- Email: [email protected]
Planning and control in insurance
105
Invited Review
Planning and control in insurance H a n s van G E L D E R Delta Llovd. Amsterdam. Netherlands
Recef red July 1981
1. Introduction
Practitioners in insurance face the same problem many practical men do, they do not share in recent theoretical developments. Although they have extensive data processing facilities and vast amounts of data at their disposal they do extract relatively little information from it as they generally do not know what questions to ask. Insurance is a very traditional industry to which planning, budgetting and control have come only lately [30]. A survey of the 'traditional' OR/MS-literature (Management Science, Interfaces, Journal" of the O.R. Soc. or Operations Research) confirms the critical opinion stated by the present author as most journals carry none or at most just a few articles denoted to insurance. This is all the more surprising to the uninitiated as one realizes that 'arithmetic' is the tool of the trade in insurance. Actuarial mathematics originated towards the end of the 17th century, when E. Halley's and Johan de Wit's mortality tables permitted the mathematical treatment and calculation of annuity values for the first time. The underlying model expressed in mathematical language at that time has been so closely adherent to in life insurance-the classical field of application for actuarial mathematics--that actuarial theory in this classical sense has often been characterized as closed upon itself. In recent decades a stimulus has come from general (non-life), property, casualty and accident insurance. This development has been made possiNot~,h-Hoiland Publishing Company European Journal of Operational Research 9 (1982) 105-113
ble by advances in probability theory and mathematical statistics. This all has lead to what is usually called (collective) Risk Theory. In the preface of his excellent book Btihlmann [21] characterizes this branch of actuarial mathematics an undertaking "... to solve the technical problems of all branches of insurance and that it concerns itself particularly with the operational problems of the insurance enterprise". This claim sounds very familiar to any knowledgeable student of O R / M S and is--to the present authormeven more preposterous than the statements made in this respect by members of the latter fraternity. As Btihlmann writes "... to solve the technical problem..." he is undoubtedly right that modern actuarial mathematics have made considerable theoretical contributions towards modeling important problems in insurance. Btihlmann made the following remark in the preface of his earlier mentioned book: "Characteristic of the present stage of development, however, is the fact that the current profusion of scientific publications in the field of actuarial mathematics deals above all with detached individual problems". Is it this detachment that actuarial attention is focused on the kisk Theoretical problems and that made them turn their back to various Managerial problems outside the narrowly defined confines of the stochastic model of a risk business? Many important management decisions have to be made in various stages of planning and control as well as in the day to day operations of an insurance company that are not addressed at all by Risk Theory. This all may sound like a harsh condemnation of the actuaries. It is not, because equally right would be the statement that OR/MS, Marketing, Production, Personnel or Finance have
i
037%2217/82/0000-0000/$02.75 © 1982 North-Holland
tl. van Gelder / Plmming and control in insurance
106
not addressed the problem in a coherent manner. This is not to say that neither Risk Theorists nor students of the other disciplines have not contributed, but that very few if any integrated analytical studies on the insurance industry have been published. May be the problem lies somewhere else? With the practitioners of insurance? As aid before insurance is a traditional industry where business folklore is held in high regard. Few of the senior-managers have quantitative-analytical background while those that have are generally employed in life insurance. Many of the seniormanagers have had a training in the legal discipline, while in some countries the road from the bottom to the top, all through the ranks still seems the most appropriate. Anyhow one thing seems clear; the worlds of theorists and practitioners are in insurance at least as far apart as in any other industry. The Risk Theorists most certainly have drawn up an impressive and elegant theory of the insurance cartier as a 'Risk Business" based on F ( x ) , the distribution function of the total amount of claims. If P~ and (7, are the premium earned and the sum of claim amount incurred in the time interval (0, t) respectively than only the difference of the two defining functions is essential for the mathematical analyses of the risk. Risk Theory does just that and generally--but not necessarily--in the study of risk processes Pt is assumed to be deterministic while Ct is a stochastic function. Every risk is characterized by a claims number process, generally assumed to follow a Poisson distribution function, and a claim amount process. The joint outcome of these two functions is oc
