Non Linear Dependence Structures: a Copula Opinion Approach in Portfolio Optimization


 Julia Sanders
 2 years ago
 Views:
Transcription
1 Non Linear Dependence Structures: a Copula Opinion Approach in Portfolio Optimization Jean Damien Villiers ESSEC Business School Master of Sciences in Management Grande Ecole September
2 Non Linear Dependence Structures: a Copula Opinion Approach in Portfolio Optimization Abstract The recent financial crisis brought out the Risk on/risk off notion, characterized by periods where the market is not driven any more by the fundamental analysis. Consequently, a question about the effectiveness of the classic portfolio optimization models has risen: the strong rise of cross asset correlations suggests integrating new market anomalies into models, such as the non normality of returns or the non linear dependence structures. The aim of this paper is to provide a methodology to implement a copula approach model, derived from the Black Littermann methodology. The model outperforms the Black Littermann model, and reduces sensitively the risk of our portfolio. INTRODUCTION Markowitz proved in 1954 that an appropriate pool of assets could reduce the systematic risk of a portfolio without impacting its return expectancy. However, several criticisms have been made on this model, discussing the simplicity of the optimization based on only mean and variance, the normality of returns assumed, and its operational non stability making the model hardly usable in practice. Since then, several models have emerged. The Black Littermann model is one of them, widely used by asset managers: based on both Bayesian markets forecasts and mean variance optimization, it solves the issue of stability on expected returns, and proposes a way to rely both on fundamental and quantitative analysis. Despite the democratization of this methodology, several weaknesses still remain: the normality of asset returns and their linear codependence modeled by a variance covariance matrix are not well adapted for the stressed scenarios we ve observed over the previous years. Since the bankruptcy of Lehman Brothers a new constraint must be taken into account: we ve observed a sharp rise of correlations between assets historically uncorrelated, and it seems necessary to widen the possibilities of the Black Littermann model to take these anomalies into account. In this research paper, we propose a methodology based on codependence structures and marginal distribution functions to model the market, keeping the strengths of the Black Littermann method. More particularly, we define marginal distribution functions for each type of asset, and quantify the correlations between them through copulas. Finally, we make our asset pooling according to the optimization of Omega ratio, a risk measure depending on the whole distribution of returns. 2
3 METHODOLOGY The first objective of the model is to simulate the expected future evolution of the market ( Posterior Market Distribution ), mixing the results of a quantitative model based on historical data ( Prior Market Distribution ) and the forecasts of an investor ( Views Distribution ). The cumulative distribution functions are then averaged according to weights depending on the confidence levels the investor has on his views (See Appendix 1). The Prior Market Distribution is the result of Monte Carlo simulations based on independent marginal distributions for each asset, and copulas to model the interdependency. As we intend to quantify non linear dependencies between assets in cases of highly stressed markets, it is necessary to use marginal distributions adapted to leptokurtic returns. We assume log normal distributions for FX and Credit products, and use the Heston model for Commodities and Equities products, a stochastic volatility model widely used for its calibration easy to implement, defined by the following stochastic differential equations: = " + = ( ) " + where is the stock price, the instantaneous variance, "# are Wiener processes with correlation ρ, μ is the rate of return of the asset, θ is the long term expected variance, κ the rate at which V t reverts to θ, and ξ is the volatility of volatility. The dependence structure between assets is then modeled by the Student copula (t copula), extracted from the bivariate Student distribution, according to the Sklar s theorem (see Appendix 2):, ;, =, (, ) with ρ the correlation coefficient and, the bivariate Student distribution with a correlation matrix depending on ρ and k, the degrees of freedom. The choice of this copula is justified by its cumulative distribution function easy to implement and the presence of lower and upper tails dependence, enabling to model non linear dependence of rare events (see Appendix 3). Both copula and marginal distributions are then calibrated on historical data with a maximum likelihood algorithm, and the Prior Market Distribution can finally be built with Monte Carlo simulations, and stored in a matrix M "# where J is the number of simulations and N the number of assets. 3
