Safety margins for unsystematic biometric risk in life and health insurance


 Pauline Shaw
 1 years ago
 Views:
Transcription
1 Safety margins for unsystematic biometric risk in life and health insurance Marcus C. Christiansen June 1, th Conference in Actuarial Science & Finance on Samos
2 Seite 2 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Introduction What is the problem?
3 Seite 3 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 biometric risk / demographic risk calculated number actual number of deaths, disabilities, lapses, etc. risk parts unsystematic risk: deviations from expected values estimation risk: estimations deviate because of finite sample sizes systematic risk: demographic changes not anticipated today
4 Seite 4 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 dealing with unsystematic biometric risk define L = future liabilities of a portfolio of m (independent) policies then because of the law of large numbers / diversifiability E(L) = equivalence premium yet the insurer charges the premium E(L) + safety margin = conservative premium E (L) = conservative premium
5 Seite 5 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 EXAMPLE pure endowments homogeneous portfolio of m pure endowment policies survival benefit of 1 after n years policies start at age x remaining lifetimes Tx 1,..., Tx m discounting factor v with P(T i x > n) = n p x actual number of survival benefits m 1 T i x >n Binomial(m, np x ) i=1 equivalence premium E(L) = E (v m n i=1 1 T i x >n ) = v n m n p x
6 Seite 6 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 EXAMPLE pure endowments explicit safety margin factor: choose s expl such that ( P s expl }{{} safety loading v n m n p x equivalence premium conservative premium v n m 1 T i x >n i=1 liabilities ) implicit safety margin factor: choose s impl such that ( P s }{{} impl np x 1 m ) 1 }{{} m T i x >n safety loading survival probability i=1 conservative survival probability both approaches are equivalent here empirical survival probability α }{{} confidence level α }{{} confidence level
7 Seite 7 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 EXAMPLE long term care homogeneous portfolio of m long term care policies state space {a, c 1, c 2, c 3, l, d} yearly care annuities of R 1 < R 2 < R 3 Markovian process Xt i gives state of policyholder i at time t discounted liabilities L = m ω i=1 s {1,2,3} t=x v t x R s 1 X i t =c s? ( Normal)
8 Seite 8 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 EXAMPLE long term care explicit safety margin factor: choose s expl such that ( ) P s }{{} expl α safety loading E(L) }{{} equivalence premium conservative premium L }{{} liabilities implicit safety margin factors: choose s impl jk,t ( P s impl jk,t }{{} safety loading P(Xt+1 i = k X t i = j) transition probability conservative transition probability such that {X i t+1 = k, X i t = j} } {Xt i = j} {{ } empirical transition probability (1) relation is unclear (2) confidence levels have different meanings (3) solution of implicit approach not unique ) j, k, t α
9 Seite 9 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 (1) relation is unclear sensitivity analysis (Linnemann (1993), Dienst (1995), Christiansen (2008),...) (2) confidence levels have different meanings topdown approach: choose P (Xt+1 i = k X t i = j) = s impl jk,t P(Xt+1 i = k X t i = j) such that ( ) P E (L) L α (Bühlman (1985), Pannenberg (1997),...) (3) solution of implicit concept not unique twostate case: allocation principles (Pannenberg (1997),... ) multistate case:?