(I) k=O
the generalized Poisson function, where Pk = Prob(v = k) when v is a random variable denoting the number of claims and C~.(x) is the conditional probability that when the number of claims is exactly k the sum of these k claims is <~x. If it is assumed that the claim amounts are mutually independent, the function C~.(x) is the k th convolution of the distribution function C(x). Among the standard text books of Risk Theory
are tile works of Biihlmann [21], Seal [46] and Beard, Pentikiiinen and Pesonen [7]. The generalized Poisson function is basic to all Risk Theoretical work. One of the problems with " this function in practice is that it's usefulness for numerical computations is restricted; specific assumptions with regard to the distribution function of the claim amount of one claim, or a very small number for n are required. Finding approximations and computational methods for practical applications has been one of the main endeavours of the theory [43,45]. From the basic--generalized Poisson--function the Risk Theorists develop their theory and it's quantitative-analytical and computational methods for such important insurance processes as: Premium calculation, Reinsurance, Reserve calculation, Solvency requirements. One of the distinct weaknesses of their theory is that almost all of them restrict themselves almost always to the risk process without considering other and equally important processes in insurance that have a direct bearing on the parameters of the risk process. One of such ancillary processes is the financial intermediary function of insurance.
2. A macro view of an insurance company
Discussing problems in OR/MS, or planning and control--terms that to the present author have a very broad meaning--one generally has the choice of following an analytical or a synthetic approach. That is to say: either start from the whole and divide that into it's most important constitutant parts, or start with the parts and construct the system which they make up. Topdown versus bottom-up. An insurance company is a financial intermediary that offers it's liabilities to the public and invests the money received into suitable assets'. A t = L, + Kt;
(2)
assets equals liabilities plus equity (capital and surplus). In banking L, is formed by client deposits and credit balances in current account. In insurance the liabilities are created by underwriting insurance business [23,30,35]. Investment income is as essential to an insurance company as any other variable in the
tt. van Gelder / Planning and control m insurance
business. To ignore this could easily lead to wrong conclusions with regard to other Plements in the process or their interaction. The ratio L , / K ,
(3)
would be called the leverage, solvency or capital adequacy. Not so in insurance; there we generally use the ratio
(4)
e,/r,
as the the solvency ratio; premium (Pt) over capital (K,). But as
(s)
gP=L
premium times a 'fund generating factor' equals liabilities [23,30,35]. As premiums are paid in advance and expenses and claims are, paid gradually later on, in subsequent periods, the insurer holds some funds to invest. If we assume that expenses in any period are paid out of current premium income than we can write the following formula where P, is the net premium earned in period ¢ after dedueth.,fl of expenses: t
L,=
2 ¢=t--i
t
P,-
2
2
ity of ruin in Risk Theoretical terms). To the contrary each of the constraints by itself could worsen the situation, but used in combination may do the trick. In [23,30,35] a portfolio approach is used. The objective in [35] is to derive an efficient frontier in mean-variance space:
Min
VAR(Pp)
a,jP,
(6)
T=t--i j=T
where Y.~=~apyP, is the total amount of claims incurred on the premium of period ~- and where ~=,a~j is the socalled 'claims ratio' of period and the a~j are the fractions of the total claims in "r being settled and payed in periodj (j~> ~); a~j = 0 for j --, oo. (6) is a retrospective view of the funds generating process. A prospective view formulates the liabilities as the net present value of all claims payments to be made from premium received up to time t plus premium received in advance at t but not yet earned. In most countries financial intermediaries, and thus insurers, operate under a number of restrictions and guidelines set out by the supervisory authorities. Kahane [36] studied the implications of the customary restrictions on leverage ( L / K ) as well of constraints upon the balance sheet composition. Analyzing side effects on the efficient frontier in mean-variance space he concludes that neither restriction on its own realizes the objective of lowering the probability of bankruptcy (probabil-
(7)
.) .j = I ..... m
where
6
=
--- = r,,, =
wj = j j
= =
w,, =
return on equity; ~ / k o, return on either classes of insurance business written or on risky investments, tildes denote stochastic variables, return on riskfree investment, Pj /ko; insurance leverage or Aj/ko; investment leverage, l ..... n; lines of insurance written, n + l, .... m - l; risky investments, A,,,/ko; riskfree investment,
subject to E ( Pp) : E ( Pp) *,
(8)
wj~>O,
(9)
I!
