- Email: [email protected]
On an integral equation for discounted compound-annuity distributions.
Abstracts and Reviews analysis are used to characterize special groups of customers and market sectors. The main chapters are: 1. Problem, study objectives, definitions. 2. Market segmentation as a prerequisite for the efficient use of marketing instruments in financial services. 3. Empirical study for the design of types of sophisticated customers of financial services, and aggregated basic findings. 4. Application of cluster analysis for the identification of types of customers. 5. Description and exploration of types of customers. 6. Recommendations for realization. (W.R. Heilmann) Keywords: Financial Services, Group of Customers, Market Segmentation.
MZO: PROBABILITY THEORY AND MATHEMATICAL STATISTICS IN INSURANCE, GENERAL AND MISCELLANEOUS 053010 (MlO) Approximating point process likelihoods with GLIM. Berman M., Turner T.R., CSIRO, Sydney, Applied Statistics, Vol. 41, nr. I, 1992, pp. 31-38. This paper shows how approximate maximum likelihood estimation for fairly general point processes on the line can be performed with GLIM. The is based on a weighted sum approximation approximation to an integral in the likelihood. Various examined. The weighting schemes are briefly methodology is illustrated with an example, and its extension to Poisson processes in higher dimensions is briefly described. (Authors) Keywords: GLIM, Likelihood, Point Process, Poisson Process, Weights. 053011 (MlO, E24, B85) Modelling the probability of leaving unemployment: competing risks models with flexible base-line hazards. Narendranathan W., Stewart M.B., University of Warwick, Coventry, UK, Appliedstatistics, Vol. 42, nr. I, 1993, pp. 63-83. Unemployment durations are generally modelled by using survival analysis. In the past, in Britain, all such studies have not only used very restrictive parametric specifications of the hazard functions, most commonly Weibull in form, but also only modelled unemployment durations without distinguishing the nature of the exit.
147
These restrictions potentially bias the estimated effects, particularly those ofthe time varying economic variables and the base-line hazard. When the authors use semiparametric methods to estimate models with completely unrestricted base-line hazards, they find the restrictions implied by the Weibull specification to be rejected for Britain. The authors then use a competing risks model to distinguish exit into a job from other exits. They find that the single-risk model of exit understates the effects of income in and out of work on the probability of entering a job. (Authors) Keywords: Competing Risks Models, Hazard Functions, Heterogeneity, Survival Analysis, Unemployment Durations, 053012 (MlO, Mll, B40) A model for distributions of injuries in autoaccidents. Consul P.C., University of Calgary, Canada, Bulletin of the Swiss Association of Actuaries, Vol. 90, nr. I, 1990, pp. 161-168. The generalized Poisson distribution (GPD), see Consul (1989), given by a random variable Y with probability mass function P( Y=y)=O(8 +yh)Y-‘e -e-y’/y! for y=O,1,2,... and 0 for y>m when h0, max( -l,-8/4)IhO when 1~0, has been applied to six observed data sets on injuries in auto-accidents in Belgium, Zaire, Switzerland, Germany and Great Britain. It has been shown that the GPD fits all the data sets reasonably well. Plausible interpretations have been given to the estimated values of the parameters 8 and h for the different countries based upon the road conditions and traffic intensity. It has been suggested that a more detailed study of observed data sets for recent years be made by the insurance companies. (Author) Keywords: Generalized Poisson Distribution, Claims Frequency in Motor Insurance. 053013 (MlO) On an integral equation for discounted compoundannuity distributions. Ramsay C.M., University of Nebraska, Lincoln NE, U.S.A., Astin Bulletin, Vol. 19, nr. 2, 1989, pp. 191198. The author considers a risk generating claims for a period of N consecutive years (after which it expires), N being an integer valued random variable. Let X, denote the total claims generated in the kth year, k> 1.
148
Abstracts and Reviews
The Xk’s are assumed to be independent and identically distributed random variables, and are paid at the end of the year. The aggregate discounted claims generated by the risk until it expires is defined as.S,v(v)=~~z, v “Xk where v is the discount factor. An integral equation similar to that given by PANJER (1981) is developed for the pdf of SN(v) . This is accomplished by assuming that N belongs to a new class of discrete distributions called annuity distributions. The probabilities in annuity distributions satisfy the following recursion: P,, = P,_,
, for n=1,2,...
a++ I
probabilities e.g. for death or disablement even in the case of small portfolios two estimating procedures, which have already been developed in the past and which draw additional information from other small, “similar” portfolios, are briefly presented. The one which is applicable to the presented data, is described in a more detailed way. In the main part this estimating procedure is applied to real existing portfolios of disability riders. The resulting probabilities of disablement are critically explained with respect to the underlying estimation. (Author) Keywords:
Probabilities,
where a,, is the present value of an n-year immediate annuity. (Author)
053016 (MlO, B21)
Keywords:
Annuity
Kopfschadeu
Aggregate
Discounted
Distributions,
Integral
Equation,
HIV, AIDS and the approximate Haberman
functions, S., Journal
annuities
calculation
of life
and net premiums.
of the Institute of Actuaries,
Vol.
