A linear algebraic method for pricing temporary life annuities


 Kristina Nelson
 1 years ago
 Views:
Transcription
1 A linear algebraic method for pricing temporary life annuities P. Date (joint work with R. Mamon, L. Jalen and I.C. Wang) Department of Mathematical Sciences, Brunel University, London
2 Outline Introduction Cash flow valuation problems in actuarial science
3 Outline Introduction Cash flow valuation problems in actuarial science 1. A linear algebraic approach to random cash flow pricing 2. Approximate solutions
4 Outline Introduction Cash flow valuation problems in actuarial science 1. A linear algebraic approach to random cash flow pricing 2. Approximate solutions Applications
5 Introduction: pricing an annuity With no mortality risk, the present value of cash flow f (e.g. purchased temporary life annuity) payable at a future times t i, at time t k,0 k < i < N is p k = N 1 i=k+1 f i i j=k+1 (1 + r j ) r i is one period interest rate in [t i 1,t i ).
6 Introduction: pricing an annuity With no mortality risk, the present value of cash flow f (e.g. purchased temporary life annuity) payable at a future times t i, at time t k,0 k < i < N is p k = N 1 i=k+1 f i i j=k+1 (1 + r j ) r i is one period interest rate in [t i 1,t i ). Example: f = 1, N = 1, p 0 = r 1.
7 Pricing an annuity (cont d) With no interest rate risk but with force of mortality risk, the same present value is p k = N 1 i=k+1 f i i j=k+1 (1 + λ j ). λ i is force of mortality in [t i 1,t i ) λ i = probability that annuitant alive at t i 1 survives till t i.
8 Pricing an annuity (cont d) With both interest rate and mortality risk, p k = N 1 i=k+1 f i i j=k+1 (1 + λ j)(1 + r j ), E(p k ) =? Conventional approaches: 1. Use affine term structure models for both interest rate and force of mortality; 2. More complex models, with simulationbased pricing.
9 Pricing an annuity (cont d) Our approach: 1. Expand p k into a polynomial in λ j, r j ; base an approximation of E(p k ) on the moments of λ j, r j. 2. E(r j ), E(r i r j ) are available (forward curve data  no parametric model needed). Forward mortality information may be available.
10 Pricing an annuity (cont d) Our approach: 1. Expand p k into a polynomial in λ j, r j ; base an approximation of E(p k ) on the moments of λ j, r j. 2. E(r j ), E(r i r j ) are available (forward curve data  no parametric model needed). Forward mortality information may be available. 3. Strong results on convergence of approximation + very good simulation results. 4. Extends to various contracts (mortgage insurance, annuities, f contingent on something else).
11 A linear algebraic approach With discounting factor θ i = r i + λ i + r i λ i, the relationship p 1 = f 2, running present value at t θ 2 p 0 = f 1 f θ 1 (1 + θ 2 )(1 + θ 1 ), = f 1 + p θ 1, may also be written as (1 + θ 2 )p 1 + 0p 0 = f 2, ( 1)p 1 + (1 + θ 1 )p 0 = f 1.
12 A linear algebraic approach (cont d) 1. Thus p = [ p 1 p 0 ] solves a system of linear equations ([ ] [ ]) θ2 0 p = f. 0 θ 1 2. If f are temporary life annuity payments, E(p 0 ) is the lump sum purchase price. 3. More involved formulation possible for insurance contracts; leads to random terms of both sides. 4. For future θ i unknown and random, we have a problem of inverting a structured matrix with random entries.
13 Random matrix inversion problem For a series of cash flows f i,0 i N, we can write f = (Q + Θ)p p is N vector of running present values, f is N vector of cash flows. with θ i = r i + λ i + r i λ i. [Q] ij = 1 if i = j = 1 if i = j + 1 = 0 otherwise [Θ] ij = θ N i+1 if i = j = 0 otherwise, This means the value of the series of cash flows at time t k is [(Q + Θ) 1 f] k.
14 Approximate inversion: scalar case For θ i = θ < 1, f 1 + θ = ( 1 θ + θ 2 ) f M ( θ) i f. i=0 A similar result holds if θ is bounded by unity, but random. Does an analogous expansion hold if θ i θ j,i j?
