From Ruin Theory to Solvency in NonLife Insurance


 Nelson Lewis
 1 years ago
 Views:
Transcription
1 From Ruin Theory to Solvency in NonLife Insurance Mario V. Wüthrich RiskLab ETH Zurich & Swiss Finance Institute SFI January 23, 2014 LUH Colloquium Versicherungs und Finanzmathematik Leibniz Universität Hannover
2 Aim of this presentation We start from Lundberg s thesis (1903) on ruin theory and modify his model step by step until we arrive at today s solvency considerations surplus process 1
3 CramérLundberg model Consider the surplus process (C t ) t 0 given by where L t = N t i=1 c 0 0 π > 0 Y i 0 C t = c 0 + πt N t i=1 initial capital, premium rate, Y i, Harald Cramér homogeneous compound Poisson claims process, satisfying the net profit condition (NPC): π > E[L 1 ]. 2
4 Ultimate ruin probability The ultimate ruin probability for initial capital c 0 0 is given by [ ] [ ] ψ(c 0 ) = P inf C t < 0 t R + C 0 = c 0 = P c0 inf C t < 0, t R + i.e. this is the infinite time horizon ruin probability. 10 Under (NPC): ψ(c 0 ) < 1 for all c surplus process 3
5 Lundberg s exponential bound Assume (NPC) and that the Lundberg coefficient γ > 0 exists. Then, we have exponential bound ψ(c 0 ) exp{ γc 0 }, for all c 0 0 (large deviation principle (LDP)). This is the lighttailed case, i.e. for the existence of γ > 0 we need exponentially decaying survival probabilities of the claim sizes Y i, because we require E[exp{γY i }] <. Filip Lundberg 4
6 Subexponential case Von Bahr, Veraverbeke, Embrechts investigate the heavytailed case. In particular, for Y i i.i.d. Pareto(α > 1) and (NPC): ψ(c 0 ) const c α+1 0 as c 0. Heavytailed case provides a much slower decay Paul Embrechts gamma surplus process Pareto surplus process 5
7 Discrete time ruin considerations 10 Insurance companies cannot continuously control their surplus processes (C t ) t gamma surplus process They close their books and check their surplus on a yearly time grid. Consider the discrete time ruin probability [ ] [ ] P c0 inf C n < 0 P c0 inf C t < 0 n N 0 t R + = ψ(c 0 ). This leads to the study of the random walk (C n c 0 ) n N0 for (discrete time) accounting years n N 0. 6
8 Oneperiod ruin problem Insured buy oneyear nonlife insurance contracts: why bother about ultimate ruin probabilities? gamma surplus process Moreover, initial capital c 0 0 needs to be readjusted every accounting year. Consider the (discrete time) oneyear ruin probability [ ] [ ] P c0 [C 1 < 0] P c0 inf C n < 0 P c0 inf C t < 0 n N 0 t R + = ψ(c 0 ). This leads to the study of the surplus C 1 = c 0 + π N 1 i=1 Y i at time 1. 7
9 Oneperiod problem and real world considerations Why do we study so complex models when the real world problem is so simple? gamma surplus process Total asset value at time 1: A 1 = c 0 + π. Total liabilities at time 1: L 1 = N 1 i=1 Y i. C 1 = c 0 + π N 1 i=1 Y i = A 1 L 1??? 0. (1) There are many modeling issues hidden in (1)! We discuss them step by step. 8
10 ValueatRisk (VaR) risk measure C 1 = A 1 L 1??? 0. ValueatRisk on confidence level p = 99.5% (Solvency II): choose c 0 minimal such that P c0 [C 1 0] = P [A 1 L 1 ] = P [L 1 c 0 π 0] p. Freddy Delbaen Choose other (normalized) risk measures ϱ : M L 1 (Ω, F, P) R and study ϱ(l 1 A 1 ) = ϱ (L 1 c 0 π)??? 0, where implies SOLVENCY w.r.t. risk measure ϱ. 9
11 Asset return and financial risk (1/2) Initial capital at time 0: c 0 0. Premium received at time 0 for accounting year 1: π > 0. Total asset value at time 0: a 0 = c 0 + π > 0. This asset value a 0 is invested in different assets k {1,..., K} at time 0. asset classes cash and cash equivalents debt securities (bonds, loans, mortgages) real estate & property equity, private equity derivatives & hedge funds insurance & reinsurance assets other assets 10
12 Asset return and financial risk (2/2) Choose an asset portfolio x = (x 1,..., x K ) R K at time 0 with initial value a 0 = K k=1 x k S (k) 0, where S (k) t is the price of asset k at time t. This provides value at time 1 A 1 = K k=1 x k S (k) 1 = a 0 (1 + w R 1 ), for buy & hold asset strategy w = w(x) R K and (random) return vector R 1. ϱ(l 1 A 1 ) = ϱ (L 1 a 0 (1 + w R 1 ))??? 0. where implies solvency w.r.t. risk measure ϱ and business plan (L 1, a 0, w). 11
