lCARE - localising Conditional AutoRegressive Expectiles Wolfgang - PowerPoint PPT Presentation

lCARE - localising Conditional AutoRegressive Expectiles Wolfgang Karl Härdle Xiu Xu Andrija Mihoci Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. – Center for Applied Statistics and Economics Humboldt–Universität zu Berlin Brandenburg University of Technology lvb.wiwi.hu-berlin.de case.hu-berlin.de irtg1792.hu-berlin.de b-tu.de

Motivation 1-1 Motivation ⊡ Risk Exposure Measure tail events ◮ Conditional autoregressive expectile (CARE) model ◮ Expectiles ⊡ Time-varying parameters Time-varying parameters in CARE ◮ Parameter Dynamics Interval length reflects the structural changes in economy ◮ 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Motivation 1-2 Objectives ⊡ Localising CARE Models Local parametric approach (LPA) ◮ Balance between modelling bias and parameter variability ◮ ⊡ Tail Risk Dynamics Estimation windows with varying lengths ◮ Time-varying expectile parameters ◮ 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Motivation 1-3 Econometrics and Risk Management Econometrics ⊡ Modelling bias vs. parameter variability ⊡ Interval length and economic variables Risk Management ⊡ Parameter dynamics and structural changes ⊡ Measuring tail risk 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Motivation 1-4 Risk Exposure An investor observes daily DAX returns from 20050103 to 20141231 and estimates the underlying risk exposure via expectiles (e.g., 1% and 5%) over a one-year time horizon. Modelling strategies (a) Data windows fixed on an ad hoc basis (b) Adaptively selected data intervals: time-varying parameters 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Motivation 1-5 Portfolio Protection An investor decides about the daily allocation into a stock portfolio (DAX). Goal: a proportion of the initial portfolio value (100) is preserved at the end of a horizon, i.e., the target floor equals 90. Decision at day t : multiple of the difference between the portfolio value and the discounted floor up to t is invested into the stock portfolio (DAX), the rest into a riskless asset. Multiplier m selection: constant or time-varying (lCARE) Constant m 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Motivation 1-6 Research Questions How to account for time-varying parameters in tail event risk measures estimation? What are the typical data interval lengths assessing risk more accurately, i.e., striking a balance between bias and variability? How well does the lCARE technique perform in practice? 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Outline 1. Motivation � 2. Conditional Autoregressive Expectile (CARE) 3. Local Parametric Approach (LPA) 4. Empirical Results 5. Applications 6. Conclusions 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Conditional Autoregressive Expectile (CARE) 2-1 Conditional Autoregressive Expectile ⊡ Taylor (2008), Kuan et al. (2009), Engle and Manganelli (2004) CAViaR ⊡ Random variable Y (e.g. returns), identically distributed, y t , t = 1 , ..., n ⊡ CARE specification conditional on information set F t − 1 � � y t = e t ,τ + ε t ,τ 0 , σ 2 ε τ ∼ AND ε,τ , τ � � 2 + α 3 ,τ � � 2 y + y − e t ,τ = α 0 ,τ + α 1 ,τ y t − 1 + α 2 ,τ t − 1 t − 1 � � ⊤ Expectile e t ,τ at τ ∈ ( 0 , 1 ) , θ τ = α 0 ,τ , α 1 ,τ , α 2 ,τ , α 3 ,τ , σ 2 ◮ ε,τ Returns: y + t − 1 = max { y t − 1 , 0 } , y − t − 1 = min { y t − 1 , 0 } ◮ 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Conditional Autoregressive Expectile (CARE) 2-2 Parameter Estimation ⊡ Data calibration with time-varying intervals ⊡ Observed returns Y = { y 1 , . . . , y n } ⊡ Quasi maximum likelihood estimate (QMLE) � θ I ,τ = arg max θ τ ∈ Θ ℓ I ( Y ; θ τ ) ℓ I ( · ) I = [ t 0 − v , t 0 ] - interval of ( v + 1 ) observations at t 0 ◮ ℓ I ( · ) - quasi log likelihood ◮ 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Conditional Autoregressive Expectile (CARE) 2-3 Estimation Quality ⊡ Mercurio and Spokoiny (2004), Spokoiny (2009) τ by QMLE � ⊡ Quality of estimating true parameter vector θ ∗ θ I ,τ in terms of Kullback-Leibler divergence; R r ( θ ∗ τ ) - risk bound � � r � � � ℓ I ( Y ; � � � θ I ,τ ) − ℓ I ( Y ; θ ∗ ≤ R r ( θ ∗ E θ ∗ τ ) τ ) � θ ∗ R r Gaussian Regression τ τ ⊡ ’Modest’ risk, r = 0 . 