15 Convergent approximation to the solution: vector case Theorem Suppose that θ i satisfy P( Q 1 Θ < 1) = 1, f i satisfy max i f i < γ, γ < and the inverse of (Q + Θ) exists with probability 1. Define p = (Q + Θ) 1 f, p M = M ( Q 1 Θ) i Q 1 f (1) i=0 with ( Q 1 Θ) 0 = I. Then (i) p M p with probability 1. (ii) ( ) lim E p M p 2 2 = 0. M
16 What does p M look like? It may be shown that [Q 1 Θ] ij = θ N j+1 if i j, = 0 otherwise, j [Q 1 f] j = f N i. i=1 e.g. for N = 3, θ Q 1 Θ = θ 3 θ 2 0 θ 3 θ 2 θ 1 f 3 Q 1 f = f 3 + f 2 f 3 + f 2 + f 1
17 How good is the approximation? With probability 1, p M p Q 1 Θ M+1 1 Q 1 Θ Q 1 f holds for any induced norm, provided Q 1 Θ < 1. If θ i < θ max w.p.1, N(N + 1) Q 1 Θ 2 θ max. 2 Further, given any ǫ > 0, a matrix norm s.t. Q 1 Θ < θ max + ǫ. Nonconservative bounds are difficult; but the algorithm seems to work well in practice.
18 Putting it all together Finally, Given moments E(r i ), E(λ i ), E(r i r j ), E(λ i λ j ), we can construct a second order approximation for ( N ) f i E i i=1 j=1 (1 + λ, j)(1 + r j ) } {{ } an element from vector E( 2 i=0 ( Q 1 Θ) i (Q 1 f)) in closedform.
19 Putting it all together Finally, Given moments E(r i ), E(λ i ), E(r i r j ), E(λ i λ j ), we can construct a second order approximation for ( N ) f i E i i=1 j=1 (1 + λ, j)(1 + r j ) } {{ } an element from vector E( 2 i=0 ( Q 1 Θ) i (Q 1 f)) in closedform. E(r i ) (forward curve), E(r i r j ) (forward curve correlations) may be known, or found from affine term structure model. E(λ i ), E(λ i λ j ) may be found from an affine term structure model for force of mortality.
20 Simulation Experiments Extensive numerical simulation experiments conducted for annuity valuation, insurance premium (Date et al, 2010), general cash flows (Date et al 2007). The method is computationally very cheap and shown to be quite accurate. This requires joint moments of interest rates and mortality up to M th order. These moments may be known (forward curve and historic correlations  no interest rate model needed for M = 2.) or they may be inferred from a given affine term structure model (e.g. for mortality).
21 Main References P. Date, R. Mamon, L. Jalen and I.C. Wang, A linear algebraic method for pricing temporary life annuities and insurance policies, Insurance: Mathematics and Economics, vol. 47, pages , P. Gaillardetz, Valuation of life insurance products under stochastic interest rates, Insurance: Mathematics and Economics, vol. 42, pages , P. Date, R. Mamon and I.C. Wang, Valuation of cash flows under random rates of interest: a linear algebraic approach, Insurance: Mathematics and Economics, vol. 41, pages 84 95, L. Ballotta and S. Haberman, The fair valuation problem of guaranteed annuity options: the stochastic mortality environment case, Insurance: Mathematics and Economics, vol. 38, pages , 2006.
Cofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
More informationMaking use of netting effects when composing life insurance contracts
Making use of netting effects when composing life insurance contracts Marcus Christiansen Preprint Series: 2113 Fakultät für Mathematik und Wirtschaftswissenschaften UNIVERSITÄT ULM Making use of netting
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationTABLE OF CONTENTS. GENERAL AND HISTORICAL PREFACE iii SIXTH EDITION PREFACE v PART ONE: REVIEW AND BACKGROUND MATERIAL
TABLE OF CONTENTS GENERAL AND HISTORICAL PREFACE iii SIXTH EDITION PREFACE v PART ONE: REVIEW AND BACKGROUND MATERIAL CHAPTER ONE: REVIEW OF INTEREST THEORY 3 1.1 Interest Measures 3 1.2 Level Annuity
More informationACTUARIAL MATHEMATICS FOR LIFE CONTINGENT RISKS
ACTUARIAL MATHEMATICS FOR LIFE CONTINGENT RISKS DAVID C. M. DICKSON University of Melbourne MARY R. HARDY University of Waterloo, Ontario V HOWARD R. WATERS HeriotWatt University, Edinburgh CAMBRIDGE
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationFundamentals of Actuarial Mathematics
Fundamentals of Actuarial Mathematics S. David Promislow York University, Toronto, Canada John Wiley & Sons, Ltd Contents Preface Notation index xiii xvii PARTI THE DETERMINISTIC MODEL 1 1 Introduction
More informationTRANSACTIONS OF SOCIETY OF ACTUARIES 1952 VOL. 4 NO. 10 COMPLETE ANNUITIES. EUGENE A. RASOR* Ann T. N. E. GREVILLE
TRANSACTIONS OF SOCIETY OF ACTUARIES 1952 VOL. 4 NO. 10 COMPLETE ANNUITIES EUGENE A. RASOR* Ann T. N. E. GREVILLE INTRODUCTION I N GENERAL, a complete annuity of one per annum may be defined as a curtate
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More information1. Datsenka Dog Insurance Company has developed the following mortality table for dogs:
1 Datsenka Dog Insurance Company has developed the following mortality table for dogs: Age l Age l 0 2000 5 1200 1 1950 6 1000 2 1850 7 700 3 1600 8 300 4 1400 9 0 Datsenka sells an whole life annuity
More informationO MIA009 (F2F) : GENERAL INSURANCE, LIFE AND
No. of Printed Pages : 11 MIA009 (F2F) kr) ki) M.Sc. ACTUARIAL SCIENCE (MSCAS) N December, 2012 0 O MIA009 (F2F) : GENERAL INSURANCE, LIFE AND HEALTH CONTINGENCIES Time : 3 hours Maximum Marks : 100
More information2. Annuities. 1. Basic Annuities 1.1 Introduction. Annuity: A series of payments made at equal intervals of time.
2. Annuities 1. Basic Annuities 1.1 Introduction Annuity: A series of payments made at equal intervals of time. Examples: House rents, mortgage payments, installment payments on automobiles, and interest
More informationINSTITUTE AND FACULTY OF ACTUARIES EXAMINATION
INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION 27 April 2015 (pm) Subject CT5 Contingencies Core Technical Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and examination
More informationOn Simulation Method of Small Life Insurance Portfolios By Shamita Dutta Gupta Department of Mathematics Pace University New York, NY 10038
On Simulation Method of Small Life Insurance Portfolios By Shamita Dutta Gupta Department of Mathematics Pace University New York, NY 10038 Abstract A new simulation method is developed for actuarial applications
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationLEVELING THE NET SINGLE PREMIUM.  Review the assumptions used in calculating the net single premium.
LEVELING THE NET SINGLE PREMIUM CHAPTER OBJECTIVES  Review the assumptions used in calculating the net single premium.  Describe what is meant by the present value of future benefits.  Explain the mathematics
More informationDuring the analysis of cash flows we assume that if time is discrete when:
Chapter 5. EVALUATION OF THE RETURN ON INVESTMENT Objectives: To evaluate the yield of cash flows using various methods. To simulate mathematical and real content situations related to the cash flow management
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationPlease write your name and student number at the spaces provided:
MATH 3630 Actuarial Mathematics I Final Examination  sec 001 Monday, 10 December 2012 Time Allowed: 2 hours (6:008:00 pm) Room: MSB 411 Total Marks: 120 points Please write your name and student number
More informationSecuritization of Longevity Risk in Reverse Mortgages
Securitization of Longevity Risk in Reverse Mortgages Emiliano A. Valdez, PhD, FSA Michigan State University joint work with L. Wang and J. Piggott XXIII CNSF s International Seminar, Mexico City, Nov
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationCase Study Modeling Longevity Risk for Solvency II
Case Study Modeling Longevity Risk for Solvency II September 8, 2012 Longevity 8 Conference Waterloo, Ontario Presented by Stuart Silverman Principal & Consulting Actuary Background SOLVENCY II New Minimum
More informationSome Observations on Variance and Risk
Some Observations on Variance and Risk 1 Introduction By K.K.Dharni Pradip Kumar 1.1 In most actuarial contexts some or all of the cash flows in a contract are uncertain and depend on the death or survival
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationValuation of the Surrender Option in Life Insurance Policies
Valuation of the Surrender Option in Life Insurance Policies Hansjörg Furrer Marketconsistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2010 Valuing Surrender Options Contents A. Motivation and
More informationSchool of Natural Sciences and Mathematics
School of Natural Sciences and Mathematics Actuarial Science (M.S.) (6 minimum hours) The Master of Science in Actuarial Science (AS) Program at the University of Texas at Dallas is administered through
More informationFurther Topics in Actuarial Mathematics: Premium Reserves. Matthew Mikola
Further Topics in Actuarial Mathematics: Premium Reserves Matthew Mikola April 26, 2007 Contents 1 Introduction 1 1.1 Expected Loss...................................... 2 1.2 An Overview of the Project...............................
More informationReduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:
Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in
More informationMay 2012 Course MLC Examination, Problem No. 1 For a 2year select and ultimate mortality model, you are given:
Solutions to the May 2012 Course MLC Examination by Krzysztof Ostaszewski, http://www.krzysio.net, krzysio@krzysio.net Copyright 2012 by Krzysztof Ostaszewski All rights reserved. No reproduction in any
More informationFundamentals of Actuarial Mathematics. 3rd Edition
Brochure More information from http://www.researchandmarkets.com/reports/2866022/ Fundamentals of Actuarial Mathematics. 3rd Edition Description:  Provides a comprehensive coverage of both the deterministic
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationBonusmalus systems and Markov chains
Bonusmalus systems and Markov chains Dutch car insurance bonusmalus system class % increase new class after # claims 0 1 2 >3 14 30 14 9 5 1 13 32.5 14 8 4 1 12 35 13 8 4 1 11 37.5 12 7 3 1 10 40 11
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationMultistate transition models with actuarial applications c
Multistate transition models with actuarial applications c by James W. Daniel c Copyright 2004 by James W. Daniel Reprinted by the Casualty Actuarial Society and the Society of Actuaries by permission
More informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 17 th November 2011 Subject CT5 General Insurance, Life and Health Contingencies Time allowed: Three Hours (10.00 13.00 Hrs) Total Marks: 100 INSTRUCTIONS TO
More informationThe basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23
(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, highdimensional
More informationRisk Management and Payout Design of Reverse Mortgages
1 Risk Management and Payout Design of Reverse Mortgages Daniel Cho, Katja Hanewald and Michael Sherris School of Risk & Actuarial ARC Centre of Excellence in Population Ageing Research (CEPAR) University
More informationParameter and Model Uncertainties in Pricing Deepdeferred Annuities
Outline Parameter and Model Uncertainties in Pricing Deepdeferred Annuities Min Ji Towson University, Maryland, USA Rui Zhou University of Manitoba, Canada Longevity 11, Lyon, 2015 1/32 Outline Outline
More informationChapter 2: Systems of Linear Equations and Matrices:
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More informationFinancial Engineering g and Actuarial Science In the Life Insurance Industry
Financial Engineering g and Actuarial Science In the Life Insurance Industry Presentation at USC October 31, 2013 Frank Zhang, CFA, FRM, FSA, MSCF, PRM Vice President, Risk Management Pacific Life Insurance
More informationMATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
More information1 Cashflows, discounting, interest rate models
Assignment 1 BS4a Actuarial Science Oxford MT 2014 1 1 Cashflows, discounting, interest rate models Please hand in your answers to questions 3, 4, 5 and 8 for marking. The rest are for further practice.
More informationChapter 6. Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams
Chapter 6 Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams 1. Distinguish between an ordinary annuity and an annuity due, and calculate present
More information1 Portfolio mean and variance
Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a oneperiod investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring
More informationTime Value of Money Concepts
BASIC ANNUITIES There are many accounting transactions that require the payment of a specific amount each period. A payment for a auto loan or a mortgage payment are examples of this type of transaction.
