Non-parametric inference of risk measures

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Spring 2012 Non-parametric inference of risk measures Jae Youn Ahn University of Iowa Copyright 2012 Jae Youn Ahn This dissertation is available at Iowa Research Online: Recommended Citation Ahn, Jae Youn. "Non-parametric inference of risk measures." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Statistics and Probability Commons

2 NON-PARAMETRIC INFERENCE OF RISK MEASURES by Jae Youn Ahn An Abstract Of a thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Statistics in the Graduate College of The University of Iowa May 2012 Thesis Supervisor: Assistant Professor Nariankadu D. Shyamalkumar

3 1 ABSTRACT Responding to the changes in the insurance environment of the past decade, insurance regulators globally have been revamping the valuation and capital regulations. This thesis is concerned with the design and analysis of statistical inference procedures that are used to implement these new and upcoming insurance regulations, and their analysis in a more general setting toward lending further insights into their performance in practical situations. The quantitative measure of risk that is used in these new and upcoming regulations is the risk measure known as the Tail Value-at-Risk (T-VaR). In implementing these regulations, insurance companies often have to estimate the T-VaR of product portfolios from the output of a simulation of its cash flows. The distributions for the underlying economic variables are either estimated or prescribed by regulations. In this situation the computational complexity of estimating the T-VaR arises due to the complexity in determining the portfolio cash flows for a given realization of economic variables. A technique that has proved promising in such settings is that of importance sampling. While the asymptotic behavior of the natural non-parametric estimator of T-VaR under importance sampling has been conjectured, the literature has lacked an honest result. The main goal of the first part of the thesis is to give a precise weak convergence result describing the asymptotic behavior of this estimator under importance sampling. Our method also establishes such a result for the natural non-parametric estimator for the Value-at-Risk, another popular risk measure, under weaker assumptions than those used in the literature. We also report on a simulation study conducted to examine the quality of these asymptotic approximations in small samples. The Haezendonck-Goovaerts class of risk measures corresponds to a premium principle that is a multiplicative analog of the zero utility principle, and is thus of significant

4 2 academic interest. From a practical point of view our interest in this class of risk measures arose primarily from the fact that the T-VaR is, in a sense, a minimal member of the class. Hence, a study of the natural non-parametric estimator for these risk measures will lend further insights into the statistical inference for the T-VaR. Analysis of the asymptotic behavior of the generalized estimator has proved elusive, largely due to the fact that, unlike the T-VaR, it lacks a closed form expression. Our main goal in the second part of this thesis is to study the asymptotic behavior of this estimator. In order to conduct a simulation study, we needed an efficient algorithm to compute the Haezendonck-Goovaerts risk measure with precise error bounds. The lack of such an algorithm has clearly been noticed in the literature, and has impeded the quality of simulation results. In this part we also design and analyze an algorithm for computing these risk measures. In the process of doing we also derive some fundamental bounds on the solutions to the optimization problem underlying these risk measures. We also have implemented our algorithm on the R software environment, and included its source code in the Appendix. Abstract Approved: Thesis Supervisor Title and Department Date

5 NON-PARAMETRIC INFERENCE OF RISK MEASURES by Jae Youn Ahn A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Statistics in the Graduate College of The University of Iowa May 2012 Thesis Supervisor: Assistant Professor Nariankadu D. Shyamalkumar

6 Copyright by JAE YOUN AHN 2012 All Rights Reserved

7 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Jae Youn Ahn has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Statistics at the May 2012 graduation. Thesis Committee: Nariankadu D. Shyamalkumar, Thesis Supervisor Jerome Pansera Ralph P. Russo Elias S.W. Shiu Qihe Tang J. Tyler Leverty

8 ACKNOWLEDGMENTS I feel privileged to have had the supervision of Professor Nariankadu D. Shyamalkumar. It always gave me great pleasure to talk with him, and I am very grateful for the time and enthusiasm that he devoted to honing my research skills, and helping me complete my thesis to the best of my ability. I also want to express my sincere gratitude to: Professor Elias S.W. Shiu for his overall academic guidance during my Ph.D. studies, and particularly for his valuable advice on my dissertation; Professor Qihe Tang for providing timely academic and professional support throughout my doctoral studies, and particularly for the illuminating discussions on Haezendonck-Goovaerts risk measures; to Professor Ralph P. Russo for his continued encouragement of my research, and the wonderful opportunity to co-work with him; to Professors Jerome Pansera and J. Tyler Leverty for kindly agreeing to serve on my doctoral committee, and whose advice helped shape my thesis to its final form. I feel a very deep sense of gratitude towards all the faculty members of the department - I have immensely benefitted by the many valuable academic opportunities provided by the department. Especially, I thank Professor Richard L. Dykstra for carefully introducing me to probability theory, and Professors Dale Zimmerman and Joseph B. Lang for their academic help during my entire stay at Iowa. I also owe sincere thanks to my fellow graduate students, Dr. Hee Seok Nam, Dr. Bangwon Ko, Dr. Xuemiao Hao, Dr. Fei Su, Jun Yang, Zhongyi Yuan and Dowan Kim, who by sharing their thoughts and knowledge made my Ph.D. studies so much more enjoyable. The administrative staff of my department, Tammy Siegel, Margie Ebert and Dena Miller, always kindly provided me with timely valuable support. I also would like to express my immense gratitude to Mr. Robert H. Taylor for supporting my doctoral studies through the Robert H. Taylor Award in Actuarial Stochastics. Finally and most importantly, I would like to say to my parents, Taehwawn Ahn ii

