General approach to the optimal portfolio risk management

Transcription

1 General approach to the optimal portfolio risk management Zinoviy Landsman, University of Haifa Joint talk with Udi Makov and Tomer Shushi This presentation has been prepared for the Actuaries Institute 05 ASTIN and AFIR/ERM Colloquium. The Institute Council wishes it to be understood that opinions put forward herein are not necessarily those of the Institute and the Council is not responsible for those opinions.

2 Outline. Mean-variance risk measure. Generalization based on Sharpe ratio 3. Main results on optimal portfolio selection (OPS) 4.Important examples: generalized Sharpe ratio and classical Sharpe ratio and OPS 5.Combination of linear and function of square root of quadratic functional 6. Translation-invariant and positive-homogeneous risk measures and OPS 7. Tail mean-variance principle and OPS

3 -The success of classical MV optimal portfolio theory is mostly because the model reduces to QP, which is quite convenient for the technical realization and for the case when short selling is possible, even provides the analytic solution (Merton (97)) - We suggest the generalization of MV-model which reduces to more complicated CFP concave fractional programming 3

4 Return on the portfolio R R= xr + xr + + xr Mean-Variance approach n n E( R) λvar( R) max n i= x i = MV ( R) = E( R) + λvar( R) inf 4

5 Another approach to OPS referenced to Sharpe Rf ER ( ) R f S = Var( R) risk free rate max How to combine these two approaches and generalize maximally? We suggest the following functional per ( ( ) f =t max v( Var( R)) 5

6 t( x) strictly increased, p( x) concave t'( x), p'( x) 0 6

7 Important examples Generalized Sharpe ratio t( x) = x, p( x) = x R, v( x) = x β, β / f ER ( ) R f f= ( Var ( R β )) max β = / classical Sharpe ratio Another important functional tx ( ) = log x, px ( ) = exp( x), vx ( ) = exp( λsx ( )), λ > 0 7

8 f = E( R) λs( Var( R)) max sx ( ) defined on [0, ) function, differentiable and positive on (0, ) Landsman and Makov (03) 8

9 Important examples f = E( R) λ( Var( R)), β / β = classical Mean Variance principle Standard deviation premium principle f = E( R) λ Var( R), Tail mean-variance premium principle f = E R + Var R + Var R ( ) α ( ) α ( ), β 9

10 Portfolio management R = (,..., ) R R n T R n = j= xr j j, n j= x j = 0

11 This image cannot currently be displayed. This image cannot currently be displayed. f Main result p( µ x) = t v( T x Σx) T max Bx=c rank( B) = m Partitions µ = ( µ, µ ), µ = ( µ,..., µ ), µ = ( µ,..., µ ) T T T T T n m n m+ n Σ Σ= Σ Σ Σ B= ( B ) B

12 v'( x) p'( x) Let u( x) = and u( x) = vx ( ) px ( ) Theorem. If equation T T u( f0 + b w ) w = u( µ B c+ Q ( DΣ Σ ) B c+ wb ) has the solution w*. Then the optimization problem has the solution x* = z + w* z where

13 z Σ B ( BΣ B ) T T c = = ( T Q, T Q D ) T z Generalized Sharpe ratio ER ( ) R f f= ( Var ( R β )) max t( x) = x, p( x) = x R, v( x) = x β, β / f 3

14 β u( x) =, u( x) = x x R f The main equation reduces to quadratic equation T T β ( µ Bc+ Q ( DΣ Σ ) Bc+ wb Rf ) w = ( f0 + b w ) solutions: w * *, w w* = arg max( f, f ) w * * w x* = z + w* z 4

15 Classical Sharpe ratio β = / The main equation reduces to equation of degree T T ( µ Bc+ Q ( DΣ Σ ) Bc+ wb Rf ) w = ( f0 + b w ) w = f µ B c+ Q D B c-r * 0 T T ( Σ Σ) f x* = z + w* z 5

16 Combination of linear and function of square root of quadratic functional Recall T per ( ( ) p( ) f =t = µ x t max T v( Var( R)) v( x Σx) If tx ( ) = log x, px ( ) = exp( x), vx ( ) = exp( λsx ( )), λ > 0 f = x λs( x Σx) max µ T T 6

17 Let s ( x ) = s '( x ) > 0, x (0, ) Theorem. If equation 0 ws ( f + b w ) = λ has the positive solution w* this solution is unique and the problem T f( x) = µ x+ λs( x Σx) inf T Bx=c 7

18 has the unique solution x* = z + w* z 8

19 Application to the optimal portfolio selection sx ( ) = x f T ( x) = µ x+ λ x Σx inf T Translation invariant and positive homogeneous risk measures ) ρ( X + α) = ρ( X) + α, α const ) ρ( cx) = cρ( X), c > 0 const 9