13 Insurance claim (liability) modeling (1/2) MAIN ISSUE: modeling of insurance claim L 1 = N 1 i=1 Y i. Insurance claims are neither known nor can immediately be settled at occurrence! premium π accident date reporting date claims closing insurance period (accounting year 1) claims cash flows X = ( X 1, X 2, ) time Insurance claims of accounting year 1 generate insurance liability cash flow X: X = (X 1, X 2,...) with X t being the payment in accounting year t. Question: How is the cash flow X related to the insurance claim L 1? 12
14 Insurance claim (liability) modeling (2/2) premium π accident date reporting date claims closing insurance period (accounting year 1) claims cash flows X = ( X 1, X 2, ) time Main tasks: cash flow X = (X 1, X 2,...) modeling, cash flow X = (X 1, X 2,...) prediction, cash flow X = (X 1, X 2,...) valuation, using all available relevant information: exactly here the oneperiod problem turns into a multiperiod problem. 13
15 Bestestimate reserves Choose a filtered probability space (Ω, F, P, F) with filtration F = (F t ) t N0 and assume cash flow X is Fadapted. 1st attempt to define L 1 (interpretation of Solvency II): L 1 = X 1 + s 2 P (1, s) E [X s F 1 ], where E [X s F 1 ] is the bestestimate reserve (prediction) of X s at time 1; P (1, s) is the zerocoupon bond price at time 1 for maturity date s. Note that L 1 is F 1 measurable, i.e. observable w.r.t. F 1 (information at time 1). 14
16 1st attempt to define L 1 L 1 = X 1 + s 2 P (1, s) E [X s F 1 ]. (2) Issue: Solvency II asks for economic balance sheet, but L 1 is not an economic value. (a) Risk margin is missing: any riskaverse risk bearer asks for such a (profit) margin. (b) Zerocoupon bond prices and claims cash flows X s, s 2, may be influenced by the same risk factors and, thus, there is no decoupling such as (2). 15
17 2nd attempt to define L 1 Choose an appropriate stateprice deflator ϕ = (ϕ t ) t 1 and L 1 = X 1 + s 2 1 ϕ 1 E [ϕ s X s F 1 ]. ϕ = (ϕ t ) t 1 is a strictly positive, a.s., and Fadapted. ϕ = (ϕ t ) t 1 reflects price formation at financial markets, in particular, P (1, s) = 1 ϕ 1 E [ϕ s F 1 ]. Hans Bühlmann If ϕ s and X s are positively correlated, given F 1, then L 1 X 1 + s 2 P (1, s) E [X s F 1 ]. 16
18 Solvency at time 0 Choose a filtered probability space (Ω, F, P, F) such that it caries the random vectors ϕ (stateprice deflator), R 1 (returns of assets) and X (insurance liability cash flows) in a reasonable way. The business plan (X, a 0, w) is solvent w.r.t. the risk measure ϱ and stateprice deflator ϕ if ϱ(l 1 A 1 ) = ϱ X 1 + s 2 1 E [ϕ s X s F 1 ] a 0 (1 + w R 1 ) 0. ϕ 1 Thus, it is likely (measured by ϱ and ϕ) that the liabilities L 1 are covered by assets A 1 at time 1 in an economic balance sheet. 17
19 Acceptability arbitrage The choice of the stateprice deflator ϕ and the risk measure ϱ cannot be done independently of each other: ϕ describes the risk reward; ϱ describes the risk punishment. Assume there exist acceptable zerocost portfolios Y with E[ϕ Y] = 0 and ϱ Y 1 + s 2 1 E [ϕ s Y s F 1 ] < 0. ϕ 1 P. Artzner Then, unacceptable positions can be turned into acceptable ones just by loading on more risk = acceptability arbitrage. Reasonable solvency models (ϕ, ϱ) should exclude acceptability arbitrage, see Artzner, Delbaen, Eisele, KochMedina. 18
20 Asset & liability management (ALM) The business plan (X, a 0, w) is solvent w.r.t. risk measure ϱ and stateprice deflator ϕ if ϱ(l 1 A 1 ) = ϱ X 1 + s 2 1 ( E [ϕ s X s F 1 ] a w ) R 1 0. ϕ 1 ALM optimize this business plan (X, a 0, w): Which asset strategy w R K minimizes the capital a 0 = c 0 + π and we still remain solvent? This is a nontrivial optimization problem. Of course, we need to exclude acceptability arbitrage, which may also provide restrictions on the possible asset strategies w = eligible assets. 19
21 Summary of modeling tasks Provide reasonable stochastic models for R 1, X and ϕ (yield curve extrapolation). What is a reasonable profit margin for risk bearing expressed by ϕ? Which risk measure(s) ϱ should be preferred? ( Noacceptability arbitrage!) Modeling is often split into different risk modules: (financial) market risk insurance risk (underwriting and reserve risks) credit risk operational risk Issue: dependence modeling and aggregation of risk modules. Aggregation over different accounting years and lines of business? 20