4 The Views Distribution is another input of the model, expressed as a combination of two matrix:  Two vectors contain the lower and upper bounds of the forecasts on returns. For example, if we consider the vectors lb=(0%, 1%) et ub=(2%, 4%), the investor expresses 2 forecasts, in the ranges [0%: 2%] and [1%: 4%]. The assets are assumed to follow uniform distributions between these bounds. In this research paper, the forecasts are voluntarily subjective views on the market and are not the result of any model.  A Pick matrix ℳ"# where K is the number of views and N the number of assets, giving the weights on the assets affected by the kth view (lower and upper bounds vectors above) The Posterior Market Distribution: we finally define a vector ℳ" with weights between 0% and 100%, whether we trust or not our forecasts. 0% would mean a model fully relying on the Views Distribution. We estimate this vector using confidence intervals. More precisely, we deduce the confidence level of each forecast from its variance, which leads to build a volatility model. We choose the heteroscedastic model E Garch(1,1), described by the following equations: = + ù = log = + log + [ ]+ where and are the return and the variance of the considered asset. Once we have calibrated each view, we estimate the variance of the residuals which follow a Gaussian distribution, and the confidence level from the confidence interval: 1, = "#$( + ) where and are respectively the return and the forecast for each asset. We finally apply this for a Gaussian distribution: =, +, where, is the Gaussian cumulative distribution function, with parameters (0, ), and T a defined threshold. We finally build the final Posterior Market Distribution, which is a weighted average of the Prior Market Distribution and Views Distribution, according to the following formula:,, + (1 ) (, ) where is Prior cumulative distribution function, the Views cumulative distribution function, the confidence in the kth view, and W the sorted returns of the J simulations. Quant Awards 2013 September
5 OPTIMIZATION CRITERIA: THE OMEGA RATIO Once we ve computed the Posterior Market Distribution, we have J simulations of the possible evolution of the market, according to both quantitative ( Prior ) and fundamental ( Views ) analysis. We choose the parametric ratio Omega as criteria to optimize our portfolio and select allocation weights: Ω H = 1 F x dx F x dx Where F is the CDF of the portfolio returns, and H the targeted return. Example of Cumulative Distribution Function: The Omega ratio, introduced by Keating and Shadwick, is interpreted as the ratio of performances over the threshold H divided by the performances under the threshold. If the average return of the portfolio is higher than the defined threshold, the ratio will tend to increase. Consequently, we optimize our portfolio increasing this ratio as much as possible. The strength of this ratio is to take into account more than two moments of the distribution, since it is derived from the CDF. This risk measure is consequently more precise than the Sharpe Ratio. From an algorithm point of view, we are facing a multiple parameter optimization problem, which is solved by a simplex method: the Nelder Mead optimization technique. 5
6 INPUTS OF THE MODEL To build the Prior Market Distribution we use the daily closing prices of cross asset single stocks and index, between 14/11/2007 and 12/11/2011. ""#$ %#&'#" ()$)"$&#)*+,'.#& "#$ %&'()*+),)./%0++)1*)234)/).&.)56.,57)078 *+,1* G.507',)HI>: %&'()*+),).5J)K/HIJ)/).&.)56.,57)078 *+,KJ+2504 LDAF LDAF)%=AACD? L&+8%+7 0?M<#BN<?B)O:"F<)E:<FCB %":";),)D:;)C?MP)L:"F< +G&592074QHII GCO$)JC<AF):<FCB %":";)LADR"A)GJ +G*LGJ82074QHII 6*)8<RB)2ADE"A)'S4 T9*)6+*0Q T9*9& *)8<RB)2$":F)'S4 T9*)6*%0Q T9*9.&.2504 LADR"A).CB"?# 8T)LADR"A).CB"?# 8T.0.( N<:OC?O)*":U<B)6V=CBC<# *0)6N<:OC?O)*":U<B# *6*W'32504 DNNDFCBC<# 8T),%)X)DNNDFCBC<#).5 8T,%.5 The Views Distribution is chosen voluntarily subjective but could be also the result of econometric forecasts, or any external model. The lower and upper bounds are the same for each forecast but we consider 6 different economic scenarios, implying different weights on the 11 assets: "#$% &#'()*+#,. /00()*+#, " #" /5 " $" 567 " #$" 681 " %" 9:1 " &" " &" " &" =<*;B,C"C(3 " %" ;<*;B,C"C(3 " %" 1#DD#.C"C(3 " &" 8#A. " &" Pick Matrix: "#$%&'() "#$%&'("%)*+,)(&. /0 /0 /0 /0 /0 /0 /0 1/0 2/0 1/0 /0,($3'$(456 /0 /0 /0 /0 /0 /0 /0 2/0 2/0 7/0 /0 8)9"3 /0 /0 /0 /0 7/0 /0 /0 2/0 2/0 /0 /0 :&%;"%&'(" /0 /0 /0 1/0 1/0 /0 /0 /0 2/0 /0 /0 :&%;#$%&'(" <=0 /0 7=0 /0 /0 /0 1/0 /0 /0 /0 1/0 >94966'(" /0 =/0 7/0 /0 /0 /0 /0 /0 /0 /0 1/0 6
