Applications of constrained non parametric smoothing methods in computing financial risk

Applications of Constrained Non-parametric Smoothing Methods in Computing Financial Risk Chung To (Charles) Wong, BCom (Melb), GradDip (RMIT) Submit for the degree of Doctor of Philosophy School of Mathematical Sciences, Queensland University of Technology, Brisbane December, 2008 Abstract The aim of this thesis is to improve risk measurement estimation by incorporating extra information in the form of constraint into completely non-parametric smoothing techniques A similar approach has been applied in empirical likelihood analysis The method of constraints incorporates bootstrap resampling techniques, in particular, biased bootstrap This thesis brings together formal estimation methods, empirical information use, and computationally intensive methods In this thesis, the constraint approach is applied to non-parametric smoothing estimators to improve the estimation or modelling of risk measures We consider estimation of Value-at-Risk, of intraday volatility for market risk, and of recovery rate densities for credit risk management Firstly, we study Value-at-Risk (VaR) and Expected Shortfall (ES) estimation VaR and ES estimation are strongly related to quantile estimation Hence, tail estimation is of interest in its own right We employ constrained and unconstrained kernel density estimators to estimate tail distributions, and we estimate quantiles from the fitted tail distribution The constrained kernel density estimator is an application of the biased bootstrap technique proposed by Hall & Presnell (1998) The estimator that we use for the constrained kernel estimator is the Harrell-Davis (H-D) quantile estimator We calibrate the performance of the constrained and unconstrained kernel density estimators by estimating tail densities based on samples from Normal and Student-t distributions We find a significant improvement in fitting heavy tail distributions using the constrained kernel estimator, when used in conjunction with the H-D quantile estimator We also present an empirical study demonstrating VaR and ES calculation A credit event in financial markets is defined as the event that a party fails to pay an obligation to another, and credit risk is defined as the measure of uncertainty of such events Recovery rate, in the credit risk context, is the rate of recuperation when a credit event occurs It is defined as Recovery rate = − LGD, where LGD is the rate of loss given default From this point of view, the recovery rate is a key element both for credit risk management and for pricing credit derivatives Only the credit risk management is considered in this thesis To avoid strong assumptions about the form of the recovery rate density in current approaches, we propose a non-parametric technique incorporating a mode constraint, with the adjusted Beta kernel employed to estimate the recovery density function An encouraging result for the constrained Beta kernel estimator is illustrated by a large number of simulations, as genuine data are very confidential and difficult to obtain Modelling high frequency data is a popular topic in contemporary finance The intraday volatility patterns of standard indices and market-traded assets have been well documented in the literature They show that the volatility patterns reflect the different characteristics of different stock markets, such as double U-shaped volatility pattern reported in the Hang Seng Index (HSI) We aim to capture this intraday volatility pattern using a non-parametric regression model In particular, we propose a constrained function approximation technique to formally test the structure of the pattern and to approximate the location of the anti-mode of the U-shape We illustrate this methodology on the HSI as an empirical example Keywords: Constraint Method; Expected Shortfall; Non-parametric approach; Recovery rate density; Intraday Volatility; Risk Management; Value-at-Risk Contents Introduction 1.1 Risk 1.2 Risk management 1.2.1 Value-at-Risk 1.2.2 Recovery rate 1.2.3 Intraday Volatility Constraint methods for risk management 1.3.1 Constraint 