Extreme Events in Finance E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Handbooks in Financial Engineering and Econometrics
- Sprache: Englisch
A guide to the growing importance of extreme value risk theory, methods, and applications in the financial sector Presenting a uniquely accessible guide, Extreme Events in Finance: A Handbook of Extreme Value Theory and Its Applications features a combination of the theory, methods, and applications of extreme value theory (EVT) in finance and a practical understanding of market behavior including both ordinary and extraordinary conditions. Beginning with a fascinating history of EVTs and financial modeling, the handbook introduces the historical implications that resulted in the applications and then clearly examines the fundamental results of EVT in finance. After dealing with these theoretical results, the handbook focuses on the EVT methods critical for data analysis. Finally, the handbook features the practical applications and techniques and how these can be implemented in financial markets. Extreme Events in Finance: A Handbook of Extreme Value Theory and Its Applications includes: * Over 40 contributions from international experts in the areas of finance, statistics, economics, business, insurance, and risk management * Topical discussions on univariate and multivariate case extremes as well as regulation in financial markets * Extensive references in order to provide readers with resources for further study * Discussions on using R packages to compute the value of risk and related quantities The book is a valuable reference for practitioners in financial markets such as financial institutions, investment funds, and corporate treasuries, financial engineers, quantitative analysts, regulators, risk managers, large-scale consultancy groups, and insurers. Extreme Events in Finance: A Handbook of Extreme Value Theory and Its Applications is also a useful textbook for postgraduate courses on the methodology of EVTs in finance.
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Veröffentlichungsjahr: 2016
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Table of Contents
Cover
Financial Engineering and Econometrics
Title Page
Copyright
About the Editor
About the Contributors
Chapter 1: Introduction
1.1 Extremes
1.2 History
1.3 Extreme value theory
1.4 Statistical Estimation of Extremes
1.5 Applications in Finance
1.6 Practitioners' points of view
1.7 A broader view on modeling extremes
1.8 Final words
1.9 Thank You Note
References
Chapter 2: Extremes Under Dependence—Historical Development and Parallels with Central Limit Theory
2.1 Introduction
2.2 Classical (I.I.D.) Central Limit and Extreme Value Theories
2.3 Exceedances of Levels,
k
th Largest Values
2.4 CLT and EVT for Stationary Sequences, Bernstein's Blocks, and Strong Mixing
2.5 Weak Distributional Mixing for EVT,
D
(
u
n
), Extremal Index
2.6 Point Process of Level Exceedances
2.7 Continuous Parameter Extremes
References
Chapter 3: The Extreme Value Problem in Finance: Comparing the Pragmatic Program with the Mandelbrot Program
3.1 The Extreme Value Puzzle in Financial Modeling
3.2 The Sato Classification and the Two Programs
3.3 Mandelbrot's Program: A Fractal Approach
3.4 The Pragmatic Program: A Data-driven Approach
3.5 Conclusion
Acknowledgments
References
Chapter 4: Extreme Value Theory: An Introductory Overview
4.1 Introduction
4.2 Univariate Case
4.3 Multivariate Case: Some Highlights
Further reading
Acknowledgments
References
Chapter 5: Estimation of the Extreme Value Index
5.1 Introduction
5.2 The Main Limit Theorem Behind Extreme Value Theory
5.3 Characterizations of the Max-Domains of Attraction and Extreme Value Index Estimators
5.4 Consistency and Asymptotic Normality of the Estimators
5.5 Second-order Reduced-bias Estimation
5.6 Case Study
5.7 Other Topics and Comments
References
Chapter 6: Bootstrap Methods in Statistics of Extremes
6.1 Introduction
6.2 A Few Details on EVT
6.3 The Bootstrap Methodology in Statistics of Univariate Extremes
6.4 Applications to Simulated Data
6.5 Concluding Remarks
Acknowledgments
References
Chapter 7: Extreme Values Statistics for Markov Chains with Applications to Finance and Insurance
7.1 Introduction
7.2 On the (pseudo) Regenerative Approach for Markovian Data
7.3 Preliminary Results
7.4 Regeneration-based Statistical Methods for Extremal Events
7.5 The Extremal Index
7.6 The Regeneration-Based Hill Estimator
7.7 Applications to Ruin Theory and Financial Time Series
7.8 An Application to the CAC40
7.9 Conclusion
References
Chapter 8: Lévy Processes and Extreme Value Theory
8.1 Introduction
8.2 Extreme Value Theory
8.3 Infinite Divisibility and Lévy Processes
8.4 Heavy-tailed Lévy Processes
8.5 Semi-heavy-tailed Lévy Processes
8.6 Lévy Processes and Extreme Values
8.7 Conclusion
References
Chapter 9: Statistics of Extremes: Challenges and Opportunities
9.1 Introduction
9.2 Statistics of Bivariate Extremes
9.3 Models Based on Families of Tilted Measures
9.4 Miscellanea
References
Chapter 10: Measures of Financial Risk
10.1 Introduction
10.2 Traditional Measures of Risk
10.3 Risk Estimation
10.4 “Technical Analysis” of Financial Data
10.5 Dynamic Risk Measurement
Properties of
10.6 Open Problems and Further Research
10.7 Conclusion
Acknowledgment
References
Chapter 11: On the Estimation of the Distribution of Aggregated Heavy-Tailed Risks: Application to Risk Measures
11.1 Introduction
11.2 A Brief Review of Existing Methods
11.3 New Approaches: Mixed Limit Theorems
11.4 Application to Risk Measures and Comparison
11.5 Conclusion
References
Chapter 12: Estimation Methods for Value at Risk
12.1 Introduction
12.2 General Properties
12.3 Parametric Methods
12.4 Nonparametric Methods
12.5 Semiparametric Methods
12.6 Computer Software
12.7 Conclusions
Acknowledgment
References
Chapter 13: Comparing Tail Risk and Systemic Risk Profiles for Different Types of U.S. Financial Institutions
13.1 Introduction
13.2 Tail Risk and Systemic Risk Indicators
13.3 Tail Risk and Systemic Risk Estimation
13.4 Empirical Results
13.5 Conclusions
References
Chapter 14: Extreme Value Theory and Credit Spreads
14.1 Preliminaries
14.2 Tail Behavior of Credit Markets
14.3 Some Multivariate Analysis
14.4 Approximating Value at Risk for Credit Portfolios
14.5 Other Directions
References
Chapter 15: Extreme Value Theory and Risk Management in Electricity Markets
15.1 Introduction
15.2 Prior Literature
15.3 Specification of VR Estimation Approaches
15.4 Empirical Analysis
15.5 Conclusion
Acknowledgment
References
Chapter 16: Margin Setting and Extreme Value Theory
16.1 Introduction
16.2 Margin Setting
16.3 Theory and Methods
16.4 Empirical Results
16.5 Conclusions
Acknowledgment
References
Chapter 17: The Sortino Ratio and Extreme Value Theory: An Application to Asset Allocation
17.1 Introduction
17.2 Data Definitions and Description
17.3 Performance Ratios and Their Estimations
17.4 Performance Measurement Results and Implications
17.5 Concluding Remarks
Acknowledgments
References
Chapter 18: Portfolio Insurance: The Extreme Value Approach Applied to the CPPI Method
18.1 Introduction
18.2 The CPPI Method
18.3 CPPI and Quantile Hedging
18.4 Conclusion
References
Chapter 19: The Choice of the Distribution of Asset Returns: How Extreme Value Can Help?1
19.1 Introduction
19.2 Extreme Value Theory
19.3 Estimation of the Tail Index
19.4 Application of Extreme Value Theory to Discriminate Among Distributions of Returns
19.5 Empirical Results
19.6 Conclusion
References
Chapter 20: Protecting Assets Under Non-Parametric Market Conditions
20.1 Investors' “Known Knowns”
20.2 Investors' “Known Unknowns”
20.3 Investors' “Unknown Knowns”
20.4 Investors' “Unknown Unknowns”
20.5 Synthesis
References
Chapter 21: EVT Seen by a Vet: A Practitioner's Experience on Extreme Value Theory
