Forecasting Volatility of Hang Seng Index and its Application on Reserving for Investment Guarantees. Herbert Tak-wah Chan Derrick Wing-hong Fung
|
|
- Bryce Gallagher
- 6 years ago
- Views:
Transcription
1 Forecasting Volatility of Hang Seng Index and its Application on Reserving for Investment Guarantees Herbert Tak-wah Chan Derrick Wing-hong Fung
2 This presentation represents the view of the presenters and does not represent our employer.
3 Objective This presentation aims to explain the 2006 ASHK Annual Best Paper Forecasting Volatility of Hang Seng Index and its Application on Reserving for Investment Guarantees jointly written by Herbert Chan and Derrick Fung, which can be downloaded at
4 The paper The paper can be generally divided into 2 parts (1) Forecast Hang Seng Index (HSI) daily volatility based on historical HSI data and validation of these forecasted daily volatilities (2) Discuss the application of forecasted volatility on reserving for investment guarantees
5 Flow Chart HSI Data Fit data into GARCH GARCH Fitting Obtain satisfactory results for data fitting Forecasting Obtain forecasted HSI daily volatilities Validation Accuracy of the forecasted HSI daily volatilities are validated by VAR and Binomial Test Application Application on reserving for investment guarantees
6 HSI Data Daily return of HSI price Historical volatility Realized volatility Implied volatility
7 Daily Return We define daily return R n at day n as logarithmic return: R n = 100 (lnp n -lnp n-1 ) where P n is the closing price of HSI at day n Why not R n = 100 (P n /P n-1-1)?
8 Daily Return lnp n -lnp n-1 = ln(p n /P n-1 ) = ln(1+r n ) = R n R n2 /2 + R n3 /3 R n4 /4 + (Taylor series) ~ R n for small values of R n
9 Daily Return Logarithmic return is commonly used in financial literature because of its additive nature Cumulative Return at day n = 100 (lnp n lnp 0 ) = R n + R n R 1
10 Historical Volatility Historical volatility at day n is defined as the variance of daily returns in the preceding 30 transaction days
11 Historical Volatility Historical volatility at day n: 1 29 n m = n 29 ( R m R where R is the mean of return in the preceding 30 transaction days A total of 1235 historical volatilities are calculated 15 February 1999 to 20 April )
12 Realized Volatility Realized Volatility is the variance of 5-minute returns within a day Record 5-minute returns of HSI R n,d = 100 (lnp n,d lnp n,d-1 ) where P n,d is the asset price at trading day n, at the 5-minute mark d.
13 Realized Volatility Realized volatility at day n is defined as: where day n ~ σ R n 2 n 54 1 = 52 d = 2 ( ) 2 R R 55 n, d n is the mean of 5-minute return at
14 Realized Volatility A total of 1235 realized volatilities are calculated 15 February 1999 to 20 April 2004
15 Implied Volatility Using Black-Scholes model, implied volatility is calculated from options whose underlying asset is HSI It is calculated on a daily basis and is obtained from Hong Kong Exchange and Clearing Limited 1235 implied volatilities are obtained 15 February 1999 to 20 April 2004
16 HSI Data 1235 historical volatilities, realized volatilities and implied volatilities are obtained from 15 February 1999 to 20 April 2004
17 Flow Chart HSI Data Fit data into GARCH GARCH Fitting Obtain satisfactory results for data fitting Forecasting Obtain forecasted HSI daily volatilities Validation Accuracy of the forecasted HSI daily volatilities are validated by VAR and Binomial Test Application Application on reserving for investment guarantees
18 GARCH model Financial studies show that stable periods and volatile periods tend to be protracted, resulting in clusters GARCH model can capture these volatility clusters
19 GARCH model Generalized autoregressive conditional heteroskedasticity (GARCH) model was developed by Bollerslev (1986) R σ n 2 n = μ σ ++ = ω n α ε R n 2 n 1 + βσ 2 n 1 ε n ~ NID(0,1)
20 GARCH model GARCH model with historical volatility, realized volatility or implied volatility ~ NID(0,1) ε n = + = n n n n n n n s R R γ βσ α ω σ ε σ μ
21 Flow Chart HSI Data Fit data into GARCH GARCH Fitting Obtain satisfactory results for data fitting Forecasting Obtain forecasted HSI daily volatilities Validation Accuracy of the forecasted HSI daily volatilities are validated by VAR and Binomial Test Application Application on reserving for investment guarantees
22 Estimation Simple GARCH Observations:1235 Standard Approx Parameter DF Estimate Error t Value Pr > t μ ω α β < <.0001
23 Fitting When HSI data are fitted into the GARCH model, empirical results show satisfactory model fitting performance.
24 Flow Chart HSI Data Fit data into GARCH GARCH Fitting Obtain satisfactory results for data fitting Forecasting Obtain forecasted HSI daily volatilities Validation Accuracy of the forecasted HSI daily volatilities are validated by VAR and Binomial Test Application Application on reserving for investment guarantees
25 Forecasting HSI Volatilities After fitting HSI data from 1999 to 2004, one day ahead volatilities are forecasted by the GARCH model.