9 and Junghae Kim, as well as to my brother Jaewon Ahn, that they have been a big source of inspiration throughout my life, and without their support and encouragement it would have been impossible to complete my doctoral studies. iii

10 ABSTRACT Responding to the changes in the insurance environment of the past decade, insurance regulators globally have been revamping the valuation and capital regulations. This thesis is concerned with the design and analysis of statistical inference procedures that are used to implement these new and upcoming insurance regulations, and their analysis in a more general setting toward lending further insights into their performance in practical situations. The quantitative measure of risk that is used in these new and upcoming regulations is the risk measure known as the Tail Value-at-Risk (T-VaR). In implementing these regulations, insurance companies often have to estimate the T-VaR of product portfolios from the output of a simulation of its cash flows. The distributions for the underlying economic variables are either estimated or prescribed by regulations. In this situation the computational complexity of estimating the T-VaR arises due to the complexity in determining the portfolio cash flows for a given realization of economic variables. A technique that has proved promising in such settings is that of importance sampling. While the asymptotic behavior of the natural non-parametric estimator of T-VaR under importance sampling has been conjectured, the literature has lacked an honest result. The main goal of the first part of the thesis is to give a precise weak convergence result describing the asymptotic behavior of this estimator under importance sampling. Our method also establishes such a result for the natural non-parametric estimator for the Value-at-Risk, another popular risk measure, under weaker assumptions than those used in the literature. We also report on a simulation study conducted to examine the quality of these asymptotic approximations in small samples. The Haezendonck-Goovaerts class of risk measures corresponds to a premium principle that is a multiplicative analog of the zero utility principle, and is thus of significant iv

11 academic interest. From a practical point of view our interest in this class of risk measures arose primarily from the fact that the T-VaR is, in a sense, a minimal member of the class. Hence, a study of the natural non-parametric estimator for these risk measures will lend further insights into the statistical inference for the T-VaR. Analysis of the asymptotic behavior of the generalized estimator has proved elusive, largely due to the fact that, unlike the T-VaR, it lacks a closed form expression. Our main goal in the second part of this thesis is to study the asymptotic behavior of this estimator. In order to conduct a simulation study, we needed an efficient algorithm to compute the Haezendonck-Goovaerts risk measure with precise error bounds. The lack of such an algorithm has clearly been noticed in the literature, and has impeded the quality of simulation results. In this part we also design and analyze an algorithm for computing these risk measures. In the process of doing we also derive some fundamental bounds on the solutions to the optimization problem underlying these risk measures. We also have implemented our algorithm on the R software environment, and included its source code in the Appendix. v

12 TABLE OF CONTENTS LIST OF TABLES viii LIST OF FIGURES ix CHAPTER 1 INTRODUCTION Prologue Risk Measures and Insurance Regulation Haezendonck-Goovaerts Risk Measures and T-VaR Notation LARGE SAMPLE BEHAVIOR OF THE T-VAR AND VAR ESTIMA- TORS UNDER IMPORTANCE SAMPLING Introduction Main Results and Discussion Notation and Standing Assumptions Main Result for the Quantile Estimator Main Result for the T-VaR Estimator Some Further Results Proofs of the Main Results Weak Convergence of the VaR Estimator Weak Convergence of the T-VaR Estimator Auxiliary Results BOUNDS OF ORLICZ QUANTILES AND THEIR APPLICATION TO THE COMPUTING ALGORITHM OF HAEZENDONCK-GOOVAERTS RISK MEASURES Introduction Bounds for Orlicz Quantiles Simplified Algorithm The Algorithm Sub-Algorithm Algorithm Implementation Details Performance of the Algorithm Proofs and Auxiliary Results ASYMPTOTIC THEORY FOR THE EMPIRICAL HAEZENDONCK- GOOVAERTS RISK MEASURES Introduction Strong Consistency vi

13 4.3 Weak Convergence Results Simulation Study Effect of Sampling Distribution and Level Effect of Young Function Concluding Remarks REFERENCES APPENDIX vii

14 LIST OF TABLES Table % Nominal Confidence Interval for VaR Using Ordinary and Importance Sampling % Nominal Confidence Interval for T-VaR Using Ordinary and Importance Sampling Distribution of No. of Grid Points Excluding x 999 and x 1000 that Require at Least 2 Newton-Raphson Steps Distribution of Newton-Raphson Steps at x Sub-Algorithm: Stopping Rule and Output Sub-Algorithm: Error Bounds for ˆπ X Calculation of π F Using the Algorithm with Ψ(x) = x 2 and an Exponential F ( ) Calculation of π F in Example 11 Using Implementation Detail 1 (H X (x 7 ) = 0) Convergence of Newton-Raphson Steps, x + Ĥn X (h(x)), at x 5 and x 6 in Example 11 (π X (x 5 ) = π X (x 6 ) = 20) Estimation of the Haezendonck-Goovaerts Risk Measure with Young Function, Ψ(x) = x2 +x 2 I(x > 0): Varying Sampling Distributions Estimation of Haezendonck-Goovaerts Risk Measure for Exponential Distribution: Different Young Functions viii

