Risk Reward Optimisation for Long-Run Investors: an Empirical Analysis

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3 Risk Reward Optimisation for Long-Run Investors: an Empirical Analysis M. Gilli University of Geneva and Swiss Finance Institute E. Schumann University of Geneva AFIR / LIFE Colloquium 2009 München, 6 11 September 2009 Gilli & Schumann Risk and reward 2

4 introduction portfolio optimisation: allocate wealth among n A assets Gilli & Schumann Risk and reward 3

5 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient Gilli & Schumann Risk and reward 3

6 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw Gilli & Schumann Risk and reward 3

7 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) Gilli & Schumann Risk and reward 3

8 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) 2: estimate ˆμ = 1 T ι R 3: estimate ˆΣ = 1 T R ( I 1 T ιι ) R Gilli & Schumann Risk and reward 3

9 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) 2: estimate ˆμ = 1 T ι R 3: estimate ˆΣ = 1 T R ( I 1 T ιι ) R 4: find weights w that maximise w ˆμ γ 2 w ˆΣw Gilli & Schumann Risk and reward 3

10 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) 2: estimate ˆμ = 1 T ι R 3: estimate ˆΣ = 1 T R ( I 1 T ιι ) R 4: find weights w that maximise w ˆμ γ 2 w ˆΣw Gilli & Schumann Risk and reward 4

11 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) 2: estimate ˆμ = 1 T ι R 3: estimate ˆΣ = 1 T R ( I 1 T ιι ) R 4: find weights w that maximise w ˆμ γ 2 w ˆΣw estimation problems: Jobson and Korkie (1980), Jorion (1985), Jorion (1986), Best and Grauer (1991), Chopra et al. (1993), Board and Sutcliffe (1994) and many others Gilli & Schumann Risk and reward 4

12 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) 2: estimate ˆμ = 1 T ι R 3: estimate ˆΣ = 1 T R ( I 1 T ιι ) R 4: find weights w that maximise w ˆμ γ 2 w ˆΣw Gilli & Schumann Risk and reward 5

13 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) 2: estimate ˆμ = 1 T ι R 3: estimate ˆΣ = 1 T R ( I 1 T ιι ) R 4: find weights w that maximise w ˆμ γ 2 w ˆΣw theoretical concerns: Artzner et al. (1999), Pedersen and Satchell (1998), Pedersen and Satchell (2002) Gilli & Schumann Risk and reward 5

14 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) 2: estimate ˆμ = 1 T ι R 3: estimate ˆΣ = 1 T R ( I 1 T ιι ) R 4: find weights w that maximise w ˆμ γ 2 w ˆΣw Gilli & Schumann Risk and reward 6

15 introduction portfolio optimisation: allocate wealth among n A assets Markowitz (1952), Markowitz (1959): select portfolio that is mean variance efficient find weights w that maximise w μ γ 2 w Σw 1: collect historical data R (T n A ) 2: estimate ˆμ = 1 T ι R 3: estimate ˆΣ = 1 T R ( I 1 T ιι ) R 4: find weights w that maximise w ˆμ γ 2 w ˆΣw aim of research: test alternative risk measures & objective functions empirically test alternative estimation and scenario generation methods Gilli & Schumann Risk and reward 6

16 outline alternative objective functions Gilli & Schumann Risk and reward 7

17 outline alternative objective functions data/optimisation Gilli & Schumann Risk and reward 7

18 outline alternative objective functions data/optimisation empirical results Gilli & Schumann Risk and reward 7

19 alternative objective functions: building blocks w μ }{{} reward γ 2 w Σw }{{} risk Gilli & Schumann Risk and reward 8

20 alternative objective functions: building blocks w μ }{{} reward γ 2 w Σw }{{} risk replace reward and risk by alternative functions min w risk reward Gilli & Schumann Risk and reward 8

21 alternative objective functions: building blocks w μ }{{} reward γ 2 w Σw }{{} risk replace reward and risk by alternative functions min w risk reward = Φ Gilli & Schumann Risk and reward 8

22 alternative objective functions: building blocks based on distribution of portfolio returns 100% 80% 60% 40% 20% 0% 4% 2% 0% 2% 4% returns Gilli & Schumann Risk and reward 9

23 alternative objective functions: building blocks based on distribution of portfolio returns moments (variance, skewness,...) conditional moments (expected shortfall,...), partial moments (semivariance,...) quantiles (VaR,...), corresponding probabilities (shortfall probability,...) Gilli & Schumann Risk and reward 9

24 alternative objective functions: building blocks based on distribution of portfolio returns moments (variance, skewness,...) conditional moments (expected shortfall,...), partial moments (semivariance,...) quantiles (VaR,...), corresponding probabilities (shortfall probability,...) based on trajectory of portfolio wealth drawdown (D), time under water,... Gilli & Schumann Risk and reward 9