Y, gjwj+l=
t
107
j= I
rlt|
E wj.
(10)
j=n+ I
(10) is essentially (2) divided by k; the balance sheet equation. Kahane [35] employs Sharp's Single Index Technique and does not suggest rules for selecting an operating point on the efficient frontier. In [23] such rules are suggested by introducing Utility Theory--originally introduced by Borch [10,11,12,13,14,15] into Risk Theory--and ruin probability [7,21,22], a classical measure of solvency in Risk Theory. In general Risk Theory distinguishes between probability of ruin during a finite and during an infinite period. Assuming that the insurer's capital funds amounts at to,tl,t 2..... t,, to ko,kl,k 2..... k,,. Then all of the k t (except for k 0, which is known with certainty) are random variables. The function k, is the random walk of the company's capital. The probability sought is
~,,(ko)=e{K~ ~<0forsome
l <~k<~n]ko}.
(11)
Monte Carlo simulation has been employed by Seal [47] for the calculation of ~k,,(ko). This method makes is possible to change F(x) --(1) the distribution function of period results--
H. can Gelder / Plamaing and control in insurance
108
over successive periods if the law of change is known or can be estimated. In [7] a case is presented where trend, short and long term, oscillations in the basic probabilities are operative. Straub [48] introduced reliability theory to the ruin problem, from which the notion of 'failure rate' is intuitively appealing in the context of ruin probability. The probability of ruin in an infinite time interval is an application of Gambler's Ruin to insurance where the company plays against the portfolio. Then it can be shown that 6,,ko <~e -Rr°
(12)
or
~/,,(ko) =C(ko) e -RK°
(13)
and C( k I) ~- 1. When the distribution function of one claim is Poisson we can solve R. i + (i + x)
dS(c).
(14)
Then R = - - - 2 ~ - m / a ~ where A is the so called safety loading in the premium, m the expected value of one claim and a, the variance around m. Ruin probability as an analytical method to the stability and solvency problem of an inrurance company incorporates an unrealistic property. Profits are continuously accumulated--no dividend payments--and capital stock grows indefinitely with time. The dividend criterion as an alternative has some definitely more realistic characteristics. What is the optimum value of k 0 (k~)--or what are the optimum values of K,(K*)--in order to maximize the expected net present value (ENPV) of future dividend payments, which are made whenever K t > Kg' (or K*), then a dividend (d, = K*, d>~O) is paid, before ruin (K, ~<0) sets in. K** is an alternative absorbing barrier. (K, <~K**) is a more general formulation of K t ~<0. Borch has written very interesting papers on the dividend criterion for stability of an insurer [17,18]. • A third and more general criterion for evaluating objectives is the utility criterion. [10,11, 12,13,14,15]. In itself not a new criterion but only a different way of measuring and discounting effects. Using the ruin probability as an objective, which value would be more suitable? Is ( = 10-4 for one
year preferable to ( = 10 -3 for 10 years or to ( = 10-2 for infinity? Concerning dividend. What is to be preferred a stable dividend stream with a low ENPV; an erratic dividend with a higher ENPV or first for a number of years no dividend at all and then a very high dividend with a very high ENPV. Utility theory can be of help here in incorporating the risk attitudes in this problem. The link between Utility theory and ruin probability is employed in [23]. A critique on all these methods is implicitly aired by Pentikiiinen in [44] and more explicitly in [30]. The objectives of management are more diversified and wider then ruin probabilities below a very small number or dividends to share holders who play only a minor part in most managerial thinking. In [30] it is argued that organizational theory of the last three decades has made it clear that organizations are made up of individuals with at least some (but not more) goals in common. Organizational objectives are the product of the objectives of the dominant participants, constrained by the objectives of the other participants in the organization or the ,mvironment. Continuity is generally one of the main