I 19, Part II, 1992, pp. 345-364.
The paper presents an application of the Markov chain as a tool for the calculation of life contingencies functions (e.g. assurance, annuity, net premium, policy value functions) arising from a multi-state model which represents the transmission and development of HIV and AIDS. The transmission model advocated by the Institute of Actuaries AIDS Working Party is modified and simplified and then applied to derive explicit formulae for these standard life contingencies functions. This investigation allows a thorough review of the properties of these functions to be conducted and assists with the calculation of premiums and reserves in the (Author) presence of HIV and AIDS. Keywords:
HIV, Insurance
aber
den
wahren
und
den
rechnungsmiihigen
in der Krankheitskosten-Versicherung
(On the true and the calculated per head claim in health insurance). Behne J., Siegen, Blatter der Deutsche Gesellschaftfiir
Claims.
053014 (MlO, BlO) insurance
Estimation.
nI
Functions,
Markov
Chain.
Versicherungsmathematik,
BandXYI,
Heft I, aprilI993,
pp. 133-140. Formulating the equivalence principle in terms of random variables and expectations one is able to interpret the usual method of private health insurance as statistical estimators of the involved unknown parameters. Thereby, the difference between “true” values (expected values) and calculated estimates will be (Author) clarified. Keywords:
Health Insurance,
Equivalence
Principle.
053017 (MlO, BlO) Zusammengesetzte Ausscheideordnungeu, Modelle konkurrierenden unabhgngige mit Risiken, Wahrscheinlichkeiten.
Einige
Zusammenhiinge.
(Multiple decrement life tables, competing risk models, independent probabilities - some relations). Milbrodt H., Dortmund, Blatter der Deutsche Gesellschaft
fur
Versicherungsmathematik,
Band
XX
053015 (MlO, BlO)
Heft 4, Oktober 1992, pp. 397-404.
Bestimmung
Independent probabilities in the sense of Karup may be obtained via partial failure rates, regardless whether the continuous model or the discrete model of insurance mathematics is employed. To see this, instead of the usual partial failure rates, cumulative partial failure rates have to be used which describe the effect of a combination of causes aggregated up to a certain time. In order to relate these considerations to well-known results from survival analysis, it is pointed out that multiple decrement life tables and independent probabilities as concepts of insurance mathematics on
versicherungsmathematischer
Ausscheidewahrscheinlichkeiten mehrerer einander lihnlicher Bestande mit Anwendungen aus der Berufsunfiihigkeits-Zusatzversicherung (Estimation
of actuarial decrement probabilities of several similar portfolios with applications to the field of supplementary disability insurance). Fillers J., Duisburg, Blatter der Deutsche Gesellschaft fir Versicherungsmathematik, 1993, pp. 11.5-131.
Considering
Band XXI, Heft I, april
the question of a reliable estimation of
MZO: PROBABILITY THEORY AND MATHEMATICAL STATISTICS IN INSURANCE, GENERAL AND MISCELLANEOUS 053010 (MlO) Approximating point process likelihoods with GLIM. Berman M., Turner T.R., CSIRO, Sydney, Applied Statistics, Vol. 41, nr. I, 1992, pp. 31-38. This paper shows how approximate maximum likelihood estimation for fairly general point processes on the line can be performed with GLIM. The is based on a weighted sum approximation approximation to an integral in the likelihood. Various examined. The weighting schemes are briefly methodology is illustrated with an example, and its extension to Poisson processes in higher dimensions is briefly described. (Authors) Keywords: GLIM, Likelihood, Point Process, Poisson Process, Weights. 053011 (MlO, E24, B85) Modelling the probability of leaving unemployment: competing risks models with flexible base-line hazards. Narendranathan W., Stewart M.B., University of Warwick, Coventry, UK, Appliedstatistics, Vol. 42, nr. I, 1993, pp. 63-83. Unemployment durations are generally modelled by using survival analysis. In the past, in Britain, all such studies have not only used very restrictive parametric specifications of the hazard functions, most commonly Weibull in form, but also only modelled unemployment durations without distinguishing the nature of the exit.