More informationSafety margins for unsystematic biometric risk in life and health insurance
Safety margins for unsystematic biometric risk in life and health insurance Marcus C. Christiansen June 1, 2012 7th Conference in Actuarial Science & Finance on Samos Seite 2 Safety margins for unsystematic
More informationMA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam
MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am  :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationThe Fair Valuation of Life Insurance Participating Policies: The Mortality Risk Role
The Fair Valuation of Life Insurance Participating Policies: The Mortality Risk Role Massimiliano Politano Department of Mathematics and Statistics University of Naples Federico II Via Cinthia, Monte S.Angelo
More informationTHE ANALYTIC HIERARCHY PROCESS (AHP)
THE ANALYTIC HIERARCHY PROCESS (AHP) INTRODUCTION The Analytic Hierarchy Process (AHP) is due to Saaty (1980) and is often referred to, eponymously, as the Saaty method. It is popular and widely used,
More informationJANUARY 2016 EXAMINATIONS. Life Insurance I
PAPER CODE NO. MATH 273 EXAMINER: Dr. C. BoadoPenas TEL.NO. 44026 DEPARTMENT: Mathematical Sciences JANUARY 2016 EXAMINATIONS Life Insurance I Time allowed: Two and a half hours INSTRUCTIONS TO CANDIDATES:
More informationNimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff
Nimble Algorithms for Cloud Computing Ravi Kannan, Santosh Vempala and David Woodruff Cloud computing Data is distributed arbitrarily on many servers Parallel algorithms: time Streaming algorithms: sublinear
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More information26. Determinants I. 1. Prehistory
26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinateindependent
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationFacilitating OnDemand Risk and Actuarial Analysis in MATLAB. Timo Salminen, CFA, FRM Model IT
Facilitating OnDemand Risk and Actuarial Analysis in MATLAB Timo Salminen, CFA, FRM Model IT Introduction It is common that insurance companies can valuate their liabilities only quarterly Sufficient
More informationFeatured article: Evaluating the Cost of Longevity in Variable Annuity Living Benefits
Featured article: Evaluating the Cost of Longevity in Variable Annuity Living Benefits By Stuart Silverman and Dan Theodore This is a followup to a previous article Considering the Cost of Longevity Volatility
More informationLecture Notes on Actuarial Mathematics
Lecture Notes on Actuarial Mathematics Jerry Alan Veeh May 1, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects of the
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationNonlinear Algebraic Equations Example
Nonlinear Algebraic Equations Example Continuous Stirred Tank Reactor (CSTR). Look for steady state concentrations & temperature. s r (in) p,i (in) i In: N spieces with concentrations c, heat capacities
More informationTraditional, investment, and risk management actuaries in the life insurance industry
Traditional, investment, and risk management actuaries in the life insurance industry Presentation at California Actuarial Student Conference University of California, Santa Barbara April 4, 2015 Frank
More informationMathematics of Life Contingencies MATH 3281
Mathematics of Life Contingencies MATH 3281 Life annuities contracts Edward Furman Department of Mathematics and Statistics York University February 13, 2012 Edward Furman Mathematics of Life Contingencies
More informationEEV, MCEV, Solvency, IFRS a chance for actuarial mathematics to get to mainstream of insurance value chain
EEV, MCEV, Solvency, IFRS a chance for actuarial mathematics to get to mainstream of insurance value chain dr Krzysztof Stroiński, dr Renata Onisk, dr Konrad Szuster, mgr Marcin Szczuka 9 June 2008 Presentation
More informationVersion 3A ACTUARIAL VALUATIONS. Remainder,Income, and Annuity Examples For One Life, Two Lives, and Terms Certain
ACTUARIAL VALUATIONS Remainder,Income, and Annuity Examples For One Life, Two Lives, and Terms Certain Version 3A For Use in Income, Estate, and Gift Tax Purposes, Including Valuation of Pooled Income
More informationHeriotWatt University. M.Sc. in Actuarial Science. Life Insurance Mathematics I. Tutorial 5
1 HeriotWatt University M.Sc. in Actuarial Science Life Insurance Mathematics I Tutorial 5 1. Consider the illnessdeath model in Figure 1. A life age takes out a policy with a term of n years that pays
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More informationMatrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
