Translation invariant and positive homogeneous risk measures and portfolio management. Zinoviy Landsman Department of Statistics, University of Haifa

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1 ranslation invariant and positive homogeneous risk measures and portfolio management 1 Zinoviy Landsman Department of Statistics, University of Haifa

2 -It is wide-spread opinion the use of any positive homogeneous and translation invariant risk measure reduces to the same portfolio management as mean-variance risk measure does, which is euivalent to Markowitz portfolio -We show that generally speaking it is not true, because these measures reduce to a different optimization problem. 2

3 -We provide the explicit closed form solution of this problem -he results will be demonstrated with the data of stocks from NASDAQ/Computer. 3

4 ranslation invariant and positive homogeneous risk measures 1) ρ( X + α) = ρ( X) + α, α const 2) ρ( cx) = cρ( X), c > 0 const 4

5 Examples VaR BASEL II ρ ( X) = VaR ( X) Expected Short Fall 5 = inf{ x F ( x) } ρ ( X) = ES ( X) X = E( X X > VaR ( X )) ail VaR, ail conditional expectation (CE) Conditional VaR (CVaR)

6 y y = F( x) = 1 F( x) 1 { 6 x ( ) ( ) 1 VaR X F( xdx ) ( ) ( ) + 1 ES ( X) = x 1 F( x) dx x x x

7 Panjer and Jing (2001) H urlimann (2001) L. and Valdez (2003, 2005) 7

8 SD-premium ρ( X) = E( X) + λsd( X) Distorted risk measures ρ ( X) = g( F( x)) dx, 0 Denneberg (1994) and Wang (1996) Coherent risk measures Artzner, Delbean, Eber, and Heath (1999) 8

9 ail SDP Furman and Landsman (2006) ρ( X ) = EX ( X> VaR( X)) + α VX ( X> VaR( X)) ail VaR ail Expectation ail Variance 9

10 ES ( X) = arg inf E(( X c) X > VaR ( X)) c 2 V( X X > VaR ( X)) 10 2 = inf E(( X c) X > VaR ( X)) c

11 Portfolio management X = ( X,..., X ) N ( μ, Σ) 1 n n P n = x X j j j= 1, n j= 1 x j = 1 ρ( P) inf 11

12 ρ ρ ρ ( P) = ( x X ) = ( x X μ x+ μ x) = ρ( + x X μ x) μ x Z + ρ x X μ x = μ x VaR ( x X) VaR ( ) x X + ρ x X μ x = μ x VaR ( x X) N(0,1) VaR ( ) x X 12 = μ x + ρ( Z) x Σx inf

13 ranslation invariant and positive homogeneous risk measure reduces to minimization of linear plus root of uadratic functional f ( x) = μ x+ λ x Σx inf 1x= 1 Z N(0,1) λ = Z inf VaR ( P). ρ 13 ( Z) λ = ES ( Z) inf ES ( P). Any translation invariant and positive homogeneous risk measure

14 Classical mean - variance optimization ( x) = E( P) + λvar( P) = μ x+ λx Σx inf subject to 1x= 1 linear + uadratic functional Exact solution (see Boyle et.al (1998)) Σ 1 1 μ Σ 1 x * = + Σ 1-Σ 1 Σ 1 1 2λ Σ μ

15 Minimization of linear plus root of uadratic functional We found the explicit closed form solution for minimization problem under system of euality constraints (L(2007)) f = + Σ Bx = c ( x) μ x λ x x inf 15 B= matrix m n, c - vector

16 he special case for only one constraint m=1 f ( x) = μ x + λ x Σ x inf 1x = 1 σ σ Σ σ Σ = 11, nn 16 Σ ( n 1) ( n 1) 11

17 Define Q ( n 1) ( n 1) 1σ 1 σ 11 σ 11 nn 1 1 =Σ + matrix Lemma Σ > 0 Q > 0. Δ = ( μn μ1,..., μn μn 1). heorem 1. L (2007a). If 17 λ > Δ Q 1 Δ,

18 the minimization problem has the finite exact solution Σ 1 1 x 1 1 Σ 1 ( λ Δ Q Δ)( 1 Σ 1) = + ( Δ Q, Q Δ) 3. Value-at-Risk and Expected Short Fall λ = ES ( Z) = VaR ( Z) VaR ( Z ) (1 F( x)) dx VaR ( Z) = Z.