15 LIST OF FIGURES Figure 2.1 Distributions with Atypical Weak Limits for the Empirical 50%-level VaR and/or T-VaR Density of the Standardized Empirical 50%-level T-VaR from F Densities of Standardized Quantile Estimators for Pareto (0.02, 5) Densities of the Standardized T-VaR Estimator for Pareto (0.02, 5) Distributions with Jumps : Understanding Assumption C π F ( ), when X Ber(0.025) and α = 95% Flowchart for the Simplified Algorithm No. of Newton-Raphson Iterations at Grid Points x j, j = 853,..., Flowchart of the Sub-Algorithm Flowchart of the Algorithm Convergence of Newton-Raphson Steps, x + N n x (h(x)), at x 5 and x 6 in Example 11 (π F (x 6 ) = π F (x 6 ) = 20) Distribution for which d H (I Fn, I F ) Epigraph of the Convex Programming Problem Associated with the Computation of π F Ratio of Asymptotic Standard Deviations of Parametric and Non-Parametric Estimators Estimated Densities of Empirical Haezendonck-Goovaerts Risk Measures - Varying Levels and Sampling Distributions Estimated Densities of Empirical Haezendonck-Goovaerts Risk Measures - Varying Young Function ix

16 1 CHAPTER 1 INTRODUCTION 1.1 Prologue The volatility of global financial markets has dramatically increased especially since the collapse of the Bretton-Woods system in the early 1970s. This accompanied with social and other economic changes, changes in taxation, and increasingly informed public had sparked a rise in the need for financial stability in the mid-1970s. It is in this environment that the first universal life insurance product was introduced, and ever since then the life insurance and annuity industry has continued to offer innovative products not only to cater to the changing needs of the public but also due to competitive pressures from blurring of boundaries between banking and insurance. While an important feature of these new products was to offer more choices of investment vehicles through special accounts, this soon created a need for guarantees to be included in them to cater to the need for increased financial security. Cumulatively these changes have resulted in the insurance company liabilities to be a lot more volatile than a few decades back, not to mention the assets. The increased volatility of both the assets and liabilities of insurance companies have rendered then existing insurance regulations for both valuation and capital globally obsolete. These regulations were largely ad hoc formulaic prescriptions written in an era of less volatility in financial markets, less competition, less correlation between assets in different markets and sectors, and less computational prowess. In the past decade, especially with insurance company liabilities increasingly containing exotic financial options and insurance guarantees, the inadequacy of such regulations have manifested themselves in out-sized insurance company losses. This new insurance environment has driven changes in regulation which leverages upon increased computational resources available to actuaries, and increased awareness among actuaries of modern finance, to adequately

17 2 value insurance liabilities and allocate capital for them. This thesis is concerned with design and analysis of statistical inference procedures to be used in implementing the new and upcoming insurance regulations, and their analysis in a more general setting towards lending further insights into their performance in practical situations. 1.2 Risk Measures and Insurance Regulation New insurance regulations are formulated using quantitative measures of risk called risk measures. Mathematically speaking, risk measures are real valued functionals defined on the space of distribution functions. The two most popular risk measures in the financial world are Value-at-Risk (VaR) and Tail Value-at-Risk (T-VaR). Towards defining these risk measures, let F ( ) denote the distribution function of the loss random variable associated with a certain financial portfolio, and let α be a real number in the interval (0, 1). The α-level VaR for the distribution F ( ), which we denote by q α (F ), is defined by q α (F ) := inf{x : F (x) α}, and the α-level T-VaR for F ( ), which we denote by T-VaR α (F ), is defined by T-VaR α (F ) := 1 1 α 1 α q s (F )ds, α (0, 1). T-VaR 1 is becoming a risk measure of choice especially in the North American insurance world, very much like VaR is for the trading desks and the banking sector at large. In the case of variable annuities, the adoption of the C-3 Phase II revision to the regulatory risk based capital model in 2005, and the implementation of the analogous principles based reserving methodology (AG VACARVM) in 2009 by the NAIC together made the T-VaR a key risk measure. Now with the Life Reserves Work Group and the Life Capital Work Group (C3WG) of the American Academy of Actuaries working on an analogous reserve and capital methodology for life insurance products, and the possibility of Principles 1 A slight variant of T-VaR is called the Conditional Tail Expectation (CTE). CTE agrees with the T-VaR in all cases except when the corresponding quantile is a mass point.

18 3 Based Reserves (PBR) being made effective in 2014, T-VaR is poised to become a risk measure of immense importance for the whole of the life industry. In regulatory computing of the T-VaR, for either capital or reserving purpose, there are two related but distinct practical settings which require statistical estimation of the T-VaR. The first, and common setting in the life sector, involves estimating the T-VaR of a portfolio from the output of a simulation of its cash flows - the distributions for the underlying economic variables are either estimated or prescribed by regulations. In this situation the computational complexity of estimating the T-VaR arises due to the complexity in determining the portfolio cash flows for a given realization of economic variables. The second setting is from the non-life sector, and is the one which involves estimation of the T-VaR using a small random sample from an unknown distribution. In both settings there is the need for inference procedures with good small sample performance (arising for reasons either of computational complexity or paucity of data). While statistical inference for VaR has been well studied in the academic literature, inference for T-VaR has received relatively little attention. Among articles dealing with inference for T-VaR the majority focuses on designing inference procedures for specific parametric models. Parametric models are classes of distributions indexed by a finite dimensional parameter, and such inference procedures guarantee good performance only when the unknown distribution belongs to the specific parametric model. Hence use of such inference procedures introduces model risk, a risk that is desirably avoided by use of non-parametric inference procedures. Non-parametric inference procedures perform well for a wide class of distributions, and a better understanding of such procedures is needed in practice. Variance reduction techniques like importance sampling are one of the most promising ways to improve efficiency in estimating risk measures from simulated cashflows. While asymptotic behavior of the natural non-parametric estimator of T-VaR under importance sampling was conjectured in Manistre and Hancock (2005), the literature lacked