7 Finally, the confidence vector is the output of the E Garch(1,1) model, using confidence intervals at 99%: "#$#%&"'(")$*+&# "#$%&'#(' "#$%&'("%)*+,)(&. /0123,($4'$(567 80/23 9):"4 /0;23 <&%="%&'(" 10>23 10A23 RESULTS To measure the improvement of the copula approach, we define as benchmark the output of the Black Littermann model, assuming similar views and economic scenarios. The model is run for 100,000 simulations and the ratio Omega is optimized with a threshold of 1%. We finally get the following basket allocation: ""#$%&'#( )"%$*+,'&&./%((+)($0/%.* 1#23"%+42'('#(+22.#%$0 52.%6 1%70 " #" " 859 $" #" %#" 9:;5 &" ##" %'" :<+1=>?:9 $" (" '$" &" " >B+,41, )" '" *" (" #" &"?B+>C8:9:>5 $" (" %(" >B+>C8:9:>5 #" #(" %#)" 14BB4 " +" %*" <4,? $" #," %#," 9#&%" #$$" #$$" The backtesting of the model is made out of the calibration window, between December 2011 and September The portfolio weights are kept unchanged during this period. We get a higher return from 5,43% to 8,39% whilst the volatility and the Omega ratio of our portfolio strongly decrease. ""#$%&'#()*#+," ,&./()#()&0,) 1,/'#+ ((.%"'2,+) 3#"%&'"'&4 5*,6%)%&'#) 7#8,9&)/,&./( "#$%&'())*+,#&*$.,#+% / :;<"#&=;((:&>;;+:#$. 702? / :'60,/)/,&./( 7
8 CONCLUSION This research paper examines whether the modeling of non linear term structure dependences between assets can reduce the systematic risk of a portfolio, without affecting its return. The model presented outperforms the Black Littermann benchmark, both on return and risk measures. Compared to the Black Littermann model, we mainly highlight three improvements:  The use of copulas enables to model the Risk on Risk off phenomenon: in case of high stressed scenarios, the fundamental stock analysis is less effective and a copula opinion pooling would be more reliable.  The simulation of assets based on marginal distributions combined with the Monte Carlo method widens the scope, letting the possibility to choose a different diffusion process for each asset.  The confidence matrix is one of the hardest parameter to estimate in the Black Littermann model. We propose a way to compute it, based on a volatility model, letting to the investor the forecasts on returns as only subjective inputs. The model presented enables to capture different Market phenomenon as the leptokurticity of returns, the non linearity of correlations, the heteroscedasticity and asymmetry of the volatility, whilst we let the possibility to take into account any fundamental analysis on stocks. For practical reasons, we ve chosen to apply the model in a very specific case (Student Copula, Heston and EGARCH diffusion process ), but the model can be seen as a generic methodology with further possible improvements (implementation of Jump process, calibration on Market prices, discussion about the copula ). In any case, we suggest measuring the effectiveness of a model with a risk ratio depending on the whole distribution of returns, such as the Omega ratio we ve presented. 8
9 APPENDIX Appendix 1: The cumulative distribution function of the Posterior Market Distribution is the weighted average between the Prior Market Distribution and the View Distribution. Appendix 2: A copula is a multivariate cumulative distribution function whose marginal distributions are uniform random variables on [0,1]. If C is a copula on R, we can find a random vector (U,, U ) such as: P U u,, U u = C(u,, u ) P U u = u for all i [0,1] A copula can be determined by the previous definition, or using pre existing multivariate distribution. In this latter case, we use the Skal s theorem, explaining the link between the copula C and the multivariate distribution F, depending on its marginal univariate distributions and. Sklar s Theorem: 9
10 Let F be a bivariate distribution with marginal distribution and. The copula C associated is defined by: C u, u = C F x, F x = F F u, F u = F(x, x ) C is unique when the margins and are continuous. Appendix 3: Source : «Nonlinear Term Structure Dependence: Copula Functions, Empirics, and Risk Implications Markus Junker, Alex Szimayer, Niklas Wagner Lower Tail dependence: (,) A copula C has a lower tail dependence if : = lim exists and ]0,1] Upper Tail dependence: (,) A copula C has a upper tail dependence if : = lim exists and ]0,1] 10
11 REFERENCES Markus Junker, Alex Szimayer, Niklas Wagner, 2004, Nonlinear Term Structure Dependence: Copula Functions, Empirics, and Risk Implications Attilio Meucci, 2005, Beyond Black Litterman: Views on Non Normal Markets Attilio Meucci, 2006, Beyond Black Litterman in Practice: a Five Step Recipe to Input Views on non Normal Markets Michael Stein, 2008, Copula Opinion Pooling in Asset Allocation Attilio Meucci, 2011, A New Breed of Copulas for Risk and Portfolio Management Arthur Charpentier, 2010, Copules et risques corrélés, Journées d Études Statistique Fischer Black and Rober Litterman, 1992, Global Portfolio Optimization, Financial Analysts Journal Con Keating and William F. Shadwick, 2002, An Introduction to Omega S.J. Kane and M.C. Bartholomew Biggs, M. Cross and M. Dewar, OPTIMIZING OMEGA Joanna Gatz, 2007, Properties and Applications of the Student T Copula 11
Contents. List of Figures. List of Tables. List of Examples. Preface to Volume IV
Contents List of Figures List of Tables List of Examples Foreword Preface to Volume IV xiii xvi xxi xxv xxix IV.1 Value at Risk and Other Risk Metrics 1 IV.1.1 Introduction 1 IV.1.2 An Overview of Market
