Lecture 1 of 4-part series. Spring School on Risk Management, Insurance and Finance European University at St. Petersburg, Russia.
|
|
- Alison Hunt
- 6 years ago
- Views:
Transcription
1 Principles and Lecture 1 of 4-part series Spring School on Risk, Insurance and Finance European University at St. Petersburg, Russia 2-4 April 2012 s University of Connecticut, USA page 1
2 s Outline s 5 page 2
3 s The allocation of capital is the term typically referring to the subdivision of a company s aggregate capital across its various constituents: lines of business its subsidiaries product types within lines of business territories, e.g. distribution channels types of risks: e.g. market, credit, pricing/underwriting, operational Company is typically involved in the financial services industry e.g. banks, insurance companies. A very important component of : identifying, measuring, pricing and controlling risks page 3
4 s The purpose of capital Knowing how much capital you need for your overall business is a key aspect of ERM. is the amount set aside, usually in excess of assets backing all liabilities, so that the firm: could withstand and absorb unexpected losses from all risks it is facing; would remain solvent with high probability; and is able to cover obligations to its customers as promised. Economic capital vs regulatory capital: Economic capital is usually calculated based on true market value (or economic) terms. Regulatory capital is usually calculated on the basis of prescribed guidelines by regulatory authorities. page 4
5 for capital computations We will assume that how we measure capital is known and given. s Requires understanding all aspects of risks (or losses) the company is facing. modeling the distribution of losses understanding expectation and variation of these losses understanding possible inter-dependencies of these losses Some well known risk measures may be used: Value-at-Risk or percentile or VaR Conditional tail expectation or Tail-VaR If X is the random loss, then ρ[x] is some risk measure. page 5
6 s - quick review A risk measure is a mapping ρ from a set Γ of real-valued random variables defined on (Ω, F, P) to R: ρ : Γ R : X Γ ρ[x]. Let X, X 1, X 2 Γ. Some well known properties that risk measures may or may not satisfy: Law invariance: If P[X 1 x] = P[X 2 x] for all x R, ρ[x 1 ] = ρ[x 2 ]. Monotonicity: X 1 X 2 implies ρ[x 1 ] ρ[x 2 ]. Positive homogeneity: For any a > 0, ρ[ax] = aρ[x]. Translation invariance: For b R, ρ[x + b] = ρ[x] + b. Subadditivity: ρ[x 1 + X 2 ] ρ[x 1 ] + ρ[x 2 ]. page 6
7 s Some important concepts Conditional Tail Expectation (CTE): (sometimes called TailVaR) CTE p [X] = E [ X X > F 1 X (p)], p (0, 1). In general, not subadditive, but it is so for continuous random variables. Comonotonic sum: S c = n (0, 1). The Fréchet bounds: where i=1 F 1 X i (U) where U is uniform on L F (u 1,..., u n ) C(u 1,..., u n ) U F (u 1,..., u n ), Fréchet lower bound: L F = max ( n i=1 u i (n 1), 0 ), and Fréchet upper bound: U F = min(u 1,..., u n ). page 7
8 Some special distributions Distribution density f X (x) Quantile Q p [X] CTE p [X] s Normal 1 2πσ e 1 2 Gamma ( x µ σ β α Γ(α) x α 1 e βx ) 2 µ + Φ 1 (p)σ µ + φ(φ 1 (p)) p σ no explicit form ) F X (x p;α+1,β F X (x p;α,β) α β ( 1 Lognormal 2πσx e 1 log(x) µ ) 2 ( ) 2 σ e µ+φ 1 (p)σ e µ+σ2 /2 Φ σ Φ 1 (p) 1 p Pareto ab a (x+b) a+1 b [ (1 p) 1/a 1 ] a a 1 Q p[x] + b a 1 page 8
9 s Illustrative case study For purposes of showing illustrations, we will consider an insurance company with five lines of business: auto insurance - property damage auto insurance - liability household or homeowners insurance professional liability other lines of business We will measure loss on a per premium basis and denote the random variable by S for the entire company and X i for the i-th line of business, i = 1, 2, 3, 4, 5. are described in the subsequent slides. page 9