21.1 What has the vet done?
21.2 Why Use EVT?
21.3 What EVT could additionally bring to the party?
21.4 A final thought
References
Chapter 22: The Robotization of Financial Activities: A Cybernetic Perspective
22.1 An Increasingly Complex System
22.2 Human Error
22.3 Concretely, What Do We Need to Do To Transform A Company Into A Machine?
References
Chapter 23: Two Tales of Liquidity Stress
23.1 The French Money Market Fund Industry. How history has Shaped a Potentially Vulnerable Framework
23.2 The 1992–1995 Forex Crisis
23.3 Four Mutations Paving the Way for Another Meltdown
23.4 The Subprime Crisis Spillover. How Some MMFs were Forced to Lock and Some Others Not
23.5 Conclusion. What Lessons can be Drawn from these Two Tales?
Further Reading
Chapter 24: Managing Operational Risk in the Banking Business – An Internal Auditor Point of View
Further Reading
References
Annexes
Chapter 25: Credo Ut Intelligam
25.1 Introduction
25.2 “Anselmist” Finance
25.3 Casino or Dance Hall?
25.4 Simple-Minded Diversification
25.5 Homo Sapiens Versus Homo Economicus
Acknowledgement
References
Chapter 26: Bounded Rationalities, Routines, and Practical as well as Theoretical Blindness: On the Discrepancy Between Markets and Corporations
26.1 Introduction: Expecting the Unexpected
26.2 Markets and Corporations: A Structural and Self-Disruptive Divergence of Interests
26.3 Making A Step Back From A Dream: On People Expectations
26.4 How to Disentangle People From A Unilateral Short-Term Orientation?
References
Name Index
Subject Index
Financial Engineering and Econometrics
End User License Agreement
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Guide
Cover
Table of Contents
Begin Reading
List of Illustrations
Chapter 4: Extreme Value Theory: An Introductory Overview
Figure 4.1 A natural disaster: Lisbon earthquake in 1755 —engraving “Aardbeeving te Lissabon in den Jaare 1755” by Reinier Vinkeles and François Bohn at Biblioteca Nacional Digital de Portugal, open source.
Source:
Nacional Digital de Portugal, http://purl.pt/13102. Public domain.
Figure 4.2 Financial disasters: Black Monday in 1987 and the financial collapse of 2007–2008.
Figure 4.3 Densities of one normal distribution () and one heavy-tailed distribution (). Interest is on both tails (a) and on the right tail (b) In log returns in finance, for instance, most of the observations are central, but it is exactly those extreme values (extremely low and/or extremely high) that constitute the focus for risk managers.
Figure 4.4 Convergence of the sample maximum to a degenerate distribution on the right endpoint (a and b) for ; convergence of the normalized maximum to a nondegenerate distribution, max Weibull, for suitable constants and (c and d), for .
Figure 4.5 For r.v.'s , distribution of (a) and of (b), , , for , comparatively to the limit law Gumbel (
fast convergence
).
Figure 4.6 For r.v.'s , distribution of (a) and of (b), , , for , comparatively to the limit law Gumbel (
very slow convergence
).
Figure 4.7 Max-stable distributions.
Figure 4.8 AM or Gumbel parametric method.
Figure 4.9 POT parametric methodology (a) and PORT semiparametric methodology (b).
Figure 4.10 LO parametric methodology (a) and largest intermediate order statistics semiparametric methodology (b).
Figure 4.11 Annual maximal river discharges of the Meuse river from 1911 to 1995.
Figure 4.12 Large claims (
USD
) of SOA Group Medical Insurance Large Claims Database, 1991.
Figure 4.13 SOA insurance data: sample path for POT high quantile estimates, , versus -largest.
Figure 4.14 Asymptotic variances for EVI estimators , , and .
Figure 4.15 SOA insurance data: sample paths for semiparametric EVI estimates, , as in (4.16), (4.18), and (4.19), versus -largest.
Figure 4.16 SOA insurance data: sample paths for semiparametric high quantiles estimates, as in (4.22) and (4.23), versus -largest.
Chapter 5: Estimation of the Extreme Value Index
Figure 5.1 Negative weekly returns of a European bank from January 1990 till December 2013 (in %).
Figure 5.2 Pareto QQ-plot (5.10) of the negative weekly returns with regression line anchored in with slope .
Figure 5.3 Hill estimates for the negative weekly returns as a function of .
Figure 5.4 Hill (full line), reduced-bias Hill (dash dotted line), and POT maximum likelihood (dotted line) estimates for the negative weekly returns as a function of .
Figure 5.5 Hill (full line) and POT maximum likelihood (dash dotted line) estimates based on the negative weekly returns up to August 2008.
Figure 5.6 Generalized Pareto QQ-plot (5.14) based on the negative weekly returns up to August 2008.
Chapter 6: Bootstrap Methods in Statistics of Extremes
Figure 6.1 Hill plots, denoted , associated with unit Pareto samples of size , from the model , for and .
