ebook img

Download Entire Volume - International Actuarial Association PDF

148 Pages·2015·6.89 MB·English
by  
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Download Entire Volume - International Actuarial Association

ISSN 0515-0361 Volume 32, No 2 November 2002 A S T I N B U L L E T I N The Journal of the ASTIN and AFIR Sections of the International Actuarial Association CONTENTS EDITORIAL H. NNAMLHOB New Math for Life Actuaries 209 :SROTIDE A. SNRIAC ASTIN Bulletin Online 212 Andrew Cairns Paul Embrechts ARTICLES .S WANG A lasrevinU Framework rof gnicirP laicnaniF and ecnarusnI sksiR 213 :SROTIDE-OC W. HORLIMANN Stephen Philbrick lacitylanA Bounds rof Two ksiR-ta-eulaV slanoitcnuF 235 John Ryan .S ,NESSUMSA .F ,MARVA M. LEBASU Edangian Approximations for Finite-Horizon Ruin Probabilities 267 K.TH. ,SSEH A. ,DLAWEIL K.D. TDIMHCS LAIROTIDE "DRAOB An Extension of Panjer's Recursion 283 Alois Gisler D.C.M. ,NOSKCID H.R. SRETAW The Distribution of the Time to Ruin in the Classical Risk Model 299 Marc Goovaerts .P OCIREV Mary Hardy Bonus-Malus Systems: "Lack of Transparency" and Adequacy Measure 315 Ole Hesselager WORKSHOP Christian Hipp I.M.E ORIEDROC Transition Intensities for a Model for Permanent Health Insurance 319 Jean Lemaire Gary Parker BOOK REVIEW Jukka Rantala A. SNRIAC K. Sandmann and EJ. Sch6nbucher (ed~): Advances in Finance and Axel Reich Stochasticg Essays in Honour of Dieter Sondermann 347 Robert Reitano MISCELLANEOUS Shann Wang The Actuarial Meetings in Cancun 349 International AFIR Colloquium 2003 350 Invitation to the ASTIN Colloquium 2003 in Berlin/Germany 351 PEETERS EDITORIAL POLICY ASTIN BULLETIN started in 1958 as a journal providing an outlet for actuarial studies in non-life insurance. Since then a well-established non-life methodology has resulted, which is also applicable to other fields of insurance. For that reason AST1N BULLETIN has always published papers written from any quantitative point of view - whether actuarial, econometric, engineering, mathematical, statistical, etc. - attacking theoretical and applied problems in any field faced with elements of insurance and risk. Since the foundation of the AFIR section of IAA, i.e. since 1988, ASTIN BULLETIN has opened its editorial policy to include any papers dealing with financial risk. We especially welcome papers opening up new areas of interest to the international actuarial profession. AST1N BULLETIN appears twice a year (May and November). Details concerning submission of manuscripts are given on the inside back cover. MEMBERSHIP ASTIN and AFIR are sections of the International Actuarial Association (IAA). Membership is open automatically to all IAA members and under certain conditions to non-members also. Applications for membership can be made through the National Correspondent or, in the case of countries not represented by a national correspondent, through a member of the Committee of ASTIN. Members of ASTIN and AFIR receive ASTIN BULLETIN as part of their annual subscription. SUBSCRIPTION AND BACK ISSUES Subscription price: 80 Euro. Payments should be sent to Peeters, Bondgenotenlaan 153, B-3000 Leuven, Belgium. To place your order or make an inquiry please contact: Peeters, Journals Department, Bondgenoten- laan 153, B-3000 Leuven, Belgium or e-mail: [email protected] Orders are regarded as binding and payments are not refundable. Subscriptions are accepted and entered by the volume. Claims cannot be made 4 months after publication or date of order, whichever is later. INDEX TO VOLUMES 1-27 The Cumulative Index to Volumes 1-27 is also published for AST1N by Peeters at the above address and is available for the price of 08 Euro. Copyright © 2002 SRETEEP NEW MATH FOR LIFE ACTUARIES The fact The model used in the technique of the Life Actuary is built on i) probabilities of insured events, e.g. death, survival, disablement, just to mention the most common examples for such events, ii) time value of money. The model has been in use since more than 200 years (1762 Old Equitable) hence its practicability has been proved. Beyond that it has been the basis for insurance legislation all over the world. Hence the model is iron cast by law to be the right one. The challenge All models are wrong. Still we all know that they can nevertheless serve their purpose if they are applied with care. That means, if we add what the prac- titioners call "experience" and what the academics call "understanding of the limitations of the model" they become very valuable and useful tools. The question hence arises by how much a useful model may be wrong. My point is that for dealing with the time value of money (item ii) above the classical actu- arial technique is nowadays so far off economic reality that it needs to be fundamentally revised. Here