10 s Line of Loss Premium business distribution share Parameters Mean Variance Auto (PD) Gamma 30% α = 360, β = Auto (liab) Lognormal 20% µ = 0.362, σ = Household Gamma 15% α = 56.25, β = Prof liab Pareto 15% a = 6.92, b = Other Lognormal 20% µ = 0.784, σ = auto (PD) auto (liab) household prof liab other auto (PD) 1.00 auto (liab) household prof liab other correlation between lines of business page 10
11 s Graph of densities - by lines of business density fxi(x) auto (PD) auto (liab) household prof liab other x page 11
12 Distribution of the aggregate loss Distribution of aggregate loss s Density loss per premium mean SD median min max VaR 0.95 [S] CTE 0.95 [S] page 12
13 Stand-alone capitals s Line of business VaR 0.95 [X i ] CTE 0.95 [X i ] Auto (PD) Auto (liab) Household Prof liab Other page 13
14 Insurance company with multiple lines of business auto property damage K 1 s insurance company K auto liability K 2 household K 3 professional liability K 4 other lines K 5 page 14
15 Proportional capital allocation s Many well-known allocation formulas fall into a class of proportional allocations. Members of this class are obtained by first choosing a risk measure ρ and then attributing the capital K i = γ i ρ [X i ] to each business unit i, i = 1,..., n. The factor γ i is chosen such that the full allocation requirement is satisfied. This gives rise to the proportional allocation principle: K i = K n j=1 ρ[x j] ρ[x i], i = 1,..., n. page 15
16 s Covariance capital allocation Because of its popularity, we also consider here for purposes of early illustrations this allocation using covariance. The covariance is based on the fact that when we have an aggregate loss that is a weighted sum such as then it is easy to see that S = n c j X j, j=1 [ n ] Var[S] = Cov c j X j, S = j=1 n c j Cov[X j, S] In some sense, this is a special case of the proportional allocation formula with the factor γ i chosen that gives rise to the covariance allocation principle: j=1 K i = c icov[x i, S] K, i = 1,..., n. Var[S] page 16
17 Proportional and covariance allocation results s proportional allocation covariance allocation based on based on Line of business VaR CTE VaR CTE Auto (PD) Auto (liab) Household Prof liab Other Total page 17
18 Results of covariance vs proportional allocations other s prof liab household auto (liab) auto (PD) covariance proportional page 18
19 s Dhaene, J., Tsanakas, A., Valdez, E.A., and S. Vanduffel (2012). Optimal capital allocation principles. Journal of Risk and Insurance, 79(1), Sweeting, P. (2011). Financial, International Series on Actuarial Science, Cambridge University Press. Tang, A. and E.A. Valdez (2006). Economic capital and the aggregation of risks using copulas. Proceedings of the 28th International Congress of Actuaries, Paris, France. page 19
Lecture 3 of 4-part series. Spring School on Risk Management, Insurance and Finance European University at St. Petersburg, Russia.
Principles and Lecture 3 of 4-part series Spring School on Risk, Insurance and Finance European University at St. Petersburg, Russia 2-4 April 2012 University of Connecticut, USA page 1 Outline 1 2 3 4
More informationLecture 4 of 4-part series. Spring School on Risk Management, Insurance and Finance European University at St. Petersburg, Russia.
Principles and Lecture 4 of 4-part series Spring School on Risk, Insurance and Finance European University at St. Petersburg, Russia 2-4 April 2012 University of Connecticut, USA page 1 Outline 1 2 3 4
More informationCapital Allocation Principles
Capital Allocation Principles Maochao Xu Department of Mathematics Illinois State University mxu2@ilstu.edu Capital Dhaene, et al., 2011, Journal of Risk and Insurance The level of the capital held by
More informationMathematics in Finance
Mathematics in Finance Steven E. Shreve Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA shreve@andrew.cmu.edu A Talk in the Series Probability in Science and Industry
More informationSOLVENCY AND CAPITAL ALLOCATION
SOLVENCY AND CAPITAL ALLOCATION HARRY PANJER University of Waterloo JIA JING Tianjin University of Economics and Finance Abstract This paper discusses a new criterion for allocation of required capital.
