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Transcript of 1 Translation invariant and positive homogeneous risk measures and portfolio management Zinoviy...
11
Translation invariant and
positive homogeneous risk measures
and portfolio management
Zinoviy LandsmanDepartment of Statistics,
University of Haifa,E-mail: [email protected]
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.
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Translation invariant and Translation invariant and positivepositive
homogeneous risk measures homogeneous risk measures
1) ( ) ( ) ,X X const
2) ( ) ( ), 0cX c X c const
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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
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Panjer (2002 (
Landsman and Valdez (2003, 2005(
H¨urlimann (2001)
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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)
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Tail SDP
( ) ( | ( )) ( | ( ))q qX E X X VaR X STD X X VaR X
Furman and L (2006)
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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
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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
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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
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The special case for only one constraintm=1
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Tnn
( 1) ( 1) 11 n n
( ) inf
1
x x x x
1 x
T T
T
f
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( 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
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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
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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
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( , )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
z l
z B l
2929
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
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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