19 heorem 2 If VaR-optimal portfolio solution is finite, the ES-optimal portfolio solution is automatically finite Consider a portfolio of 10 stocks from NASDAQ/Computers 19

20 20

21 he loss on the portfolio n j j j=1 L= R= x X, VaR L μ z ( ) = x+ x Σx inf x 1= 1 ES ( L) = μ x+ λ x Σx inf x 1= 1 From heorem 1: Lower bound for solution B 1 = Δ Q Δ =

22 X Nn ( μ, Σ ) VaR optimal portfolio finite > ES optimal portfolio finite > =

23 Weights o o o - minvar - mines Adobe Compuware NVDIA Staples VeriSign Sandisk Microsoft Citrix Intuit Symantec Companies

24 Closeness between VaR and ES optimal portfolios d = Distance between portfolios arg min VaR ( L ), arg min ES ( L ) R 1 x 1 x n (arg min VaR ( L) arg min ES ( L)) (arg min VaR ( L) arg min ES ( L)) Denote 1 x 1 x 1 x 1 x 1 1/ 2 2 Δ ( 1 Δ 2 C = ( 1 Σ 1) Δ Q + 1 Q ) 1 24

25 heorem / 1 2 λ 2 2 / 1 2 d = [( z B ) ( B ) ] C 0, 1 Boundary 25

26 Elliptical family X E n ( g ) μ,σ,. n c ( ) n 1 ( ) 1 f x = g ( ) n μ Σ μ, X 2 Σ x x density generator g ( u) = exp( u) N ( μ, Σ) n n 26

27 Property BX+ c ~ E ( B μ + c,bσb, g ) guarantees m ρ( x X ) = μ x+ λ x Σx m λ = ρ ( Z), Z ~ E (0,1, g )

28 p g u u k t p ( ) = 1 + / (μ, Σ, ) n n, p n Multivariate Generalized t-distribution (GS) f ( x) = μ x + λ x Σ x inf 1x = 1 t1 (0,1, p1 ) VaR optimal ES ( ) ES optimal 28

29 Figure 1. Lower boundary of VaR and ES - probability level for GS. -boundary bound.var.norm 29 -bound.es.norm

30 Closeness between VaR and ES optimal portfolios Figure 2. VaR and ES optimal portfolios for Norm. Weights o o o - minvar - mines Adobe Compuware NVDIA Staples VeriSign Sandisk Microsoft Citrix Intuit Symantec Companies

31 Distance between portfolios d = arg min VaR ( L ), arg min ES ( L ) R 1 x 1 x n (arg min VaR ( L) arg min ES ( L)) (arg min VaR ( L) arg min ES ( L)) 1 x 1 x 1 x 1 x Denote δ = λ z = ES ( Z) VaR ( Z) 1 1/ 2 2 Δ ( 1 Δ 2 C = ( 1 Σ 1) Δ Q + 1 Q ) 1 31

32 heorem 4 X E ( g ) n μ,σ,. n 2 2 / / 1 2 1) d = [( z B ) ( λ B ) ] C 0, 1 2) 1 2 dlog g 2 1( 2 z ) Let z l 1 dx l 3 2l 3 2 d = [ + ( α ) + O(( α ) )] C, 2 2 z B l 1 2 ( l 2) ( l 2) 2 2 α δ (2 z + δ ) 2l 3 =. z B ( l 2)

33 3) l = 2 p, p = p ( n 1) / 2, and 33 d If X 1 1 t 2 2 μ pk n np,,σ;,, ( p1 1) p1 3/ 4 = [( + ( α ) 3 2 z B 2p1 1 2 ( p1 1/ 2) ( p1 1) p 34 / + O C. 1 2 (( α ) )] 2 ( p1 1) ( ) 4) If X N μ,σ, l =, and n 1 1 [ ( )] d 2 α O α C z B =

34 Figure 3. Distance and it's approximation; GS with power parameter p=1.6 distance approximation o o o o - distance level

35 Figure 4. Changing distance with increasing of power parameter p of GS distance distance for norm p

36 Conclusion -Problem of minimization of translation invariant and positive homogeneous risk measures has exact close form solution -his solution can be easy realized and used for analyzing the influence of parameters of underlying distribution on the portfolio selecting 36

37 -he solution is different with that of mean-variance except the case when the portfolio s expected mean is certain 37

38 References Boyle, P.P, Cox, S.H,...[et. al]. Financial Economics: With Applications to Investments, Insurance and Pensions, Actuarial Foundation, Schaumburg (1998) Landsman, Z. (2007a). Minimization of the root of a uadratic functional under an affine euality constraint. Journal of Computational and Applied Math. o appear Landsman, Z. (2007b). Minimization of the root of a uadratic functional under a system of euality constraint with application in portfolio management, Journal of Computational and Applied Math. 38 o appear