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Risk Measures Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com Reference: Chapter 8
More informationOptimal capital allocation principles
MPRA Munich Personal RePEc Archive Optimal capital allocation principles Jan Dhaene and Andreas Tsanakas and Valdez Emiliano and Vanduffel Steven University of Connecticut 23. January 2009 Online at http://mpra.ub.uni-muenchen.de/13574/
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More informationAggregating Economic Capital
Aggregating Economic Capital J. Dhaene 1 M. Goovaerts 1 M. Lundin 2. Vanduffel 1,2 1 Katholieke Universiteit Leuven & Universiteit van Amsterdam 2 Fortis Central Risk Management eptember 12, 2005 Abstract
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 information2 Modeling Credit Risk
2 Modeling Credit Risk In this chapter we present some simple approaches to measure credit risk. We start in Section 2.1 with a short overview of the standardized approach of the Basel framework for banking
More informationMeasures of Contribution for Portfolio Risk
X Workshop on Quantitative Finance Milan, January 29-30, 2009 Agenda Coherent Measures of Risk Spectral Measures of Risk Capital Allocation Euler Principle Application Risk Measurement Risk Attribution
More informationMultivariate longitudinal data analysis for actuarial applications
Multivariate longitudinal data analysis for actuarial applications Priyantha Kumara and Emiliano A. Valdez astin/afir/iaals Mexico Colloquia 2012 Mexico City, Mexico, 1-4 October 2012 P. Kumara and E.A.
More informationNormal Distribution. Notes. Normal Distribution. Standard Normal. Sums of Normal Random Variables. Normal. approximation of Binomial.
Lecture 21,22, 23 Text: A Course in Probability by Weiss 8.5 STAT 225 Introduction to Probability Models March 31, 2014 Standard Sums of Whitney Huang Purdue University 21,22, 23.1 Agenda 1 2 Standard
More informationLecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
More informationRisk Aggregation with Dependence Uncertainty
Risk Aggregation with Dependence Uncertainty Carole Bernard (Grenoble Ecole de Management) Hannover, Current challenges in Actuarial Mathematics November 2015 Carole Bernard Risk Aggregation with Dependence
More informationP VaR0.01 (X) > 2 VaR 0.01 (X). (10 p) Problem 4
KTH Mathematics Examination in SF2980 Risk Management, December 13, 2012, 8:00 13:00. Examiner : Filip indskog, tel. 790 7217, e-mail: lindskog@kth.se Allowed technical aids and literature : a calculator,
More informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
More informationWhat was in the last lecture?
What was in the last lecture? Normal distribution A continuous rv with bell-shaped density curve The pdf is given by f(x) = 1 2πσ e (x µ)2 2σ 2, < x < If X N(µ, σ 2 ), E(X) = µ and V (X) = σ 2 Standard
More informationCapital allocation: a guided tour
Capital allocation: a guided tour Andreas Tsanakas Cass Business School, City University London K. U. Leuven, 21 November 2013 2 Motivation What does it mean to allocate capital? A notional exercise Is
More informationAn application of capital allocation principles to operational risk
MPRA Munich Personal RePEc Archive An application of capital allocation principles to operational risk Jilber Urbina and Montserrat Guillén Department of Economics and CREIP, Universitat Rovira i Virgili,
More informationCAT Pricing: Making Sense of the Alternatives Ira Robbin. CAS RPM March page 1. CAS Antitrust Notice. Disclaimers
CAS Ratemaking and Product Management Seminar - March 2013 CP-2. Catastrophe Pricing : Making Sense of the Alternatives, PhD CAS Antitrust Notice 2 The Casualty Actuarial Society is committed to adhering
More informationFinancial Risk Management
Financial Risk Management Professor: Thierry Roncalli Evry University Assistant: Enareta Kurtbegu Evry University Tutorial exercices #4 1 Correlation and copulas 1. The bivariate Gaussian copula is given
More informationContinuous Distributions
Quantitative Methods 2013 Continuous Distributions 1 The most important probability distribution in statistics is the normal distribution. Carl Friedrich Gauss (1777 1855) Normal curve A normal distribution
More informationThe Normal Distribution
The Normal Distribution The normal distribution plays a central role in probability theory and in statistics. It is often used as a model for the distribution of continuous random variables. Like all models,
More informationAll Investors are Risk-averse Expected Utility Maximizers. Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel)
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) First Name: Waterloo, April 2013. Last Name: UW ID #:
More informationThe Statistical Mechanics of Financial Markets
The Statistical Mechanics of Financial Markets Johannes Voit 2011 johannes.voit (at) ekit.com Overview 1. Why statistical physicists care about financial markets 2. The standard model - its achievements
More informationRisk Aggregation with Dependence Uncertainty
Risk Aggregation with Dependence Uncertainty Carole Bernard GEM and VUB Risk: Modelling, Optimization and Inference with Applications in Finance, Insurance and Superannuation Sydney December 7-8, 2017
More informationThe Use of the Tukey s g h family of distributions to Calculate Value at Risk and Conditional Value at Risk
The Use of the Tukey s g h family of distributions to Calculate Value at Risk and Conditional Value at Risk José Alfredo Jiménez and Viswanathan Arunachalam Journal of Risk, vol. 13, No. 4, summer, 2011
More informationMTH6154 Financial Mathematics I Stochastic Interest Rates
MTH6154 Financial Mathematics I Stochastic Interest Rates Contents 4 Stochastic Interest Rates 45 4.1 Fixed Interest Rate Model............................ 45 4.2 Varying Interest Rate Model...........................