More informationPricing of a worst of option using a Copula method M AXIME MALGRAT
Pricing of a worst of option using a Copula method M AXIME MALGRAT Master of Science Thesis Stockholm, Sweden 2013 Pricing of a worst of option using a Copula method MAXIME MALGRAT Degree Project in Mathematical
More informationMarket Risk Analysis. Quantitative Methods in Finance. Volume I. The Wiley Finance Series
Brochure More information from http://www.researchandmarkets.com/reports/2220051/ Market Risk Analysis. Quantitative Methods in Finance. Volume I. The Wiley Finance Series Description: Written by leading
More informationTailDependence an Essential Factor for Correctly Measuring the Benefits of Diversification
TailDependence an Essential Factor for Correctly Measuring the Benefits of Diversification Presented by Work done with Roland Bürgi and Roger Iles New Views on Extreme Events: Coupled Networks, Dragon
More informationC: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)}
C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)} 1. EES 800: Econometrics I Simple linear regression and correlation analysis. Specification and estimation of a regression model. Interpretation of regression
More informationAn introduction to ValueatRisk Learning Curve September 2003
An introduction to ValueatRisk Learning Curve September 2003 ValueatRisk The introduction of ValueatRisk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk
More informationLDA at Work: Deutsche Bank s Approach to Quantifying Operational Risk
LDA at Work: Deutsche Bank s Approach to Quantifying Operational Risk Workshop on Financial Risk and Banking Regulation Office of the Comptroller of the Currency, Washington DC, 5 Feb 2009 Michael Kalkbrener
More informationCONTENTS. List of Figures List of Tables. List of Abbreviations
List of Figures List of Tables Preface List of Abbreviations xiv xvi xviii xx 1 Introduction to Value at Risk (VaR) 1 1.1 Economics underlying VaR measurement 2 1.1.1 What is VaR? 4 1.1.2 Calculating VaR
More informationData Preparation and Statistical Displays
Reservoir Modeling with GSLIB Data Preparation and Statistical Displays Data Cleaning / Quality Control Statistics as Parameters for Random Function Models Univariate Statistics Histograms and Probability
More informationAnalysis of Financial Time Series
Analysis of Financial Time Series Analysis of Financial Time Series Financial Econometrics RUEY S. TSAY University of Chicago A WileyInterscience Publication JOHN WILEY & SONS, INC. This book is printed
More informationA constant volatility framework for managing tail risk
A constant volatility framework for managing tail risk Alexandre Hocquard, Sunny Ng and Nicolas Papageorgiou 1 Brockhouse Cooper and HEC Montreal September 2010 1 Alexandre Hocquard is Portfolio Manager,
More informationQuantitative Methods for Finance
Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain
More informationExample 1: Calculate and compare RiskMetrics TM and Historical Standard Deviation Compare the weights of the volatility parameter using,, and.
3.6 Compare and contrast different parametric and nonparametric approaches for estimating conditional volatility. 3.7 Calculate conditional volatility using parametric and nonparametric approaches. Parametric
More informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationAN ACCESSIBLE TREATMENT OF MONTE CARLO METHODS, TECHNIQUES, AND APPLICATIONS IN THE FIELD OF FINANCE AND ECONOMICS
Brochure More information from http://www.researchandmarkets.com/reports/2638617/ Handbook in Monte Carlo Simulation. Applications in Financial Engineering, Risk Management, and Economics. Wiley Handbooks
More informationMonte Carlo Methods and Models in Finance and Insurance
Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Monte Carlo Methods and Models in Finance and Insurance Ralf Korn Elke Korn Gerald Kroisandt f r oc) CRC Press \ V^ J Taylor & Francis Croup ^^"^ Boca Raton
More informationMarket Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk
Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day
More informationBlackLitterman Return Forecasts in. Tom Idzorek and Jill Adrogue Zephyr Associates, Inc. September 9, 2003
BlackLitterman Return Forecasts in Tom Idzorek and Jill Adrogue Zephyr Associates, Inc. September 9, 2003 Using BlackLitterman Return Forecasts for Asset Allocation Results in Diversified Portfolios
More informationJohn W Muteba Mwamba s CV
John W Muteba Mwamba s CV Emails: johnmu@uj.ac.za moi175@hotmail.com arg@analyticsresearch.net Location: Johannesburg, South Africa Personal Website: www.analyticsresearch.net Education University of Johannesburg
More information11. Time series and dynamic linear models
11. Time series and dynamic linear models Objective To introduce the Bayesian approach to the modeling and forecasting of time series. Recommended reading West, M. and Harrison, J. (1997). models, (2 nd
More informationJava Modules for Time Series Analysis
