Probability finance and insurance

"ze Leung Lai Hailiang Yang Siu Pang Yung PROBABILITY, FINANCE AND INSURANCE Proceedings of a Workshop at the University of Hong Kong PROBABILITY, FINANCE AND INSURANCE This page is intentionally left blank PROBABILITY, FINANCE AND INSURANCE Proceedings of a Workshop at the University of Hong Kong Hong Kong 15 - 17 July 2002 editors Tze Leung Lai Stanford University, USA Hailiang Yang Siu Pang Yung The University of Hong Kong, China Y|j5 World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library PROBABILITY, FINANCE AND INSURANCE Proceedings of a Workshop at the University of Hong Kong Copyright © 2004 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 981-238-853-2 Printed in Singapore PREFACE This volume contains about half of the papers that were presented at the Workshop on Probability with Applications to Finance and Insurance on July 15-17, 2002 at The University of Hong Kong Thirty-one invited speakers from mainland China, Hong Kong, Taiwan, Singapore, Australia and the United States gave talks at the workshop, which brought together many leading researchers in probability theory, stochastic processes, mathematical finance and actuarial science The participants discussed important current issues and exchanged ideas, and there was fruitful interaction between researchers from different fields The workshop was supported by the Institute of Mathematical Research at The University of Hong Kong We are particularly grateful to Professor Ngaiming Mok, the Institute's Director, for suggesting the idea of a workshop of this kind and for his generous support The workshop was also sponsored by the Hong Kong Mathematical Society We also thank Professor Eric Chang of the Department of Finance, Dr Kai Wang Ng of the Department of Statistics and Actuarial Science, Professor Man Keung Siu and the staff of the Department of Mathematics for their help and support We are deeply grateful to Professor Inchi Hu and Dr Bing-Yi Jing of the Hong Kong University of Science and Technology and Professor Qi-Man Shao of the University of Oregon and the National University of Singapore for working closely with us on the Organizing Committee in planning and making arrangements for the workshop We want to express our gratitude to the authors for their careful preparation of the manuscripts, which have made publication of this volume possible, and to Professors Gerold Alsmeyer, Beda Chan, Wai Ki Ching, Wai Keung Li, Qihe Tang and Lixing Zhu for their help in the refereeing process Thanks are also due to Ms Ada Lai and Mimi Lui for their valuable administrative and technical assistance in the preparation of the Proceedings T L Lai, H Yang and S.P Yung Stanford and Hong Kong V LIST OF PARTICIPANTS Andrew Carverhill The University of Hong Kong Hong Kong Eric C Chang The University of Hong Kong Hong Kong Kani Chen Hong Kong University of Science and Technology Hong Kong Ngai-Hang Chan The Chinese University of Hong Kong Hong Kong Sung Nok Chiu Hong Kong Baptist University Hong Kong Kwok Pui Choi National University of Singapore Singapore August Chow Office of the Commissioner of Insurance Hong Kong Cheng-Der Fuh Institute of Statistical Science Academia Sinica Taipei Fu Zhou Gong Institute of Applied Mathematics Chinese Academy of Sciences Beijing Minggao Gu The Chinese University of Hong Kong Hong Kong Inchi Hu Hong Kong University of Science and Technology Hong Kong Yaozhong Hu University of Kansas Lawrence, KS 66045 vu Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei Bing-Yi Jing Hong Kong University of Science and Technology Hong Kong Yue-Kuen Kwok Hong Kong University of Science and Technology Hong Kong Tze Leung Lai Stanford University Stanford, CA 94305 Wai Keung Li The University of Hong Kong Hong Kong Szu-Lang Liao Banking and Financial Markets National Cheng-Chi University Taipei Tiong Wee Lim National University of Singapore Singapore Shiqing Ling Hong Kong University of Science and Technology Hong Kong Ngaiming Mok The University of Hong Kong Hong Kong Kai Wang Ng The University of Hong Kong Hong Kong Qi-Man