25 350 alternative objective functions: building blocks based on distribution of portfolio returns moments (variance, skewness,...) conditional moments (expected shortfall,...), partial moments (semivariance,...) quantiles (VaR,...), corresponding probabilities (shortfall probability,...) 200 Dec97 Feb98 Apr98 May98 Jul98 25 based on trajectory of portfolio wealth Dec97 Feb98 Apr98 May98 Jul98 Gilli & Schumann Risk and reward 9

26 alternative objective functions: building blocks based on distribution of portfolio returns moments (variance, skewness,...) conditional moments (expected shortfall,...), partial moments (semivariance,...) quantiles (VaR,...), corresponding probabilities (shortfall probability,...) based on trajectory of portfolio wealth drawdown (D), time under water,... objective function: do as you please Gilli & Schumann Risk and reward 9

27 alternative objective functions: building blocks based on distribution of portfolio returns moments (variance, skewness,...) conditional moments (expected shortfall,...), partial moments (semivariance,...) quantiles (VaR,...), corresponding probabilities (shortfall probability,...) based on trajectory of portfolio wealth drawdown (D), time under water,... objective function: do as you please Gilli & Schumann Risk and reward 9

28 alternative objective functions: partial moments capture non-symmetrical returns Bawa (1975); Fishburn (1977): r = r }{{} d + (r r d ) + }{{} desired return upside (r d r) + }{{} downside Gilli & Schumann Risk and reward 10

29 alternative objective functions: partial moments capture non-symmetrical returns Bawa (1975); Fishburn (1977): r = r }{{} d + (r r d ) + }{{} desired return upside (r d r) + }{{} downside P + γ (r d ) = 1 T P γ (r d ) = 1 T (r r d ) γ, r>r d (r d r) γ. r<r d example: semi-variance Gilli & Schumann Risk and reward 10

30 alternative obj. functions: conditional moments capture non-symmetrical returns: r = r }{{} d + (r r d ) + }{{} desired return upside (r d r) + }{{} downside C + γ (r d ) = C γ (r d ) = 1 #{r > r d } 1 #{r < r d } (r r d ) γ, r>r d (r d r) γ, r<r d example: Expected Shortfall Gilli & Schumann Risk and reward 11

31 alternative obj. functions: conditional moments capture non-symmetrical returns: r = r }{{} d + (r r d ) + }{{} desired return upside (r d r) + }{{} downside conditional vs partial moments P + γ (r d ) = C + γ (r d )P + 0 (r d ) }{{} π of r>r d P γ (r d ) = C γ (r d )P 0 (r d ) }{{} π of r<r d Gilli & Schumann Risk and reward 11

32 alternative objective functions: quantiles Q q = CDF 1 (q) = min{r CDF(r) q}, example: VaR Gilli & Schumann Risk and reward 12

33 alternative objective functions: examples reward risk constant C 1 (Q q ) minimise Expected Shortfall for qth quantile constant Q 0 minimise maximum loss 1 n S r P 2 (r d ) Sortino ratio P + 1 (r d ) P 2 (r d ) Upside Potential ratio P + 1 (r d ) P 1 (r d ) Omega for threshold r d 1 n S r Dmax Calmar ratio C + γ (Q p ) C δ (Q q) Rachev Generalised ratio for exponents γ and δ Gilli & Schumann Risk and reward 13

34 estimation Gilli & Schumann Risk and reward 14

35 estimation x 100% 80% 60% 40% 20% 0% 4% 2% 0% 2% 4% returns Gilli & Schumann Risk and reward 14

36 estimation x 100% 80% 60% 40% 20% 0% 4% 2% 0% 2% 4% returns empirical distribution of portfolio returns (order statistics r [1] r [2] r [T] ) Gilli & Schumann Risk and reward 14

37 estimation bootstrapping returns (r B ) from a simple regression model: r it = α i +β i r Mt + +ε it i = 1,...,n A t = 1,...,T Gilli & Schumann Risk and reward 15

38 estimation bootstrapping returns (r B ) from a simple regression model: r it = α i +β i r Mt + +ε it i = 1,...,n A t = 1,...,T regressors: indices, PCA... Gilli & Schumann Risk and reward 15

39 optimisation min x Φ(x) Gilli & Schumann Risk and reward 16

40 optimisation min Φ(x) x j J x jp 0j = v 0 Gilli & Schumann Risk and reward 16

41 optimisation min x Φ(x) j J x jp 0j = v 0 x inf j x j x sup j j J K inf #{J} K sup. (x = numbers of shares, A = all assets, J = assets included in portfolio) Gilli & Schumann Risk and reward 16