26 Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 1235 data
27 Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 1 to
28 Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 2 to
29 Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 500 to
30 Forecasting-GARCH Use the method rolling window for forecasting 1 day ahead forecasting 736 to 1235
31 Flow Chart HSI Data Fit data into GARCH GARCH Fitting Obtain satisfactory results for data fitting Forecasting Obtain forecasted HSI daily volatilities Validation Accuracy of the forecasted HSI daily volatilities are validated by VAR and Binomial Test Application Application on reserving for investment guarantees
32 Forecasted Volatility Validation Compare actual volatility with forecasted volatility from day 736 to day 1235? However, actual volatility is not observable!
33 Value at Risk approach Since volatilities are not observable, we use the value at risk approach to validate the accuracy of forecasted volatilities In the GARCH model, return is assumed to be normally distributed with mean and variance 2 σ n μ n
34 Value at Risk approach μ n σ n μn
35 Value at Risk approach Now we have 500 forecasted daily volatilities. If we assume forecasted daily volatility and estimated mean at day n are accurate there should be around 25, i.e. 5% of 500, of them falling into the colored area In other words, if we observe around 25 falling into the colored area, we can conclude that those 500 forecasted daily volatilities and estimated means are accurate.
36 Value at Risk approach The new problem is: are we safe to conclude that forecasted daily volatilities and estimated means are accurate if we observe 24 falling into the colored area? What if 23, 26 or 27 falling into the colored area? What is our tolerance limit?
37 Binomial Test H 0 :forecasted daily volatility and estimated mean are accurate H 1 :forecasted daily volatility and estimated mean are not accurate Rejection Rule: With n=500, p=0.05, np(1-p)=23.75, α =5%, we reject H 0 if test statistic< np 1.96 np(1 p) =15.4 Or test statistic> np np (1 p) =34.6 Conclusion: If the number of returns falling into the colored area is between 16 and 34, we fail to reject H 0 at 5% significance level
38 Binomial Test Results GARCH Simple With historical volatility With realized volatility With implied volatility No. of returns falling into the colored area p-value
39 Validation Result Based on value at risk approach and results of binomial test, it is found that the forecasted volatilities from the GARCH model are accurate.
40 Flow Chart HSI Data Fit data into GARCH GARCH Fitting Obtain satisfactory results for data fitting Forecasting Obtain forecasted HSI daily volatilities Validation Accuracy of the forecasted HSI daily volatilities are validated by VAR and Binomial Test Application Application on reserving for investment guarantees
41 Application of Forecasted Volatilities Reserving for investment guarantees Asset management Option pricing Calculation of VaR for asset portfolios
42 Reserving for investment guarantees For illustrative purposes, the remaining slides demonstrate a very simple simulation method of reserving for investment guarantees Conditional tail expectation (CTE), lapse rates, management expenses, mortality, withdrawal, future contributions, interest rate, etc., are not discussed in the simple simulation method
43 Reserving for investment guarantees Actuaries shall pay due regard to GN7 and AGN8 issued by the OCI and ASHK respectively for reserving principles, modeling process, calibration standard and modeling constraints
44 HSI price at worst case scenario Now we know the daily return follows properties in GARCH model and forecasted volatilities are accurate, we can simulate the most adverse HSI price by using random generator.