Shao University of Oregon Eugene OR 97403-1217 Elias Sai Wan Shiu University of Iowa Iowa City, Iowa 52242-1409 Man Keung Siu The University of Hong Kong Hong Kong Chun Su University of Science and Technology of China Hefei Qiying Wang Australian National University Canberra, ACT, 0200 Australia Ching-Zong Wei Institute of Statistical Science Academia Sinica Taipei Michael Wong The Chinese University of Hong Kong Hong Kong Liming Wu Wuhan University Wuhan Kai-Nan Xiang Hunan Normal University Changsha Hailiang Yang The University of Hong Kong Hong Kong Xiao-Guang Yang Institute of System Science Chinese Academy of Sciences Beijing Yi Ching Yao Institute of Statistical Science Academia Sinica Taipei Zhiliang Ying Columbia University New York, NY 10027 Siu Pang Yung The University of Hong Kong Hong Kong Prank Zhang Morgan Stanley New York Lixin Zhang Zhejiang University Hangzhou Xunyu Zhou The Chinese University of Hong Kong Hong Kong CONTENTS Preface v List of Participants vi Limit theorems for moving averages Tze Leung LAI On large deviations for moving average processes 15 Liming WU Recent progress on self-normalized limit theorems 50 Qi-Man SHAO Limit theorems for independent self-normalized sums 69 Bing-Yi JING Phase changes in random recursive structures and algorithms 82 Hsien-Kuei HWANG Iterated random function system: convergence theorems 98 Cheng-Der FUH Asymptotic properties of adaptive designs via strong approximations 112 Li-Xin ZHANG Johnson-Mehl tessellations: asymptotics and inferences 136 Sung Nok CHIU Rapid simulation of correlated defaults and the valuation of basket default swaps 150 Zhifeng ZHANG, Kin PANG, Peter COTTON, Chak WONG and Shikhar RANJAN Optimal consumption and portfolio in a market where the volatility is driven by fractional Brownian motion 164 Yaozhong HU MLE for change-point in ARMA-GARCH models with a changing drift 174 Shiqing LING Dynamic protection with optimal withdrawal 195 Hans U GERBER and Elias Sai Wan SHIU Ruin probability for a model under Markovian switching regime 206 Hailiang YANG and G YIN Heavy-tailed distributions and their applications 218 Chun SU and Qihe TANG The insurance regulatory regime in Hong Kong (with an emphasis on the actuarial aspect) 237 August CHOW ix 228 (1979) and Rozovski (1993) See also the monographs Vinogradov (1994), Gnedenko and Korolev (1996) and Meerschaert and Schemer (2001) Recently, the precise large deviations for the randomly indexed sum (random sum), JV(t) ^w(t) = ^ xk, t > o, fc=l have been investigated by many researchers such as Cline and Hsing (1991), Kluppelberg and Mikosch (1997), Mikosch and Nagaev (1998) and Tang et al (2001), among others Here the random index N(t) is a nonnegative and integer-valued process, independent of the sequence {X, Xk,k > 1} We always suppose that the mean function \(t) —» oo as t —+ oo Most recently, Ng et al (2003) established the large deviations result for a socalled prospective-loss process under a very mild condition on the counting process N(t) We cite here the following result of Theorem 3.1 in Kluppelberg and Mikosch (1997); see also Proposition 7.1 in Mikosch and Nagaev (1998) for a further extension: Proposition 4.1 (Kluppelberg and Mikosch (1997)) Let the common d.f F S ERV(—a, — (3) for some < a < /? < oo If the process N(t) satisfies that N(t)/X(t) —>p Assumption A and that for some e > and any S > 0, Y^ (1 + e)k¥{N(t) > k) = o(l), Assumption B fc>(l+(5)A(t) then for any > 0, we have F (SN{t) - n\(t) > x) ~ X(t)F(x) uniformly for x > 7A(t) (4.2) As verified in Kluppelberg and Mikosch (1997), homogenous Poisson process satisfies both assumptions A and B above Some applications of result (4.2) to insurance and finance can be found in Chapter of Embrechts et al (1997) and some of the above-mentioned references Lately, Su et al (2001) and Tang et al (2001) weakened Assumptions A and B into one as E ((N(t)f+e V(t)>(i+«)A(t))) = O (A(i)) Assumption C 229 for some e > and any S > This is a much weaker condition, which can be satisfied, for example, by the general renewal process and the compound renewal process Recently, we have made some progress on precise large deviations Theorem 4.1 (Ng et al (2004)) If F GC has a finite expectation /x, then for any fixed > 0, it holds that P (Sn - n/j, > x) ~ nF(x) uniformly for x > jn (4.3) Using the method of Kliippelberg and Mikosch (1997), we can extend Theorem 4.1 to the random case To this end, we introduce the following notation Let X be a r.v with a d.f F supported on (—00,00) For any y > we set F.