaR.norm

Figure 1. Lower boundary of

VaR and ES q- probability level for GST .

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4 .Closeness between VaR and ES optimal portfolios

Figure 2. VaR and ES optimal portfolios for Norm.

Companies

We

igh

ts

2 4 6 8 10

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

AdobeCompuware

NVDIA

StaplesVeriSign

Sandisk

Microsoft

Citrix

Intuit

Symantec

o o o - minVaR

______ - minES

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Distance between portfolios

1 1 1 1(arg min ( ) arg min ( )) (arg min ( ) arg min ( ))

T T T T

Tq q q q

d

VaR L ES L VaR L ES L

x x x x

1 1arg min ( ), arg min ( )

T T

nq qVaR L ES L R

x x

( ) ( )q q q q qz ES Z VaR Z Denote

1 1 2 2 1 2( ) )( TT TC Q Q 11 1 1

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Theorem 3 n nE g X

2 2 1 2 2 2 1 21) [( ) ( ) ] 0 1q qd z B B C q 21

1 221

log ( )2) q

q q

d g zLet z l

dx

22 22 2

1 1 1 2 3 2 3[ ( ) (( ) )]

1 2 ( 2) ( 2)q

q

l ld O C

l l lz B

2 2 2

(2 ) 2 3

( 2)q q q

qq

z l

z B l

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1 1

3)

2 ( 1) 2,

n pn p kIf

l p p p n and

X t

31 1

3 22 21 1 1

212

1

( 1) 3 41 1 1[( ( )

2 1 2 ( 1 2) ( 1)

3 4(( ) )]

( 1)

q

q

q

p pd

p p pz B

pO C

p

4) If ,n l and X N2

2 2

1 1[ ( )]2

q q

q

d O Cz B

3030

level q

dis

tan

ce

0.90 0.92 0.94 0.96 0.98 1.00

05

10

15

______ - approximation

o o o o - distance

Figure 3. Distance and it's approximation ; GST with power parameter p=1.6

3131

p

dis

tan

ce

2 4 6 8 10 12

01

23

4

distance for norm

Figure 4. Changing distance with increasing of power parameter p of GST

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Conclusion

-Problem of minimization of translation invariant and positive homogeneous risk measures has exact close form solution

-This solution can be easy realized and used for analyzing the influence of

parameters of underlying distribution on the portfolio selecting

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-The solution is different with that of mean-variance except the

case when the portfolio’s expected mean is certain

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References

Boyle, P.P, Cox, S.H,...[et. al]. Financial Economics : With Applications to Investments,

Insurance and Pensions, Actuarial Foundation, Schaumburg (1998)

Landsman, Z. (2007). Minimization of the root of a quadratic functional under an affine equality constraint.

Journal of Computational and Applied Math. To appear

Landsman, Z. (2007). Minimization of the root of a quadratic functional under a system equality constraint

with application in portfolio management, EURANDOM, report 2007-012