22 Dynamic considerations Are we happy with the above considerations? Not entirely! Liability runoff is a multiperiod problem: We also want sensible dynamic behavior. This leads to the consideration of multiperiod problems and supermartingales. 21
Embedded Value in Life Insurance
Embedded Value in Life Insurance Gilberto Castellani Università di Roma La Sapienza Massimo De Felice Università di Roma La Sapienza Franco Moriconi Università di Perugia Claudio Pacati Università di Siena
More informationModelling the Claims Development Result for Solvency Purposes
Modelling the Claims Development Result for Solvency Purposes Michael Merz, Mario V. Wüthrich Version: June 10, 008 Abstract We assume that the claims liability process satisfies the distributionfree
More informationInfinite Mean Models and the LDA for Operational Risk
Infinite Mean Models and the LDA for Operational Risk Johanna Nešlehová RiskLab, Department of Mathematics ETH Zurich Rämistrasse 101 CH8092 Zurich, Switzerland Tel.: +41 44 632 3227 email: johanna@math.ethz.ch
More informationQuantifying Risk in the Electricity Business: A RAROCbased Approach
Quantifying Risk in the Electricity Business: A RAROCbased Approach M.Prokopczuk, S.Rachev and S.Trück October 11, 2004 Abstract The liberalization of electricity markets has forced the energy producing
More informationOneyear reserve risk including a tail factor : closed formula and bootstrap approaches
Oneyear reserve risk including a tail factor : closed formula and bootstrap approaches Alexandre Boumezoued R&D Consultant Milliman Paris alexandre.boumezoued@milliman.com Yoboua Angoua NonLife Consultant
More informationProperty & Casualty Simulation Game. Reinsurance. Peter Liebwein Keith Purvis. riva 20002005 Version 2.1E
Property & Casualty Simulation Game Reinsurance Peter Liebwein Keith Purvis riva 20002005 Version 2.1E Contents Reinsurance Basics 2 What is Reinsurance?...2 Why Do Insurance Companies Need Reinsurance?...2
More informationCEIOPS Preparatory Field Study for Life Insurance Firms. Summary Report
CEIOPSFS08/05 S CEIOPS Preparatory Field Study for Life Insurance Firms Summary Report 1 GENERAL OBSERVATIONS AND CONCLUSIONS 1.1 Introduction CEIOPS has been asked to prepare advice for the European
More informationST334 ACTUARIAL METHODS
ST334 ACTUARIAL METHODS version 214/3 These notes are for ST334 Actuarial Methods. The course covers Actuarial CT1 and some related financial topics. Actuarial CT1 which is called Financial Mathematics
More informationMIT Sloan Finance Problems and Solutions Collection Finance Theory I Part 1
MIT Sloan Finance Problems and Solutions Collection Finance Theory I Part 1 Andrew W. Lo and Jiang Wang Fall 2008 (For Course Use Only. All Rights Reserved.) Acknowledgements The problems in this collection
More informationAN INSTITUTIONAL EVALUATION OF PENSION FUNDS AND LIFE INSURANCE COMPANIES
AN INSTITUTIONAL EVALUATION OF PENSION FUNDS AND LIFE INSURANCE COMPANIES DIRK BROEDERS, AN CHEN, AND BIRGIT KOOS Abstract. This paper analyzes two different types of annuity providers. In the first case,
More informationQuantifying Risk in the Electricity Business: A RAROCbased Approach
Quantifying Risk in the Electricity Business: A RAROCbased Approach Marcel Prokopczuk a, Svetlozar T. Rachev a,b,2, Gero Schindlmayr c and Stefan Trück a,1 a Institut für Statistik und Mathematische Wirtschaftstheorie,
More informationSCOR Papers. Securitization, Insurance, and Reinsurance 1. 1. Introduction. Abstract. July 2009 N 5
July 2009 N 5 SCOR Papers Securitization, Insurance, and Reinsurance 1 By J. David Cummins and Philippe Trainar 2 May 8, 2009 Abstract This paper considers strengths and weaknesses of reinsurance and securitization
More informationStop Loss Reinsurance
Stop Loss Reinsurance Stop loss is a nonproportional type of reinsurance and works similarly to excessofloss reinsurance. While excessofloss is related to single loss amounts, either per risk or per
More informationOn Numéraires and Growth Optimum Portfolios an expository note
On Numéraires and Growth Optimum Portfolios an expository note Theo K. Dijkstra Department of Economics, Econometrics & Finance University of Groningen Groningen, The Netherlands t.k.dijkstra@rug.nl working
More informationDOES REINSURANCE NEED REINSURERS?