{y) = liminf ^ M x—00 F(x) and then define F*(y) = limsup ^ M i-»oo F(x) n = M{J!£M-.V>l\ to,*£M, (4.4) (4.5) j- = su P (-^M :y > a = _ lim is^M (4.6) [ logy J y^oo logy In the terminology of Bingham et al (1987), here the quantities j £ and J ^ are the upper and lower Matuszewska indices of the function f(x) — (F(x)) , x > Without any confusion, we simply call JJ^ the upper/lower Matuszewska index of the d.f F The latter equalities in (4.5) and (4.6) are due to Theorem 2.1.5 in Bingham et al (1987) Trivially, for any d.f F it holds that < J ^ < j j < 00 But from inequality (2.1.9) in Theorem 2.1.8 in Bingham et al (1987) we see that F € V if and only if J j < 00 Moreover, if F € TZ-a for some a > 0, then J ^ = a For more details of the Matuszewska indices, see Chapter 2.1 of Bingham et al (1987), Cline and Samorodnitsky (1994) and Tang and Tsitsiashvili (2003) Theorem 4.2 (Ng et al (2004)) If F eC has a finite expectation \i and {N(t),t>0} satisfies EAT p (i)I ( N ( t ) > ( + ) A ( t ) ) = O (A(i)) Assumption D for some p > j £ and any > 0, then, the large deviations result (4-2) holds 230 Now we propose a new application for Theorem 4.2 In the context of insurance risk theory, the claim sizes Xk, k > 1, are often assumed to be i.i.d nonnegative r.v.'s with common d.f F Their occurrence times o^, fc > 1, constitute an ordinary renewal counting process N{t)=sup{n:an 0, (4.7) with X(t) = EN(t) < oo for any t > With er0 = 0, we can write the interarrival times by Yfc = ak — o~k-i, k > 1, which therefore form a sequence of i.i.d nonnegative r.v.'s These are standard assumptions in the ordinary renewal model Now we consider the model in Denuit et al (2002) In their model, the fcth insurance policy, k > 1, is associated with a Bernoulli variable Ik, and the variable Ik has a common expectation < q < 1, where q is the claim occurrence probability of the fcth policy, k > Then, the total claim amount up to time t is N(t) s!{t) = j2XkI^ t ^°- (4-8) fc=l We assume that the three sequences {Xk,k > 1}, {Yk,k > 1}, and {Ik, k > 1} are mutually independent and that the sequence {Ik, k > 1} is negatively associated Therefore, Si (t) is a non-standard random sum unless the parameter q = l Generally speaking, a sequence {Ik, k > 1} is said to be negatively associated (NA) if, for any disjoint finite subsets A and B of {1,2, • • •} and any coordinatewise monotonically increasing functions / and g, the inequality Cav\f(Ii]i £ A), < ? ( J j ; j e J ) ] < holds whenever the moment involved exists For details about the notation of NA, please refer to Alam and Saxena (1981), Joag-Dev and Proschan (1983), and Shao (2000), among others We proved the following result: Theorem 4.3 (Ng et al (2004)) If F & C has a finite expectations \x, then it holds for any > and all x > jX(t) that V(Si(t) - M\{t) > x) ~ q\{t)F(x) The finite time ruin probability The following renewal risk model has been extensively investigated in the literature since it was introduced by Sparre Anderson (1957) As described in Section 4.3, the costs of claims Xi} i > 1, form a sequence of i.i.d., 231 nonnegative r.v.'s with common d.f F, and the inter-occurrence times Yi,i > 1, form another sequence of i.i.d nonnegative r.v.'s which are not degenerate at We assume that the sequences {Xi, i > 1} and {Yi, i > 1} are mutually independent The locations of the successive claims constitute a renewal process defined by (4.7) The surplus process of the insurance company is then expressed as N(t) x R(t) =x + ct-^2 i, * > 0, i=l where x > denotes the initial surplus, c > denotes the constant premium rate, and Y^t=i %% — by convention If Y\ is exponentially distributed, hence