Java Modules for Time Series Analysis Agenda Clustering Nonnormal distributions Multifactor modeling Implied ratings Time series prediction 1. Clustering + Cluster 1 Synthetic Clustering + Time series
More informationMeasuring downside risk of stock returns with timedependent volatility (DownsideRisikomessung für Aktien mit zeitabhängigen Volatilitäten)
Topic 1: Measuring downside risk of stock returns with timedependent volatility (DownsideRisikomessung für Aktien mit zeitabhängigen Volatilitäten) One of the principal objectives of financial risk management
More informationCorrelation Effects in Stock Market Volatility
Correlation Effects in Stock Market Volatility Chris Lee Jackson Newhouse Alex Livenson Eshed OhnBar Advisors: Alex Chen and Todd Wittman Abstract: Understanding the correlation between stocks may give
More informationStephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer
Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of DiscreteTime Stochastic
More informationJohn W Muteba Mwamba s CV
John W Muteba Mwamba s CV Emails: johnmu@uj.ac.za moi175@hotmail.com arg@analyticsresearch.net Location: Johannesburg, South Africa Personal Website: www.analyticsresearch.net Education University of Johannesburg
More informationPortfolio insurance a comparison of naive versus popular strategies
Jorge Costa (Portugal), Raquel M. Gaspar (Portugal) Insurance Markets and Companies: Analyses and Actuarial Computations, Volume 5, Issue 1, 2014 Portfolio insurance a comparison of naive versus popular
More informationCopula Concepts in Financial Markets
Copula Concepts in Financial Markets Svetlozar T. Rachev, University of Karlsruhe, KIT & University of Santa Barbara & FinAnalytica* Michael Stein, University of Karlsruhe, KIT** Wei Sun, University of
More informationAnalysis of Financial Time Series with EViews
Analysis of Financial Time Series with EViews Enrico Foscolo Contents 1 Asset Returns 2 1.1 Empirical Properties of Returns................. 2 2 Heteroskedasticity and Autocorrelation 4 2.1 Testing for
More informationAPPLYING COPULA FUNCTION TO RISK MANAGEMENT. Claudio Romano *
APPLYING COPULA FUNCTION TO RISK MANAGEMENT Claudio Romano * Abstract This paper is part of the author s Ph. D. Thesis Extreme Value Theory and coherent risk measures: applications to risk management.
More informationDOWNSIDE RISK IMPLICATIONS FOR FINANCIAL MANAGEMENT ROBERT ENGLE PRAGUE MARCH 2005
DOWNSIDE RISK IMPLICATIONS FOR FINANCIAL MANAGEMENT ROBERT ENGLE PRAGUE MARCH 2005 RISK AND RETURN THE TRADEOFF BETWEEN RISK AND RETURN IS THE CENTRAL PARADIGM OF FINANCE. HOW MUCH RISK AM I TAKING? HOW
More informationIndividual contributions to portfolio risk: risk decomposition for the BETFI index
Individual contributions to portfolio risk: risk decomposition for the BETFI index Marius ACATRINEI Institute of Economic Forecasting Abstract The paper applies Euler formula for decomposing the standard
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationL10: Probability, statistics, and estimation theory
L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian
More informationVolatility modeling in financial markets
Volatility modeling in financial markets Master Thesis Sergiy Ladokhin Supervisors: Dr. Sandjai Bhulai, VU University Amsterdam Brian Doelkahar, Fortis Bank Nederland VU University Amsterdam Faculty of
More informationEDF CEA Inria School Systemic Risk and Quantitative Risk Management
C2 RISK DIVISION EDF CEA Inria School Systemic Risk and Quantitative Risk Management EDF CEA INRIA School Systemic Risk and Quantitative Risk Management Regulatory rules evolutions and internal models
More informationMälardalen University
Mälardalen University http://www.mdh.se 1/38 Value at Risk and its estimation Anatoliy A. Malyarenko Department of Mathematics & Physics Mälardalen University SE72 123 Västerås,Sweden email: amo@mdh.se
More informationChapter 6. Modeling the Volatility of Futures Return in Rubber and Oil
Chapter 6 Modeling the Volatility of Futures Return in Rubber and Oil For this case study, we are forecasting the volatility of Futures return in rubber and oil from different futures market using Bivariate
More informationCalculating Interval Forecasts
Calculating Chapter 7 (Chatfield) Monika Turyna & Thomas Hrdina Department of Economics, University of Vienna Summer Term 2009 Terminology An interval forecast consists of an upper and a lower limit between
More informationThird Edition. Philippe Jorion GARP. WILEY John Wiley & Sons, Inc.
2008 AGIInformation Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Third Edition Philippe Jorion GARP WILEY John Wiley & Sons, Inc.
More informationINTRODUCTORY STATISTICS
INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore
More informationCost of Capital and Corporate Refinancing Strategy: Optimization of Costs and Risks *
Cost of Capital and Corporate Refinancing Strategy: Optimization of Costs and Risks * Garritt Conover Abstract This paper investigates the effects of a firm s refinancing policies on its cost of capital.