is a program for revision. The ideas are not new. One finds them in several scientific papers, some also published in the ASTIN Bulletin. My aim is to communicate these ideas more widely such that they will be applied and used. 1) Recognize that Life Assurance is always expressed in units. In private insurance (social insurance needs another definition of units) these units can always be thought of as units of a (possibly dynamic) portfolio. Such units can contain guarantees, one only needs to add the appropriate (Euro- pean) options to the portfolio. As a special case, if the units are defined as the unit of the local currency, the portfolio is made up by Zero Coupon bonds plus options for the contractual guarantees. 2) Understand that in the language of units one unit always remains one unit. This has the great advantage that we can use the classical actuarial formu- lae at interest rate zero. This is the first discovery: The technical interest rate which good actuaries have always handled with suspicion has disap- peared! The second discovery is even more convenient: The present value (in local currency) defined as the value of a corresponding (called replicat- ing) portfolio can always (in principle) be read in todays newspaper! (at least if it has a reasonable financial section). ASTIN BULLETIN. Vol. 32, No. 2, 2002, pp. 209-211 210 SNAH NNAMLH3(B OT the non initiated the above may sound too simple and possibly more risky than how Life Assurance typically operates now. Let me address these two points. Simplicity In this new approach the actuary still needs to construct life tables, multiple decrement tables, probabilities for disablement etc. The only difference in his work is the fact that he can apply all his actuarial techniques calculating at interest rate zero. The more difficult part is the construction (in thought) of the portfolio which replicates the assured unit. Maybe some of us know already how to do that. If you don't, ask a Financial Engineer, but insist that he com- municates to you exactly how this portfolio needs to be constructed and man- aged. He must in principle come up with the exact instructions how the Invest- ment Department (if they wanted) could invest in the replicating portfolio. Hence -- in all fairness -- the suggested new treatment of the time value of money is indeed conceptually simple to understand, but the practical implementation has its intricacies. So, nobody should fear to lose his or her job, if the new ideas should be applied in practice. Riskiness The financial risk (which is the most important for any Life Assurer) derives in the new approach from the fact that in reality Life Assurers often do not invest in the replicating portfolio. They may act so in the hope to make more profit by investing differently. If you replace "replicating portfolio" by the tradition- ally named "matching assets strategy", the same remark applies for the classi- cal actuarial model. If however with the new approach the Life Insurer invests exactly in the replicating portfolio then the Financial Risk is Zero. The classi- cal actuarial model -- even under a matching assets strategy -- can never achieve this since it does not account for contractual guarantees (most impor- tant in practice the guarantee of the technical interest rate). It is exactly this fact, which urges for a change in our cherished traditional actuarial model. How could the change happen? )1 My appeal is first to my colleagues teaching actuarial science: Change the basis of your teaching along the principles outlined above. For the young generation of actuaries and financial engineers the new approach should become the obvious one. Therefore my title NEW MATH which could also be read NUMAT namely NUMeraire based Actuarial Teaching (Numeraire = Value of unit) NEW MATH FOR LIFE ACTUARIES 211 By the way, the time needed to explain the "numeraire" you can gain by omitting "compound interest". 2) To the actuarial consultants it should be obvious how to use the proposed approach for calculating Embedded Value. This would then also provide a common basis for such calculations hence lead to identical results -- inde- pendent of who of us performs this calculation. 