N(t) is a homogenous Poisson process, the model above is just the classical risk model, which is also called the compound Poisson risk model We define, as usual, the time to ruin of this model with initial surplus x > as T(X) = inf {t > : R(t) < | fl(0) = x} Hence, the probability of ruin within finite time t > can be defined as a bivariate function V>(z; t) = P (T(Z) < t | R{0) =x), (5.1) and the ultimate ruin probability can be defined as ip(x; oo) = lim ip(x; t) = P ( r ( i ) < oo | R{0) = x) t—*oo In order for the ultimate ruin not to be certain, it is natural to assume that the safety loading condition /x = cEYi — MX\ > holds Under the assumption that the equilibrium d.f Fe defined by (1.1) is subexponential, Veraverbeke (1979) and Embrechts and Veraverbeke (1982) established a celebrated asymptotic relation for the ultimate ruin probability that tf>(x;oo)~- r°° — / F(u)du (5.2) As universally admitted, the study of the finite time ruin probability is more practical, but much harder, than that of the infinite time ruin probability The problem of finding accurate approximations to the finite time ruin probability for the renewal model has a long history and during the period many methods have been developed In this section we aim at a simple asymptotic relation for the finite time ruin probability ip(x;t) as x —> oo with special request that this asymptotic 232 relation should be uniform for t in a relevant infinite time interval Under some mild assumptions on the distribution functions of the heavy-tailed claim size X\ and of the inter-occurrence time Y\, applying a recent result by Tang (2004) on the tail probability of the maxima over a finite horizon inspired by Korshunov (2002), we obtained the following result: Theorem 5.1 (Tang (2003)) Consider the renewal risk model with the safety loading condition If F € C with its upper Matuszewska index j £ and EYf < oo for some p > Jf£ + 1, then for arbitrarily given to > such that A(to) > 0, it holds uniformly for t £ [to, oo] that T/>(x;t)~- / F{u)du M Jx Moreover, relation (5.3) can be rewritten as (5.3) (x; t) ~ - / ~F(u)du M Jx if the horizon t is restricted to a smaller region [t(x),oo] for arbitrarily given function t(x) —> oo, where A = 1/EYi The uniformity of the asymptotic relation (5.3) allows us to flexibly vary the horizon t as the initial surplus x increases For example, under the conditions of Theorem 5.1, from (5.3) we can obtain that the relation ip{x;t(x))~- F{u)du (5.4) A* Jx holds for any real function t(x) such that to < t(x) < oo If we specify the d.f F G 7?._Q for some a > 1, a more explicit formula can be derived immediately, which extends Corollary 1.6(a) of Asmussen and Kluppelberg (1996) to the renewal model The uniformity of relation (5.3) also makes it possible to change the horizon t into a random variable as long as it is independent of the risk system, as done by Asmussen et al (2002) and Avram and Usabel (2003) In fact, we have the following consequence of Theorem 5.1: Corollary 5.1 (Tang (2003)) Let the conditions of Theorem 5.1 be valid and let T be an independent r.v with a d.f H satisfying E T < oo and P(T > to) = for some 10 > such that A(t0) > Then it holds that •0(x;T)~EA(T)-B(x) (5.5) 233 It is clear that ip (i; T) = P ( max Y" (Zj - c9i) > x ) (5.6) i=l A similar result as (3.3) has recently been established by Foss and Zachary (2003) However, we remark t h a t our Corollary 5.1 is never a special case of theirs since the r.v N(T) in (5.6) cannot be explained as a stopping time in their sense References Alam, K., Saxena, K.M.L., 1981 Positive dependence in multivariate distributions Comm Statist A-Theory Methods 10, no 12, 1183-1196 Asmussen, S., Avram, F., Usabel, M., 2002 o Erlangian approximations for 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17 Daley, D.J., 2001 The moment index of minima J Appl Probab 38A, 3 36 18 Denuit, M., Lefevre, C , Utev, S., 2002 Measuring the impact of dependence between claims occurrences Insurance Math Econom 30, no 1, 1-19 19 Embrechts, P., Goldie, C M , 1982 On convolution tails Stoch Proc Appl 13, 