More informationAsymmetric Correlations and Tail Dependence in Financial Asset Returns (Asymmetrische Korrelationen und TailDependence in Finanzmarktrenditen)
Topic 1: Asymmetric Correlations and Tail Dependence in Financial Asset Returns (Asymmetrische Korrelationen und TailDependence in Finanzmarktrenditen) Besides fat tails and timedependent volatility,
More informationHEDGE FUND RETURNS WEIGHTEDSYMMETRY AND THE OMEGA PERFORMANCE MEASURE
HEDGE FUND RETURNS WEIGHTEDSYMMETRY AND THE OMEGA PERFORMANCE MEASURE by Pierre Laroche, Innocap Investment Management and Bruno Rémillard, Department of Management Sciences, HEC Montréal. Introduction
More informationInternet Appendix to False Discoveries in Mutual Fund Performance: Measuring Luck in Estimated Alphas
Internet Appendix to False Discoveries in Mutual Fund Performance: Measuring Luck in Estimated Alphas A. Estimation Procedure A.1. Determining the Value for from the Data We use the bootstrap procedure
More informationQUANTITATIVE FINANCIAL ECONOMICS
Ill. i,t.,. QUANTITATIVE FINANCIAL ECONOMICS STOCKS, BONDS AND FOREIGN EXCHANGE Second Edition KEITH CUTHBERTSON AND DIRK NITZSCHE HOCHSCHULE John Wiley 8k Sons, Ltd CONTENTS Preface Acknowledgements 2.1
More informationProbability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
More informationTechnical Report. Value at Risk (VaR)
Technical Report On Value at Risk (VaR) Submitted To: Chairman of Department of Statistics University of Karachi MUNEER AFZAL EP042922 BS. Actuarial Sciences & Risk Management Final Year (Last Semester)
More informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationUsing least squares Monte Carlo for capital calculation 21 November 2011
Life Conference and Exhibition 2011 Adam Koursaris, Peter Murphy Using least squares Monte Carlo for capital calculation 21 November 2011 Agenda SCR calculation Nested stochastic problem Limitations of
More informationSystematic risk modelisation in credit risk insurance
Systematic risk modelisation in credit risk insurance Frédéric Planchet JeanFrançois Decroocq Ψ Fabrice Magnin α ISFA  Laboratoire SAF β Université de Lyon  Université Claude Bernard Lyon 1 Groupe EULER
More informationOpen Access Asymmetric Dependence Analysis of International Crude Oil Spot and Futures Based on the Time Varying CopulaGARCH
Send Orders for Reprints to reprints@benthamscience.ae The Open Petroleum Engineering Journal, 2015, 8, 463467 463 Open Access Asymmetric Dependence Analysis of International Crude Oil Spot and Futures
More informationOutline. Random Variables. Examples. Random Variable
Outline Random Variables M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno Random variables. CDF and pdf. Joint random variables. Correlated, independent, orthogonal. Correlation,
More informationCopula Simulation in Portfolio Allocation Decisions
Copula Simulation in Portfolio Allocation Decisions Gyöngyi Bugár Gyöngyi Bugár and Máté Uzsoki University of Pécs Faculty of Business and Economics This presentation has been prepared for the Actuaries
More informationOptimization applications in finance, securities, banking and insurance
IBM Software IBM ILOG Optimization and Analytical Decision Support Solutions White Paper Optimization applications in finance, securities, banking and insurance 2 Optimization applications in finance,
More informationStatistics Graduate Courses
Statistics Graduate Courses STAT 7002Topics in StatisticsBiological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.
More informationImplementing an AMA for Operational Risk
Implementing an AMA for Operational Risk Perspectives on the Use Test Joseph A. Sabatini May 20, 2005 Agenda Overview of JPMC s AMA Framework Description of JPMC s Capital Model Applying Use Test Criteria
More informationUnivariate and Multivariate Methods PEARSON. Addison Wesley
Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston
More informationUsing Duration Times Spread to Forecast Credit Risk
Using Duration Times Spread to Forecast Credit Risk European Bond Commission / VBA Patrick Houweling, PhD Head of Quantitative Credits Research Robeco Asset Management Quantitative Strategies Forecasting
More informationThe Methodology for the MidTerm Forecasting of the Financial Results of Firms Belikov V.V., Gataullina R.I., Tregub I.V.