19 4 an honest result. The main goal of chapter 2 is to give a precise weak convergence result describing the asymptotic behavior of this estimator under importance sampling. Interestingly, our method also establishes such a result for the natural non-parametric estimator for VaR under importance sampling, weakening assumptions required by Glynn (1996). We also report on a simulation study conducted to examine the quality of asymptotic approximations in small samples. The results in this chapter have appeared in Ahn and Shyamalkumar (2011). 1.3 Haezendonck-Goovaerts Risk Measures and T-VaR In Haezendonck and Goovaerts (1982) a rather natural premium calculation principle which is multiplicatively equivalent to the zero utility principle is suggested. Towards defining this premium principle we first define a normalized Young function, say Ψ( ), as strictly increasing convex function on R + satisfying Ψ(0) = 0 and Ψ(1) = 1. For a risk X (with distribution F ) in { ( )] } X X [Ψ E < for every h > 0, h this premium principle allocates a premium defined as the unique positive solution h of [ ( )] X E Ψ = 1. h The risk measure corresponding to this premium principle, proposed in Goovaerts et al. (2004), is called the Haezendonck-Goovaerts risk measure, a class of risk measures indexed by normalized Young functions. The Haezendonck-Goovaerts risk measure for X at the α-level, for α in the interval (0, 1), is defined as π s X := inf x R {x + Hs X (x)}, where HX s (x) is defined as the unique positive solution h of [ ( )] (X x)+ E Ψ = 1 α, h

20 5 for x < ess sup(x) and defined as 0 otherwise. Clearly the Haezendonck-Goovaerts risk measure is of significant academic interest due to it arising from a well motivated premium principle. This is evidenced by the growing literature devoted to studying its properties, which includes Bellini and Rosazza Gianin (2008a), Bellini and Rosazza Gianin (2008b), Bellini and Rosazza Gianin (2011), Nam et al. (2011), Krätschmer and Zähle (2011), and Tang and Yang (2012). From a practical point of view our interest in this class of risk measures arose primarily from the fact that the T-VaR is a member of this class. More specifically, since Rockafellar and Uryasev (2002) have shown that { T-VaR α (F ) = inf x + 1 <x< 1 α E [ } ] (X x) +, we have T-VaR α (F ) equals π s X for Ψ(x) = x - in a sense T-VaR is a minimal member of the class of Haezendonck-Goovaerts risk measures. In Bellini and Rosazza Gianin (2008b) the natural non-parametric estimator for π s X was suggested, but analysis of its asymptotic behavior was elusive. This was largely due to the fact that, unlike the case of T-VaR, this estimator lacks a closed form expression. Our main goal in the second part of this thesis is to study the asymptotic behavior of this estimator from the point of interest that such a study will lend further insights into the statistical inference for the T-VaR. In order to conduct a simulation study of this non-parametric estimator we needed an efficient algorithm to compute the Haezendonck-Goovaerts risk measure with precise error bounds. The lack of such an algorithm was clearly noticed in Bellini and Rosazza Gianin (2008b), and this impeded the quality of their simulation results. In Chapter 3 we design and analyze an algorithm for computing the Haezendonck-Goovaerts risk measure without imposing any constraints on the normalized Young function. In the process of doing so, we also derive some fundamental bounds on the solutions to the optimization problem underlying these risk measures. In Chapter 4 we report on our asymptotic results, and conduct a a simulation study (using the algorithm of Chapter 3)

21 6 to examine the quality of asymptotic approximations in small samples. The Appendix contains the source code of a package implementing our algorithm on the R software environment. 1.4 Notation We will use X, X 1, X 2,... to denote a sequence of independent and identically distributed random variables on our underlying probability space (Ω, F, P ) with X X Ψ. Also, by F ( ) we will denote their common distribution function, unless specified otherwise. By F n ( ) we will denote the empirical distribution function of the random sample of size n consisting of X 1,..., X n, i.e. F n (x) := 1 n n I (,x] (X i ), x R. i=1 For a sequence of random variables {Z i } i 1, by Z n d Z (resp., Z n p Z) we denote the convergence in distribution (resp., probability) to Z, as n. We also call the convergence in distribution as the weak convergence. For a distribution function F ( ), by q + α (F ) we denote inf{x : F (x) > α}. Note that q α (F ) q + α (F ) with equality if and only if F ( ) is strictly increasing at q α (F ). In a sense the set of α level quantiles of F ( ) equals [q α (F ), q + α (F )]. For x R, (x) + or x + equals x for non-negative x, and equals zero for negative x. We let N denote the set of positive integers and N 0 the set of non-negative integers.