10 Seite 10 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Basic Modeling
11 Seite 11 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 cumulative transition intensity matrix Markovian jump processes Xt 1,..., X t m deterministic initial states Xx 1 =... = Xx m transition probability matrix ( ) p(s, t) := P(X t = k X s = j) (j,k) S 2, = a x s t ω transition intensity matrix µ(t) := d dτ τ=t p(t, τ) cumulative transition intensity matrix q(t) = (x,t] µ(s)ds NelsonAalen estimator for q jk (t) Q (m) 1 jk (t) := jk (u) (x,t] I (m) j (u ) dn(m) I (m) j (u )= number of policies in state j at time u N (m) jk (u)= number of jumps from j to k till time u
12 Seite 12 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 portfolio liability (m policies): L = L (m) prospective reserve (at starting age x in initial state a) E(L (1) ) =: V (q) = v(x, t) p aj (x, t) db j (t) j S (x,ω] + v(x, t) b jk (t) p aj (x, t ) dq jk (t) (j,k) J LEMMA: L (m) = m V (Q (m) ) (x,ω] economic implication of random fluctuations L (m) = m V (Q (m) ) true loss = m V (q) mean loss + m(v (Q (m) ) V (q)) unsystematic risk
13 Seite 13 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 PROBLEM explicit confidence estimate: find s such that P ( s m V (q) m V (Q (m) ) ) α implicit confidence estimate: depending on the sign of the sumatrisk find lower bound: choose s such that P ( Q (m) q s ) α upper bound: choose s such that P ( Q (m) q + s ) α topdown confidence estimate: find s such that (a) functional ( confidence condition: ) P m V (Q (m) ) m V (q ± s) α (b) risk proportional: ds jk (t) proportional to uncertainty in dq (m) jk (t)
14 Seite 14 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 asymptotic distribution
15 Seite 15 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Theorem (cf. Andersen et al. (1991)) Let q be absolutely continuous with density function µ, and let P(X t = j) > 0 for all j S and t (x, ω). Then m ( Q (m) jk q jk )(j,k) J d (U jk ) (j,k) J, m, on [x, ω], where (U jk ) (j,k) J is a vector of stochastically independent Gaussian processes with zero mean and covariance functions Cov(U jk (s), U jk (t)) = (x,s t] 1 P(X u = j) dq jk(u). Interpretation: uncertainty in the number of transitions jk at t dq (m) jk (t) dq jk (t) d 1 m du jk (t)
16 Seite 16 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Theorem (C., 2011) Let q n = q n + ε n h n with q n q 0, h n h 0, ε n 0, and the functions q, q n, and q n of uniformly bounded variation. Then we have for all 0 s T 1 ( V (qn ) V (q n ) ) D q V (h) ε n 0, n, where D q V (h) is a supremum norm continuous linear mapping in h that equals D q V (h) = v(x, t) p aj (x, t ) R jk (t) dh(t), (j,k) J (x,ω] interpreted for h not of bounded variation by formal integration by parts.
17 Seite 17 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 limit theorem for the NelsonAalen estimator + Hadamard differentiability of V + functional delta method =... Theorem (C., 2011) Under the assumptions of the previous theorems we have m ( V (Q (m) ) V (q) ) d D q V (U), m on [x, ω], where U = (U jk ) (j,k) J is a vector of independent Gaussian processes. and D q V (U) = v(x, t) p aj (x, t) R jk (t) du jk (t) (j,k) J (x,ω] (defined by integration by parts).
18 Seite 18 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Interpretation L (m) E(L (m) ) = mv (Q (m) ) mv (q) d m (j,k) (x,ω] total unsystematic risk 1 v(x, u) p aj (x, u) R jk (u) du jk (u) m weighting factor independent risk sources additive decomposition to independent risk contributions
19 Seite 19 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 risk proportional allocation
20 Seite 20 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 according to the Hadamard differentiability of q V (q) mv (q + s) mv (q) m (j,k) total margin (x,ω] v(x, u) p aj (x, u) R jk (u) ds jk (u) weighting factor partial margin Definition: γriskproportional given that risk measure γ is additive for independent risks, let asymptotically in m ( 1 ) γ v(x, u) p aj (x, u) R jk (u) du jk (u) m = const v(x, u) p aj (x, u) R jk (u) ds jk (u)
21 Seite 21 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Theorem (C., 2011) Safety margin s meets ( the functional confidence ) condition P m V (Q (m) ) m V (q ± s) α is γriskproportional asymptotically in m if (a) γ( ) = Var( ) and ds jk (u) = u α v(x, u) R m Var(L jk(u) dq jk (u) (1) ) (b) γ( ) = E Q ( ) and ds jk (u) = const m E Q (du jk (u)) Remark RamlauHansen (1988), Linnemann (1993), etc. sign ds jk (u) = sign R jk (u)
22 Seite 22 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 numerical examples
23 Seite 23 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Corollary: For γ( ) = Var( ) we have ( µ jk (t) = µ u ) α jk(t) 1 + m Var(L (1) ) v(x, t) R jk(t) In the following we calculate the safety margin factor functions ( u ) α t 1 + m Var(L (1) ) v(x, t) R jk(t)
24 Seite 24 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Example: pure endowment policies homogeneous portfolio of m = 1000 policies starting age x = 35 and contract period of n = 30 years yearly constant premium, yearly interest rate of 2.25% topdown method with confidence level of 0.95 DAV 2004 R: constant line below 1
25 Seite 25 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Example: temporary life policies homogeneous portfolio of m = 1000 policies yearly constant premium... DAV 2004 T: constant line above 1
26 Seite 26 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Example: endowment policies survival benefit two times the death benefit...