More informationEquilibrium computation: Part 1
Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationInstitute of Actuaries of India Subject ST2 Life Insurance
Institute of Actuaries of India Subject ST2 Life Insurance For 2015 Examinations Aim The aim of the Life Insurance Specialist Technical subject is to instil in successful candidates principles of actuarial
More informationDepartment of Chemical Engineering ChE101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI
Department of Chemical Engineering ChE101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Solving a System of Linear Algebraic Equations (last updated 5/19/05 by GGB) Objectives:
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 informationBiometrical worstcase and bestcase scenarios in life insurance
Biometrical worstcase and bestcase scenarios in life insurance Marcus C. Christiansen 8th Scientific Conference of the DGVFM April 30, 2009 Solvency Capital Requirement: The Standard Formula Calculation
More informationSESSION/SÉANCE : 37 Applications of Forward Mortality Factor Models in Life Insurance Practice SPEAKER(S)/CONFÉRENCIER(S) : Nan Zhu, Georgia State
SESSION/SÉANCE : 37 Applications of Forward Mortality Factor Models in Life Insurance Practice SPEAKER(S)/CONFÉRENCIER(S) : Nan Zhu, Georgia State University and Illinois State University 1. Introduction
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationActuarial Speak 101 Terms and Definitions
Actuarial Speak 101 Terms and Definitions Introduction and Caveat: It is intended that all definitions and explanations are accurate. However, for purposes of understanding and clarity of key points, the
More informationChapter 13 Introduction to Nonlinear Regression( 非 線 性 迴 歸 )
Chapter 13 Introduction to Nonlinear Regression( 非 線 性 迴 歸 ) and Neural Networks( 類 神 經 網 路 ) 許 湘 伶 Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) LR Chap 10 1 / 35 13 Examples
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationManual for SOA Exam MLC.
Chapter 6. Benefit premiums. Extract from: Arcones Fall 2010 Edition, available at http://www.actexmadriver.com/ 1/24 Nonlevel premiums and/or benefits. Let b k be the benefit paid by an insurance company
More informationMATH1510 Financial Mathematics I. Jitse Niesen University of Leeds
MATH1510 Financial Mathematics I Jitse Niesen University of Leeds January May 2012 Description of the module This is the description of the module as it appears in the module catalogue. Objectives Introduction
More informationAnnuities and decumulation phase of retirement. Chris Daykin UK Government Actuary Chairman, PBSS Section of IAA
Annuities and decumulation phase of retirement Chris Daykin UK Government Actuary Chairman, PBSS Section of IAA CASH LUMP SUM AT RETIREMENT CASH INSTEAD OF PENSION > popular with pension scheme members
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationTHE MATHEMATICS OF LIFE INSURANCE THE NET SINGLE PREMIUM.  Investigate the component parts of the life insurance premium.
THE MATHEMATICS OF LIFE INSURANCE THE NET SINGLE PREMIUM CHAPTER OBJECTIVES  Discuss the importance of life insurance mathematics.  Investigate the component parts of the life insurance premium.  Demonstrate
More informationOnline Appendix to The Cost of Financial Frictions for Life Insurers
Online Appendix to The Cost of Financial Frictions for Life Insurers By Ralph S.J. Koijen and Motohiro Yogo October 22, 2014 Koijen: London Business School, Regent s Park, London NW1 4SA, United Kingdom
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationLiquidity premiums and contingent liabilities
Insights Liquidity premiums and contingent liabilities Craig Turnbull Craig.Turbull@barrhibb.com The liquidity premium the concept that illiquid assets have lower prices than equivalent liquid ones has
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationAnnuities. Lecture: Weeks 911. Lecture: Weeks 911 (STT 455) Annuities Fall 2014  Valdez 1 / 43
Annuities Lecture: Weeks 911 Lecture: Weeks 911 (STT 455) Annuities Fall 2014  Valdez 1 / 43 What are annuities? What are annuities? An annuity is a series of payments that could vary according to:
More informationFrom Factorial Designs to Hilbert Schemes
From Factorial Designs to Hilbert Schemes Lorenzo Robbiano Università di Genova Dipartimento di Matematica Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationRandom graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.
More information