Figure 6.2 Bootstrap adaptive EVI estimates, and , as a function of , in , for (a) and (b).
Figure 6.3 PORT-Hill/Hill and PORT-MVRB/MVRB adaptive EVI estimates (a), the -estimates (b), and the RMSE estimates (c) for the generated Student- sample.
Chapter 7: Extreme Values Statistics for Markov Chains with Applications to Finance and Insurance
Figure 7.1 Splitting a reflected random walk, with an atom at {0}; vertical lines corresponds to regeneration times, at which the chain forgets its past. A block is a set of observations between two lines (it may be reduced to {0} in some case).
Figure 7.2 Cramér–Lundberg model with a dividend barrier at (where the chain is reflected); ruin occurs at when the chain goes below 0.
Figure 7.3 Splitting the embedded chain of a Cramér–Lundberg model with a dividend barrier. Vertical lines corresponds to regeneration times (when the chain attains the barrier ). The blocks of observations between two vertical lines are independent.
Figure 7.4 Splitting a Smooth Exponential Threshold Arch time series, with , , , and . (a) Estimator of the transition density. (b) Visit of the chain to the small set and the level sets of the transition density estimator: the optimal small set should contain a lot of points in a region with high density. (c) Number of regenerations according to the size of the small set, optimal for . (d) Splitting (vertical bars) of the original time series, with horizontal bars corresponding to the optimal small set.
Figure 7.5 Estimator (continuous line) and bootstrap confidence interval (dotted lines) of the extremal index , for a sequence of high values of the threshold (seen as a quantile of the -coordinate). True value of .
Figure 7.6 Simulation of the SETAR-ARCH process for , , , and , exhibiting strong volatility and large excursions.
Figure 7.7 Estimator (continuous line) and confidence intervals (dotted lines) of the extremal index as a function of the quantile level , for a sequence of high values of the threshold (seen as a quantile of the -coordinate). True value of close to 0.5.
Figure 7.8 Bootstrap distribution of the pseudoregenerative Hill estimator (smoothed with a Gaussian kernel), based on bootstrap replications. Mode around 2.8.
Figure 7.9 Log returns of the CAC40, from 10/07/1987 to 06/16/2014.
Chapter 9: Statistics of Extremes: Challenges and Opportunities
Figure 9.1 (a) Example of a spectral density. (b) Spectral surface from a predictor-dependent beta family, with , for .
Figure 9.2 Scatter plots presenting two configurations of data (predictor, pseudo-angles): one (a) where there are sample pseudo-angles per each observed covariate and another (b) where to each observed covariate may correspond a single pseudo-angle.
Figure 9.3 Scedasis density estimates. The solid line represents the beta kernel estimate from (9.16), whereas the dashed line represents the estimate from (9.15). (a) Daily Standard and Poor's index from 1988 to 2007; the gray rectangles correspond to contraction periods in the US economy. (b) Simulated data illustration from , for , with and ; the grey line represents the true scedasis , for , and , for .
Chapter 10: Measures of Financial Risk
Figure 10.1 Standard deviations of daily log-returns of the S&P 500 index during the period from January 1960 – October 1987. The value of the standard deviation on 16.10.1987 is close to those on 16.10.1982, 16.10.1970, and 16.10.1962, and hardly can serve an indicator of a possible crash.
Figure 10.2 The SP 500 index during the period from May 1987 to October 1987. On Monday, October 19, 1987, the index fell by 20.5%.
Figure 10.3 Open-High-Low-Close price bar. Figure 10.5 is an example of a price chart with Open-High-Low-Close price bars. Many web-cites offer displaying price charts as a sequence of price bars.
Figure 10.5 Prices of the Akamai stock in November 2008 – March 2009. In early December an investor might have concluded that the price was in wave 2. With vertex 0 identified when the price was near $9.50, the risk of holding a “long” position was around $2 per share. The potential profit (if the position held till the end of wave 3) was around $5.
Figure 10.4 Basic Elliott wave for an up-trend (see also see Figure 10.5). The use of straight lines is, of course, a simplification (we consider an up-trend): each wave has “inside” a set of smaller scale waves, each “sub-wave” has again a set of even smaller waves inside, etc.
Figure 10.6 Standard deviations of the S&P 500 index (a) and (b) in 14.05.1987–19.10.1987. The standard deviation decreases, suggesting lower risk; exhibits spikes, indicating high level of risk.
Figure 10.7 The S&P 500 index in March 2007 – March 2008 had a level of resistance at 1406. After breaking through the level of resistance in January 2008, the index lost half of its value (see Figure 10.8).
Figure 10.8 S&P 500 index in April 2004–June 2009. After reaching 1565 in October 2007, the index fell to 678 in March 2009.
Figure 10.9 Standard deviation (a), measure (b), and the combined measure (c) of the S&P 500 index for the period November 2007–March 2008.
Chapter 11: On the Estimation of the Distribution of Aggregated Heavy-Tailed Risks: Application to Risk Measures
Figure 11.1 (a) plot of the S&P 500 daily returns from 1987 to 2007, plotted against the Gaussian one (same scaling) that appears as a straight line. (b) plot of the S&P 500 monthly returns from 1987 to 2007, plotted against the Gaussian one (same scaling) that appears as a straight line.
Figure 11.2 (a) plot of the S&P 500 monthly returns from 1987 to 2013, plotted against the Gaussian one (same scaling) that appears as a straight line. (b) plot of the S&P 500 monthly returns from 1791 to 2013, plotted against the Gaussian one (same scaling) that appears as a straight line.
Figure 11.3 Plots of the function defined in (11.35), for various , at given . (a) Case . (b) Case . (c) Case .
Chapter 12: Estimation Methods for Value at Risk
Figure 12.1 Black Monday crash on October 19, 1987. The Dow Jones stock index crashed down by 22.6% (by 508 points). Overall the stock market lost $0.5 trillion.
Figure 12.2 Japan stock price bubble near the end of 1989. A loss of $2.7 trillion in capital. A recovery happened after mid-1990.
Figure 12.3 Dot-com bubble (the NASDAQ index) during 1999 and 2000. The bubble burst on March 10, 2000. The peak on that day was $5048.62. There is a recovery after 2002. Never recovered to attain the peak.
Figure 12.4 Asian financial crisis (Asian dollar index) in July 1997. Not fully recovered even in 2011.
Figure 12.5 Black Wednesday crash of September 16, 1992. (a) shows the exchange rate of Deutsche mark to British pounds. (b) shows the UK interest rate on the day.