goals of most organizations and therefore a favourable balance of output over input is a necessity. Pentikiiinen [44] model is attractive that it takes many aspects of an insurance company into account. It allows for variations in the basic probabilities; introduces reflecting barrier, upper capital level, the excess being paid out in dividends, an absorbing (winding up) barrier at which special measures have to be taken to continue the company. He introduces investment income, promotional expenses, differences between expected and actual cost. His model can be used for various purposes; solvency testing, determination of fluctuation reserves, reinsurance decisions, rate making, business planning in general and finally as a teaching tool. It is obvious that capital funds are not only required to absorb the risk resulting from the randomness in the Risk Theoretical model, but also to form a buffer stock when unexpected and unforseeable losses occur because interest rates rise faster than expected or premiums can not rize with inflation and rising costs of claims' acquisition and administration [42].
!!. van Gelder / Plamting and control m insurance
3. Parameters and decision variables in insurance After the macro view we now embark on a course which will lead us along the subsystems of our insurance model; an image of an insurance company in real life.
3.1. Premium calculation Below eq. (1) we have briefly touched upon the claims process. P = (1 + y ) P , ,
(15)
where P
Pr
= gross premium = risk premium = proportional loading to cover cost of acquisition, administration, overheads, miscellaneous expenses and profit.
Premium calculation is based on the assumption that a contingent claims experience can be compensated by fixed payements, called premiums. Premium then is calculated on the basic of an 'equivalence principle'; net premium equals the expected value of claims [21]. So called premium calculation principles have been introduced by Btihlmann [211 and a good discussion on it can be found in [31 ]. On the premium a contingency, or safety loading is required which is either proportional to the expected value of claims {(1 + h ) E ( S ) }, proportional to standard deviation {ao(S)} or to the variance {fla2(S) } of claims. There is a relationship between 'ris~< premium' and 'collective premium' but a difference as well. The risk premium is essentially unknown as the collective premium is calculated from the claims statistics. Only when the collective is strictly homogeneous we know the (true) risk premium. If we observe a risk long enough we may approximate the (true) risk premium through experience rating also called the Theory of Credibility [5,6,38,39,201. In general the original (collective) premium is (1 + h ) P and actual amount of claims in a year is ~, then a bonus or 'profit sharing' can be agreed upon:
B=k{(i +X)P-¢}.
(16)
109
In general B~>0, although in the Bonus-Malus systems of automobile insurance B < 0 is not uncommon. In case of (16) the cost of the insurance--the 'net' premium--is then (! + h ) P - B; the credibility premium. The credibility premium is stepwize approximation of the risk premium by a function dependent on risk experience [2,21,32,33]. Experience rating is a very powerful instrument that has important marketing implications. An insurance company that does not engage in selective marketing and does not discriminate in pricing between good and bad risks may end up loosing its good customers while charging a premium rate that is inadequate to compensate for the risks of customers of dubious quality. "Bad risks drive out good risks," when the same premium is charged. Finding 'correct' premium rates for all sub groups in the market is an important problem for practical insurance management. Due to the stochastic nature of insurance a larger portfolio leads--ceteris paribus--to reduction in the variance of outcomes and thus to a reduction in the safety loading. That correct or attractive premiums are not always identical to actuarial fair premiums and that the principle of equivalence can be ignored in that respect is shown in a attractive study by Borch [19], a prolific writer who has introduced many concepts familiar to O R / M S students to Risk Theory and insurance.