45 We now forecast most adverse HSI price 100 days later (1) 100 random numbers between (0,1) are generated by a random generator and considered to be F(x) of returns from day 1 to 100 (2) Mean of return μ n for day 1 to 100 is assumed to be the same and estimated by the GARCH model by fitting historical data into the model (3) Daily volatility for day 1 is forecasted by the GARCH model by fitting historical data into the model
46 We now forecast most adverse HSI price 100 days later (4) As we have the cumulative distribution function, estimated mean and forecasted daily volatility for day 1, we can determine daily return for day 1, hence the HSI price at day 1 (5) Daily volatility for day 2 is forecasted by the GARCH model by fitting historical data and simulated data for day 1 into the model (6) HSI price at day 2 is simulated (7) Similarly, HSI price at day 100 is simulated
47 We now forecast most adverse HSI price 100 days later (8) Repeat step 1 to 7 for 2000 times, we get 2000 simulated HSI price at day 100 (9) The most adverse HSI price at day 100 with 99% level of confidence is the 21 st simulated price in ascending order among those 2000 simulated prices
48 Reserving for Investment Guarantees A simple provision (ignoring mortality, lapse, expense, interest rate, etc.) for investment guarantees wholly invested in HSI is: Reserve = Guaranteed benefit most adverse HSI price at 99% confidence interval
49 Flow Chart HSI Data Fit data into GARCH GARCH Fitting Obtain satisfactory results for data fitting Forecasting Obtain forecasted HSI daily volatilities Validation Accuracy of the forecasted HSI daily volatilities are validated by VAR and Binomial Test Application Application on reserving for investment guarantees
50 Q&A Section
Some Simple Stochastic Models for Analyzing Investment Guarantees p. 1/36
Some Simple Stochastic Models for Analyzing Investment Guarantees Wai-Sum Chan Department of Statistics & Actuarial Science The University of Hong Kong Some Simple Stochastic Models for Analyzing Investment
More informationFinancial Times Series. Lecture 6
Financial Times Series Lecture 6 Extensions of the GARCH There are numerous extensions of the GARCH Among the more well known are EGARCH (Nelson 1991) and GJR (Glosten et al 1993) Both models allow for
More information12. Conditional heteroscedastic models (ARCH) MA6622, Ernesto Mordecki, CityU, HK, 2006.
12. Conditional heteroscedastic models (ARCH) MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Robert F. Engle. Autoregressive Conditional Heteroscedasticity with Estimates of Variance
More informationAssicurazioni Generali: An Option Pricing Case with NAGARCH
Assicurazioni Generali: An Option Pricing Case with NAGARCH Assicurazioni Generali: Business Snapshot Find our latest analyses and trade ideas on bsic.it Assicurazioni Generali SpA is an Italy-based insurance
More informationMarket Risk and Model Risk of Financial Institutions Writing Derivative Warrants: Evidence from Taiwan and Hong Kong
Market Risk and Model Risk of Financial Institutions Writing Derivative Warrants: Evidence from Taiwan and Hong Kong Huimin Chung Department of Finance and Applications Tamkang University, Taipei 106,
More informationAnalytical Finance 1 Seminar Monte-Carlo application for Value-at-Risk on a portfolio of Options, Futures and Equities
Analytical Finance 1 Seminar Monte-Carlo application for Value-at-Risk on a portfolio of Options, Futures and Equities Radhesh Agarwal (Ral13001) Shashank Agarwal (Sal13002) Sumit Jalan (Sjn13024) Calculating
More informationWeek 2 Quantitative Analysis of Financial Markets Hypothesis Testing and Confidence Intervals
Week 2 Quantitative Analysis of Financial Markets Hypothesis Testing and Confidence Intervals Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg :
More information**BEGINNING OF EXAMINATION** A random sample of five observations from a population is:
**BEGINNING OF EXAMINATION** 1. You are given: (i) A random sample of five observations from a population is: 0.2 0.7 0.9 1.1 1.3 (ii) You use the Kolmogorov-Smirnov test for testing the null hypothesis,
More informationCross-Sectional Distribution of GARCH Coefficients across S&P 500 Constituents : Time-Variation over the Period
Cahier de recherche/working Paper 13-13 Cross-Sectional Distribution of GARCH Coefficients across S&P 500 Constituents : Time-Variation over the Period 2000-2012 David Ardia Lennart F. Hoogerheide Mai/May
More informationTests for Two Variances
Chapter 655 Tests for Two Variances Introduction Occasionally, researchers are interested in comparing the variances (or standard deviations) of two groups rather than their means. This module calculates
More informationBinomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
More informationRisk Management and Time Series
IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh Risk Management and Time Series Time series models are often employed in risk management applications. They can be used to estimate
More informationcontinuous rv Note for a legitimate pdf, we have f (x) 0 and f (x)dx = 1. For a continuous rv, P(X = c) = c f (x)dx = 0, hence
continuous rv Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a b, P(a X b) = b a f (x)dx.