C The Journal of Risk and Insurance, 2006, Vol. 73, No. 1, 153168 DOES REINSURANCE NEED REINSURERS? Guillaume Plantin ABSTRACT The reinsurance market is the secondary market for insurance risks. It has
More informationIASB/FASB Meeting Week beginning 16 May 2011. 1. This paper discusses the accounting for reinsurance and asks the boards to decide whether to:
IASB/FASB Meeting Week beginning 16 May 2011 IASB Agenda reference 3J FASB Agenda Staff Paper reference 68J Contact(s) Jennifer Weiner jmweiner@fasb.org +1 (203) 9565305 Adam Lindemuth alindemuth@fasb.org
More informationCHAPTER 26 VALUING REAL ESTATE
1 CHAPTER 26 VALUING REAL ESTATE The valuation models developed for financial assets are applicable for real assets as well. Real estate investments comprise the most significant component of real asset
More informationEuropean Commission. 31 January 2002 Contract no: ETD/2000/BS3001/C/44. KPMG This report contains 157 pages Appendices contain 16 pages mk/nb/552
European Commission Study into the methodologies for prudential supervision of reinsurance with a view to the possible establishment of an EU framework 31 January 2002 Contract no: ETD/2000/BS3001/C/44
More informationINTRODUCTION RESERVING CONTEXT. CHRIS DAYKIN Government Actuary GAD, United Kingdom
CHRIS DAYKIN Government Actuary GAD, United Kingdom INTRODUCTION In discussing the principles of reserving for annuity business, attention needs to be paid to the purpose of the reserving exercise. It
More informationStarr Insurance & Reinsurance Limited and Subsidiaries
Starr Insurance & Reinsurance Limited and Subsidiaries Financial Statements Table of Contents Page Independent Auditors Report 1 Financial Statements Consolidated Balance Sheet 2 Consolidated Statement
More informationInvesting for the Long Run when Returns Are Predictable
THE JOURNAL OF FINANCE VOL. LV, NO. 1 FEBRUARY 2000 Investing for the Long Run when Returns Are Predictable NICHOLAS BARBERIS* ABSTRACT We examine how the evidence of predictability in asset returns affects
More informationAsset Allocation with Annuities for Retirement Income Management
Asset Allocation with Annuities for Retirement Income Management April 2005 Paul D. Kaplan, Ph.D., CFA Vice President, Quantitative Research Morningstar, Inc. 225 West Wacker Drive Chicago, IL 60606 The
More informationBINOMIAL OPTION PRICING
Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex optionpricing
More informationINSURANCE AND REINSURANCE
LIFE ASSURANCE RISKS By Mamadou FAYE Deputy Managing Director, Millénium Assurances Internationales, Abidjan Article 328 of the insurance code, which lists out the main classes of insurance, defines life
More informationDISCUSSION PAPER PI1303
DISCUSSION PAPER PI1303 Adjusted Money s Worth Ratios In Life Annuities Jaime Casassus and Eduardo Walker February 2013 ISSN 1367580X The Pensions Institute Cass Business School City University London
More informationDraft: Life insurance mathematics in discrete time
1 Draft: Life insurance mathematics in discrete time Tom Fischer Darmstadt University of Technology, Germany Lecture at the METU Ankara, Turkey April 1216, 2004 2 A recent version of the lecture notes
More informationA new strategy to guarantee retirement income using TIPS and longevity insurance
Financial Services Review 18 (2009) 53 68 A new strategy to guarantee retirement income using TIPS and longevity insurance S. Gowri Shankar Business Administration Program, University of Washington, Bothell,
More informationBuy and Maintain: A smarter approach to credit portfolio management
FOR INSTITUTIONAL/WHOLESALE OR PROFESSIONAL CLIENTS ONLY NOT FOR RETAIL USE OR DISTRIBUTION INVESTMENT GLOBAL INSURANCE SOLUTIONS Buy and Maintain: A smarter approach to credit portfolio management SEPTEMBER
More informationA PRACTICAL GUIDE TO THE CLASSIFICATION OF FINANCIAL INSTRUMENTS UNDER IAS 32 MARCH 2013. Liability or equity?
A PRACTICAL GUIDE TO THE CLASSIFICATION OF FINANCIAL INSTRUMENTS UNDER IAS 32 MARCH 2013 Liability or equity? Important Disclaimer: This document has been developed as an information resource. It is intended
More informationFinancialfacts. London Life participating life insurance. Accountability Strength Performance
2014 Financialfacts London Life participating life insurance Accountability Strength Performance This guide provides key financial facts about the management, strength and performance of the London Life
More information