22 7 CHAPTER 2 LARGE SAMPLE BEHAVIOR OF THE T-VAR AND VAR ESTIMATORS UNDER IMPORTANCE SAMPLING 2.1 Introduction In response to the need for statistical inference procedures for the T-VaR for regulatory compliance purposes, and for a better understanding of their performance, there has been a surge in the actuarial literature of papers dealing with statistical inference of the T-VaR and related risk measures, see for example Jones and Zitikis (2003), Manistre and Hancock (2005), Kaiser and Brazauskas (2006), Kim and Hardy (2007), Brazauskas et al. (2008), Ko et al. (2009), and Russo and Shyamalkumar (2010). Nevertheless, only Manistre and Hancock (2005) discusses the use of variance reduction techniques for the estimation of the T-VaR, and this is the area of focus for this chapter. Our interest in establishing asymptotic convergence results for the empirical T-VaR and quantile under importance sampling arises mainly because we see importance sampling as one of the potent practical strategies to not only get better point estimators but also confidence intervals, see for example Manistre and Hancock (2005), and Glasserman et al. (2000). The main contribution of this chapter is that we establish asymptotic normality of the T-VaR and VaR estimators under importance sampling. In the case of VaR, as discussed later, our result improves upon an earlier result of Glynn (1996), whereas there is no published result for the case of the T-VaR. However, we note that our results have been suggested and supported by heuristics derived from the use of influence functions in Manistre and Hancock (2005), one of the earlier articles on the estimation of T-VaR in the actuarial literature. While a theoretical result justifying the use of a methodology is undoubtedly of interest, its practical value is amplified if it prevents the use of the methodology in cases where against expectations the methodology fails. Through the first two examples, for expository ease dealing with the case of ordinary sampling, we motivate the practical need for theoretical results establishing asymptotic normality of

23 8 the T-VaR and VaR estimators under importance sampling. The following non-pathological ordinary sampling example shows that the existence of influence function in the case of the T-VaR falls short of establishing convergence of the empirical T-VaR to normality, and also that the formula derived for the asymptotic variance through the use of influence function could be misleading. The use of influence function for VaR is not similarly prone to misuse as the asymptotic variance formula for the VaR is proportional to the reciprocal of the density evaluated at the quantile, and known results for weak convergence of empirical quantiles under ordinary sampling (for example, see Reiss (1989)) require only that the density evaluated at the quantile be positive (a) Distribution Function F 1 (b) Distribution Function F 2 (c) Distribution Function F 3 Figure 2.1: Distributions with Atypical Weak Limits for the Empirical 50%-level VaR and/or T-VaR Example 1 Let F 1 ( ) denote the distribution function of the equal mixture of U(0, 1) and U(2, 3); Figure 2.1a contains a graph of this distribution. One can think of F 1 ( ) as being the distribution of a random variable X which is distributed as U(0, 1) in Regime 1, is distributed as U(2, 3) in Regime 2, and with equal probabilities for the two regimes. We observe that F 1 ( ) is a continuous distribution with a density (in other words, absolutely

24 9 continuous). We note that for F 1 ( ), the 50%-level VaR and the 50%-level T-VaR are 1 and 2.5, respectively, under any of the above definitions. Also, it is easy to check that the influence function for the 50%-level T-VaR at F 1, denoted by IF T-VaR;F1 ( ), is given by { 1.5 x < 1, IF T-VaR;F1 (x) := x x > 2, on the complement of [1, 2], which is a set of probability 1 under F 1 ( ). It is easy to check that [ ] ( ) E IF T-VaR;F1 (X) = 1/2 and Var IF T-VaR;F1 (X) = 7/6, as follows. First we define T (F 1 ) = xdg α [F 1 (x)] = 3 q α(x) 2x df 1 (x), and F 1,y,ɛ (x) = (1 ɛ)f 1 (x) + ɛ y (x). For the case of y < 1, let ɛ > 0 to be small enough such that y < 1 2ɛ. Then we have 1 ɛ T (F 1,y,ɛ (x)) = = 1 1 2ɛ 1 ɛ 1 1 2ɛ 1 ɛ = 1 ɛ 2 2x df 1,y,ɛ (x) x df 1,y,ɛ (x) 2x [(1 ɛ) df 1 (x) + ɛd y (x)] + [ 1 ( ) ] 2 1 2ɛ + (1 ɛ) 5 1 ɛ 2. For the case of y > 2, let ɛ > 0 to be small enough so that y > 1 2ɛ 1 ɛ T (F 1,y,ɛ (x)) = = = (1 ɛ) 2 2x df 1,y,ɛ (x) + 2x df 1,y,ɛ (x) ɛ 1 ɛ 1 2ɛ 1 ɛ 1 2ɛ 1 ɛ 2x df 1,y,ɛ (x) (1 ɛ)x dx + 2yɛ x df 1,y,ɛ (x) > 2. Then we have 2x [(1 ɛ) df 1 (x) + ɛd y (x)]

25 10 Then using above two equations, we can easily compute IF T-VaR;F1 (X). We note that the above influence function is different from the one derived from the general version given on pg. 133 of Manistre and Hancock (2005). It is worth mention that if one uses the latter influence function then the above mean and variance are 0 and 29/12, respectively. In Lemma 1 it is shown that the standardized 50%-level T-VaR converges to the non-normal distribution of Z I(Z 2<0) ( Z2 where Z 1, Z 2 are i.i.d. N(0, 1) random variables. The density plots of this limiting distribution as well as that of the standardized T-VaR from a sample of size 50 are given in Figure 2.2. Also, the mean of this limiting distribution is 1/ 2π and its variance is 17/12 1/(2π), and these clearly differ from those derived above using either of the influence functions. 2 ), Asymptotic Limit Sample Size 50 Figure 2.2: Density of the Standardized Empirical 50%-level T-VaR from F 1