27 Seite 27 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Example: disability policies homogeneous portfolio of m = 1000 policies constant yearly disability annuities and premiums state space {active, disabled, dead} active to dead active to disabled DAV 1997: constant line at 1.076
28 Seite 28 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 disabled to active DAV 1997: constant line at 0.79 disabled to dead DAV 1997: constant line at 0.78
29 Seite 29 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Conclusion decomposition of unsystematic biometric risk: independent addends with respect to transitions (j, k) and times t risk proportional allocation of margins variance principle: extends the classical theory
30 Seite 30 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 safety margins for transition estimation risk
31 Seite 31 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 transition estimation risk GIVEN: data of m policyholders TASK: estimate q q }{{} true distribution economic implication = } Q {{ (m) } + q Q (m) estimated distribution estimation risk V (q) = V (Q (m) ) + V (Q (m) ) V (q) mean loss estimated mean loss estimation risk
32 Seite 32 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 estimation risk Problem: find s such that (a) functional ( confidence condition: ) P V (Q (m) ± s) V (q) α (b) risk proportional: ds jk (t) proportional to uncertainty in dq (m) jk (t)
33 Seite 33 Safety margins for unsystematic biometric risk Marcus C. Christiansen June 1, 2012 Theorem (C., 2011) Safety margin s meets the functional confidence condition is γriskproportional asymptotically in m if (a) γ( ) = Var( ) and u ds jk (t) = α v(x, t) R m Var(L jk(t; q) dq jk (t) (1) ;q) (b) γ( ) = E Q ( ) and ds jk (t) = const m E Q (du jk (t)) Corollary We can substitute (a) by ds jk (t) = 1 m Var(L (1) ; Q (m) ) v(x, t) R jk(t; Q (m) ) dq (m) (t) jk
Safety margins for unsystematic biometric risk in life and health insurance
Safety margins for unsystematic biometric risk in life and health insurance Marcus C. Christiansen Preprint Series: 2004 Fakultät für Mathematik und Wirtschaftswissenschaften UNIVERSITÄT ULM Safety margins
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 informationDecomposition of life insurance liabilities into risk factors theory and application
Decomposition of life insurance liabilities into risk factors theory and application Katja Schilling University of Ulm March 7, 2014 Joint work with Daniel Bauer, Marcus C. Christiansen, Alexander Kling
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 informationSolution. Let us write s for the policy year. Then the mortality rate during year s is q 30+s 1. q 30+s 1
Solutions to the May 213 Course MLC Examination by Krzysztof Ostaszewski, http://wwwkrzysionet, krzysio@krzysionet Copyright 213 by Krzysztof Ostaszewski All rights reserved No reproduction in any form
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 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 informationDisability insurance: estimation and risk aggregation
Disability insurance: estimation and risk aggregation B. Löfdahl Department of Mathematics KTH, Royal Institute of Technology May 2015 Introduction New upcoming regulation for insurance industry: Solvency
More informationOn the decomposition of risk in life insurance
On the decomposition of risk in life insurance Tom Fischer HeriotWatt University, Edinburgh April 7, 2005 fischer@ma.hw.ac.uk This work was partly sponsored by the German Federal Ministry of Education
More informationDecomposition of life insurance liabilities into risk factors theory and application to annuity conversion options
Decomposition of life insurance liabilities into risk factors theory and application to annuity conversion options Joint work with Daniel Bauer, Marcus C. Christiansen, Alexander Kling Research Training
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 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 informationHattendorff s theorem for nonsmooth continuoustime Markov models II: Application
Insurance: Mathematics and Economics 26 (2000) 1 14 Hattendorff s theorem for nonsmooth continuoustime Markov models II: Application Hartmut Milbrodt Mathematisches Institut, Universität zu Köln, Weyertal
More informationManual for SOA Exam MLC.