The Methodology for the MidTerm Forecasting of the Financial Results of Firms Belikov V.V., Gataullina R.I., Tregub I.V. Abstract. The article is dedicated to the midterm forecasting of the Company financial
More informationHandbook in. Monte Carlo Simulation. Applications in Financial Engineering, Risk Management, and Economics
Handbook in Monte Carlo Simulation Applications in Financial Engineering, Risk Management, and Economics PAOLO BRANDIMARTE Department of Mathematical Sciences Politecnico di Torino Torino, Italy WlLEY
More informationAPPLIED MISSING DATA ANALYSIS
APPLIED MISSING DATA ANALYSIS Craig K. Enders Series Editor's Note by Todd D. little THE GUILFORD PRESS New York London Contents 1 An Introduction to Missing Data 1 1.1 Introduction 1 1.2 Chapter Overview
More informationPortfolio Management for institutional investors
Portfolio Management for institutional investors June, 2010 Bogdan Bilaus, CFA CFA Romania Summary Portfolio management  definitions; The process; Investment Policy Statement IPS; Strategic Asset Allocation
More informationDOES IT PAY TO HAVE FAT TAILS? EXAMINING KURTOSIS AND THE CROSSSECTION OF STOCK RETURNS
DOES IT PAY TO HAVE FAT TAILS? EXAMINING KURTOSIS AND THE CROSSSECTION OF STOCK RETURNS By Benjamin M. Blau 1, Abdullah Masud 2, and Ryan J. Whitby 3 Abstract: Xiong and Idzorek (2011) show that extremely
More informationChapter 1 INTRODUCTION. 1.1 Background
Chapter 1 INTRODUCTION 1.1 Background This thesis attempts to enhance the body of knowledge regarding quantitative equity (stocks) portfolio selection. A major step in quantitative management of investment
More informationNEXT GENERATION RISK MANAGEMENT and PORTFOLIO CONSTRUCTION
STONYBROOK UNIVERSITY CENTER FOR QUANTITATIVE FINANCE EXECUTIVE EDUCATION COURSE NEXT GENERATION RISK MANAGEMENT and PORTFOLIO CONSTRUCTION A fourpart series LED BY DR. SVETLOZAR RACHEV, DR. BORYANA RACHEVAIOTOVA,
More informationMonte Carlo Simulations of the multivariate distributions with different marginals
Monte Carlo Simulations of the multivariate distributions with different marginals Mária Bohdalová Comenius University, Faculty of Management, Department of Information Systems email: maria.bohdalova@fm.uniba.sk
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (40 points) Answer briefly the following questions. 1. Describe
More informationComputer Handholders Investment Software Research Paper Series TAILORING ASSET ALLOCATION TO THE INDIVIDUAL INVESTOR
Computer Handholders Investment Software Research Paper Series TAILORING ASSET ALLOCATION TO THE INDIVIDUAL INVESTOR David N. Nawrocki  Villanova University ABSTRACT Asset allocation has typically used
More informationPortfolio Distribution Modelling and Computation. Harry Zheng Department of Mathematics Imperial College h.zheng@imperial.ac.uk
Portfolio Distribution Modelling and Computation Harry Zheng Department of Mathematics Imperial College h.zheng@imperial.ac.uk Workshop on Fast Financial Algorithms Tanaka Business School Imperial College
More informationGraduate Programs in Statistics
Graduate Programs in Statistics Course Titles STAT 100 CALCULUS AND MATR IX ALGEBRA FOR STATISTICS. Differential and integral calculus; infinite series; matrix algebra STAT 195 INTRODUCTION TO MATHEMATICAL
More informationBetter decision making under uncertain conditions using Monte Carlo Simulation
IBM Software Business Analytics IBM SPSS Statistics Better decision making under uncertain conditions using Monte Carlo Simulation Monte Carlo simulation and risk analysis techniques in IBM SPSS Statistics
More informationUsing Patterns of Volatility in Calculating VaR
Using Patterns of Volatility in Calculating VaR Professor Radu Titus MARINESCU, Ph.D. Artifex University of Bucharest Lecturer Mădălina Gabriela ANGHEL, Ph.D. Artifex University of Bucharest Daniel DUMITRESCU,
More informationThe Delta Method and Applications
Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and
More informationMonte Carlo Simulation
1 Monte Carlo Simulation Stefan Weber Leibniz Universität Hannover email: sweber@stochastik.unihannover.de web: www.stochastik.unihannover.de/ sweber Monte Carlo Simulation 2 Quantifying and Hedging
More information(Part.1) FOUNDATIONS OF RISK MANAGEMENT
(Part.1) FOUNDATIONS OF RISK MANAGEMENT 1 : Risk Taking: A Corporate Governance Perspective Delineating Efficient Portfolios 2: The Standard Capital Asset Pricing Model 1 : Risk : A Helicopter View 2:
More informationPETROLEUM ECONOMICS AND MANAGEMENT
PETROLEUM ECONOMICS AND MANAGEMENT APPLIED GRADUATE STUDIES PEM1 Business Accounting PEM2 Organizational Behavior PEM3 Strategic Marketing & Management PEM4 Energy Economics & Development PEM5 Energy Geopolitics
More informationRecent Developments in Stress Testing Market Risk
Recent Developments in Stress Testing Market Risk Gerald Krenn Financial Markets Analysis and Surveillance Division Paper presented at the Expert Forum on Advanced Techniques on Stress Testing: Applications
More informationMeasuring the tracking error of exchange traded funds: an unobserved components approach
Measuring the tracking error of exchange traded funds: an unobserved components approach Giuliano De Rossi Quantitative analyst +44 20 7568 3072 UBS Investment Research June 2012 Analyst Certification
More informationMARKET RISK MEASUREMENT. Lecture 2