26 11 It is easier to see the occurrence of the non-normality in the weak limit of the standardized empirical T-VaR when sampling from the distribution F 2 of Figure 2.1b. This distribution is an equal mixture of U(0, 1) and the degenerate distribution at 2; in other words, if X F 1 then min(x, 2) F 2. We note that under this alternate distribution the median continues to be 1 whereas the 50%-level T-VaR reduces to 2. As the empirical T-VaR cannot exceed 2, the limiting distribution of the standardized empirical T-VaR cannot be normal. In fact, proceeding along the lines of Lemma 1 one can show that the limiting distribution is in fact the distribution of min(z, 0) where Z is a standard normal random variable. In the above example, an argument could be made that since it is easy to see that the median will oscillate and hence is inconsistent, one would be suspicious of the convergence of the standardized 50%-level T-VaR. But on the other hand a risk averse practitioner may feel that the result may as well require the n consistency of the median, and that this is not the case is shown by the next example. In fact, in the case of ordinary sampling, the elegant proof of the Theorem 3.1 of Brazauskas et al. (2008) makes it clear that the consistency of the α-level quantile is all that is required (apart from the finiteness of the second moment) for the asymptotic normality of the standardized α-level T-VaR. Example 2 Let F 3 denote the distribution given by 0 x < 0, 1 F 3 (x) = (1 (1 2 x)2 ) 0 < x 1, 1 2 (1 + (x 1)2 ) 1 < x 2, 1 x > 2. The median of F 3 clearly equals 1, and the 50%-level T-VaR equals 5/3. From the plot of F 3 in Figure 2.1c, or otherwise, it is clear that F 3 has a point of increase at the median, and as F 3 has a bounded support it has a finite second moment. These observations along

27 12 with Theorem 3.1 of Brazauskas et al. (2008) imply that the standardized empirical 50%- level T-VaR has a normal limiting distribution. Now, we will explore the behavior of the sample median, and for expository convenience we will assume that the sample size is 2n + 1, for some n 1. It is easy to see that the sample median has the same distribution as F 1 4 (U n+1:2n+1 ) with F 1 4 (U n+1:2n+1 ) = ( 1 U ) 2 n+1:2n+1 2 ( ) U n+1:2n U n+1:2n+1 1/2, U n+1:2n+1 > 2, where U n+1:2n+1 is the sample median of a 2n + 1 random sample from U(0, 1). From the above representation and asymptotic normality of central order statistics from U(0, 1) (for example, see Reiss (1989)), it is easy to see that [2n + 1] 1/4 ( F 1 4 (U n+1:2n+1 ) 1 ) d sgn(z) Z, where Z is a standard normal random variable. Importantly, we have the median converging at the much slower n 1/4 rate, whereas, as observed above, the sample 50%-level T-VaR converges to normality at the usual n rate. We end by noting that a careful practitioner can often be able to create checks to avoid falling into traps left in the absence of theoretical results, and in our context the variance formula verification process of Manistre and Hancock (2008) is a pertinent such example. In the next section we present our notation, key assumptions, and our main results. Also in the next section, we embed our main results in the literature, and include simulation studies to lend insight into the sample sizes at which these asymptotic results become meaningful. In the subsequent section we present the proofs of our main results. Section 2.4 contains all of the intermediate results and their proofs. 2.2 Main Results and Discussion In this section we begin with notation, then present our main result for the VaR estimator and compare it with that of Glynn (1996), and finally present our main result

28 13 for the T-VaR estimator. After each result, through simulation examples we try to lend insight into the small sample sizes required for these estimators to have an approximate normal distribution, and the accompanying benefit in terms of better true coverage levels for asymptotic confidence intervals Notation and Standing Assumptions First we define symbols and notation for this chapter. We will find it sometimes convenient to alternatively denote q α (F ) by ξ α and the α-level T-VaR of F ( ) by c α (F ). In the case that the distribution F is continuous at q α (F ), we have c α (F ) = E (Y Y > q α (F )), for Y distributed as F ( ). By F ( ) we denote the importance sampling distribution. By the very nature of importance sampling, we require that F ( ) is absolutely continuous with respect to F ( ), that is every set of zero probability under F ( ) is also a set of zero probability under F ( ). By L( ) we denote the Radon-Nikodym derivative of F ( ) with respect to F ( ); if both F ( ) and F ( ) have densities f( ) and f ( ), respectively, then L( ) can be taken to be f( )/f ( ). Unlike in other chapters, in this chapter, we use X, X 1, X 2,..., X n for independent and identically distributed random variables with a common distribution F ( ). The order statistics of the sample are represented by X 1:n, X 2:n,..., X n:n, with X 1:n and X n:n being the sample minimum and maximum, respectively. The empirical distribution function corresponding to the n-sample from F is denoted by Fn and is defined by ( ) n 1 Fn (y) := I (,y] (X i ). n i=1 The importance sampling estimator for q α (F ) based on an n-sample from F ( ) that we

29 14 consider is denoted by Q α (n, F ; F ), which is defined as { } Q α (n, F ; F ) := inf x : 1 L (X i ) (1 α). (2.1) n X i >x The above estimator along with some alternate estimators for the quantile have been studied in Glynn (1996), and it is shown there that for α close to 1 the estimator defined above is the preferred one. We restrict ourselves to Q α (n, F ; F ) while noting that our methods extend to the alternates considered in Glynn (1996). The importance sampling estimator for c α (F ) based on an n-sample from F ( ) that we consider is denoted by C α (n, F ; F ). C α (n, F ; F ) is defined by [ n C α (n, F 1 ; F ) := X i L(X i )I (X i > Q α (n, F ; F )) n(1 α) i=1 + Q α (n, F ; F ) ( n(1 α) )] n L(X i )I (X i > Q α (n, F ; F )). i=1 (2.2) Note that while Q α (n, F ; F ) and C α (n, F ; F ) can assume the value, the probability that they do so converges to zero with increasing sample size by the law of large numbers. One could alternatively work with max{q α (n, F ; F ), X 1:n } as an estimator of q α (F ), to avoid the estimator taking the value, with the weak convergence results for this alternate estimator agreeing with that of Q α (n, F ; F ). An alternate estimator for c α (F ) would be { max C α (n, F ; F ), n i=1 X } il(x i ) n i=1 L(X. i) We end this sub-section by stating and discussing the assumptions on F and F that are common to our results for both the quantile and T-VaR estimators; these have been grouped together in the following as Assumption S:

30 15 Assumption S: F and F have the following properties: S1. F is absolutely continuous with respect to F. S2. F is continuous at q α (F ). S3. L( ) is a function with finite variation on compacts, and it has finite negative variation on (y, ), for all y R. S4. L( ) is right continuous. As noted above, Assumption S1 is essential for importance sampling. Assumption S2 in view of Assumption S1 is essentially requiring F ( ) to be continuous at q α (F ). Note that if F ( ) is continuous at q α (F ), there is no reason to choose an F ( ) not satisfying Assumption S2. Now the condition that F ( ) be continuous at q α (F ) is clearly required for a normal asymptotic limit as otherwise it can be shown that asymptotically the quantile estimator would be at least q α (F ). Assumption S3 is a mild condition which serves to exclude distributions which lead to highly oscillating L( ) that have graphs of infinite length on finite intervals; also, since L( ) is non-negative the second part of Assumption S3 is similarly non-restrictive in nature. In view of Assumption S2, one could choose L( ) to be right continuous, and as a result of S3, L( ) will have well defined left limits Main Result for the Quantile Estimator In the case of the sample quantile, an estimator of the quantile under ordinary sampling, the weak convergence result is now part of the folklore (for example, see Reiss (1989)). In the case of importance sampling, Theorem 1 of Glynn (1996) establishes weak convergence results for some natural candidates for the quantile estimator, of which Q α (n, F ; F ) is the preferred one for actuarial applications (as interesting values for α tend to be close to 1). Since the definition of Q involves the sum of the likelihood on a subset of the sample, it is natural to expect that the right moment condition would be to require the finiteness of the second moment of L(X). Theorem 1 of Glynn (1996)

31 16 instead requires the finiteness of the third moment of L(X), mainly due to the use of the Berry-Esseen theorem in their proof. Our main result for the quantile estimator in a sense is an attempt to close this gap in the moment condition. Apart from Assumption S, for the weak convergence of our quantile estimator, we will need F ( ) and F ( ) to satisfy these two additional requirements: Assumption V: F and F have the following properties: V1. F has a positive first derivative at q α (F ). V2. There exists a λ (0, 1/2] such that (1 F (x )) 1/2 λ d L (x) <, y R. (2.3) (y, ) We note that Assumption V1 is very much expected from the result for the weak convergence of the sample quantile. The other requirement is our moment condition. Lemma 2 shows that our requirement is at most slightly stronger than the second moment condition (i.e., E F L 2 (X) < ), and weaker than requiring E F L (2+δ) (X) <, for some δ > 0. In particular, it is weaker than requiring a finite third moment as in Glynn (1996). Our main result for Q α (n, F ; F ) is the following: Theorem 1 Suppose that F and F together satisfy Assumptions S and V. Then ( n (Qα (n, F d ; F ) q α (F )) N 0, E ( F L 2 (X)I ) ) (ξα, )(X) (1 α) 2, (2.4) f 2 (ξ α ) as n. The following example is presented to both give the reader a sense of the sample size required for the above asymptotic result to be meaningful, and to demonstrate that, for small sample sizes, the true coverage levels of the asymptotic confidence intervals are much closer to the nominal under importance sampling. It is worth mention that in

32 17 this example our moment condition in fact proves to be no more stringent than the ideal condition. Example 3 We consider a liability distributed as a Pareto(0.02,5), where the distribution of twoparameter Pareto(σ,β) with inverse scale parameter σ and shape parameter β has the hazard rate function given by µ(x) = { 0 x 0, σβ 1+σx x > 0. The two-parameter Pareto family has been considered in Manistre and Hancock (2005) and Kim and Hardy (2007), and which is also studied in Ko et al. (2009). Moreover, the two-parameter Pareto family is commonly used in actuarial modeling. Pareto(0.02,5) has mean 12.5 and standard deviation Moreover, it has a.95-level (.99-level, resp.) VaR of (75.59, resp.) and a.95-level (.99-level, resp.) T-VaR of ( , resp.). For importance sampling distribution we (for expository ease) restrict the choice of the sampling density within the subclass of Pareto(σ,β) distribution where σ is held constant at 0.02 and β is allowed to take an arbitrary value in (0, ). Such a choice results in L(x) = ( ) 5 ( x) (β 5), x 0. β It is easy to check that the moment condition in the CLT for Q α (n, F ; F ) of Glynn (1996), namely that E F (L 3 ) <, is satisfied if and only if β < 7.5. The moment condition for our Theorem 1 given in (2.3) requires β < 10, and Assumption V2. is then satisfied for λ (0, 5/β 1/2). It is worth mention that finiteness of the asymptotic variance Q α (n, F ; F ) given in Theorem 1 is attained if and only if β < 10. Hence, in this example, Theorem 1 attains the ideal moment condition of requiring only the finiteness of asymptotic variance, alike the case of the central limit theorem for independently and