263-278 20 Embrechts, P., Goldie, C.M., Veraverbeke, N., 1979 Subexponentiality and infinite divisibility Z Wahrsch Verw Gebiete 49, no 3, 335-347 21 Embrechts, P., Kliippelberg, C., Mikosch, T., 1997 Modelling Extremal Events for Insurance and Finance Berlin: Springer 22 Embrechts, P., Veraverbeke, N., 1982 Estimates for the probability of ruin with special emphasis on the possibility of large claims Insurance Math Econom 1, no 1, 55-72 23 Foss, S., Zachary, S., 2003 The maximum on a random time interval of a random walk with long-tailed increments and negative drift Ann Appl Probab 13, no 1, 37-53 24 Gnedenko, B.V., Korolev, V.Y., 1996 Random Summation Limit Theorems and Applications CRC Press 25 Goldie, C M , Kliippelberg, C , 1998 Subexponential distributions A practical Guide to Heavy-Tails: Statistical Techniques and Applications Eds Adler, R., Feldman, R., Taqqu, M.S Birkhauser, Boston 26 Heyde, C C , 1967 a A contribution to the theory of large deviations for sums of independent random variables Z Wahrscheinlichkeitstheorie und Verw Gebiete 7, 303-308 27 Heyde, CC, 1967;, On large deviation problems for sums of random variables which are not attracted to the normal law Ann Math Statist 38, 1575-1578 28 Heyde, CC, 1968 On large deviation probabilities in the case of attraction to a nonnormal stable law Sankhya 30, 253-258 29 Jelenkovic, P.R., Lazar, A.A., 1999 Asymptotic results for multiplexing subexponential on-off processes Adv Appl Prob , 394-421 30 Joag-Dev, K., Proschan, F., 1983 Negative association of random variables with applications Ann Statist 11, 286-295 31 Kaas, R., Tang, Q., 2003 Note on the tail behavior of random walk maxima with heavy tails and negative 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vectors Heavy tails in theory and practice Wiley Series in Probability and Statistics: Probability and Statistics John Wiley & Sons, Inc., New York 38 Mikosch, T., Nagaev, A.V., 1998 Large deviations of heavy-tailed sums with applications in insurance Extremes 1, no 1, 81-110 39 Nagaev, A.V., 1969 a - Integral limit theorems for large deviations when Cramer's condition is not fulfilled I, II Theory Prob Appl 14, 51-64, 193208 40 Nagaev, A.V., 1969;, Limit theorems for large deviations when Cramer's conditions are violated (In Russian) Fiz-Mat Nauk 7, 17-22 41 Nagaev, S.V., 1973 Large deviations for sums of independent random variables In Trans Sixth prague conf on Information Theory, Random Processes and Statistical Decision Functions Academic, Prague 657-674 42 Nagaev, S.V., 1979 Large deviations of sums of independent random variables Ann Prob 7, 745-789 43 Ng, K., Tang, Q., 2004 Asymptotic behavior of sums of subexponential random variables on the real line J Appl Prob no 1, to appear 44 Ng, K., Tang, Q., Yang, H., 2002 Maxima of sums of heavy-tailed random variables Astin Bull 32, no 1, 43-55 45 Ng, K., Tang, Q., Yan, J., Yang, H., 2003 Precise large deviations for the prospective-loss process J Appl Probab 40, no 2, 391-400 46 Ng, K., Tang, Q., Yan, J., Yang, H., 2004 Precise large deviations for sums of random variables with consistently varying tails J in Appl Probab no 1, to appear 47 Rozovski, L.V., 1993 Probabilities of large deviations on the whole axis Theory Probab Appl 38, 53-79 48 Schlegel, S., 1998 Ruin probabilities in perturbed risk models The interplay between insurance, finance and control (Aarhus, 1997) Insurance Math Econom 22, no 1, 93-104 49 Sgibnev, M.S., 1996 On the distribution of the maxima of partial sums Stat Prob Letters 28, 235-238 50 Shao, Q.M., 2000 A comparison theorem on moment inequalities between negatively associated and independent random variables J Theoret Probab 13, no 2, 343-356 51 Sparre Anderson, E., 1957 On the collective theory of risk in the case of contagious between the claims In: Transactions of the XV International Congress of Actuaries 52 Su, C , Hu, Z., 2003 On Two Broad Classes of Heavy-Tailed Distributions 236 Submited to Statistica Sinica 53 Su, C , Tang, Q., 2003 Characterizations on heavy-tailed distributions by means of hazard rate Acta Math Appl Sinica (English Ser.) 19, no 1, 135-142 54 Su, C , Tang, Q., Chen, Y., Liang, H., 2002 Two broad classes of heavytailed distributions and their applications to insurance Obozrenie Prikladnoi i Promyshlenni Matematiki, to appear 55 Su, C , Tang, Q., Jiang, T., 2001 A contribution to large deviations for heavy-tailed random sums Sci China Ser A 44, no 4, 438-444 56 Tang, Q., 2001 Extremal values of risk processes for insurance and finance: with special emphasis on the possibility of large claims Doctoral Thesis, University of Science and Technology of China 57 Tang, Q., 2002 An asymptotic relationship for ruin probabilities under 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of a random walk Stochastic Processes Appl 5, no 1, 27-37 64 Vinogradov, V., 1994 Refined large deviation Limit theorems Longman, Harlow T H E I N S U R A N C E REGULATORY R E G I M E IN HONG KONG (WITH A N EMPHASIS ON THE A C T U A R I A L ASPECT) AUGUST CHOW Office of the Commissioner of Insurance 21/F Queensway Government Offices 66 Queensway, Hong Kong E-mail: augustchow@oci.gov.hk The Insurance Regulatory Regime in Hong Kong In this article, I would like to share with you the insurance regulatory regime in Hong Kong, and in particular some of the actuarial applications for purposes of monitoring the life insurance industry in Hong Kong today Outline This article will cover areas: First, to enable you to have a better understanding as to what we do, I would like to give you an overview of the life insurance market and our insurance regulatory philosophy in Hong Kong Then, I would like to cover the Appointed Actuary System that we have adopted in Hong Kong and the roles played by the actuaries in the market place And finally, I would like to provide you with an example of an application on Reserving for Investment Guarantees with respect to the retirement scheme business sold in Hong Kong An Overview of the Hong Kong Life Insurance Market Compared with other territories in the region, Hong Kong is a free and open city with many insurers competing in the market, and has the highest number of insurers per capita However, the penetration ratio, which is defined as 100% times the number of policies per capita, is only 70% in Hong Kong which is relatively 237 238 low as compared to over 200% in the United States and Japan Therefore, there is still ample room for life insurance to prosper further in Hong Kong Under our prudential regulatory regime, insurers in Hong Kong have the freedom to set premium rates and design products Number of Authorized Life Insurers by Place of Incorporation as of 30 June 2002 At the end of June 2002, there were 48 life insurers and 18 composite insurers authorized to write long term business in and from Hong Kong About 1/3 of them (21) were incorporated locally in Hong Kong, with the remaining 2/3 incorporated in overseas (10 in Bermuda, in UK, in the US and the rest from China, Canada, Switzerland, Isle of Man etc) Figure Number of authorized life insurers by place of incorporation as at 30 June 2002 View of the Hong Kong Insurance Regulatory Framework The insurance industry in Hong Kong is regulated by the Insurance Companies Ordinance It provides a legislative framework for the prudential supervision of the insurance industry in Hong Kong The objectives of the Insurance Authority are to protect the interests of the policyholders and to maintain the general stability of the market place In addition to being an insurance regulator, the Insurance Authority is a proactive market enabler that we welcome and stand ready to facilitate 239 entry of newcomers and a referee of insurers As mentioned earlier, we adopt a prudential supervisory approach rather than a close supervision, with emphasis on self-control mechanism and the appropriate checks and balances Appointed Actuary System in Hong Kong At the beginning of 2001, we had introduced a fully fledged Appointed Actuary System for the life insurance business in Hong Kong The adoption of this system is to strengthen the monitoring of the financial condition of life insurers and is also to be in