3) It is not nay intention to make any proposals regarding legal and/or account- ing rules. They rely on a process of consensus hence changes will always take time. Still from a risk management point of view one has to notice that the numeraire which has value 1 at time zero and value 1( + i) n at each time n is -- theoretically not replicable. HANS BIHLMANN PS My aim was to communicate ideas. Some fine points have been omitted on purpose -- e.g. how to deal with surrender values. Such guarantees having the character of an American option can also be dealt with by the approach as outlined in my editorial, but need a technically different treatment than those guarantees which can be replicated by European options. A UNIVERSAL FRAMEWORK FOR PRICING FINANCIAL AND INSURANCE RISKS BY SHAUN S. WANG ABSTRACT This paper presents a universal framework for pricing financial and insurance risks. Examples are given for pricing contingent payoffs, where the underlying asset or liability can be either traded or not traded. The paper also outlines an application of the framework to prescribe capital allocations within insurance companies, and to determine fair values of insurance liabilities. INTRODUCTION Currently there is a pressing need for a universal framework for the deter- mination of the fair value of financial and insurance risks. In the insurance industry, this need is evident in the Society of Actuaries' "Symposium on Fair Value of Liabilities", and in the Casualty Actuarial Society's "Risk Premium Project" and "Task Force on Fair Valuing P/C Insurance Liabilities". In the financial services industry, this pressing need is evidenced by the recent Basel Accords on regulatory risk management that require fair value, analogous to market prices, to be applied to all assets or liabilities, whether traded or not, on or off the balance sheet. In light of all these current events, this paper addresses a very timely subject. The paper is comprised of three parts, summarized as follows: Part One: The Framework introduces a new transform and correlation mea- sure that extends CAPM to pricing all kinds of assets and liabilities, having any type of probability distribution, whether traded or underwritten, in finance or insurance. This transform is just as easily applied to contingent payoffs that are co-monotone with their underlying assets or liabilities. In its simplest form, the new transform relies on a parameter called the "market price of risk", extending a familiar concept in CAPM to risks with non-normal distributions. The "market price of risk" can either be applied to, 1 SCOR, One Pierce Place, Itasca, IL 60143, E-mail: [email protected] NITSA ,NITELLUB .loV ,23 .oN ,2 ,2002 .pp 432-312 214 NUAHS .S GNAW or implied from, a distribution, in order to arrive at a "risk-adjusted price" for the underlying risk in question. The "market price of risk" increases continuously with duration, and is consistent at each horizon date between an underlying and its co-monotone contingent payoff. When the return for an underlying asset has a normal distribution, the new transform replicates the CAPM price for that underlying asset, and recovers the Black-Scholes price for options on that underlying asset. Part Two: Examples of Pricing Contingent Payoffs illustrates the application of the new framework to pricing call options on traded stocks, and to pricing weather derivatives. Part Three: Capital Allocation & Fair Values of Liabilities illustrates the appli- cation of the new framework to insurance company capital allocations, and to the determination of fair values of insurance liabilities. In particular, it addresses a challenging issue concerning the long-term duration of liabilities. Also, the framework is equally applicable to primary insurance business and excess-of-loss reinsurance when calculating fair values of liabilities. TRAP ONE. THE KROWEMARF Capital Asset Pricing Model CAPM is a set of predictions concerning equilibrium expected returns on assets. Classic CAPM assumes that all investors have the same one-period horizon, and asset returns have multivariate normal distributions. For a fixed time horizon, let ~R and RM be the rate-of-return for asset i and the market portfo- lio M, respectively. Classic CAPM asserts that ERh.l= r + fli{ERM-r}, where r is the risk-free rate-of-return and Cov Ri , MR ~i-- 2 G M is the beta of asset i. Assuming that asset returns are normally distributed and the time horizon is one period (e.g., one year), a key concept in financial economics is the market price of risk: i2 _ ERi- r 71 i In asset portfolio management, this is also called the Sharpe Ratio, after Wil- liam Sharpe. A FRAMEWORK FOR PRICING FINANCIAL AND INSURANCE RISKS 215 In terms of market price of risk, CAPM can be restated as follows: ~i El~.l-r CovRi,RM ERM-F = i7¢ : i~¢ MT~ (7 M -- Pi, M" ~M~t where Pi, M is the linear correlation coefficient between Ri and R M. In other words, the market price of risk for asset i is directly proportional to the cor- relation coefficient between asset i and the market portfolio M. CAPM provides powerful insight regarding the risk-return relationship, where only systematic risk deserves an extra risk premium in an efficient market. However, CAPM and the concept of "market price of risk" were developed under the assumption of multivariate normal