More informationStatistical Tables Compiled by Alan J. Terry
Statistical Tables Compiled by Alan J. Terry School of Science and Sport University of the West of Scotland Paisley, Scotland Contents Table 1: Cumulative binomial probabilities Page 1 Table 2: Cumulative
More informationTests for One Variance
Chapter 65 Introduction Occasionally, researchers are interested in the estimation of the variance (or standard deviation) rather than the mean. This module calculates the sample size and performs power
More informationEvaluating the Accuracy of Value at Risk Approaches
Evaluating the Accuracy of Value at Risk Approaches Kyle McAndrews April 25, 2015 1 Introduction Risk management is crucial to the financial industry, and it is particularly relevant today after the turmoil
More informationJohn Hull, Risk Management and Financial Institutions, 4th Edition
P1.T2. Quantitative Analysis John Hull, Risk Management and Financial Institutions, 4th Edition Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Chapter 10: Volatility (Learning objectives)
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationResearch Article The Volatility of the Index of Shanghai Stock Market Research Based on ARCH and Its Extended Forms
Discrete Dynamics in Nature and Society Volume 2009, Article ID 743685, 9 pages doi:10.1155/2009/743685 Research Article The Volatility of the Index of Shanghai Stock Market Research Based on ARCH and
More informationLecture 5a: ARCH Models
Lecture 5a: ARCH Models 1 2 Big Picture 1. We use ARMA model for the conditional mean 2. We use ARCH model for the conditional variance 3. ARMA and ARCH model can be used together to describe both conditional
More informationModelling stock index volatility
Modelling stock index volatility Răduță Mihaela-Camelia * Abstract In this paper I compared seven volatility models in terms of their ability to describe the conditional variance. The models are compared
More informationChapter 9: Sampling Distributions
Chapter 9: Sampling Distributions 9. Introduction This chapter connects the material in Chapters 4 through 8 (numerical descriptive statistics, sampling, and probability distributions, in particular) with
More information. 13. The maximum error (margin of error) of the estimate for μ (based on known σ) is:
Statistics Sample Exam 3 Solution Chapters 6 & 7: Normal Probability Distributions & Estimates 1. What percent of normally distributed data value lie within 2 standard deviations to either side of the
More information1. You are given the following information about a stationary AR(2) model:
Fall 2003 Society of Actuaries **BEGINNING OF EXAMINATION** 1. You are given the following information about a stationary AR(2) model: (i) ρ 1 = 05. (ii) ρ 2 = 01. Determine φ 2. (A) 0.2 (B) 0.1 (C) 0.4
More informationLecture Stat 302 Introduction to Probability - Slides 15
Lecture Stat 30 Introduction to Probability - Slides 15 AD March 010 AD () March 010 1 / 18 Continuous Random Variable Let X a (real-valued) continuous r.v.. It is characterized by its pdf f : R! [0, )
More informationThe Great Moderation Flattens Fat Tails: Disappearing Leptokurtosis
The Great Moderation Flattens Fat Tails: Disappearing Leptokurtosis WenShwo Fang Department of Economics Feng Chia University 100 WenHwa Road, Taichung, TAIWAN Stephen M. Miller* College of Business University
More informationValue at Risk with Stable Distributions
Value at Risk with Stable Distributions Tecnológico de Monterrey, Guadalajara Ramona Serrano B Introduction The core activity of financial institutions is risk management. Calculate capital reserves given
More informationSampling & populations
Sampling & populations Sample proportions Sampling distribution - small populations Sampling distribution - large populations Sampling distribution - normal distribution approximation Mean & variance of
More informationSYSM 6304 Risk and Decision Analysis Lecture 2: Fitting Distributions to Data
SYSM 6304 Risk and Decision Analysis Lecture 2: Fitting Distributions to Data M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 5, 2015
More informationVolatility in the Indian Financial Market Before, During and After the Global Financial Crisis
Volatility in the Indian Financial Market Before, During and After the Global Financial Crisis Praveen Kulshreshtha Indian Institute of Technology Kanpur, India Aakriti Mittal Indian Institute of Technology
More informationFORECASTING PERFORMANCE OF MARKOV-SWITCHING GARCH MODELS: A LARGE-SCALE EMPIRICAL STUDY
FORECASTING PERFORMANCE OF MARKOV-SWITCHING GARCH MODELS: A LARGE-SCALE EMPIRICAL STUDY Latest version available on SSRN https://ssrn.com/abstract=2918413 Keven Bluteau Kris Boudt Leopoldo Catania R/Finance
More informationImpact of Derivatives Expiration on Underlying Securities: Empirical Evidence from India
Impact of Derivatives Expiration on Underlying Securities: Empirical Evidence from India Abstract Priyanka Ostwal Amity University Noindia Priyanka.ostwal@gmail.com Derivative products are perceived to
More informationNo, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
More informationA gentle introduction to the RM 2006 methodology
A gentle introduction to the RM 2006 methodology Gilles Zumbach RiskMetrics Group Av. des Morgines 12 1213 Petit-Lancy Geneva, Switzerland gilles.zumbach@riskmetrics.com Initial version: August 2006 This
More informationProperties of financail time series GARCH(p,q) models Risk premium and ARCH-M models Leverage effects and asymmetric GARCH models.