Figure 12.6 Value at risk illustrated.
Chapter 13: Comparing Tail Risk and Systemic Risk Profiles for Different Types of U.S. Financial Institutions
Figure 13.1 Joint bank crashes: historical versus simulated (Gaussian) return pairs.
Figure 13.2 Expected shortfall E (
X
− 50%|
X
> 50%): cross sectional averages over 1-year rolling window.
Figure 13.3 Tail-
β
: cross-sectional average over 1-year rolling window.
Figure 13.4 Tail risk (ES(
X
> 50%)) and extreme systemic risk (tail-
β
) pre-crisis versus crisis for all institutions and for industry groups.
Figure 13.5 Tail risk (ES(
X
> 50%)) versus extreme systemic risk (tail-
β
) for full-sample, pre-crisis, and crisis.
Chapter 14: Extreme Value Theory and Credit Spreads
Figure 14.1 Citigroup 5-year CDS spread. Daily history of the 5-year CDS spread for Citigroup senior debt. For periods when CDS traded points upfront, trading levels have been translated into spread terms. Note the extreme volatility in spreads observed during the crisis period in late 2008/early 2009.
Figure 14.2 Quantile–quantile plot of daily changes in Citigroup CDS spread. Quantile–quantile plot of observed daily changes in the 5-year CDS spread for Citigroup senior debt. The deviation from normality in both left and right tails is evident.
Figure 14.3 Return volatility versus initial spread for selected credits, post-crisis data. This chart shows, for 50 selected issuers, the relationship between the 5-year CDS spread on June 1, 2009 and the annualized volatility of daily returns for the period June 1, 2009 to February 24, 2014. Note that the initial spread was a good predictor of subsequent return volatility: the relationship was approximately linear.
Figure 14.4 Left tail index versus initial spread for selected credits, post-crisis data. This chart shows, for the same 50 issuers shown in Figure 14.3, the relationship between the 5-year CDS spread on June 1, 2009 and the left tail index estimated using daily CDS total returns for the period June 1, 2009 to February 24, 2014. Except for a few high-quality issuers with low initial spread, the observations are distributed around a horizontal line. That is, the estimated tail index appears to be independent of the initial spread level.
Figure 14.5 Distribution of tail index estimates for selected credits, post-crisis data. This histogram shows the distribution of left tail index estimates for the same 50 issuers shown in Figure 14.3. The left tail indices mostly fall in the range 0.3–0.4; the exceptions are a few high-quality issuers with low initial spread, as shown in Figure 14.4.
Chapter 15: Extreme Value Theory and Risk Management in Electricity Markets
Figure 15.1 Time Series Plots of Prices and Returns. Data are daily peak load prices for the EEX, PWX, and PJM power markets over the sample period covering January 3, 2006 and December 20, 2013. The left panel plots daily spot prices, while the right panel plots the corresponding daily continuously compounded returns (in %). For ease of comparison, the
y
-axis in the left and right panels are truncated between 0 and 200, and between −130% and +130%, respectively.
Chapter 16: Margin Setting and Extreme Value Theory
Figure 16.1 Margin requirements for a short position and a distribution of returns. This Figure illustrates a distribution of futures returns with special emphasis on the short position. At the upper tail of the distribution, a certain margin requirement is identified. Any price movement in excess of this margin requirement, given by the shaded area, represents a violation of this by the investor.
Figure 16.2 Fat-tailed and normal distributions. This Figure illustrates the tail distribution of futures returns with fat tails (dashed curve) and a normal distribution (solid curve). The fat tails ensure more probability mass in the tails than the normal distribution.
Figure 16.3
Q–Q
plot of DAX log returns series. This Figure plots the quantile of the empirical distribution of the DAX futures index returns against the normal distribution. The plot shows whether the distribution of the DAX returns matches a normal distribution. The straight line represents a normal quantile plot, whereas the curved line represents the quantile plot of the empirical distribution of the DAX contract. If the full set of DAX returns followed a normal distribution, then its quantile plot should match the normal plot and also be a straight line. The extent to which these DAX returns diverge from the straight line indicates the relative lack of normality.
Chapter 17: The Sortino Ratio and Extreme Value Theory: An Application to Asset Allocation
Figure 17.1 REIT and S&P 500 daily returns. 6555 observations from January 1987 to December 2012.
Figure 17.2 REIT and S&P 500 return left-tail histograms, as represented by bars, with normal distribution overlay, as represented by line.
Figure 17.3 REIT returns plotted against S&P 500 returns. Correlation coefficient between the REIT and S&P 500 is 0.63.
Figure 17.4 Plots of S&P 500 and REIT portfolios with various exceedances used for GPD estimation. GPD PDF generated from (17.5) . Panel A: S&P 500 and Panel B: REIT.
Figure 17.5 Sortino ratio values using (17.3) versus various MAR and REIT weights. Panel A: Plots of 1111 Sortino ratio values.
X
-axis represents weight in REIT portfolio,
Z
-axis represents Minimal Acceptable Return (MAR), and
Y
-axis represents calculated Sortino ratio provided REIT weight and MAR. Panel B: Selected Sortino ratios from Panel A.
Figure 17.6 Sortino ratio values using (17.6) versus various MAR and REIT weights. Panel A: Plots of 1111 Sortino ratio values.
X
-axis represents weight in REIT portfolio,
Z
-axis represents Minimal Acceptable Return (MAR), and
Y
-axis represents calculated Sortino ratio provided REIT weight and MAR. Panel B: Selected Sortino ratios from Panel A.
Chapter 18: Portfolio Insurance: The Extreme Value Approach Applied to the CPPI Method
Figure 18.1 The CPPI method.
Figure 18.2 Maxima (20 days).
Figure 18.3 Maxima (60 days).
Chapter 19: The Choice of the Distribution of Asset Returns: How Extreme Value Can Help?1
Figure 19.1 Evolution of the S&P 500 index over the period January 1950–December 2015.
Figure 19.2 Evolution of the S&P 500 index return over the period January 1950–December 2015.
Chapter 20: Protecting Assets Under Non-Parametric Market Conditions
Figure 20.1
Market conditions matrix
, based on investors' actual behavior and anticipations.
Figure 20.2 Usual market impact of a quarterly release.
Figure 20.3 Capstone Turbine Corp.'s daily quote data for the period covering March through May 2014.
Chapter 22: The Robotization of Financial Activities: A Cybernetic Perspective
Figure 22.1 Self-learning mechanism.