1.3.2 Background 1.4 Aim of this thesis 1.5 Structure of this thesis 1.3 Estimation of Value-at-Risk and Expected Shortfall 2.1 2.2 Introduction 2.1.1 Background 2.1.2 Description of the Problem 11 Methodology 13 2.2.1 Constrained Kernel Estimator 13 2.3 Value-at-Risk 19 2.4 Simulation Study 21 2.4.1 Densities Investigated 22 i 2.4.2 Choice of Kernel Function and Bandwidth 22 2.4.3 Measure of Discrepancy: Mean Integrated Squared Error 26 2.4.4 Simulation Result for the Quantile Estimators 29 2.4.5 Convergence of the Constrained Kernel estimator 40 2.4.6 Test of quantile estimation using CKE 41 2.5 Expected Shortfall 47 2.5.1 Simulation Results for the ES Estimators 50 2.6 Empirical Study 53 2.6.1 Dataset 54 2.6.2 Risk factor 54 2.6.3 Value-at-Risk 57 2.6.4 Empricial VaR and ES Estimation 58 2.6.5 Confidence intervals for VaR 61 2.6.6 Confidence intervals for ES 64 2.7 Backtesting 66 2.7.1 Dataset 66 2.7.2 Backtesting: Result 67 2.8 Conclusion 75 Estimation of recovery rate density 77 3.1 Introduction 77 3.1.1 Credit Risk and Recovery Rate 77 3.1.2 Background 78 3.1.3 Aim and structure of this chapter 80 3.2 Methodology 81 3.2.1 Overview of methodology 81 3.2.2 The Beta kernel estimator 81 3.2.3 The constrained Beta kernel estimator 82 ii 3.2.4 Objective function 82 3.2.5 Constraint 83 3.2.6 First derivative of the Beta kernel estimator 86 3.2.7 Optimisation 89 3.3 Bandwidth selection 89 3.4 Simulation Study 90 3.5 3.4.1 Mode Estimation 91 3.4.2 Density Estimation 105 Conclusion 119 Modelling intraday volatility patterns 4.1 4.2 4.3 120 Introduction 120 4.1.1 High frequency volatility 120 4.1.2 Background 121 4.1.3 Aim and structure of this chapter 122 Regression estimator 122 4.2.1 Nadaraya-Watson estimator 123 4.2.2 Distance measure 123 4.2.3 The U-shape constraint 124 4.2.4 First derivative of linear estimator 124 4.2.5 Optimisation for constrained Nadaraya-Watson estimator 125 4.2.6 Bandwidth selection 125 Constrained function approximation (CFA) 128 4.3.1 Computational aspects of the CFA 129 4.3.2 The Initialisation procedure 130 4.3.3 The Adding procedure 132 4.3.4 The Optimisation procedure 134 4.3.5 Model diagnostics 136 iii 4.3.6 Anti-mode estimation 141 4.4 Simulation 141 4.4.1 Simulation results 145 4.5 Empirical Study 153 4.5.1 Dataset 153 4.5.2 Empirical result 155 4.6 Conclusion 167 Conclusion 168 A Empirical Study: All Ordinaries Index 172 A.0.1 Dataset 172 A.0.2 Risk factor 173 A.0.3 Value-at-Risk 174 A.0.4 Empricial VaR and ES Estimation 175 A.0.5 Confidence intervals for VaR 179 A.0.6 Confidence intervals for ES 181 A.1 Backtesting 181 A.1.1 Dataset 181 A.1.2 Backtesting: Result 183 B Counter Example 192 iv Abbreviation Chapter • CKE: Constrained Kernel Estimator • EQ: Empirical Quantile Estimator • ES: Expected Shortfall • EVT: Extreme Value Theory • GPD: Generalised Pareto Distribution • HSI: Hang Seng Index • ISE: Integrated Squared Error • K-L: Kaigh-Lachenbruch Estimator • MSP bandwidth: Maximal Smoothing Principle • NKE bandwidth: Na¨ıve Kernel Estimator • NR bandwidth: • POT: • SJ: Normal Reference bandwidth peaks-over-threshold Sheather and Jones’s bandwidth • VaR: Value-at-Risk Chapter • B-kernel: • CB-kernel: Beta kernel Constrained Beta kernel • EDM: Empirical Density Mode • ERM: Empirical Relationship Mode • GM: Grenander Mode • HSM: Half Sample Mode • MSE: Mean Square Error • ROT: Rule of Thumb • SP: Semi-Parametric • SPM: Standard Parametric Mode • T-Gauss: Transformed Gaussian Chapter • AAE: Average Absolute Error • CFA: Constrained Function Approximation • LLS: Linear Least Squares • LR: • NW: Left to Right Nadaraya-Watson • PI bandwidth: Plug-in bandwidths • RD: Random • SSE: Sum of the Squared Errors Extreme Value Theory and constrained kernel estimators According to the previous simulation result of the Normal and Student-t distributions, in many cases, the naive kernel estimator are underestimates quantiles compared with the other estimators On the other hand, the ratios of the actual loss returns that exceed the estimated VaR using the naive kernel estimator are smaller than those from the other estimators Also, Figures A.6b and A.5b show that the