RMOR MSc MARKET RISK MEASUREMENT Lecture 2 Measurements for Market Risk Control 1 2.1 VaRBased Trading Limits A risk adjusted performance measure (RAPM) can be applied to measure the performance of a
More informationApplication of Quantitative Credit Risk Models in Fixed Income Portfolio Management
Application of Quantitative Credit Risk Models in Fixed Income Portfolio Management Ron D Vari, Ph.D., Kishore Yalamanchili, Ph.D., and David Bai, Ph.D. State Street Research and Management September 263,
More informationTarget Strategy: a practical application to ETFs and ETCs
Target Strategy: a practical application to ETFs and ETCs Abstract During the last 20 years, many asset/fund managers proposed different absolute return strategies to gain a positive return in any financial
More informationImplementing BlackLitterman using an Equivalent Formula and Equity Analyst Target Prices
Implementing BlackLitterman using an Equivalent Formula and Equity Analyst Target Prices Leon Chen, Zhi Da and Ernst Schaumburg February 9, 2015 Abstract We examine an alternative and equivalent Black
More informationBusiness Valuation under Uncertainty
Business Valuation under Uncertainty ONDŘEJ NOWAK, JIŘÍ HNILICA Department of Business Economics University of Economics Prague W. Churchill Sq. 4, 130 67 Prague 3 CZECH REPUBLIC ondrej.nowak@vse.cz http://kpe.fph.vse.cz
More informationSTOCK MARKET VOLATILITY AND REGIME SHIFTS IN RETURNS
STOCK MARKET VOLATILITY AND REGIME SHIFTS IN RETURNS ChiaShang James Chu Department of Economics, MC 0253 University of Southern California Los Angles, CA 90089 Gary J. Santoni and Tung Liu Department
More informationSYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation
SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline
More information2. Default correlation. Correlation of defaults of a pair of risky assets
2. Default correlation Correlation of defaults of a pair of risky assets Consider two obligors A and B and a fixed time horizon T. p A = probability of default of A before T p B = probability of default
More informationEC 6310: Advanced Econometric Theory
EC 6310: Advanced Econometric Theory July 2008 Slides for Lecture on Bayesian Computation in the Nonlinear Regression Model Gary Koop, University of Strathclyde 1 Summary Readings: Chapter 5 of textbook.
More informationMarket Risk Management for Hedge Funds
Market Risk Management for Hedge Funds Foundations of the Style and Implicit ValueatRisk Francois Due and Yann Schorderet WILEY A John Wiley & Sons, Ltd., Publication Acknowledgements xv 1 Introduction
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More informationPeer Reviewed. Abstract
Peer Reviewed William J. Trainor, Jr.(trainor@etsu.edu) is an Associate Professor of Finance, Department of Economics and Finance, College of Business and Technology, East Tennessee State University. Abstract
More informationStock Investing Using HUGIN Software
Stock Investing Using HUGIN Software An Easy Way to Use Quantitative Investment Techniques Abstract Quantitative investment methods have gained foothold in the financial world in the last ten years. This
More informationProbability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0
Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall
More informationMODEL DOSSIER FOR MARKET RISK MODELS
Directorate General Banking Supervision MODEL DOSSIER FOR MARKET RISK MODELS Structure of the Dossier used to monitor and document internal models for calculating minimum capital requirements for market
More informationReducing Active Return Variance by Increasing Betting Frequency
Reducing Active Return Variance by Increasing Betting Frequency Newfound Research LLC February 2014 For more information about Newfound Research call us at +16175319773, visit us at www.thinknewfound.com
More informationAffinestructure models and the pricing of energy commodity derivatives
Affinestructure models and the pricing of energy commodity derivatives Nikos K Nomikos n.nomikos@city.ac.uk Cass Business School, City University London Joint work with: Ioannis Kyriakou, Panos Pouliasis
More informationLean Six Sigma Training/Certification Book: Volume 1
Lean Six Sigma Training/Certification Book: Volume 1 Six Sigma Quality: Concepts & Cases Volume I (Statistical Tools in Six Sigma DMAIC process with MINITAB Applications Chapter 1 Introduction to Six Sigma,
More informationModels for Product Demand Forecasting with the Use of Judgmental Adjustments to Statistical Forecasts
Page 1 of 20 ISF 2008 Models for Product Demand Forecasting with the Use of Judgmental Adjustments to Statistical Forecasts Andrey Davydenko, Professor Robert Fildes a.davydenko@lancaster.ac.uk Lancaster
More informationSimulating Investment Portfolios
Page 5 of 9 brackets will now appear around your formula. Array formulas control multiple cells at once. When gen_resample is used as an array formula, it assures that the random sample taken from the
More informationBetter Risk Management for Improved Business Decision Making
WHITE PAPER Better Risk Management for Improved Business Decision Making SAS Risk Management for Banking Table of Contents Introduction...1 SAS Risk Management for Banking...2 Infrastructure...2 Risk Data
More informationLifecycle Investment Strategies  Myths and Facts
Lifecycle Investment Strategies  Myths and Facts Stefan Trück Centre for Financial Risk, Macquarie University Financial Risk Day 2016 Banking, Investment and Property Risk March 18, 2016 based on joint
More information