Chapter 23: Two Tales of Liquidity Stress
Figure 23.1 FRF overnight repo rate (T4M) 1990–1996.
List of Tables
Chapter 3: The Extreme Value Problem in Finance: Comparing the Pragmatic Program with the Mandelbrot Program
Table 3.1 The Sato classification, the variance issue, and the financial modeling programs
Table 3.2 Examples of Lévy processes in financial modeling
Chapter 8: Lévy Processes and Extreme Value Theory
Table 8.1 Limit laws of means and maxima
Table 8.2 Comparison of characteristic exponents
Table 8.3 Laws of maxima and maximum domain of attraction
Table 8.4 Infinitely divisible distributions and asymptotic behavior
Chapter 10: Measures of Financial Risk
Table 10.1 The S&P 500 index, its standard deviations, and on the eve of the Black Monday. While the standard deviation decreases, suggesting a reduction of the level of risk, exhibits spikes, indicating high level of risk
Chapter 11: On the Estimation of the Distribution of Aggregated Heavy-Tailed Risks: Application to Risk Measures
Table 11.1 Necessary and sufficient condition on for having ,
Table 11.2 Value of for having up to
Table 11.3 Coordinates of the maximum of (defined in (11.35)), as a function of and
Table 11.4 Approximations of extreme quantiles (95%; 99%; 99.5%) by various methods (CLT, Max, Normex, weighted normal) and associated approximative relative error to the empirical quantile , for
n
= 52, 100, 250, 500 respectively, and
Table 11.5 Approximations of extreme quantiles (95%; 99%; 99.5%) by various methods (GCLT, Max, Normex) and associated approximative relative error to the empirical quantile , for , respectively, and for
Chapter 13: Comparing Tail Risk and Systemic Risk Profiles for Different Types of U.S. Financial Institutions
Table A1 Tail risk and systemic risk for deposit banks
Table A2 Tail risk and systemic risk for category “others”
Table A3 Tail risk and systemic risk for insurance companies
Table A4 Tail risk and systemic risk measures for “Broker-dealers”
Table 13.1 Tail risk and systemic risk for different types of financial institutions
Table 13.2 Equality test for sample means across financial industry groups
Table 13.3 Rank correlation (in %) of tail risk and extreme systemic risk measures pre-crisis versus crisis
Table 13.4 Rank correlation between extreme tail risk and tail-
β
Table 13.6 Rank correlation of tail risk and systemic risk with different proxies for size
Chapter 14: Extreme Value Theory and Credit Spreads
Table 14.1 Tail index estimates for CDX HG5 and Citigroup 5-year CDS
Table 14.2 Tail index estimates for the post-crisis period
Table 14.5 Value-at-risk rule-of-thumb adjustments and the impact of 0.05 error in tail index estimate
Table 14.3 Left tail dependence coefficients for selected credits, post-crisis data
Table 14.4 Sample correlations and outer correlations for selected credits, post-crisis data
Chapter 15: Extreme Value Theory and Risk Management in Electricity Markets
Table 15.1 Descriptive statistics
Table 15.2 In-sample parameter estimates of condEVT model
Table 15.3 Out-of-sample VaR forecasting accuracy (right tail)
Table 15.4 Out-of-sample VaR forecasting accuracy (left tail)
Chapter 16: Margin Setting and Extreme Value Theory
Table 16.1 Summary statistics for stock index futures
Table 16.2 Optimal tail estimates for stock index futures
Table 16.3 Common margin requirements to cover extreme price movements
Chapter 17: The Sortino Ratio and Extreme Value Theory: An Application to Asset Allocation
Table 17.1 Descriptive statistics for portfolios with various REIT weights
Table 17.2 GPD estimates for S&P 500 and REIT portfolios. Relationship estimated is from Eq. (17.4)
Table 17.3 Optimal REIT portfolio weights
Chapter 18: Portfolio Insurance: The Extreme Value Approach Applied to the CPPI Method
Table 18.1 Descriptive statistics of daily variations
Table 18.2 Estimation results for a Fréchet distribution
Table 18.3 Numerical upper bounds on the multiple
Chapter 19: The Choice of the Distribution of Asset Returns: How Extreme Value Can Help?1
Table 19.1 Tail index and highest existing moment for different models for returns
Table 19.2 Top 20 yearly minimum and maximum daily returns in the S&P 500 index
Table 19.3 Top 20 negative and positive returns in the S&P 500 index
Table 19.4 Parametric estimates of the tail index using minimum and maximum returns
Table 19.5 Parametric estimates of the tail index using negative and positive return exceedances
Table 19.6 Nonparametric estimates of the tail index
Table 19.9 Optimal value for nonparametric estimators of the tail index
Table 19.7 Choice of the distribution of returns based on the tail index
Table 19.8 Maximum existing moment of the distribution of the S&P 500 index returns
Chapter 24: Managing Operational Risk in the Banking Business – An Internal Auditor Point of View
Table 24.1 Beta factors in standardized approach
Table 24.2 Categorization of operational risks
Table 24.3 Number of internal losses and loss amount reported by the 2008 loss data collection exercise participants
Table 24.4 Annualized loss frequencies normalized per €billion of assets
Table 24.5 Cross-bank median of distribution across severity brackets
Table 24.6 Sum and distribution of annualized loss frequencies and amounts by operational risk
Table 24.7 Sum and distribution of annualized loss frequencies and amounts by business line
Wiley Handbooks in
Financial Engineering and Econometrics
Advisory Editor
Ruey S. Tsay
The University of Chicago Booth School of Business USA
A complete list of the titles in this series appears at the end of this volume.
Extreme Events in Finance
A Handbook of Extreme Value Theory and its Applications
Edited by
François Longin
Department of Finance,ESSEC Business School,Paris, France
Copyright © 2017 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Names: Longin, François Michel, 1968- editor.
Title: Extreme events in finance : a handbook of extreme value theory and its applications / edited by François Longin.
Description: Hoboken : Wiley, 2017. | Series: Wiley handbooks in financial engineering and econometrics | Includes bibliographical references and index.
Identifiers: LCCN 2016004187| ISBN 9781118650196 (hardback) | ISBN 9781118650202 (epub)
Subjects: LCSH: Finance–Mathematical models. | Extreme value theory–Mathematical models. | BISAC: BUSINESS & ECONOMICS / Insurance / Risk Assessment & Management.