Chapter 6. Benefit premiums. Extract from: Arcones Fall 2010 Edition, available at http://www.actexmadriver.com/ 1/77 Fully discrete benefit premiums In this section, we will consider the funding of insurance
More informationPremium Calculation. Lecture: Weeks 1214. Lecture: Weeks 1214 (Math 3630) Annuities Fall 2015  Valdez 1 / 32
Premium Calculation Lecture: Weeks 1214 Lecture: Weeks 1214 (Math 3630) Annuities Fall 2015  Valdez 1 / 32 Preliminaries Preliminaries An insurance policy (life insurance or life annuity) is funded
More informationPremium calculation. summer semester 2013/2014. Technical University of Ostrava Faculty of Economics department of Finance
Technical University of Ostrava Faculty of Economics department of Finance summer semester 2013/2014 Content 1 Fundamentals Insurer s expenses 2 Equivalence principles Calculation principles 3 Equivalence
More informationPremium Calculation. Lecture: Weeks 1214. Lecture: Weeks 1214 (STT 455) Premium Calculation Fall 2014  Valdez 1 / 31
Premium Calculation Lecture: Weeks 1214 Lecture: Weeks 1214 (STT 455) Premium Calculation Fall 2014  Valdez 1 / 31 Preliminaries Preliminaries An insurance policy (life insurance or life annuity) is
More informationPremium Calculation  continued
Premium Calculation  continued Lecture: Weeks 12 Lecture: Weeks 12 (STT 456) Premium Calculation Spring 2015  Valdez 1 / 16 Recall some preliminaries Recall some preliminaries An insurance policy (life
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 informationManual for SOA Exam MLC.
Chapter 5. Life annuities. Extract from: Arcones Manual for the SOA Exam MLC. Spring 2010 Edition. available at http://www.actexmadriver.com/ 1/114 Whole life annuity A whole life annuity is a series of
More informationTABLE OF CONTENTS. 4. Daniel Markov 1 173
TABLE OF CONTENTS 1. Survival A. Time of Death for a Person Aged x 1 B. Force of Mortality 7 C. Life Tables and the Deterministic Survivorship Group 19 D. Life Table Characteristics: Expectation of Life
More information4. Life Insurance. 4.1 Survival Distribution And Life Tables. Introduction. X, Ageatdeath. T (x), timeuntildeath
4. Life Insurance 4.1 Survival Distribution And Life Tables Introduction X, Ageatdeath T (x), timeuntildeath Life Table Engineers use life tables to study the reliability of complex mechanical and
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 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 informationQuantitative Impact Study 1 (QIS1) Summary Report for Belgium. 21 March 2006
Quantitative Impact Study 1 (QIS1) Summary Report for Belgium 21 March 2006 1 Quantitative Impact Study 1 (QIS1) Summary Report for Belgium INTRODUCTORY REMARKS...4 1. GENERAL OBSERVATIONS...4 1.1. Market
More informationLiving to 100: Survival to Advanced Ages: Insurance Industry Implication on Retirement Planning and the Secondary Market in Insurance
Living to 100: Survival to Advanced Ages: Insurance Industry Implication on Retirement Planning and the Secondary Market in Insurance Jay Vadiveloo, * Peng Zhou, Charles Vinsonhaler, and Sudath Ranasinghe
More informationManual for SOA Exam MLC.