More informationComparing approximations for risk measures of sums of non-independent lognormal random variables
Comparing approximations for risk measures of sums of non-independent lognormal rom variables Steven Vuffel Tom Hoedemakers Jan Dhaene Abstract In this paper, we consider different approximations for computing
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 informationAdvanced Tools for Risk Management and Asset Pricing
MSc. Finance/CLEFIN 2014/2015 Edition Advanced Tools for Risk Management and Asset Pricing June 2015 Exam for Non-Attending Students Solutions Time Allowed: 120 minutes Family Name (Surname) First Name
More informationIntroduction to Computational Finance and Financial Econometrics Descriptive Statistics
You can t see this text! Introduction to Computational Finance and Financial Econometrics Descriptive Statistics Eric Zivot Summer 2015 Eric Zivot (Copyright 2015) Descriptive Statistics 1 / 28 Outline
More informationImplied Systemic Risk Index (work in progress, still at an early stage)
Implied Systemic Risk Index (work in progress, still at an early stage) Carole Bernard, joint work with O. Bondarenko and S. Vanduffel IPAM, March 23-27, 2015: Workshop I: Systemic risk and financial networks
More informationStatistical and Computational Inverse Problems with Applications Part 5B: Electrical impedance tomography
Statistical and Computational Inverse Problems with Applications Part 5B: Electrical impedance tomography Aku Seppänen Inverse Problems Group Department of Applied Physics University of Eastern Finland
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 informationERM (Part 1) Measurement and Modeling of Depedencies in Economic Capital. PAK Study Manual
ERM-101-12 (Part 1) Measurement and Modeling of Depedencies in Economic Capital Related Learning Objectives 2b) Evaluate how risks are correlated, and give examples of risks that are positively correlated
More informationSection B: Risk Measures. Value-at-Risk, Jorion
Section B: Risk Measures Value-at-Risk, Jorion One thing to always keep in mind when reading this text is that it is focused on the banking industry. It mainly focuses on market and credit risk. It also
More informationOptimal reinsurance strategies
Optimal reinsurance strategies Maria de Lourdes Centeno CEMAPRE and ISEG, Universidade de Lisboa July 2016 The author is partially supported by the project CEMAPRE MULTI/00491 financed by FCT/MEC through
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 informationPricing and risk of financial products
and risk of financial products Prof. Dr. Christian Weiß Riga, 27.02.2018 Observations AAA bonds are typically regarded as risk-free investment. Only examples: Government bonds of Australia, Canada, Denmark,
More informationAsset Allocation Model with Tail Risk Parity
Proceedings of the Asia Pacific Industrial Engineering & Management Systems Conference 2017 Asset Allocation Model with Tail Risk Parity Hirotaka Kato Graduate School of Science and Technology Keio University,
More informationBounds for Stop-Loss Premiums of Life Annuities with Random Interest Rates
Bounds for Stop-Loss Premiums of Life Annuities with Random Interest Rates Tom Hoedemakers (K.U.Leuven) Grzegorz Darkiewicz (K.U.Leuven) Griselda Deelstra (ULB) Jan Dhaene (K.U.Leuven) Michèle Vanmaele
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 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 informationAsymptotic methods in risk management. Advances in Financial Mathematics
Asymptotic methods in risk management Peter Tankov Based on joint work with A. Gulisashvili Advances in Financial Mathematics Paris, January 7 10, 2014 Peter Tankov (Université Paris Diderot) Asymptotic
More informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationINDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY. Lecture -5 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.
INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY Lecture -5 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture Moments of a distribubon Measures of
More informationINSTITUTE AND FACULTY OF ACTUARIES SUMMARY
INSTITUTE AND FACULTY OF ACTUARIES SUMMARY Specimen 2019 CP2: Actuarial Modelling Paper 2 Institute and Faculty of Actuaries TQIC Reinsurance Renewal Objective The objective of this project is to use random
More informationRisk based capital allocation
Proceedings of FIKUSZ 10 Symposium for Young Researchers, 2010, 17-26 The Author(s). Conference Proceedings compilation Obuda University Keleti Faculty of Business and Management 2010. Published by Óbuda
More informationOperational Risk Aggregation
Operational Risk Aggregation Professor Carol Alexander Chair of Risk Management and Director of Research, ISMA Centre, University of Reading, UK. Loss model approaches are currently a focus of operational
More informationLecture 22. Survey Sampling: an Overview
Math 408 - Mathematical Statistics Lecture 22. Survey Sampling: an Overview March 25, 2013 Konstantin Zuev (USC) Math 408, Lecture 22 March 25, 2013 1 / 16 Survey Sampling: What and Why In surveys sampling
More informationGeneral Notation. Return and Risk: The Capital Asset Pricing Model
Return and Risk: The Capital Asset Pricing Model (Text reference: Chapter 10) Topics general notation single security statistics covariance and correlation return and risk for a portfolio diversification
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 informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationA class of coherent risk measures based on one-sided moments
A class of coherent risk measures based on one-sided moments T. Fischer Darmstadt University of Technology November 11, 2003 Abstract This brief paper explains how to obtain upper boundaries of shortfall
More informationProbability Weighted Moments. Andrew Smith
Probability Weighted Moments Andrew Smith andrewdsmith8@deloitte.co.uk 28 November 2014 Introduction If I asked you to summarise a data set, or fit a distribution You d probably calculate the mean and
More informationRisk measures: Yet another search of a holy grail
Risk measures: Yet another search of a holy grail Dirk Tasche Financial Services Authority 1 dirk.tasche@gmx.net Mathematics of Financial Risk Management Isaac Newton Institute for Mathematical Sciences
More informationLecture notes on risk management, public policy, and the financial system. Credit portfolios. Allan M. Malz. Columbia University
Lecture notes on risk management, public policy, and the financial system Allan M. Malz Columbia University 2018 Allan M. Malz Last updated: June 8, 2018 2 / 23 Outline Overview of credit portfolio risk
More informationTests for Two ROC Curves
Chapter 65 Tests for Two ROC Curves Introduction Receiver operating characteristic (ROC) curves are used to summarize the accuracy of diagnostic tests. The technique is used when a criterion variable is
More informationMean-Variance Optimal Portfolios in the Presence of a Benchmark with Applications to Fraud Detection
Mean-Variance Optimal Portfolios in the Presence of a Benchmark with Applications to Fraud Detection Carole Bernard (University of Waterloo) Steven Vanduffel (Vrije Universiteit Brussel) Fields Institute,
More informationConditional Value-at-Risk, Spectral Risk Measures and (Non-)Diversification in Portfolio Selection Problems A Comparison with Mean-Variance Analysis
Conditional Value-at-Risk, Spectral Risk Measures and (Non-)Diversification in Portfolio Selection Problems A Comparison with Mean-Variance Analysis Mario Brandtner Friedrich Schiller University of Jena,
More informationChapter 3 Common Families of Distributions. Definition 3.4.1: A family of pmfs or pdfs is called exponential family if it can be expressed as
Lecture 0 on BST 63: Statistical Theory I Kui Zhang, 09/9/008 Review for the previous lecture Definition: Several continuous distributions, including uniform, gamma, normal, Beta, Cauchy, double exponential
More informationA Multivariate Analysis of Intercompany Loss Triangles
A Multivariate Analysis of Intercompany Loss Triangles Peng Shi School of Business University of Wisconsin-Madison ASTIN Colloquium May 21-24, 2013 Peng Shi (Wisconsin School of Business) Intercompany
More informationAll Investors are Risk-averse Expected Utility Maximizers
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) AFFI, Lyon, May 2013. Carole Bernard All Investors are
More informationPractical methods of modelling operational risk
Practical methods of modelling operational risk Andries Groenewald The final frontier for actuaries? Agenda 1. Why model operational risk? 2. Data. 3. Methods available for modelling operational risk.