20 Examples BASEL II VaR VaR ( X) = inf{ x F ( x) q} q Tail conditional expectation X TCE ( X ) = E( X X > VaR ( X )) q q Tail VaR, Expected Short Fall (ES), Conditional VaR (CVaR) 0

21 STD-premium ρ( X ) = E( X ) + λstd( X ) Distorted risk measures ρ ( X) = g( F ( x)) dx, 0 Denneberg (994) and Wang (996) Coherent risk measures Artzner, Delbean, Eber, and Heath (999)

22 Application of Theorem sx ( ) = x s ( x) = s'( x) = Recall general equation x 0 ws ( f + b w ) = λ

23 Reduces to w = f + bw λ 0 ( λ b ) w = f 0 The positive solution w f ( BΣ c B ) c λ b λ Δ Q Δ T T * 0 = = T

24 Exists iff λ > b T = Δ Q Δ Recall the general solution * = w * x z + z T ( BΣ B T ) λ T Q c Δ Δ c

25 This well conforms with L. (008), L. and Makov (0) Power function s β sx ( ) = x, β f T ( x) = µ x+ λ( x Σx) inf T β

26 Condition v( x) = s( px + qx + r) = ( px + qx + r) convex β q pr < β v ''( x) = ( px + qx + r) 0 ((β )( px + q) ( q pr)) > 0

27 Application of Theorem sx ( ) = s ( x) = s'( x) = x x β β β General equation 0 ws ( f + b w ) = λ Reduces to power equation 7

28 β cw b w f = 0 0 (*) where c = ( βλ) β For β = (*) reduces w ( λ b ) = f 0 Then the positive solution f0 w* =, λ > b = λ b Δ T 8 Q Δ

29 well conforms with TIPH case. Another special case 3 The power equation β β = cw b w f = where c = 4 ( βλ) 0 β 0 (*) Reduces to biquadratic equation cw 4 b w f = 0 0 9

30 where c = (.5 λ) 4. The unique positive solution w* = 4 b + b + cf0 c Recall the general solution * = w * x z + z 30

31 Solutions w* of power equation w* beta 3

32 Classical mean-variance case sx ( ) = x f T ( x) = µ x+ λx Σx inf T s ( x) = s'( x) = Recall general equation 0 ws ( f + b w ) = Reduces w* = λ λ 3

33 The optimal portfolio x * = z + z λ Tail mean-variance optimal portfolio selection sx ( ) = α x+ α x α s ( x) = s'( x) = + x The optimization problem reduces α f Σ Σ T T T ( x) = µ x+ λα ( x x+ αx x33 ) inf

34 Special case-tail mean-variance premium TMV ( R) = TCE ( R) + λtv ( R) q q q = µ x+ λ x Σx+ λλ x Σx T T T, q, q inf where TV ( R) = V ( R R > VaR ( R)) q is tail variance of R q 34

35 General equation 0 ws ( f + b w ) = λ reduces to a quartic equation w 4 kw 3 + ( f + ) 0 0 k 0 0 4λ w kf w + k f = 35

36 w* -unique real solution of quartic equation on the interval [0,k] Analysis of quartic equation P ( w) 4 = 4 3 w kw + ( f0 + k ) w kf 0w + k f0 = 0 4λ P kf 4 0 (0) = > 0 The real root exists in [0,k], k P ( k) = < it is unique, and we found its explicit closed form (L. (00)) λ 36

37 Application to stock data returns Consider a portfolio of 0 stocks from NASDAQ/Computers 37

38 38

39 Optimal selection 39

40 Changing portfolio with q increased MV TMV q=0.5 Weights Adobe Compuware NVDIA TMV q=0.9 Staples VeriSign Sandisk Microsoft Intuit Citrix Pure minv Symantec Companies 40

41 Conclusion -Optimal selection with generalization of Sharpe ratio allows to solve many problems This problem reduces to CFP, the solution has explicit close from and can be used for analyzing the influence of parameters of underlying distribution on the portfolio selecting 4

42 References Boyle, P.P, Cox, S.H,...[et. al]. Financial Economics: With Applications to Investments, Insurance and Pensions, Actuarial Foundation, Schaumburg (998) Landsman, Z. (008). Minimization of the root of a quadratic functional under an affine equality constraint. Journal of Computational and Applied Math. 6, Landsman, Z. (008b). Minimization of the root of a quadratic functional under a system of equality constraint with application in portfolio management, Journal of Computational and Applied Math, 4 0, -, pp

43 Landsman, Z. (00). On the tail mean variance optimal portfolio selection, Insurance Mathematics and Economics, 46, p andsman Z. and Makov U. (0). TIPH risk measures and OPS n the presence of a riskless component, IME, 0, 50,, Landsman Z. and Makov U. (03) Minimization of a function of a quadratic functional with application to optimal portfolio selection. Technical Report. ARC, University of Haifa Thank You!