Chapter 5. Life annuities Extract from: Arcones Fall 2009 Edition, available at http://www.actexmadriver.com/ 1/60 (#24, Exam M, Fall 2005) For a special increasing whole life annuitydue on (40), you
More informationA linear algebraic method for pricing temporary life annuities
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 Outline Introduction
More informationChapter 2. 1. You are given: 1 t. Calculate: f. Pr[ T0
Chapter 2 1. You are given: 1 5 t F0 ( t) 1 1,0 t 125 125 Calculate: a. S () t 0 b. Pr[ T0 t] c. Pr[ T0 t] d. S () t e. Probability that a newborn will live to age 25. f. Probability that a person age
More informationMonte Carlo Simulation
1 Monte Carlo Simulation Stefan Weber Leibniz Universität Hannover email: sweber@stochastik.unihannover.de web: www.stochastik.unihannover.de/ sweber Monte Carlo Simulation 2 Quantifying and Hedging
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationASSESSING THE RISK POTENTIAL OF PREMIUM PAYMENT OPTIONS
ASSESSING THE RISK POTENTIAL OF PREMIUM PAYMENT OPTIONS IN PARTICIPATING LIFE INSURANCE CONTRACTS Nadine Gatzert phone: +41 71 2434012, fax: +41 71 2434040 nadine.gatzert@unisg.ch Hato Schmeiser phone:
More informationStochastic Analysis of LongTerm MultipleDecrement Contracts
Stochastic Analysis of LongTerm MultipleDecrement Contracts Matthew Clark, FSA, MAAA, and Chad Runchey, FSA, MAAA Ernst & Young LLP Published in the July 2008 issue of the Actuarial Practice Forum Copyright
More informationManual for SOA Exam MLC.
Chapter 4. Life Insurance. Extract from: Arcones Manual for the SOA Exam MLC. Fall 2009 Edition. available at http://www.actexmadriver.com/ 1/44 Properties of the APV for continuous insurance The following
More informationIdentities for Present Values of Life Insurance Benefits
Scand. Actuarial J. 1993; 2: 100106 ORIGINAL ARTICLE Identities for Present Values of Life Insurance Benefits : RAGNAR NORBERG Norberg R. Identities for present values of life insurance benefits. Scand.
More informationOptimal proportional reinsurance and dividend payout for insurance companies with switching reserves
Optimal proportional reinsurance and dividend payout for insurance companies with switching reserves Abstract: This paper presents a model for an insurance company that controls its risk and dividend
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
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 informationCEIOPSDOC33/09. (former CP 39) October 2009
CEIOPSDOC33/09 CEIOPS Advice for Level 2 Implementing Measures on Solvency II: Technical provisions Article 86 a Actuarial and statistical methodologies to calculate the best estimate (former CP 39)
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 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 informationHedging Variable Annuity Guarantees
p. 1/4 Hedging Variable Annuity Guarantees Actuarial Society of Hong Kong Hong Kong, July 30 Phelim P Boyle Wilfrid Laurier University Thanks to Yan Liu and Adam Kolkiewicz for useful discussions. p. 2/4
More informationSOCIETY OF ACTUARIES. EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE WRITTENANSWER QUESTIONS AND SOLUTIONS
SOCIETY OF ACTUARIES EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE WRITTENANSWER QUESTIONS AND SOLUTIONS Questions February 12, 2015 In Questions 12, 13, and 19, the wording was changed slightly
More informationOPTIMAl PREMIUM CONTROl IN A NONliFE INSURANCE BUSINESS
ONDERZOEKSRAPPORT NR 8904 OPTIMAl PREMIUM CONTROl IN A NONliFE INSURANCE BUSINESS BY M. VANDEBROEK & J. DHAENE D/1989/2376/5 1 IN A OPTIMAl PREMIUM CONTROl NONliFE INSURANCE BUSINESS By Martina Vandebroek
More informationPractical Applications of Stochastic Modeling for Disability Insurance
Practical Applications of Stochastic Modeling for Disability Insurance Society of Actuaries Session 8, Spring Health Meeting Seattle, WA, June 007 Practical Applications of Stochastic Modeling for Disability
More informationNovember 2012 Course MLC Examination, Problem No. 1 For two lives, (80) and (90), with independent future lifetimes, you are given: k p 80+k
Solutions to the November 202 Course MLC Examination by Krzysztof Ostaszewski, http://www.krzysio.net, krzysio@krzysio.net Copyright 202 by Krzysztof Ostaszewski All rights reserved. No reproduction in
More informationManual for SOA Exam MLC.