3. 2. Market volume and market share Little has been published on the subject, particularly in Europe. For the U.S. Etgar [25] and Joskov [34] published on the subject. In [30] the present author has stated on the basis of empirical research among a great number of our independent agents and brokers that the market form (at least in Holland) is that of monopolistic competition where a quick and generous claim handling, speed and quality of administrative procedures, the willingness to accept 'somewhat more difficult risks' as well and the quality of staff are all considered to be generally more important than price. On the other hand there is a growing market share for Direct Writers, those companies that sell directly to the public without the intermediary of agents or brokers. They aim at
tl. van Gelder / Planning and control in insurance
I i0
price buyers, while the traditional companies stress convenience and quality. The marketing strategies of the future have to be based on: marketing; discrimination between distribution channels and risk groups; experience rating; Improving the quality of staff and procedures to speed up and guarantee the quality of the administrative process. Organizational structure correlated with mission; - High quality of claims handling and underwriting by commercially sensitive staff that communicate well with agents and insured and that can take operational decisions. -
S
e
l
e
c
t
i
v
e
-
3.3. Reinsurance We have before given a description of the risk process as a combination of a random claims number process and a random claims amount process. Then
where Pk is the probability that the numbers of claims is exactly k, while the conditional probability that the total amount of claims ~
The former cases are called proportional reinsurance. Non-proportional reinsurance is:
- Exces~of loss: Zo--e z if Zo > e . , Zr=
r
where Z n -- net claim retained by the original insurer, Z o = original amount, Z r = reinsurer's part of the claim. If Zr/Z o = constant we have a 'quota share' reinsurance contract Where a surplus contract is used, the allocation of sum insured (Q) between cedant ( M ) and reinsurer ( Q - M) is fixed in advance and in the event of a claim
Zr:{QZo
for Q > M, for Q~
0
ifZ0 ~
where e~ is the maximum loss retained by the • original insurer.
- Stop loss: z r = { Zr-e"r
ifZr°>e:r'ifZr<~e,r
where e~r is an upper limit for the total (accumulated) loss in a period. Reinsurance arrangements are certain functions which define the portion of claims experience to be retained by the original insurer. The ultimate goal of reinsurance is quite the same as of any other insurance contract; the only difference being that the insurer who concludes a reinsurance contract is much better informed than his client. Whether he is the rationally calculating homo stochasticus (a close relative to homo economicus) may be open to discussion; there is quite some folklore to be found in the insurance industry. "The quality of the lunch with a reinsurer is as important as his price and conditions", has been said [3C]. As all reinsurance is conducted at a price there is of course a lively interest for 'optimal reinsurance'. As with all problems of 'optimally' there is the requirement of objectives and constraints. Objectives can be formulated as related to: - cost of reinsurance, the variance of the original insures net retention, the probability of ruin, - the dividend policy, the utility, etc. Reinsurance is only one variable that effect the latter three objectives. We can conclude that the reinsurance policy is not an independent variable, but has to be chosen in relation to other decisions, as well as to objectiv¢~ and constraints. The constraints are formed by the initial state in which the company is with respect of its funding, the size and composition of its portfolio, as well as with respect of its expectations concerning the claims distribution functions of its portfolio, -
Zn~--Zo--Z
{
-
-
-
ll. van Gelder / Piamling and coJltrol m insurance
Finally there are the types of reinsurance contracts and their premium in the market. In theory it can be shown that stop loss reinsurance is the optimal solution from all reinsurance methods in the sense that it gives for a ilxed net premium P (without or with a safety |.~ading proportional to expected claims), the smallest variance V,,(a) of the companies net retention. In practice however heavy safety loadings in stop loss premiums (proportional to the variance of claims) are common as this form of reinsurance gives maximum relative variance to the reinsurer. Authors that have studied this subject extensively are among others, Borch [8,9] and Gerber [29]. Pechlivanides [41] presents a normative model for the sequential reinsurance and dividendpayment problem of an insurance company. Optimal strategies are found in closed form for a class of utility functions. The model is formulated as a discrete time dynamic program. Leuven [37] critisises the usual optimality criteria (see preceding pages) as non-operational. He proposes the criterion of profit stabilization. Maximize G as a function of the retention (t) whereby
