Modelling Joint Distribution of Returns. Dr. Sawsan Hilal space

Transcription

1 Modelling Joint Distribution of Returns Dr. Sawsan Hilal space Maths Department - University of Bahrain space October 2011

2 REWARD Asset Allocation Problem PORTFOLIO w 1 w 2 w 3 ASSET 1 ASSET 2 R 1 R 2 w 4 ASSET 3 R3 ASSET 4 R4 Modelling Joint Distribution of Portfolio Constituent Assets RISK margins dependence

3 Sklar s Theorem - Theoretical Formulation

4 Sklar s Theorem - Practical Implication How can we exploit this result for statistical modelling ( inference for the joint distribution F )? Stage 1. Margins Modelling F i for i = 1,..., d. Stage 2. Dependence Modelling C

5 Financial Returns - Univariate Stylized Facts The return on day t is R t = 100 ln{p t /P t 1 }: relative price change observations on CAC40 Crash 81 Invasion of Iraq Asian Crisis 9/ Returns 5 l l l l 10 Gulf War LTCM debacle Oil Crises Black Monday Mexican Peso Crisis Dot Com bubble Jan 69 Jan 70 Jan 71 Jan 72 Jan 73 Jan 74 Jan 75 Jan 76 Jan 77 Jan 78 Jan 79 Jan 80 Jan 81 Jan 82 Jan 83 Jan 84 Jan 85 Jan 86 Jan 87 Jan 88 Jan 89 Jan 90 Jan 91 Jan 92 Jan 93 Jan 94 Jan 95 Jan 96 Jan 97 Jan 98 Jan 99 Jan 00 Jan 01 Jan 02 Jan 03

6 Financial Returns - Univariate Stylized Facts Volatility Clustering The returns are uncorrelated but not independent! Returns Squared Returns ACF ACF Lag Lag

7 Financial Returns - Univariate Stylized Facts Heavy Tailedness The returns are not normally distributed! Density empirical normal

8 Stage 1: Margins Modelling

9 Stage 1: Margins Modelling where R t = with the conditions that µ t }{{} conditional mean + σ t }{{} volatility process X t }{{} filtered returns µ t = E t 1 (R t ) σ 2 t = E t 1 [ (Rt µ t ) 2] }{{} shocks X t F X X t are IID having mean 0 & variance 1, X t and σ t are independent at any time point.

10 Stage 1: Margins Modelling Step 1. Modelling the conditional mean µ t Auto-Regressive Moving Average (ARMA) e.g. ARMA(1,0): µ t = ψ 0 + ψ 1 R t 1 Step 2. Modelling the volatility process σ t Generalized Auto-Regressive Conditional Heteroskedastic (GARCH) e.g. GARCH(1,1): σt 2 = ω }{{} 0 + ω 1 (R }{{} t 1 µ t ) 2 + ω }{{} 2 >0 0 0 σ 2 t 1 where 0 < ω 1 + ω 2 < 1 measures degree of volatility persistence.

11 Stage 1: Margins Modelling Step 3. Modelling the distribution F X of filtered returns: It is commonly assumed to be either Normal or Student s t. Instead, we adopt a semi-parametric approach. Specifically, G } eneralizedparetodistribution {{} + E mpiricaldistributionf unction }{{} GPD EDF Tails of F X are modelled by GPD Centre of F X is modelled by EDF

12 Stage 1: Margins Modelling

13 Stage 1: Margins Modelling Theorem (Balkema & de Haan 1974 and Pickands 1975) Let u + be a high threshold. Then we have ( x u Pr [X > x X > u + ] GPD = 1 [1 + ξ + + σ + )] 1/ξ + Let u be a low threshold. Then we have Pr [X < x X < u ] GPD = [ ( u 1 + ξ )] 1/ξ x σ σ > 0 (scale parameter) and ξ R (shape parameter)

14 Stage 1: Margins Modelling F X (x) #(X t x)/n if x [u, u + ] P (1 + ξ (u x)/σ ) 1/ξ if x < u 1 P + (1 + ξ + (x u + )/σ + ) 1/ξ+ if x > u +. P = #(X t < u )/n P + = #(X t > u + )/n

15 Stage 1: Margins Modelling

16 Stage 1: Margins Modelling Realized Returns: R Filtered Returns: X Jan 69 Jan 70 Jan 71 Jan 72 Jan 73 Jan 74 Jan 75 Jan 76 Jan 77 Jan 78 Jan 79 Jan 80 Jan 81 Jan 82 Jan 83 Jan 84 Jan 85 Jan 86 Jan 87 Jan 88 Jan 89 Jan 90 Jan 91 Jan 92 Jan 93 Jan 94 Jan 95 Jan 96 Jan 97 Jan 98 Jan 99 Jan 00 Jan 01 Jan 02 Jan 03 Jan 69 Jan 70 Jan 71 Jan 72 Jan 73 Jan 74 Jan 75 Jan 76 Jan 77 Jan 78 Jan 79 Jan 80 Jan 81 Jan 82 Jan 83 Jan 84 Jan 85 Jan 86 Jan 87 Jan 88 Jan 89 Jan 90 Jan 91 Jan 92 Jan 93 Jan 94 Jan 95 Jan 96 Jan 97 Jan 98 Jan 99 Jan 00 Jan 01 Jan 02 Jan 03 Squared Realized Returns Squared Filtered Returns ACF ACF Lag Lag

17 Stage 1: Margins Modelling Lower Tail Fit 1 F(x) (on log scale) GPD Student's t Normal x (on log scale)

18 Marginal Standardization

19 Financial Returns - Multivariate Stylized Facts (1) Non-Linear Dependence: ρ is not a suitable dependence measure

20 Financial Returns - Multivariate Stylized Facts (2) Two Forms of Tail Dependence & Asymmetric Dependence

21 Stage 2: Dependence Modelling Conditional Approach (Heffernan & Tawn 2004)

22 Stage 2: Dependence Modelling Problem of Interest: Consider the conditional distribution Pr {Y j z Y i = y} for y > u (high threshold)

23 Stage 2: Dependence Modelling Asymptotic Assumption: Assume the existence of L j i (y) : R R (location function) S j i (y) : R R (scale function) such that for z R we have { lim Pr Yj L j i (y) y S j i (y) z Y i } = y = G j i (z) where G j i (z) is a non-degenerate distribution.

24 Stage 2: Dependence Modelling Statistical Model: The following non-linear regression model is applied if Y i = y > u Y j = L j i (y) + S j i (y) Z j i Identify the form of location & scale function & error distribution L(y) j i = αy for α [ 1, +1] S(y) j i = y β for β (, 1) G j i (z) it has no specific form

25 Stage 2: Dependence Modelling The one-day (January 4, 2010) joint probability of G5 economies

26 Summary Statistical Model truly multivariate facilitates extrapolation straightforward inference Financial Application asset allocation problem...

27 References books 1 Coles 2001: introduction to EVT and data analysis 2 Embrechts et al. 1997: EVT for insurance and finance articles 1 Coles et al. 1999: extremal dependence measures 2 McNeil and Frey 2000: estimating tail-related risk measures 3 Poon, Tawn, Rockinger 2004: extremal dependence in finance 4 Heffernan and Tawn 2004: conditional approach for extremes R-packages * ismev and evd by Alec Stephenson