5 III Properties of financail time series GARCH(p,q) models Risk premium and ARCH-M models Leverage effects and asymmetric GARCH models 1 ARCH: Autoregressive Conditional Heteroscedasticity Conditional
More informationFORECASTING OF VALUE AT RISK BY USING PERCENTILE OF CLUSTER METHOD
FORECASTING OF VALUE AT RISK BY USING PERCENTILE OF CLUSTER METHOD HAE-CHING CHANG * Department of Business Administration, National Cheng Kung University No.1, University Road, Tainan City 701, Taiwan
More informationModelling Joint Distribution of Returns. Dr. Sawsan Hilal space
Modelling Joint Distribution of Returns Dr. Sawsan Hilal space Maths Department - University of Bahrain space October 2011 REWARD Asset Allocation Problem PORTFOLIO w 1 w 2 w 3 ASSET 1 ASSET 2 R 1 R 2
More informationForecasting Stock Index Futures Price Volatility: Linear vs. Nonlinear Models
The Financial Review 37 (2002) 93--104 Forecasting Stock Index Futures Price Volatility: Linear vs. Nonlinear Models Mohammad Najand Old Dominion University Abstract The study examines the relative ability
More informationDownside Risk: Implications for Financial Management Robert Engle NYU Stern School of Business Carlos III, May 24,2004
Downside Risk: Implications for Financial Management Robert Engle NYU Stern School of Business Carlos III, May 24,2004 WHAT IS ARCH? Autoregressive Conditional Heteroskedasticity Predictive (conditional)
More informationFINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE MODULE 2
MSc. Finance/CLEFIN 2017/2018 Edition FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE MODULE 2 Midterm Exam Solutions June 2018 Time Allowed: 1 hour and 15 minutes Please answer all the questions by writing
More informationIntraday Volatility Forecast in Australian Equity Market
20th International Congress on Modelling and Simulation, Adelaide, Australia, 1 6 December 2013 www.mssanz.org.au/modsim2013 Intraday Volatility Forecast in Australian Equity Market Abhay K Singh, David
More informationModelling of Long-Term Risk
Modelling of Long-Term Risk Roger Kaufmann Swiss Life roger.kaufmann@swisslife.ch 15th International AFIR Colloquium 6-9 September 2005, Zurich c 2005 (R. Kaufmann, Swiss Life) Contents A. Basel II B.
More informationIndian Institute of Management Calcutta. Working Paper Series. WPS No. 797 March Implied Volatility and Predictability of GARCH Models
Indian Institute of Management Calcutta Working Paper Series WPS No. 797 March 2017 Implied Volatility and Predictability of GARCH Models Vivek Rajvanshi Assistant Professor, Indian Institute of Management
More informationFinancial Econometrics: A Comparison of GARCH type Model Performances when Forecasting VaR. Bachelor of Science Thesis. Fall 2014
Financial Econometrics: A Comparison of GARCH type Model Performances when Forecasting VaR Bachelor of Science Thesis Fall 2014 Department of Statistics, Uppsala University Oscar Andersson & Erik Haglund
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Basic Concepts and Techniques of Risk Management Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationStatistics Class 15 3/21/2012
Statistics Class 15 3/21/2012 Quiz 1. Cans of regular Pepsi are labeled to indicate that they contain 12 oz. Data Set 17 in Appendix B lists measured amounts for a sample of Pepsi cans. The same statistics
More informationAmath 546/Econ 589 Univariate GARCH Models: Advanced Topics
Amath 546/Econ 589 Univariate GARCH Models: Advanced Topics Eric Zivot April 29, 2013 Lecture Outline The Leverage Effect Asymmetric GARCH Models Forecasts from Asymmetric GARCH Models GARCH Models with
More informationChapter 8 Statistical Intervals for a Single Sample
Chapter 8 Statistical Intervals for a Single Sample Part 1: Confidence intervals (CI) for population mean µ Section 8-1: CI for µ when σ 2 known & drawing from normal distribution Section 8-1.2: Sample
More informationChapter 6 Confidence Intervals
Chapter 6 Confidence Intervals Section 6-1 Confidence Intervals for the Mean (Large Samples) VOCABULARY: Point Estimate A value for a parameter. The most point estimate of the population parameter is the
More informationModelling the stochastic behaviour of short-term interest rates: A survey
Modelling the stochastic behaviour of short-term interest rates: A survey 4 5 6 7 8 9 10 SAMBA/21/04 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Kjersti Aas September 23, 2004 NR Norwegian Computing
More informationVolatility Clustering of Fine Wine Prices assuming Different Distributions
Volatility Clustering of Fine Wine Prices assuming Different Distributions Cynthia Royal Tori, PhD Valdosta State University Langdale College of Business 1500 N. Patterson Street, Valdosta, GA USA 31698
More informationForecasting Value at Risk in the Swedish stock market an investigation of GARCH volatility models