line with the development in other developed countries Under this system, the appointed actuaries have many other responsibilities apart from the traditional work in product pricing and valuation Appointed Actuaries now have the responsibility to the company's board of directors and to the regulators To enable the Appointed Actuary can properly carry out the responsibility, the actuary is given unrestricted access to all of the company's relevant financial and non-financial data that are complete and accurate Specific Reference to Actuaries in the Insurance Regulation As mentioned earlier, one of the important duties of the appointed actuary is to monitor the financial condition of insurance company and to report them to the board of directors To fulfill this duty, the Insurance Authority has imposed regulations on the valuation of long term liabilities and the determination of solvency margin Moreover, the appointed actuary has a whistle-blowing function under our Insurance Companies Ordinance If the Appointed Actuary is aware of a situation that could threaten the company's financial condition, the Appointed Actuary has an obligation to report this to the company's management and the Board If the company does not correct the situation, the Appointed Actuary is obligated to report the situation directly to the Insurance Authority to take proper action In order to fulfill the duties of the Appointed Actuaries, the Insurance Authority has prescribed a professional standard for the Appointed Actuaries to follow The professional standard was issued by a local actuarial body, the Actuarial Society of Hong Kong (ASHK) which I would further comment later on Every year, the appointed actuary is required to issue 240 an actuarial certification indicating the compliance with this professional standard or some other standards acceptable by the Insurance Authority Roles of Actuaries in the Market Place Apart from all the regulatory responsibilities, actuaries also play a very important role in the operation of a life insurance company Some of them could include identifying and targeting profitable market segments, designing sensible product features, setting the correct premium rate, assessing and conducting the risk control through reinsurance; and managing asset liability For participating policyholders, the Appointed Actuary is also required to preserve the Policyholders' Reasonable Expectation (PRE) The Policyholders' Reasonable Expectation has often to with the amount of bonus or dividends expected to be payable to policyholders in the future Actuarial Professional Body In order to have a strong Appointed Actuary system, it is very important that the actuarial professional body provides a professional code of conduct and a standard of practice to guide its members so that its members can carry out the responsibilities given to them The Actuarial Society of Hong Kong (the ASHK) has taken up this important role The ASHK has issued a number of professional standards and guidance notes over the years, such as the PS1 which I have already mentioned, and the GN3, an additional guidance for Appointed Actuary with respect to the determination of reserves and solvency margin The ASHK also conducts training and professional development meetings such as the Appointed Actuaries Symposium held last summer 10 Actuarial Application: Reserving for Investment Guarantee Finally, I would like to talk about an actuarial application on reserving standards for investment guarantees In January 2001, the Insurance Authority issued a guidance note, GN7, which requires life insurers carrying on Class G retirement scheme insurance business with investment guarantees on capital or return to set up an adequate reserve The Class G business includes the Mandatory Provident Funds (MPF) Scheme which is a community wide scheme providing a com- 241 pulsory retirement saving vehicle for all working population in Hong Kong, except for the very low income group To better assist the members in complying with the GN7, the ASHK has recently released a draft guideline which further