Classification: LCC HG106 .E98 2016 | DDC 332.01/5195–dc23 LC record available at http://lccn.loc.gov/2016004187
Cover image courtesy of iStockphoto © Nikada
About the Editor
François Longin (ESSEC Business School)
François Longin graduated from the French engineering school Ecole Nationale des Ponts et Chaussées in 1990, and received the PhD degree in finance from HEC Paris in 1993 for his thesis “Volatility and extreme price movements in equity markets.” He then conducted research on financial markets at New York University and the London Business School. He is now Professor of Finance at ESSEC Business School and a consultant to several financial institutions and firms. He is an active member of CREAR (Center of Research in Econo-finance and Actuarial sciences on Risk) at ESSEC. His current research interests include extreme events in finance, as well as financial applications of extreme value theory in risk management and portfolio management. His works have been published in international scientific journals such as the Journal of Finance, Journal of Business, Review of Financial Studies, Journal of Banking and Finance, and the Journal of Derivatives. He is Associate Editor of the Journal of Banking and Finance and the Journal of Risk. His domains of expertise include risk management for banks, portfolio management for fund management firms, financial management for firms, and wealth management for individuals. (More information can be found on www.longin.fr.) He is also a participant in the SimTrade project, which is a pedagogical tool to help understand how financial markets work and to learn to act in financial markets, and a simulation‐based research program to improve the behavior of individuals and the statistical characteristics of financial markets. More information can be had from www.simtrade.fr.
About the Contributors
Jan Beirlant (KU Leuven University)
Jan Beirlant obtained a PhD in statistics from KU Leuven in 1984. He is currently a Professor with the Department of Mathematics, KU Leuven University. Presently, he is chairing LRisk, a center for research, training, and advice in insurance and financial risk analysis, combining all relevant KU Leuven expertise. His main research interests include extreme value methodology with emphasis on applications in insurance and finance. He has published over 100 papers in statistical research journals and has published the following books: Statistics of Extremes: Theory and Applications, with Y. Goegebeur, J. Segers, and J.L. Teugels (2004), and Reinsurance: Actuarial and Statistical Aspects, with H. Albrecher and J.L. Teugels (2016).
Chapter: Estimation of the Extreme Value Index
Patrice Bertail (University of Paris-Ouest-Nanterre la Défense)
Patrice Bertail is Professor of applied mathematics (statistics and probabilities) at the University of Paris-Ouest-Nanterre la Défense. He has been in charge of the Master's ISIFAR (Ingénierie Statistique et Informatique de la Finance, l'Assurance et du Risque) program. He is also a researcher with the MODAL'X laboratory and CREST-ENSAE. His research interests include resampling methods for dependent data, survey sampling, empirical processes and extremes, especially for Markovian data (with applications toward food risks assessment).
Chapter: Extreme Values Statistics for Markov Chains with Applications to Finance and Insurance
Philippe Bertrand (IAE Aix-en Provence)
Philippe Bertrand obtained a PhD in mathematical economics from Ecole des Hautes Etudes en Sciences Sociales and the Habilitation à diriger des recherches (HDR) from University Paris-Dauphine. He is currently a Full Professor of finance with IAE Aix-en Provence. He is also a member of the CERGAM Research Center and a member of Aix-Marseille School of Economics. He joined IAE in 2011, from the Faculté d'Economie of Aix-Marseille, where he was Professor of finance. He was formerly the head of Financial Engineering, CCF Capital Management. His research interests include portfolio management, risk and performance evaluation, and portfolio insurance, as well as financial structured products. He has published numerous articles in scientific journals such as the Journal of Banking and Finance, Finance, Geneva Risk and Insurance Review, Financial Analysts Journal, and the Journal of Asset Management. He is currently the executive president of the French Finance Association (AFFI). He has served as an associate editor of the review Bankers, Markets & Investors. He chaired the 31st Spring International Conference of the French Finance Association, held at IAE AIX, May 20–21, 2014.
Chapter: Portfolio Insurance: The Extreme Value Approach Applied to the CPPI Method
Laurent Bibard (ESSEC Business School)
Laurent Bibard has been a Professor with ESSEC Business School since 1991. He was Dean of the MBA Programs (2005–2009), and is currently a Full Professor, Management Department, and Head of the Edgar Morin Chair on Complexity. His current research interests include organizational vigilance interpreted as the organizational conditions favoring collective as well as individual mindfulness. He has been invited to many prestigious universities in Germany (Mannheim), Canada (UQAM), Japan (Keio Business School, Keio University), and others. His publications include “Management and Philosophy : What is at Stake?” (Keio Business Forum, March 2011, Vol. 28, no 1, pp. 227–243) and Sexuality and Globalization (Palgrave Macmillan, New York, 2014). His book La sagesse et le féminin (Wisdom and Feminity) was republished in Japan, at the end of 2014.
Chapter: Bounded Rationalities, Routines, and Practical as well as Theoretical Blindness: On the Discrepancy Between Markets and Corporations
Jean-François Boulier (Aviva Investors France)
Jean-François Boulier graduated from the Polytechnique and obtained a PhD in fluid mechanics. He was a researcher with CNRS in Grenoble. He started his career in finance in 1987 with Credit Commercial de France, where he headed the Research and Innovation Department, then the Market Risk Department, and subsequently became CIO of Sinopia asset management and deputy CEO. He is currently the CEO of Aviva Investors France. He joined Aviva Investors in 2008 and has held several positions: CIO in Paris, then CEO in Europe, and Global CIO for Fixed Income. Between 2002 and 2008, he was heading Euro FI at Credit Agricole Asset Management.
Chapter: EVT Seen by a Vet: A Practitioner's Experience on Extreme Value Theory
Henri Bourguinat (University of Bordeaux IV)
Henri Bourguinat is Emeritus Professor of Economics, University of Bordeaux IV. In 1974, he founded LAREFI, a research laboratory dedicated to monetary and financial economics (http://lare-efi.u-bordeaux4.fr/spip.php?article36). He is a former research director at CNRS. He is the author of sixty articles published in various journals, such as Revue Economique, Economie Appliquée, and others. He has (co)-authored eighteen books on international economics and finance. His book, Finance Internationale, has been a best seller since it was first published.
Chapter: Credo Ut Intelligam
Geoffrey Booth (Michigan State University)
Geoffrey Booth holds the Frederick S. Addy Distinguished Chair in Finance, Michigan State University. He has published more than 150 journal articles, monographs, and professional papers. Booth's work has appeared in the Journal of Finance, Review of Economics and Statistics, and Review of Financial Studies, to name but a few. His current research interests include the behavior of financial markets with special emphasis on market microstructure issues and asset allocation decisions of financial institutions.