Forecasting Value at Risk in the Swedish stock market an investigation of GARCH volatility models Joel Nilsson Bachelor thesis Supervisor: Lars Forsberg Spring 2015 Abstract The purpose of this thesis
More informationModelling volatility - ARCH and GARCH models
Modelling volatility - ARCH and GARCH models Beáta Stehlíková Time series analysis Modelling volatility- ARCH and GARCH models p.1/33 Stock prices Weekly stock prices (library quantmod) Continuous returns:
More informationFinancial Econometrics
Financial Econometrics Value at Risk Gerald P. Dwyer Trinity College, Dublin January 2016 Outline 1 Value at Risk Introduction VaR RiskMetrics TM Summary Risk What do we mean by risk? Dictionary: possibility
More informationECE 295: Lecture 03 Estimation and Confidence Interval
ECE 295: Lecture 03 Estimation and Confidence Interval Spring 2018 Prof Stanley Chan School of Electrical and Computer Engineering Purdue University 1 / 23 Theme of this Lecture What is Estimation? You
More informationECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10 Fall 2011 Lecture 8 Part 2 (Fall 2011) Probability Distributions Lecture 8 Part 2 1 / 23 Normal Density Function f
More informationFuzzy Volatility Forecasts and Fuzzy Option Values
Class of Volatility Models Fuzzy Volatility Forecasts and Fuzzy Option Values K. Thiagarajah Illinois State University, Normal, Illinois. 41st Actuarial Research Conference Montreal, Canada August 10-12,
More informationValue at Risk Ch.12. PAK Study Manual
Value at Risk Ch.12 Related Learning Objectives 3a) Apply and construct risk metrics to quantify major types of risk exposure such as market risk, credit risk, liquidity risk, regulatory risk etc., and
More informationA Quantile Regression Approach to the Multiple Period Value at Risk Estimation
Journal of Economics and Management, 2016, Vol. 12, No. 1, 1-35 A Quantile Regression Approach to the Multiple Period Value at Risk Estimation Chi Ming Wong School of Mathematical and Physical Sciences,
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
More informationFINA 695 Assignment 1 Simon Foucher
Answer the following questions. Show your work. Due in the class on March 29. (postponed 1 week) You are expected to do the assignment on your own. Please do not take help from others. 1. (a) The current
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationMATH 264 Problem Homework I
MATH Problem Homework I Due to December 9, 00@:0 PROBLEMS & SOLUTIONS. A student answers a multiple-choice examination question that offers four possible answers. Suppose that the probability that the
More informationBasis Risk and Optimal longevity hedging framework for Insurance Company
Basis Risk and Optimal longevity hedging framework for Insurance Company Sharon S. Yang National Central University, Taiwan Hong-Chih Huang National Cheng-Chi University, Taiwan Jin-Kuo Jung Actuarial
More informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationTests for Intraclass Correlation
Chapter 810 Tests for Intraclass Correlation Introduction The intraclass correlation coefficient is often used as an index of reliability in a measurement study. In these studies, there are K observations
More informationINTERTEMPORAL ASSET ALLOCATION: THEORY
INTERTEMPORAL ASSET ALLOCATION: THEORY Multi-Period Model The agent acts as a price-taker in asset markets and then chooses today s consumption and asset shares to maximise lifetime utility. This multi-period
More informationParametric and Semi-parametric models of Value-at-Risk: On the way to bias reduction
Parametric and Semi-parametric models of Value-at-Risk: On the way to bias reduction Yan Liu and Richard Luger Emory University October 24, 2006 Abstract Even if a GARCH model generates unbiased variance
More informationConditional Heteroscedasticity
1 Conditional Heteroscedasticity May 30, 2010 Junhui Qian 1 Introduction ARMA(p,q) models dictate that the conditional mean of a time series depends on past observations of the time series and the past
More informationFinancial Econometrics
Financial Econometrics Volatility Gerald P. Dwyer Trinity College, Dublin January 2013 GPD (TCD) Volatility 01/13 1 / 37 Squared log returns for CRSP daily GPD (TCD) Volatility 01/13 2 / 37 Absolute value
More informationHandbook of Financial Risk Management
Handbook of Financial Risk Management Simulations and Case Studies N.H. Chan H.Y. Wong The Chinese University of Hong Kong WILEY Contents Preface xi 1 An Introduction to Excel VBA 1 1.1 How to Start Excel
More informationChapter 1. Introduction
Chapter 1 Introduction 2 Oil Price Uncertainty As noted in the Preface, the relationship between the price of oil and the level of economic activity is a fundamental empirical issue in macroeconomics.