distributions for asset returns. CAPM has serious limitations when applied to insurance pricing when loss distributions are not normally distributed. In the absence of an active market for insurance liabilities, the underwriting beta by line of business has been dif- ficult to estimate. Option Pricing Theory Besides CAPM, another major financial pricing paradigm is modern option pricing theory, first developed by Fischer Black and Myron Scholes (1973). Some actuarial researchers have noted that the payoff functions of a Euro- pean call option and a stop-loss reinsurance contract are similar, and have pro- posed an "option-pricing" approach to pricing insurance risks. Unfortunately, the Black-Scholes formula only applies to lognormal distributions of market returns, whereas actuaries work with a large array of distributional forms. Furthermore, there are subtle differences between option pricing and actu- arial pricing (see Mildenhall, 2000). One way to better appreciate the differ- ence between "financial asset pricing" and "insurance pricing", is to recognize the difference in types of data available for pricing. Options pricing is performed in a world of Q-measure (using risk-adjusted probabilities), where the available data consists of observed market prices for related financial assets. On the other hand, actuarial pricing is conducted in a world of P-measure (using objective probabilities), where the available data consists of projected losses, whose amounts and likelihood need to be con- verted to a "fair value" price (see Panjer et al, 1998). Because of this difference, the price of an option is determined from the minimal cost of setting up a hedging portfolio, whereas the price of insurance is based on the actuarial present value of costs, plus an additional risk pre- mium for correlation risk, parameter uncertainty and cost of capital. A Universal Pricing Method Consider a financial asset or liability over a time horizon 0, 7. Let X = Xr denote its future value at time t = T, with a cumulative distribution function 216 NUAHS .S GNAW (cdf) F(x)=Pr{X< x}. In Wang (2000), the author proposed a universal pricing method based on the following transform: F * )x( : *~-l ))x(F( + ,2 (I) erehw )1( si eht standard normal evitalumuc .noitubirtsid ehT yek retemarap 2 si dellac eht tekram ecirp of ,ksir gnitcelfer eht level of citametsys risk. ehT mrofsnart (i) si won better known sa eht ,gnaW mrofsnart among laicnanif sreenigne dna risk .sreganam ehT gnaW mrofsnart saw partly deripsni yb eht krow of lareves tnenimorp lairautca ,srehcraeser gnidulcni Gary retneV ,1991( )8991 dna treboR cistuB .)9991( roF a nevig tessa X htiw cdf ,)x(F eht gnaW mrofsnart lliw ecudorp a -ksir" "detsujda cdf .)x(*F ehT mean eulav under ,)x(*F detoned yb ,X*E lliw enifed a detsujda-ksir "fair "eulav of X ta time ,T hcihw nac eb further detnuocsid to emit ,orez gnisu eht eerf-ksir tseretni .etar ehT gnaW mrofsnart si ylriaf easy to yllaciremun .etupmoc Many erawtfos segakcap evah both )1( dna )1( I- sa ni-tliub .snoitcnuf In tfosorciM ,lecxE )y()1( nac eb detaulave yb )y(TSIDSMRON dna )z(1-)1( nac eb detaulave yb -MRON SINV(z). One fortunate property of the Wang transform is that normal and lognor- mal distributions are preserved: • If F has a Normal(g, o )2 distribution, F* is also a normal distribution with *t.I = ~ - ~,o and o* = o. • If F has a lognormal(g, o )2 distribution such that In(X) - Normal (la, o2), F* is another lognormal distribution with g* = ~t- ~o and o* = o. Stock prices are often modeled by lognormal distributions, which implies that stock returns are modeled by normal distributions. Equivalent results can be obtained by applying the Wang transform either to the stock price distribution, or, to the stock return distribution. Consider an asset i on a one-period time horizon. Assume that the return ~R for asset i has a normal distribution with a standard deviation of oi. Applying the Wang transform to the distribution of Ri we get a risk-adjusted rate-of- return: E* /~. : ERi-2a .i In a competitive market, the risk-adjusted return for all assets should be equal to the risk-free rate, r. Therefore we can infer that L = (ER~ -r)loi, which is exactly the same as the market price of risk in classic CAPM. With ~ being the market price of risk for an asset, the Wang transform replicates the clas- sic CAPM. Unified Treatment of Assets & Liabilities A liability with loss variable X can be viewed as a negative asset with gain Y= -X, and vice versa. Mathematically, if a liability has a market price of risk ~,

Description:
NEW MATH FOR LIFE ACTUARIES. The fact. The model . risks. Examples are given for pricing contingent payoffs, where the underlying asset or liability can
See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.