Chapter: The Sortino Ratio and Extreme Value Theory: An Application to Asset Allocation
Eric Briys (Cyberlibris)
Eric Briys is the co-founder of www.cyberlibris.com, and a former Managing Director, Deutsche Bank Global Markets Division, London, where he headed the European Insurance Coverage Group. Prior to joining Deutsche Bank, he worked with Merrill Lynch, Lehman Brothers, Tillinghast, and The World Bank. He has held academic positions at CERAM, Concordia University, University of Montreal, and HEC Paris. He has published articles in American Economic Review, Journal of Finance, Journal of Financial and Quantitative Analysis, Journal of Risk and Insurance, Geneva Papers on Risk and Insurance Theory, the Southern Economic Journal, Journal of Risk and Uncertainty, Journal of International Money and Finance, European Economic Review, and others. He is a former Editor of Finance, the Journal of the French Finance Association, and Founding Editor of the Review of Derivatives Research. He has also (co)-authored 11 books on economics and finance.
Chapter: Credo Ut Intelligam
John Paul Broussard (Rutgers University)
John Paul Broussard is an Associate Professor of finance at Rutgers University, Camden, NJ, where he teaches investments and corporate finance courses. His research papers have been published in the Journal of Financial Economics, Financial Management, Management Science, Journal of Financial Services Research, Quarterly Review of Economics and Finance, and the European Journal of Operational Research, as well as in other journals and monographs. His current financial market research interests include extreme value applications to portfolio decision making and high-frequency trading.
Chapter: The Sortino Ratio and Extreme Value Theory: An Application to Asset Allocation
Frederico Caeiro (Nova University of Lisbon)
Frederico Caeiro received a MSc in probability and statistics in 2001, and a PhD in statistics in 2006, from the Faculty of Science, Lisbon University. He is currently an Auxiliary Professor with the Mathematics Department, Faculty of Science and Technology, Nova University of Lisbon, and a member of the Center for Mathematics and Applications. His current research interests include statistics of extremes, extreme value theory, nonparametric statistics, and computational statistical methods.
Chapter: Bootstrap Methods in Statistics of Extremes
Kam Fong Chan (University of Queensland Business School)
Kam Fong is currently a Senior Lecturer in finance with the University of Queensland Business School, University of Queensland, Australia. He has previously worked for several years as a quant at the Risk Analytics Division of the Risk Management Department, United Overseas Bank (UOB), Singapore. His research interests include modeling asset prices using various state-of-the-art econometric techniques, derivatives pricing, and risk management. He has published in various journals of international repute, including the Journal of Banking and Finance, International Journal of Forecasting, Pacific Basin Finance Journal, and the Journal of International Financial Markets, Institutions & Money.
Chapter: Extreme Value Theory and Risk Management in Electricity Markets
Stephen Chan (University of Manchester)
Stephen Chan is currently working toward the PhD degree at the University of Manchester, UK. He is the winner of an EPSRC Doctoral Prize Fellowship. His research interests include extreme value analysis, financial theory, and distribution theory. His publications include an R package and papers in Quantitative Finance.
Chapter: Estimation Methods for Value at Risk
Jean-Marie Choffray (ESSEC Business School and University of Liège)
Dr. Jean-Marie Choffray was, until recently, Senior Lecturer at ESSEC (France) and Chair Professor of Management Science at the Graduate School of Business, University of Liège (Belgium). He is the author of several books and a frequent contributor to scientific and professional journals, which includes over 70 articles. He is the recipient of a number of distinguished research awards and sits on the boards of several companies that he co-founded.
Chapter: Protecting Assets Under Non-Parametric Market Conditions
Stéphan Clémençon (Telecom ParisTech)
Stéphan Clémençon received a PhD in applied mathematics from the University Denis Diderot, Paris, France, in 2000. In October 2001, he joined the faculty of the University Paris X as an Associate Professor and successfully defended his habilitation thesis in 2006. Since October 2007, he has been a Professor and Researcher with Telecom ParisTech, the leading school in the field of information technologies in France, holding the Chair in Machine Learning. His research interests include machine learning, Markov processes, computational harmonic analysis, and nonparametric statistics.
Chapter: Extreme Values Statistics for Markov Chains with Applications to Finance and Insurance
John Cotter (University College Dublin)
John Cotter is Professor of Finance and the Chair in quantitative finance, University College, Dublin. He is also a Research Fellow with the UCLA Ziman Research Center for Real Estate. His recent professional papers include those in the Review of Financial Studies, Journal of Banking and Finance, and Journal of International Money and Finance. He is an associate editor of the Journal of Banking and Finance, Journal of International Financial Markets, Institutions and Money, and European Journal of Finance.
Chapter: Margin Setting and Extreme Value Theory
Miguel de Carvalho (Pontificia Universidad Católica de Chile)
Miguel de Carvalho is an Associate Professor of applied statistics, Pontificia Universidad Católica de Chile. Before moving to Chile, he was a postdoctoral fellow with the Swiss Federal Institute of Technology (EPFL). He is an applied mathematical statistician with a variety of interdisciplinary interests, inter alia, biostatistics, econometrics, and statistics of extremes. In addition to serving at the university, he is also a regular academic consultant of Banco de Portugal (Portuguese Central Bank). He has been on the editorial board of the Annals of Applied Statistics (IMS) and Statistics and Public Policy (ASA).
Chapter: Statistics of Extremes: Challenges and Opportunities
Thanh Thi Huyen Dinh (De Lage Landen Group)
Thanh Thi Huyen Dinh studied at Maastricht University, the Netherlands. She obtained a PhD based on her research on collateralization and credit scoring in the Vietnamese loan market and on tail risk and systemic risk of different types of financial institutions, the topic of this handbook contribution. She is currently a Global Analytics Consultant at the US division of the De Lage Landen Group (DLL), a Dutch insurance company.
Chapter: Comparing Tail Risk and Systemic Risk Profiles for Different Types of U.S. Financial Institutions
Kevin Dowd (Durham University)
Kevin Dowd is Professor of finance and economics at Durham University, UK. He has written extensively on the history and theory of free banking, central banking, financial regulation and monetary systems, financial risk management, pensions, and mortality modeling. His books include Private Money: The Path to Monetary Stability, The State and the Monetary System, Laissez-Faire Banking, Competition and Finance: A New Interpretation of Financial and Monetary Economics, Money and the Market: Essays on Free Banking, and Measuring Market Risk. He is also the co-author with Martin Hutchinson of Alchemists of Loss: How Modern Finance and Government Intervention Crashed the Financial System (Wiley, 2010).