More informationA Study of Stock Return Distributions of Leading Indian Bank s
Global Journal of Management and Business Studies. ISSN 2248-9878 Volume 3, Number 3 (2013), pp. 271-276 Research India Publications http://www.ripublication.com/gjmbs.htm A Study of Stock Return Distributions
More informationLONG MEMORY IN VOLATILITY
LONG MEMORY IN VOLATILITY How persistent is volatility? In other words, how quickly do financial markets forget large volatility shocks? Figure 1.1, Shephard (attached) shows that daily squared returns
More informationNon-Inferiority Tests for the Ratio of Two Means
Chapter 455 Non-Inferiority Tests for the Ratio of Two Means Introduction This procedure calculates power and sample size for non-inferiority t-tests from a parallel-groups design in which the logarithm
More informationLESSON 7 INTERVAL ESTIMATION SAMIE L.S. LY
LESSON 7 INTERVAL ESTIMATION SAMIE L.S. LY 1 THIS WEEK S PLAN Part I: Theory + Practice ( Interval Estimation ) Part II: Theory + Practice ( Interval Estimation ) z-based Confidence Intervals for a Population
More informationINSTITUTE AND FACULTY OF ACTUARIES. Curriculum 2019 SPECIMEN EXAMINATION
INSTITUTE AND FACULTY OF ACTUARIES Curriculum 2019 SPECIMEN EXAMINATION Subject CS1A Actuarial Statistics Time allowed: Three hours and fifteen minutes INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate
More informationTime-Simultaneous Fan Charts: Applications to Stochastic Life Table Forecasting
19th International Congress on Modelling and Simulation, Perth, Australia, 12 16 December 211 http://mssanz.org.au/modsim211 Time-Simultaneous Fan Charts: Applications to Stochastic Life Table Forecasting
More informationSTAT Chapter 7: Confidence Intervals
STAT 515 -- Chapter 7: Confidence Intervals With a point estimate, we used a single number to estimate a parameter. We can also use a set of numbers to serve as reasonable estimates for the parameter.
More informationImplied Volatility v/s Realized Volatility: A Forecasting Dimension
4 Implied Volatility v/s Realized Volatility: A Forecasting Dimension 4.1 Introduction Modelling and predicting financial market volatility has played an important role for market participants as it enables
More informationFinancial Econometrics Jeffrey R. Russell. Midterm 2014 Suggested Solutions. TA: B. B. Deng
Financial Econometrics Jeffrey R. Russell Midterm 2014 Suggested Solutions TA: B. B. Deng Unless otherwise stated, e t is iid N(0,s 2 ) 1. (12 points) Consider the three series y1, y2, y3, and y4. Match
More informationInterval estimation. September 29, Outline Basic ideas Sampling variation and CLT Interval estimation using X More general problems
Interval estimation September 29, 2017 STAT 151 Class 7 Slide 1 Outline of Topics 1 Basic ideas 2 Sampling variation and CLT 3 Interval estimation using X 4 More general problems STAT 151 Class 7 Slide
More informationTRΛNSPΛRΣNCY ΛNΛLYTICS
TRΛNSPΛRΣNCY ΛNΛLYTICS RISK-AI, LLC PRESENTATION INTRODUCTION I. Transparency Analytics is a state-of-the-art risk management analysis and research platform for Investment Advisors, Funds of Funds, Family
More informationAlexander Marianski August IFRS 9: Probably Weighted and Biased?