Chapter 4. Life Insurance. Extract from: Arcones Manual for the SOA Exam MLC. Fall 2009 Edition. available at http://www.actexmadriver.com/ 1/14 Level benefit insurance in the continuous case In this chapter,
More informationEDUCATION AND EXAMINATION COMMITTEE SOCIETY OF ACTUARIES RISK AND INSURANCE. Copyright 2005 by the Society of Actuaries
EDUCATION AND EXAMINATION COMMITTEE OF THE SOCIET OF ACTUARIES RISK AND INSURANCE by Judy Feldman Anderson, FSA and Robert L. Brown, FSA Copyright 25 by the Society of Actuaries The Education and Examination
More informationXII. RISKSPREADING VIA FINANCIAL INTERMEDIATION: LIFE INSURANCE
XII. RISSPREADIG VIA FIACIAL ITERMEDIATIO: LIFE ISURACE As discussed briefly at the end of Section V, financial assets can be traded directly in the capital markets or indirectly through financial intermediaries.
More informationQuantitative Operational Risk Management
Quantitative Operational Risk Management Kaj Nyström and Jimmy Skoglund Swedbank, Group Financial Risk Control S105 34 Stockholm, Sweden September 3, 2002 Abstract The New Basel Capital Accord presents
More informationA Shortcut to Calculating Return on Required Equity and It s Link to Cost of Capital
A Shortcut to Calculating Return on Required Equity and It s Link to Cost of Capital Nicholas Jacobi An insurance product s return on required equity demonstrates how successfully its results are covering
More informationCOMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS
COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic meanvariance problems in
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 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 informationNEDGROUP LIFE FINANCIAL MANAGEMENT PRINCIPLES AND PRACTICES OF ASSURANCE COMPANY LIMITED. A member of the Nedbank group
NEDGROUP LIFE ASSURANCE COMPANY LIMITED PRINCIPLES AND PRACTICES OF FINANCIAL MANAGEMENT A member of the Nedbank group We subscribe to the Code of Banking Practice of The Banking Association South Africa
More informationPractice Exam 1. x l x d x 50 1000 20 51 52 35 53 37
Practice Eam. You are given: (i) The following life table. (ii) 2q 52.758. l d 5 2 5 52 35 53 37 Determine d 5. (A) 2 (B) 2 (C) 22 (D) 24 (E) 26 2. For a Continuing Care Retirement Community, you are given
More informationStochastic Analysis of Life Insurance Surplus
Stochastic Analysis of Life Insurance Surplus by Natalia Lysenko B.Sc., Simon Fraser University, 2005. a project submitted in partial fulfillment of the requirements for the degree of Master of Science
More informationA distributionbased stochastic model of cohort life expectancy, with applications
A distributionbased stochastic model of cohort life expectancy, with applications David McCarthy Demography and Longevity Workshop CEPAR, Sydney, Australia 26 th July 2011 1 Literature review Traditional
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationPrivate Equity Fund Valuation and Systematic Risk
An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology
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 informationHeriotWatt University. BSc in Actuarial Mathematics and Statistics. Life Insurance Mathematics I. Extra Problems: Multiple Choice
HeriotWatt University BSc in Actuarial Mathematics and Statistics Life Insurance Mathematics I Extra Problems: Multiple Choice These problems have been taken from Faculty and Institute of Actuaries exams.