42 optimisation min x Φ(x) j J x jp 0j = v 0 x inf j x j x sup j j J K inf #{J} K sup. (x = numbers of shares, A = all assets, J = assets included in portfolio) Threshold Accepting: Dueck and Scheuer (1990), Winker (2001), Gilli and Schumann (in press), Matlab code available from Gilli & Schumann Risk and reward 16

43 data and methodology Gilli & Schumann Risk and reward 17

44 data and methodology 600 assets (EUR) from DJ STOXX (7-Jan Mar-2008) Gilli & Schumann Risk and reward 17

45 data and methodology 600 assets (EUR) from DJ STOXX (7-Jan Mar-2008) market capitalisation > Euros (248 assets) Gilli & Schumann Risk and reward 17

46 data and methodology 600 assets (EUR) from DJ STOXX (7-Jan Mar-2008) market capitalisation > Euros (248 assets) historical window (52 weeks), horizon (12 weeks) 40 rebalancings Gilli & Schumann Risk and reward 17

47 data and methodology 600 assets (EUR) from DJ STOXX (7-Jan Mar-2008) market capitalisation > Euros (248 assets) historical window (52 weeks), horizon (12 weeks) 40 rebalancings iper = 1 fcsthorz day lengthhist t 1 t 2 t 3 iper = 2 debsim Rebalance PF t 1 t 2 t 3 Gilli & Schumann Risk and reward 17

48 data and methodology 600 assets (EUR) from DJ STOXX (7-Jan Mar-2008) market capitalisation > Euros (248 assets) historical window (52 weeks), horizon (12 weeks) 40 rebalancings 10bp variable cost for long positions Gilli & Schumann Risk and reward 17

49 data and methodology 600 assets (EUR) from DJ STOXX (7-Jan Mar-2008) market capitalisation > Euros (248 assets) historical window (52 weeks), horizon (12 weeks) 40 rebalancings 10bp variable cost for long positions minimum trading size (5 000) Gilli & Schumann Risk and reward 17

50 data and methodology 600 assets (EUR) from DJ STOXX (7-Jan Mar-2008) market capitalisation > Euros (248 assets) historical window (52 weeks), horizon (12 weeks) 40 rebalancings 10bp variable cost for long positions minimum trading size (5 000) holding size constraints Gilli & Schumann Risk and reward 17

51 benchmark: minimum variance (MV) Gilli & Schumann Risk and reward 18

52 benchmark: minimum variance (MV) Σ estimated as covariance matrix ˆΣ from historical observations Gilli & Schumann Risk and reward 18

53 benchmark: minimum variance (MV) Σ estimated as covariance matrix ˆΣ from historical observations min w ˆΣw w j J w j = 1 0 w j w sup j j = 1,...,n A Gilli & Schumann Risk and reward 18

54 benchmark: minimum variance (MV) Σ estimated as covariance matrix ˆΣ from historical observations min w ˆΣw w j J w j = 1 0 w j w sup j j = 1,...,n A optimisation with maximum holding size and sector allocation constraints done with Matlab s quadprog. Gilli & Schumann Risk and reward 18

55 benchmark: minimum variance (MV) Gilli & Schumann Risk and reward 19

56 benchmark: minimum variance (MV) MV 1/N DJind Sep98 Jan00 May01 Oct02 Feb04 Jul05 Nov06 Gilli & Schumann Risk and reward 19

57 benchmark: minimum variance (MV) introducing uncertainty: Gilli & Schumann Risk and reward 19

58 benchmark: minimum variance (MV) introducing uncertainty: compute minimum-variance portfolio from jackknifed or bootstrapped time series Gilli & Schumann Risk and reward 19

59 benchmark: minimum variance (MV) Gilli & Schumann Risk and reward 20

60 benchmark: minimum variance (MV) Sep98 Jan00 May01 Oct02 Feb04 Jul05 Nov06 Apr08 Gilli & Schumann Risk and reward 20

61 benchmark: minimum variance (MV) Gilli & Schumann Risk and reward 20

62 benchmark: minimum variance (MV) Annualized return Annualized return Gilli & Schumann Risk and reward 20

63 benchmark: minimum variance (MV) Gilli & Schumann Risk and reward 21

64 benchmark: minimum variance (MV) Gilli & Schumann Risk and reward 21

65 benchmark: minimum variance (MV) Gilli & Schumann Risk and reward 21

66 benchmark: minimum variance (MV) Gilli & Schumann Risk and reward 21

67 benchmark: minimum variance (MV) % 5% reward risk median c M MV-portfolio 10% 15% 20% 0% 5% 10% 15% 20% Gilli & Schumann Risk and reward 21

68 results VaR, Expected Shortfall Gilli & Schumann Risk and reward 22

69 results VaR, Expected Shortfall MV VaR ES Annualized return MV VaR ES Annualized return Gilli & Schumann Risk and reward 22