Alexander Marianski August 2017 IFRS 9: Probably Weighted and Biased? Introductions Alexander Marianski Associate Director amarianski@deloitte.co.uk Alexandra Savelyeva Assistant Manager asavelyeva@deloitte.co.uk
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (42 pts) Answer briefly the following questions. 1. Questions
More informationTwo Hours. Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER. 22 January :00 16:00
Two Hours MATH38191 Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER STATISTICAL MODELLING IN FINANCE 22 January 2015 14:00 16:00 Answer ALL TWO questions
More informationANALYZING VALUE AT RISK AND EXPECTED SHORTFALL METHODS: THE USE OF PARAMETRIC, NON-PARAMETRIC, AND SEMI-PARAMETRIC MODELS
ANALYZING VALUE AT RISK AND EXPECTED SHORTFALL METHODS: THE USE OF PARAMETRIC, NON-PARAMETRIC, AND SEMI-PARAMETRIC MODELS by Xinxin Huang A Thesis Submitted to the Faculty of Graduate Studies The University
More informationA Fuzzy Pay-Off Method for Real Option Valuation
A Fuzzy Pay-Off Method for Real Option Valuation April 2, 2009 1 Introduction Real options Black-Scholes formula 2 Fuzzy Sets and Fuzzy Numbers 3 The method Datar-Mathews method Calculating the ROV with
More informationRetirement, Saving, Benefit Claiming and Solvency Under A Partial System of Voluntary Personal Accounts
Retirement, Saving, Benefit Claiming and Solvency Under A Partial System of Voluntary Personal Accounts Alan Gustman Thomas Steinmeier This study was supported by grants from the U.S. Social Security Administration
More informationHomework Problems Stat 479
Chapter 2 1. Model 1 is a uniform distribution from 0 to 100. Determine the table entries for a generalized uniform distribution covering the range from a to b where a < b. 2. Let X be a discrete random
More informationCourse information FN3142 Quantitative finance
Course information 015 16 FN314 Quantitative finance This course is aimed at students interested in obtaining a thorough grounding in market finance and related empirical methods. Prerequisite If taken
More informationInternet Appendix for Asymmetry in Stock Comovements: An Entropy Approach
Internet Appendix for Asymmetry in Stock Comovements: An Entropy Approach Lei Jiang Tsinghua University Ke Wu Renmin University of China Guofu Zhou Washington University in St. Louis August 2017 Jiang,
More informationTutorial 6. Sampling Distribution. ENGG2450A Tutors. 27 February The Chinese University of Hong Kong 1/6
Tutorial 6 Sampling Distribution ENGG2450A Tutors The Chinese University of Hong Kong 27 February 2017 1/6 Random Sample and Sampling Distribution 2/6 Random sample Consider a random variable X with distribution
More informationFinancial Times Series. Lecture 8
Financial Times Series Lecture 8 Nobel Prize Robert Engle got the Nobel Prize in Economics in 2003 for the ARCH model which he introduced in 1982 It turns out that in many applications there will be many
More informationCHAPTER 8. Confidence Interval Estimation Point and Interval Estimates
CHAPTER 8. Confidence Interval Estimation Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about the variability of the estimate Lower
More informationSection 7.2. Estimating a Population Proportion
Section 7.2 Estimating a Population Proportion Overview Section 7.2 Estimating a Population Proportion Section 7.3 Estimating a Population Mean Section 7.4 Estimating a Population Standard Deviation or
More informationOil Price Effects on Exchange Rate and Price Level: The Case of South Korea
Oil Price Effects on Exchange Rate and Price Level: The Case of South Korea Mirzosaid SULTONOV 東北公益文科大学総合研究論集第 34 号抜刷 2018 年 7 月 30 日発行 研究論文 Oil Price Effects on Exchange Rate and Price Level: The Case
More informationFINITE SAMPLE DISTRIBUTIONS OF RISK-RETURN RATIOS
Available Online at ESci Journals Journal of Business and Finance ISSN: 305-185 (Online), 308-7714 (Print) http://www.escijournals.net/jbf FINITE SAMPLE DISTRIBUTIONS OF RISK-RETURN RATIOS Reza Habibi*
More information