More informationASSESSING THE RISK POTENTIAL OF PREMIUM PAYMENT OPTIONS IN PARTICIPATING LIFE INSURANCE CONTRACTS
ASSESSING THE RISK POTENTIAL OF PREMIUM PAYMENT OPTIONS IN PARTICIPATING LIFE INSURANCE CONTRACTS NADINE GATZERT HATO SCHMEISER WORKING PAPERS ON RISK MANAGEMENT AND INSURANCE NO. 22 EDITED BY HATO SCHMEISER
More informationSOCIETY OF ACTUARIES. EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE QUESTIONS
SOCIETY OF ACTUARIES EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE QUESTIONS The following questions or solutions have been modified since this document was prepared to use with the syllabus effective
More informationReserving for income protection (IP) business (individual and group)
Reserving for income protection (IP) business (individual and group) 1. Please could you give us your name and contact details so that we can ensure that we get a wide spread of responses to this survey.
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 informationCAPM, Arbitrage, and Linear Factor Models
CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose meanvariance e cient portfolios. By equating these investors
More informationValuation and hedging of. life insurance liabilities. with. systematic mortality risk
Valuation and hedging of life insurance liabilities with systematic mortality risk Thomas Møller, PFA Pension, Copenhagen www.math.ku.dk/ tmoller (Joint work with Mikkel Dahl, Laboratory of Actuarial Mathematics,
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN13: 9780470860809 ISBN10: 0470860804 Editors Brian S Everitt & David
More informationEXAMINATION. 6 April 2005 (pm) Subject CT5 Contingencies Core Technical. Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE
Faculty of Actuaries Institute of Actuaries EXAMINATION 6 April 2005 (pm) Subject CT5 Contingencies Core Technical Time allowed: Three hours INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate and
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 informationPricing Frameworks for Securitization of Mortality Risk
1 Pricing Frameworks for Securitization of Mortality Risk Andrew Cairns HeriotWatt University, Edinburgh Joint work with David Blake & Kevin Dowd Updated version at: http://www.ma.hw.ac.uk/ andrewc 2
More informationMath 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 2 Solutions
Math 70/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 2 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,
More informationOn the transferability of reserves in lifelong health insurance contracts
On the transferability of reserves in lifelong health insurance contracts Els Godecharle KU Leuven This presentation has been prepared for the Actuaries Institute 2015 ASTIN and AFIR/ERM Colloquium. The
More information1. A survey of a group s viewing habits over the last year revealed the following
1. A survey of a group s viewing habits over the last year revealed the following information: (i) 8% watched gymnastics (ii) 9% watched baseball (iii) 19% watched soccer (iv) 14% watched gymnastics and
More information**BEGINNING OF EXAMINATION** The annual number of claims for an insured has probability function: , 0 < q < 1.
**BEGINNING OF EXAMINATION** 1. You are given: (i) The annual number of claims for an insured has probability function: 3 p x q q x x ( ) = ( 1 ) 3 x, x = 0,1,, 3 (ii) The prior density is π ( q) = q,
More informationMatching Investment Strategies in General Insurance Is it Worth It? Aim of Presentation. Background 34TH ANNUAL GIRO CONVENTION
Matching Investment Strategies in General Insurance Is it Worth It? 34TH ANNUAL GIRO CONVENTION CELTIC MANOR RESORT, NEWPORT, WALES Aim of Presentation To answer a key question: What are the benefit of
More informationREGS 2013: Variable Annuity Guaranteed Minimum Benefits
Department of Mathematics University of Illinois, UrbanaChampaign REGS 2013: Variable Annuity Guaranteed Minimum Benefits By: Vanessa Rivera Quiñones Professor Ruhuan Feng September 30, 2013 The author
More informationUnobserved heterogeneity; process and parameter effects in life insurance
Unobserved heterogeneity; process and parameter effects in life insurance Jaap Spreeuw & Henk Wolthuis University of Amsterdam ABSTRACT In this paper life insurance contracts based on an urnofurns model,
More informationRandom Vectors and the Variance Covariance Matrix
Random Vectors and the Variance Covariance Matrix Definition 1. A random vector X is a vector (X 1, X 2,..., X p ) of jointly distributed random variables. As is customary in linear algebra, we will write
More informationRiskminimization for life insurance liabilities
Riskminimization for life insurance liabilities Francesca Biagini Mathematisches Institut Ludwig Maximilians Universität München February 24, 2014 Francesca Biagini USC 1/25 Introduction A large number
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 informationManual for SOA Exam MLC.