provides guidance on satisfying the requirements of GN7 11 Details of the Reserving for Investment Guarantees The guidance note detailed the approaches to determine the provision such as the annual projection of assets and policy liabilities cash flow based on the stochastic scenario with 99% level of confidence If deterministic scenario or factor approach is used, a stochastic adequacy test must be performed at least once a year on the total provision on the adverse scenarios with a 99% level of confidence The provisions for investment guarantees must be revised at least quarterly to reflect current information Provisions at month-ends within a quarter may be extrapolated from the previous quarter-end values However, where there are significant fluctuations in the underlying factors that have been modeled by the stochastic analysis, the scenario testing should be done on a monthly basis 12 Stochastic Approach Under the stochastic approach, the process for completing a full stochastic investigation includes the modeling and projection of investment returns as well as the in-force business liabilities In the modeling and projection of investment returns, one needs to determine the relevant economic/market risk factors to be included in the model, the underlying investment policy of the assets and its investment returns, and the use of stochastic model to simulate the evolution of these quantities For the modeling and projection of in-force business liabilities, one needs to extract the relevant policy in-force data and to determine the assumptions used in the projection such as the mortality, lapse, withdrawal, future contributions, expenses, policy holders behaviors, management strategies Then the investment and business liabilities models are run with over 1000 economic scenarios The result of the scenarios, which is the present value of claims less the present value of income, are ranked in the order of severity of the required reserve 242 Finally, the total provision is established for the investment guarantees which is consistent with a 99% level of confidence A transitional arrangement is provided for certain Class G insurance policies which are not related to MPF schemes For these non-MPF policies, the level of reserves for guarantees is required to reach the 99% level of confidence by year 2004 It is also required that the results of stochastic testing must be traceable and reasonably reproducible for audit purposes 13 Other Actuarial Applications Apart from this, there are other actuarial applications too Due to the time constraint, I'm not going to cover them in details with you I would just mention that the Insurance Authority had requested the ASHK to look into the possibility of implementing Risk based capital and the Dynamic solvency testing for purposes of enhancing the financial monitoring of life insurers Starting from the end of 2001, the Insurance Authority also required general insurers engaging an actuary to certify reserves of certain compulsory lines of business 14 Conclusion In conclusion, actuaries and actuarial mathematics have played and will continue to play a significant role in the regulatory framework of the insurance market I hope you would agree with me that the demand for actuarial skills is getting greater in Hong Kong and particularly in China with the new opportunities upon China's entry into the WTO References GN Guidance Note On Reserving Standards For Investment Guarantees PS Professional Standard For Fellow Members Of The Actuarial Society Of Hong Kong .. .PROBABILITY, FINANCE AND INSURANCE This page is intentionally left blank PROBABILITY, FINANCE AND INSURANCE Proceedings of a Workshop at the University... presented at the Workshop on Probability with Applications to Finance and Insurance on July 15-17, 2002 at The University of Hong Kong Thirty-one invited speakers from mainland China, Hong Kong, Taiwan,... U GERBER and Elias Sai Wan SHIU Ruin probability for a model under Markovian switching regime 206 Hailiang YANG and G YIN Heavy-tailed distributions and their applications 218 Chun SU and Qihe