VaR using the naive kernel estimators are below that using the constrained kernel estimator most of the time Possible Explanations We have shown the possible dependence structure of the returns over time in Figure A.4c Also, Gaunt & Gray (2003) shows both statistical and economic evidence of the existence of the return autocorrelations on individual Australian stocks However, our simulation study is based on the independence assumption for a random sample The empirical study may fail to meet the independence assumption Thus, the dependence structure of the returns may be one of the factors to cause the variation of the ratio in Table A.5 Furthermore, Table A.5 records the frequency of actual loss returns that exceed the VaR estimation over a given period of time Hence, it also involves Poisson variation Expected Shortfall 188 (a) Time series plot of the actual return and the 98% ES estimation from 07/06/01 to 17/06/05 0.04 Actual EQ K−L H−D 0.03 0.02 0.01 Return −0.01 −0.02 −0.03 −0.04 −0.05 06/07/01 02/07/02 27/06/03 Date 22/06/04 17/06/05 (b) Time series plot of the actual return and the 98% VaR estimation from 07/06/01 to 17/06/05 0.04 Actual 5%CKE 7.5%CKE 10%CKE NKE(MSP) NKE(SJ) NKE(NR) 0.03 0.02 0.01 Return −0.01 −0.02 −0.03 −0.04 −0.05 06/07/01 02/07/02 27/06/03 Date 22/06/04 17/06/05 Figure A.7: Figures (a)-(b) give the expected shortfall of the All Ordinaries Index over 06/07/01 to 17/06/05 The estimated 98% ES is using Empirical Quantile (EQ), K-L quantile, H-D quantile, the 5%, 7.5% and 10% Constrained Kernel Estimator and the Naive Kernel Estimators with MSP bandwidth, Sheather and Jones bandwidth and Normal reference bandwidth 189 (a) Time series plot of the actual return and the 99% ES estimation from 07/06/01 to 17/06/05 0.04 Actual EQ K−L H−D 0.03 0.02 0.01 Return −0.01 −0.02 −0.03 −0.04 −0.05 06/07/01 02/07/02 27/06/03 Date 22/06/04 17/06/05 (b) Time series plot of the actual return and the 99% ES estimation from 07/06/01 to 17/06/05 0.04 Actual 5%CKE 7.5%CKE 10%CKE NKE(MSP) NKE(SJ) NKE(NR) 0.03 0.02 0.01 Return −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 06/07/01 02/07/02 27/06/03 Date 22/06/04 17/06/05 Figure A.8: Figures (a)-(b) give the expected shortfall of the All Ordinaries Index over 06/07/01 to 17/06/05 The estimated 99% ES is using Empirical Quantile (EQ), K-L quantile, H-D quantile, the 5%, 7.5% and 10% Constrained Kernel Estimator and the Naive Kernel Estimators with MSP bandwidth, Sheather and Jones bandwidth and Normal reference bandwidth 190 Figures A.7 a and A.8 a show the estimation of the 98% and 99% ES respectively of the AOI from 06/07/01 to 12/06/05 using the EQ, K-L, H-D estimators The ES using these estimators are smooth step functions as the ES using EQ, K-L and H-D quantile estimators are calculated based on Equation (2.35), which is very sensitive to the corresponding quantile estimation in Figures A.5a and A.6a The estimation of the 98% and 99% ES using the constrained and na¨ıve kernel estimators are shown in Figures A.7b and A.8b Since the estimation of the ES is based on the estimated density which is computed with the whole data set, it follows that the ES estimators over time using the constrained and na¨ıve kernel estimators are more sensitive than those using the EQ, K-L and H-D quantile estimators 191 Appendix B Counter Example When n = 1, X1 = 0.5 and b = 1, Fˆb (x) = fb (x)dx n = j=1 = = = = = fbeta (0.5, x + 1, − x)dx 0.5 dx Bf n(x + 1, − x) 0.5Γ(3) dx Γ(x + 1)Γ(2 − x) dx x(x − 1)Γ(x − 1)Γ(2 − x) sin(πx − π) dx x(x − 1)π − sin(πx) dx x(x − 1)π 192 Expanding sin(πx) and = x−1 −1 π with the Taylor expansion at 0, 1 ∞ = −1 x ∞ n=0 xn ∞ (−1)m π 2m+1 x2m+1 dx (2m + 1)! m=0 ∞ (−1)m π 2m x2m+n dx (2m + 1)! n=0 m=0 ∞ ∞ ∞ ∞ (−1)m π 2m = 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markets In other words, estimation of VaR is, in principle, based on a small number of observations lying in the tail of a portfolio’s return... Estimation of risk measures is divided into two principal approaches: parametric and non- parametric methods Parametric methods are based on strong assumptions about distributions, and non- parametric methods