More informationStochastic Loss Reserving with Bayesian MCMC Models Revised March 31
w w w. I C A 2 0 1 4. o r g Stochastic Loss Reserving with Bayesian MCMC Models Revised March 31 Glenn Meyers FCAS, MAAA, CERA, Ph.D. April 2, 2014 The CAS Loss Reserve Database Created by Meyers and Shi
More informationFinancial Risk Forecasting Chapter 4 Risk Measures
Financial Risk Forecasting Chapter 4 Risk Measures Jon Danielsson 2017 London School of Economics To accompany Financial Risk Forecasting www.financialriskforecasting.com Published by Wiley 2011 Version
More informationNormal Distribution. Definition A continuous rv X is said to have a normal distribution with. the pdf of X is
Normal Distribution Normal Distribution Definition A continuous rv X is said to have a normal distribution with parameter µ and σ (µ and σ 2 ), where < µ < and σ > 0, if the pdf of X is f (x; µ, σ) = 1
More informationDependence Modeling and Credit Risk
Dependence Modeling and Credit Risk Paola Mosconi Banca IMI Bocconi University, 20/04/2015 Paola Mosconi Lecture 6 1 / 53 Disclaimer The opinion expressed here are solely those of the author and do not
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 informationPareto-optimal reinsurance arrangements under general model settings
Pareto-optimal reinsurance arrangements under general model settings Jun Cai, Haiyan Liu, and Ruodu Wang Abstract In this paper, we study Pareto optimality of reinsurance arrangements under general model
More informationChapter 7: Portfolio Theory
Chapter 7: Portfolio Theory 1. Introduction 2. Portfolio Basics 3. The Feasible Set 4. Portfolio Selection Rules 5. The Efficient Frontier 6. Indifference Curves 7. The Two-Asset Portfolio 8. Unrestriceted
More informationModeling Partial Greeks of Variable Annuities with Dependence
Modeling Partial Greeks of Variable Annuities with Dependence Emiliano A. Valdez joint work with Guojun Gan University of Connecticut Recent Developments in Dependence Modeling with Applications in Finance
More informationLecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall Financial mathematics
Lecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall 2014 Reduce the risk, one asset Let us warm up by doing an exercise. We consider an investment with σ 1 =
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 informationThe Vasicek Distribution
The Vasicek Distribution Dirk Tasche Lloyds TSB Bank Corporate Markets Rating Systems dirk.tasche@gmx.net Bristol / London, August 2008 The opinions expressed in this presentation are those of the author
More informationSOLUTIONS 913,
Illinois State University, Mathematics 483, Fall 2014 Test No. 3, Tuesday, December 2, 2014 SOLUTIONS 1. Spring 2013 Casualty Actuarial Society Course 9 Examination, Problem No. 7 Given the following information
More informationBivariate Birnbaum-Saunders Distribution
Department of Mathematics & Statistics Indian Institute of Technology Kanpur January 2nd. 2013 Outline 1 Collaborators 2 3 Birnbaum-Saunders Distribution: Introduction & Properties 4 5 Outline 1 Collaborators
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 informationMean-Variance Portfolio Theory
Mean-Variance Portfolio Theory Lakehead University Winter 2005 Outline Measures of Location Risk of a Single Asset Risk and Return of Financial Securities Risk of a Portfolio The Capital Asset Pricing
More informationBudget Setting Strategies for the Company s Divisions
Budget Setting Strategies for the Company s Divisions Menachem Berg Ruud Brekelmans Anja De Waegenaere November 14, 1997 Abstract The paper deals with the issue of budget setting to the divisions of a
More informationOperational Risk Aggregation
Operational Risk Aggregation Professor Carol Alexander Chair of Risk Management and Director of Research, ISMA Centre, University of Reading, UK. Loss model approaches are currently a focus of operational