70 results VaR, Expected Shortfall Gilli & Schumann Risk and reward 23

71 results VaR, Expected Shortfall 0% 5% reward risk median c M MV-portfolio c Q c Q c Q c Q % 15% 20% 0% 5% 10% 15% 20% Gilli & Schumann Risk and reward 23

72 results VaR, Expected Shortfall 0% reward risk median c M MV-portfolio c Q c Q c Q c Q % 5% 5% reward risk median c M MV-portfolio c C 1 (Q 1) c C 1 (Q 5) c C 1 (Q 10) c C 1 (Q 20) % 10% 15% 15% 20% 20% 0% 5% 10% 15% 20% Gilli & Schumann Risk and reward 23

73 results partial moments Gilli & Schumann Risk and reward 24

74 results partial moments MV Γ (Prob) Ω SemiVar Annualized return MV Γ (Prob) Ω SemiVar Annualized return Gilli & Schumann Risk and reward 24

75 results partial moments Gilli & Schumann Risk and reward 25

76 results partial moments reward risk median c M MV-portfolio P + 0 P P + 1 P P + 1 P P P P P P + 2 P P + 3 P P + 4 P P + 4 P % 5% 10% 15% 20% 0% 5% 10% 15% 20% Gilli & Schumann Risk and reward 25

77 results partial moments Gilli & Schumann Risk and reward 26

78 results partial moments median annualised returns q (gains) 0 0 Gilli & Schumann p (losses) Risk and reward

79 results partial moments Gilli & Schumann Risk and reward 27

80 results partial moments reward risk median c M MV-portfolio P + 2 c 0.70 P + 3 c 0.96 c P c P % 5% 10% 15% 20% 0% 5% 10% 15% 20% Gilli & Schumann Risk and reward 27

81 results partial moments Gilli & Schumann Risk and reward 28

82 results partial moments reward risk upside deviation in % downside deviation in % c M c P0 c P1 c P1.5 c P2 c P3 c Q 1 c Q 5 c Q 10 c Q Gilli & Schumann Risk and reward 28

83 results drawdowns Gilli & Schumann Risk and reward 29

84 results drawdowns MV std DD mean DD max DD Annualised return MV std DD mean DD max DD Annualised return Gilli & Schumann Risk and reward 29

85 results drawdowns Gilli & Schumann Risk and reward 30

86 results drawdowns MV std DD std DD / mean R Annualised return MV std DD std DD / mean R Annualised return Gilli & Schumann Risk and reward 30

87 results drawdowns Gilli & Schumann Risk and reward 31

88 results drawdowns 0% 5% reward risk median c M MV-portfolio c D mean M 1 D mean c D max M 1 D max c D std M 1 D std % 15% 20% 0% 5% 10% 15% 20% Gilli & Schumann Risk and reward 31

89 conclusions Gilli & Schumann Risk and reward 32

90 conclusions alternative objective functions optimisation more difficult, but manageable Gilli & Schumann Risk and reward 32

91 conclusions alternative objective functions optimisation more difficult, but manageable add value over mean variance Gilli & Schumann Risk and reward 32

92 conclusions alternative objective functions optimisation more difficult, but manageable add value over mean variance problems sensitive to data changes Gilli & Schumann Risk and reward 32

93 conclusions alternative objective functions optimisation more difficult, but manageable add value over mean variance problems sensitive to data changes minimise risk: low variability leads to well-performing portfolios adding reward increases return but also variability and sensitivity Gilli & Schumann Risk and reward 32

94 selected references Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath (1999). Coherent Measures of Risk. Mathematical Finance 9(3), Dueck, G. and T. Scheuer (1990). Threshold Accepting. A General Purpose Optimization Algorithm Superior to Simulated Annealing. Journal of Computational Physics 90(1), Dueck, G. and P. Winker (1992). New Concepts and Algorithms for Portfolio Choice. Applied Stochastic Models and Data Analysis 8(3), Fishburn, P. C. (1977). Mean Risk Analysis with Risk Associated with Below-Target Returns. American Economic Review 67(2), Gilli, M. and E. Schumann (2009a). An Empirical Analysis of Alternative Portfolio Selection Criteria. Swiss Finance Institute Research Paper No Gilli, M. and E. Schumann (2009b). Optimal enough? COMISEF Working Paper Series No. 10. Gilli, M. and E. Schumann (in press). Portfolio optimization with Threshold Accepting : a practical guide. In S. E. Satchell (Ed.), Optimizing Optimization: The Next Generation of Optimization Applications and Theory. Elsevier. Pedersen, C. S. and S. E. Satchell (2002). On the foundation of performance measures under asymmetric returns. Quantitative Finance 2, Winker, P. (2001). Optimization Heuristics in Econometrics: Applications of Threshold Accepting. Wiley. Gilli & Schumann Risk and reward 33