5 (shorter intervals of homogeneity) ⊡ ’Conservative’ risk, r = 1 (longer intervals of homogeneity) Solomon Kullback and Richard A. Leibler on BBI: 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Local Parametric Approach (LPA) 3-1 Local Parametric Approach (LPA) ⊡ LPA, Spokoiny (1998, 2009) Time series parameters can be locally approximated ◮ Finding the interval of homogeneity ◮ Details Balance between modelling bias and parameter variability ◮ ⊡ Time series literature GARCH ( 1 , 1 ) models - Čížek et al. (2009) ◮ Realized volatility - Chen et al. (2010) ◮ Multiplicative Error Models - Härdle et al. (2015) ◮ 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Local Parametric Approach (LPA) 3-2 Interval Selection ⊡ ( K + 1 ) nested intervals with length n k = | I k | I 0 ⊂ I 1 ⊂ · · · ⊂ I k ⊂ · · · ⊂ I K � � � � θ 0 θ 1 θ k θ K Example: Daily index returns � n 0 c k � Fix t 0 , I k = [ t 0 − n k , t 0 ] , n k = , c > 1 { n k } 11 k = 0 = { 20 days , 25 days , . . . , 250 days } , c = 1 . 25 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Local Parametric Approach (LPA) 3-3 Local Change Point Detection ⊡ Fix t 0 , sequential test ( k = 1 , . . . , K ) H 0 : parameter homogeneity within I k H 1 : ∃ change point within J k = I k \ I k − 1 � � � � � � �� Y , � Y , � Y , � T k ,τ = sup ℓ A k , s θ A k , s ,τ + ℓ B k , s θ B k , s ,τ − ℓ I k + 1 θ I k + 1 ,τ s ∈ J k with A k , s = [ t 0 − n k + 1 , s ] and B k , s = ( s , t 0 ] 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Local Parametric Approach (LPA) 3-4 Critical Values, z k ,τ ⊡ Simulate z k - homogeneity of the interval sequence I 1 , . . . , I k ⊡ ’Propagation’ condition � � � � �� r � � Y ; � Y ; � ≤ ρ k R r ( θ ∗ E θ ∗ � ℓ I k θ I k ,τ − ℓ I k θ τ τ ) � τ ρ k = ρ k � K for a given significance level ρ θ τ - adaptive estimate ⊡ Check z k ,τ for (six) different θ ∗ Parameter Scenarios τ 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Local Parametric Approach (LPA) 3-5 Critical Values, z k ,τ 20 20 20 Values 10 10 10 0 0 0 20 40 60 120 20 40 60 120 20 40 60 120 Length in Days Length in Days Length in Days Figure 1: Simulated critical values across different parameter constellations Parameter Scenarios for the modest case r = 0 . 5, τ = 0 . 05 and τ = 0 . 01. LCARE_Critical_Values LCARE_Critical_Values_Th1_001 LCARE_Critical_Values_Th1_005 LCARE_Critical_Values_Th2_001 LCARE_Critical_Values_Th2_005 LCARE_Critical_Values_Th3_001 LCARE_Critical_Values_Th3_005 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Local Parametric Approach (LPA) 3-6 Critical Values, z k ,τ 400 400 400 Values 200 200 200 0 0 0 20 40 60 120 20 40 60 120 20 40 60 120 Length in Days Length in Days Length in Days Figure 2: Simulated critical values across different parameter constellations Parameter Scenarios for the conservative case r = 1, τ = 0 . 05 and τ = 0 . 01 . LCARE_Critical_Values LCARE_Critical_Values_Th1_001 LCARE_Critical_Values_Th1_005 LCARE_Critical_Values_Th2_001 LCARE_Critical_Values_Th2_005 LCARE_Critical_Values_Th3_001 LCARE_Critical_Values_Th3_005 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Local Parametric Approach (LPA) 3-7 Adaptive Estimation LPA z k ,τ - Critical Values ⊡ Compare T k ,τ at every step k with z k ,τ ⊡ Data window index of the interval of homogeneity - � k ⊡ Adaptive estimate θ τ = � � � k = max θ I ˆ k ,τ , k ≤ K { k : T ℓ,τ ≤ z ℓ,τ , ℓ ≤ k } 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014

Empirical Results 4-1 Data ⊡ Series DAX, FTSE 100 and S&P 500 returns ◮ 20050103-20141231 (2608 days) Research Data Center (RDC) - Datastream ◮ ⊡ Setup Expectile levels: τ = 0 . 05 and τ = 0 . 01 ◮ Modest ( r = 0 . 5) and conservative ( r = 1) risk cases ◮ { n k } 11 k = 0 = { 20 days , 25 days , . . . , 250 days } ◮ 0.1 lCARE - localising Conditional AutoRegressive Expectiles 0 −0.1 2006 2008 2010 2012 2014