147
These restrictions potentially bias the estimated effects, particularly those ofthe time varying economic variables and the base-line hazard. When the authors use semiparametric methods to estimate models with completely unrestricted base-line hazards, they find the restrictions implied by the Weibull specification to be rejected for Britain. The authors then use a competing risks model to distinguish exit into a job from other exits. They find that the single-risk model of exit understates the effects of income in and out of work on the probability of entering a job. (Authors) Keywords: Competing Risks Models, Hazard Functions, Heterogeneity, Survival Analysis, Unemployment Durations, 053012 (MlO, Mll, B40) A model for distributions of injuries in autoaccidents. Consul P.C., University of Calgary, Canada, Bulletin of the Swiss Association of Actuaries, Vol. 90, nr. I, 1990, pp. 161-168. The generalized Poisson distribution (GPD), see Consul (1989), given by a random variable Y with probability mass function P( Y=y)=O(8 +yh)Y-‘e -e-y’/y! for y=O,1,2,... and 0 for y>m when h
148
Abstracts and Reviews
The Xk’s are assumed to be independent and identically distributed random variables, and are paid at the end of the year. The aggregate discounted claims generated by the risk until it expires is defined as.S,v(v)=~~z, v “Xk where v is the discount factor. An integral equation similar to that given by PANJER (1981) is developed for the pdf of SN(v) . This is accomplished by assuming that N belongs to a new class of discrete distributions called annuity distributions. The probabilities in annuity distributions satisfy the following recursion: P,, = P,_,
, for n=1,2,...
a++ I
probabilities e.g. for death or disablement even in the case of small portfolios two estimating procedures, which have already been developed in the past and which draw additional information from other small, “similar” portfolios, are briefly presented. The one which is applicable to the presented data, is described in a more detailed way. In the main part this estimating procedure is applied to real existing portfolios of disability riders. The resulting probabilities of disablement are critically explained with respect to the underlying estimation. (Author) Keywords:
Probabilities,
where a,, is the present value of an n-year immediate annuity. (Author)
053016 (MlO, B21)
Keywords:
Annuity
Kopfschadeu
Aggregate
Discounted
Distributions,
Integral
Equation,
HIV, AIDS and the approximate Haberman
functions, S., Journal
annuities
calculation
of life
and net premiums.
of the Institute of Actuaries,
Vol.
I 19, Part II, 1992, pp. 345-364.
The paper presents an application of the Markov chain as a tool for the calculation of life contingencies functions (e.g. assurance, annuity, net premium, policy value functions) arising from a multi-state model which represents the transmission and development of HIV and AIDS. The transmission model advocated by the Institute of Actuaries AIDS Working Party is modified and simplified and then applied to derive explicit formulae for these standard life contingencies functions. This investigation allows a thorough review of the properties of these functions to be conducted and assists with the calculation of premiums and reserves in the (Author) presence of HIV and AIDS. Keywords:
HIV, Insurance
aber
den
wahren
und
den
rechnungsmiihigen
in der Krankheitskosten-Versicherung
(On the true and the calculated per head claim in health insurance). Behne J., Siegen, Blatter der Deutsche Gesellschaftfiir
Claims.
053014 (MlO, BlO) insurance
Estimation.
nI
Functions,
Markov
Chain.
Versicherungsmathematik,
BandXYI,
Heft I, aprilI993,
pp. 133-140. Formulating the equivalence principle in terms of random variables and expectations one is able to interpret the usual method of private health insurance as statistical estimators of the involved unknown parameters. Thereby, the difference between “true” values (expected values) and calculated estimates will be (Author) clarified. Keywords:
Health Insurance,
Equivalence
Principle.
053017 (MlO, BlO) Zusammengesetzte Ausscheideordnungeu, Modelle konkurrierenden unabhgngige mit Risiken, Wahrscheinlichkeiten.
Einige
Zusammenhiinge.
(Multiple decrement life tables, competing risk models, independent probabilities - some relations). Milbrodt H., Dortmund, Blatter der Deutsche Gesellschaft
fur
Versicherungsmathematik,
Band
XX
053015 (MlO, BlO)
Heft 4, Oktober 1992, pp. 397-404.
Bestimmung
Independent probabilities in the sense of Karup may be obtained via partial failure rates, regardless whether the continuous model or the discrete model of insurance mathematics is employed. To see this, instead of the usual partial failure rates, cumulative partial failure rates have to be used which describe the effect of a combination of causes aggregated up to a certain time. In order to relate these considerations to well-known results from survival analysis, it is pointed out that multiple decrement life tables and independent probabilities as concepts of insurance mathematics on
versicherungsmathematischer
Ausscheidewahrscheinlichkeiten mehrerer einander lihnlicher Bestande mit Anwendungen aus der Berufsunfiihigkeits-Zusatzversicherung (Estimation
of actuarial decrement probabilities of several similar portfolios with applications to the field of supplementary disability insurance). Fillers J., Duisburg, Blatter der Deutsche Gesellschaft fir Versicherungsmathematik, 1993, pp. 11.5-131.
Considering
Band XXI, Heft I, april
the question of a reliable estimation of