33 18 identically distributed data. The importance sampling distribution minimizing the asymptotic variance of the α-level quantile estimator is one with β given by ( ) log(1 α) log 2 (1 α) < 10, α (0, 1). The optimal β, calculated using the above expression, approximately equals 1.40 (resp., 0.97) for α equal to 0.95 (resp., 0.99). We use these optimal distributions to demonstrate the power of importance sampling under a good choice for the sampling distribution. On the other hand, it is now part of folklore that things can go awry for bad choices for the importance sampler. For each α in {0.95, 0.99} we simulated the sample quantile (ordinary sampling) for sample sizes 200, 1000, and 5000; we simulated the quantile estimator when sampling from the optimal Pareto importance sampler for sample sizes 8, 12, and 40; and we simulated the quantile estimator when sampling from Pareto(0.02, 8.75) for sample sizes 50, 000, 200, 000, and 5, 000, 000. We chose Pareto(0.02, 8.75) only to demonstrate that the convergence to normality of the distribution of Q α (n, F ; F ) holds even when the third moment of the likelihood is infinite, as long as the second moment is finite. We simulate 100, 000 random samples for each combination of α, sampling distribution, and sample size. The programs are run parallel on 30 processors over 10 nodes of a 22 node Beowulf cluster using the snow R package (Tierney et al. (2008, 2009)). Figure 2.3 plots the Gaussian kernel density estimator for each of the sampling distribution, three sample sizes and the two α levels. We use a Gaussian kernel density estimator, via the function density on R, to estimate the densities of the sampling distribution. The bandwidth used equals 0.9 times the minimum of the standard deviation and the interquartile range divided by 1.34 times the sample size to the negative onefifth power (i.e. Silverman s rule of thumb, see Silverman (1986)) unless the quartiles

34 19 coincide, in which case a positive value is used. The plots overall show a good degree of closeness to the standard normal density except in the case when sampling from Pareto(0.02, 8.75) with α = 0.99 where a similar degree of closeness is achieved only for a sample size of 5, 000, 000. While the dependence of the plots on the sample size and the α-level are as expected, it is remarkable that normality takes hold at significantly smaller sample sizes when sampling from the above defined optimal Pareto distributions. One practical utility of faster convergence to normality is that the confidence intervals for the quantile constructed using the asymptotic theory tend to have true coverage levels closer to their nominal confidence levels at smaller sample sizes. In Table 2.1 we have tabulated the bias in the coverage levels, i.e. the difference between the true coverage level and the nominal confidence level, for the sample quantile under ordinary sampling, and the quantile estimator when sampling from the optimal Pareto importance sampling distribution. Also tabulated are the lengths of these asymptotic confidence intervals. Note that by the true coverage level we mean the proportion of times that the confidence interval for the T-VaR contains the population T-VaR, which differs from the nominal confidence level due to the use of asymptotics in their construction. We simulated 10, 000, 000 random samples for each result in Table 2.1. It is noteworthy that while importance sampling gives a reduction in sample size by a factor of 5 (resp., 16) for similar lengths of the asymptotic confidence intervals and for α equal to 0.95 (resp., 0.99), an added benefit is the significantly reduced coverage bias (excepting the case of α = 0.99 and sample size 8) Main Result for the T-VaR Estimator The weak convergence results for the sample T-VaR (for the case of distributions which strictly increase at the pertinent quantile) has been established by Brazauskas et al. (2008) (see their Theorem 3.1). We are not aware of similar results in the literature for an estimator of the T-VaR in the case of importance sampling, and hence our Theorem 2

35 20 Ordinary Sampling : α = 0.95 Ordinary Sampling : α = Std. Normal n=200 n=1000 n= Pareto(0.02, ~1.40) IS (optimal) : α = 0.95 Pareto(0.02, ~0.97) IS (optimal) : α = Std. Normal n=8 n=12 n= Pareto(0.02, 8.75) IS : α = 0.95 Pareto(0.02, 8.75) IS : α = Std. Normal n=50,000 n=200,000 n=5,000, Figure 2.3: Densities of Standardized Quantile Estimators for Pareto (0.02, 5) can be seen as the first extension of Theorem 3.1 to importance sampling. We note that the techniques used in Brazauskas et al. (2008) are different from those employed in this chapter. For Theorem 2, apart from Assumption S we will need the requirements listed as part of Assumption C below, where L C ( ) is defined as L C (x) = xl(x), for all x R.

36 21 Sampler Sample Size 95%-level VaR 99%-level VaR Coverage Bias Coverage Bias Length 10 2 Length 10 2 (s.e ) (s.e ) Ordinary Sampling Optimal Pareto IS % (4.83) % (5.87) % (6.34) % (4.86) % (6.78) % (6.37) % (7.26) % (8.68) % (6.97) % (7.26) % (6.90) % (6.96) 0.54 Table 2.1: 95% Nominal Confidence Interval for VaR Using Ordinary and Importance Sampling Assumption C: F and F have the following properties: C1. F has a point of increase at ξ α in the sense that F (u) < F (ξ α ) < F (v), u < ξ α < v. C2. F is such that C3. There exists a λ (0, 1/2] such that (y, ) F (F 1 (s)) s sup s [0,1]\{α} s α <. (2.5) (1 F (x )) 1/2 λ d L C (x) <, y R. (2.6) We note that the first requirement above is expected from our earlier examples and the proof of Theorem 3.1 of Brazauskas et al. (2008). The second condition is a mild condition which is satisfied if F is continuous in a neighborhood of ξ α, or F is differentiable at ξ α with a strictly positive derivative at ξ α. For additional insight into this