Chapter: Margin Setting and Extreme Value Theory
Isabel Fraga Alves (University of Lisbon)
Isabel Fraga Alves obtained a PhD in statistics and computation in 1992 for her thesis “Statistical Inference in Extreme Value Models,” and the Habilitation degree in statistics and operations research in 2004, both from the University of Lisbon. She is currently an Associate Professor with the Department of Statistics and Operations Research, Faculty of Sciences, University of Lisbon. She is a past Coordinator of the Center of Statistics and Applications, University of Lisbon (2006–2009), an elected member of International Statistical Institute, and a member of the Bernoulli Society for Mathematical Statistics and Probability, Portuguese Statistical Society, and Portuguese Mathematical Society.
Chapter: Extreme Value Theory: An Introductory Overview
Ivette Gomes (University of Lisbon)
Ivette Gomes obtained a PhD in statistics from the University of Sheffield, UK, in 1978, and the Habilitation degree in applied mathematics from the University of Lisbon in 1982. She was a Full Professor with the Department of Statistics and Operations Research, Faculty of Sciences, University of Lisbon (1988–2011), and is now a Principal Researcher with the Centre for Statistics and Applications, University of Lisbon (CEAUL). Her current research interests include statistics of extremes. She is a founding member of the Portuguese Statistical Society and member of several scientific associations. She has been involved in the organization of several international conferences, including the 56th Session of ISI, 2007. Among other editorial duties, she has been the chief editor of Revstat, since 2003, and associate editor of Extremes since 2007. She is currently Vice-President of the International Statistical Institute (ISI) for the period 2015–2019.
Chapter: Bootstrap Methods in Statistics of Extremes
Philip Gray (Monash Business School)
Philip Gray is a Professor of finance with the Monash Business School, Monash University, Melbourne, Australia. His research interests include asset pricing, empirical finance, and capital markets. He also applies quantitative techniques in derivative valuation and risk management. His research has been published in journals including the Journal of Finance, Journal of Futures Markets, Journal of Banking and Finance, Journal of Business, Finance & Accounting, International Review of Finance, and International Journal of Forecasting.
Chapter: Extreme Value Theory and Risk Management in Electricity Markets
Lígia Henriques-Rodrigues (University of São Paulo)
Lígia Henriques-Rodrigues received a degree in applied mathematics and computation (probability and statistics) from the Instituto Superior Técnico (Technical University) of Lisbon in 1996, a Master's in applied mathematics (probability and statistics) from the University of Évora in 2000, and a PhD in statistics and operational research in the field of probability and statistics from the Faculty of Sciences, University of Lisbon in2009. She was as a postdoctoral fellow with the Faculty of Sciences, University of Lisbon, in 2014. She is currently an Assistant Professor with the Institute of Mathematics and Statistics, University of São Paulo, Brazil, and a Researcher at the Center of Statistics and Applications, University of Lisbon. Her research interests include extreme value theory, reduced-bias semiparametric estimation, location- and scale-invariant estimation, and resampling methodologies in statistics of extremes, with applications to life sciences, environment, risk, insurance, and finance.
Chapter: Bootstrap Methods in Statistics of Extremes
Klaus Herrmann (KU Leuven University)
Klaus Herrmann obtained a PhD in science from KU Leuven in 2015 under the supervision of Professor Irène Gijbels. He completed a research stay at the ETH Zurich RiskLab with Professor Paul Embrechts in the same year. He is currently with the Department of Mathematics, KU Leuven, as a postdoctoral researcher. His research interests include statistical and probabilistic dependence concepts and their application to financial and actuarial mathematics.
Chapter: Estimation of the Extreme Value Index
Marie Kratz (ESSEC Business School Paris – Singapore, Center of Research in Econo-finance and Actuarial Sciences on Risk – CREAR)
Marie Kratz is Professor at ESSEC Business School and Director of its risk research center, CREAR. She holds a Doctorate in Applied Mathematics (UPMC, Paris 6; carried out to a great extent at the Center for Stochastic Processes, Chapel Hill, North Carolina) & Habilitation (HDR), did a post-doc at Cornell University. Her research addresses a broad range of topics in probability and statistics, and actuarial mathematics, with a focus on extreme value theory, risk analysis and Gaussian processes. These fields find natural applications in Finance and Actuarial Sciences that she is developing at ESSEC. Marie is a Fellow (Actuaire Agrégée) of the French Institute of Actuaries. She coordinates the ESSEC-ISUP (Paris 6) Actuarial Track, as well as organizes since 2009 a fortnightly Working Group on Risk Analysis at ESSEC – Paris La Défense with Academics and Professionals. Marie is also the President of the Group ‘Banque Finance Assurance’ of SFdS (French Society of Statistics).
Chapter: On the Estimation of the Distribution of Aggregated Heavy-Tailed Risks: Application to Risk Measures
Maxime Laot (European Central Bank)
Maxime Laot obtained a MBA with a major in applied economics from the ESSEC Business School. He is a practitioner in the field of banking supervision. He has spent several years working as an internal auditor for Groupe BPCE, one of the largest French banks, assessing the level and risk management of financial, credit, and operational risks in various retail and wholesale banking institutions in France and abroad. He recently joined the
new regulatory body of the European Central Bank, and is responsible for the direct supervision of the Eurozone's largest banks.
Chapter: Managing Operational Risk in the Banking Business – An Internal Auditor Point of View
Ross Leadbetter (University of North Carolina)
Ross Leadbetter received a MSc from the University of New Zealand in 1954, a MA from Cambridge University in 1962, and a PhD (1963) from the University of North Carolina (UNC), Chapel Hill. He has also received honorary Doctorates from Lund University, Sweden (1991), and Lisbon University, Portugal (2013). He is currently Professor of statistics at UNC. Before joining UNC in 1966, he worked with the New Zealand Applied Mathematics Laboratory, Wellington, the Naval Research Laboratory, Auckland, and the Research Triangle Institute, North Carolina. His research interests include probability and statistics, stochastic processes, extremal theory, and statistical communication theory in engineering, oceanographic, and environmental applications. He has written many articles and books including Stationary and Related Stochastic Processes (with Harald Cramer) and Extremes and Related Properties of Random Sequences and Processes (with Georg Lindgren and Holger Rootzen).
Chapter: Extremes Under Dependence—Historical Development and Parallels with Central Limit Theory