Chapter 4. Life Insurance. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam MLC. Fall 29 Edition. available at http://www.actexmadriver.com/ c 29. Miguel A. Arcones.
More informationOn Bonus and Bonus Prognoses in Life Insurance
Scand. Actuarial J. 2001; 2: 126 147 On Bonus and Bonus Prognoses in Life Insurance RAGNAR NORBERG ORIGINAL ARTICLE Norberg R. On bonus and bonus prognoses in life insurance. Scand. Actuarial J. 2001;
More informationA Model of Optimum Tariff in Vehicle Fleet Insurance
A Model of Optimum Tariff in Vehicle Fleet Insurance. Bouhetala and F.Belhia and R.Salmi Statistics and Probability Department Bp, 3, ElAlia, USTHB, BabEzzouar, Alger Algeria. Summary: An approach about
More informationLife Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans
Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Challenges for defined contribution plans While Eastern Europe is a prominent example of the importance of defined
More informationAsymptotics for ruin probabilities in a discretetime risk model with dependent financial and insurance risks
1 Asymptotics for ruin probabilities in a discretetime risk model with dependent financial and insurance risks Yang Yang School of Mathematics and Statistics, Nanjing Audit University School of Economics
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationValuation of guaranteed annuity options in affine term structure models. presented by. Yue Kuen KWOK. Department of Mathematics
Valuation of guaranteed annuity options in affine term structure models presented by Yue Kuen KWOK Department of Mathematics Hong Kong University of Science & Technology This is a joint work with Chi Chiu
More informationMultistate Models in Health Insurance
Multistate Models in Health Insurance Marcus Christiansen Preprint Series: 201015 Fakultät für Mathematik und Wirtschaftswissenschaften UNIVERSITÄT ULM Multistate models in health insurance Marcus C.
More informationErmanno Pitacco. University of Trieste (Italy) ermanno.pitacco@deams.units.it 1/38. p. 1/38
p. 1/38 Guarantees and product design in Life & Health Insurance Ermanno Pitacco University of Trieste (Italy) ermanno.pitacco@deams.units.it 1/38 p. 2/38 Agenda Introduction & Motivation Weakening the
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More informationVasicek Single Factor Model
Alexandra Kochendörfer 7. Februar 2011 1 / 33 Problem Setting Consider portfolio with N different credits of equal size 1. Each obligor has an individual default probability. In case of default of the
More informationLife Assurance (Provision of Information) Regulations, 2001
ACTUARIAL STANDARD OF PRACTICE LA8 LIFE ASSURANCE PRODUCT INFORMATION Classification Mandatory MEMBERS ARE REMINDED THAT THEY MUST ALWAYS COMPLY WITH THE CODE OF PROFESSIONAL CONDUCT AND THAT ACTUARIAL
More informationInternational Stock Market Integration: A Dynamic General Equilibrium Approach
International Stock Market Integration: A Dynamic General Equilibrium Approach Harjoat S. Bhamra London Business School 2003 Outline of talk 1 Introduction......................... 1 2 Economy...........................
More informationMULTIVARIATE PROBABILITY DISTRIBUTIONS
MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models  part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK2800 Kgs. Lyngby
More information. (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i.
Chapter 3 KolmogorovSmirnov Tests There are many situations where experimenters need to know what is the distribution of the population of their interest. For example, if they want to use a parametric
More information