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 informationSOCIETY OF ACTUARIES EXAM STAM SHORT-TERM ACTUARIAL MATHEMATICS EXAM STAM SAMPLE QUESTIONS
SOCIETY OF ACTUARIES EXAM STAM SHORT-TERM ACTUARIAL MATHEMATICS EXAM STAM SAMPLE QUESTIONS Questions 1-307 have been taken from the previous set of Exam C sample questions. Questions no longer relevant
More informationBasic Data Analysis. Stephen Turnbull Business Administration and Public Policy Lecture 4: May 2, Abstract
Basic Data Analysis Stephen Turnbull Business Administration and Public Policy Lecture 4: May 2, 2013 Abstract Introduct the normal distribution. Introduce basic notions of uncertainty, probability, events,
More informationA Comparison Between Skew-logistic and Skew-normal Distributions
MATEMATIKA, 2015, Volume 31, Number 1, 15 24 c UTM Centre for Industrial and Applied Mathematics A Comparison Between Skew-logistic and Skew-normal Distributions 1 Ramin Kazemi and 2 Monireh Noorizadeh
More informationPORTFOLIO THEORY. Master in Finance INVESTMENTS. Szabolcs Sebestyén
PORTFOLIO THEORY Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Portfolio Theory Investments 1 / 60 Outline 1 Modern Portfolio Theory Introduction Mean-Variance
More informationEconomic capital allocation derived from risk measures
Economic capital allocation derived from risk measures M.J. Goovaerts R. Kaas J. Dhaene June 4, 2002 Abstract We examine properties of risk measures that can be considered to be in line with some best
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 informationEVA Tutorial #1 BLOCK MAXIMA APPROACH IN HYDROLOGIC/CLIMATE APPLICATIONS. Rick Katz
1 EVA Tutorial #1 BLOCK MAXIMA APPROACH IN HYDROLOGIC/CLIMATE APPLICATIONS Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO USA email: rwk@ucar.edu
More informationExam M Fall 2005 PRELIMINARY ANSWER KEY
Exam M Fall 005 PRELIMINARY ANSWER KEY Question # Answer Question # Answer 1 C 1 E C B 3 C 3 E 4 D 4 E 5 C 5 C 6 B 6 E 7 A 7 E 8 D 8 D 9 B 9 A 10 A 30 D 11 A 31 A 1 A 3 A 13 D 33 B 14 C 34 C 15 A 35 A
More informationLecture 6: Non Normal Distributions
Lecture 6: Non Normal Distributions and their Uses in GARCH Modelling Prof. Massimo Guidolin 20192 Financial Econometrics Spring 2015 Overview Non-normalities in (standardized) residuals from asset return
More informationThe mean-risk portfolio optimization model
The mean-risk portfolio optimization model The mean-risk portfolio optimization model Consider a portfolio of d risky assets and the random vector X = (X 1,X 2,...,X d ) T of their returns. Let E(X) =
More informationConfidence Intervals for the Difference Between Two Means with Tolerance Probability
Chapter 47 Confidence Intervals for the Difference Between Two Means with Tolerance Probability Introduction This procedure calculates the sample size necessary to achieve a specified distance from the
More informationOvernight Index Rate: Model, calibration and simulation
Research Article Overnight Index Rate: Model, calibration and simulation Olga Yashkir and Yuri Yashkir Cogent Economics & Finance (2014), 2: 936955 Page 1 of 11 Research Article Overnight Index Rate: Model,
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2011, Mr. Ruey S. Tsay. Solutions to Final Exam.
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2011, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (32 pts) Answer briefly the following questions. 1. Suppose
More informationQuantitative Risk Management
Quantitative Risk Management Asset Allocation and Risk Management Martin B. Haugh Department of Industrial Engineering and Operations Research Columbia University Outline Review of Mean-Variance Analysis
More informationReverse Sensitivity Testing: What does it take to break the model? Silvana Pesenti
Reverse Sensitivity Testing: What does it take to break the model? Silvana Pesenti Silvana.Pesenti@cass.city.ac.uk joint work with Pietro Millossovich and Andreas Tsanakas Insurance Data Science Conference,
More information