a(ti
= e(w(t))
where E(w(t)) = expected profit when retention is t, a(w(t)) = standard deviation of profit when retention is t, a = the risk attitude parameter. He demonstrates that the risk attitude is one of the determining functions in choosing the optimum level of the retention while the retention has the tendency to increase with the size of the reinsurance premium's loading which both are intuitively rather obvious. A good treatment of the subject can also be found by Wolff [49] from which it appears that the conditions of the reinsurance market are generally not in accordance with the conditions assumed under which optimal reinsurance strategies are devised. 3. 4. Cost structure of non.life insurance companies Cost of insurance companies are the cost of claims handling, acquisition, underwriting, transaction-processing, administration, housing, super-
III
vision and management. For companies that write insurance through agents we have two sets of costs: (la) Commissions payable to brokers and agents; these costs are about linearly proportional to premium volume. (b) Cost of specific services to and financing of agents; these costs are generally more or less linearly proportional to premium volume or relatively small. (2) Cost of: (a) Acquisition: field and internal staff, Marketing department, advertising and promotion; these costs are program costs and as such invariant to premium volume (at least over a certain range). (b) Underwriting, transaction processing and claims handling. These activities involve variable (to volume) cost of directly productive labor and fixed cost of indirectly productive labor, supervision and management. (c) Housing, data processing and ancillary services. For a very minor part variable with volume and for the major part fixed; at least within a reasonable range insensitive to premium volume. A very high percentage of the cost under point two are personnel costs. Insurance is a relatively labor intensive service industry. Next to salaries, pension premiums and other benefits, the cost of maintaining an adequate labor force are important. Selection, training, placement and renumeration are key variables to succes. So is the organization structure and the quality and levels of communication. Recalling the determinants of the agent's choice demonstrates that quality and motivation of staff and their communication with the intermediaries is the key factor of succes. We see within the industry a growing interest for personnel planning and management development. Particularly so as the recruitement of today determines the composition and quality of staff for the next 20 to 30 years. Literature on production and cost functions in insurance is scarce. Eisen [24] and Farney [26,27] are the only authors mentioned by Botch [13]. It is obvious that the rapid development of electronic data processing and communications techniques will offer a number of problems to insurers in the not to distant future. (i) The planning component in management
i 12
H. can Gelder
/
Planning and control in insurance
will increase in volume and importance. (2) Unless a considerable increase in business volume can be realized companies will be faced with serious social/labor problems as a result from strongly increased productivity; lay offs. (3) New insurance companies that do not have a large, generally older and strongly established labor force are in a position to introduce micro electronics rapidly, have much lower pr';.ces and will undercut traditional companies with 'older' and unfavorable cost structures.
4. In conclusion
We opened this paper with a statement that the 'traditional' O R / M S literature showed a scarcity of publications on insurance. If we had chosen for a survey on decision making under uncertainty instead we would have fared much better and in that sub-discipline lies an attractive avenue to improved decision making for planning and control in insurance. Now we had to turn to the actuarial journals. Although we criticized the Risk Theorists for an number of omissions in their models and a too strict adherence on their side to the risk process proper with the consequential neglect of other, and related problems in insurance we have to give a lot of credit to quite a number of them for the excellent treatment of many of the important issue's in insurance. A closer link between the 'traditional' actuaries and the as 'traditional' M S / O R practitioners with a good dose of Analytical Economics, Marketing, Finance and Accounting could lead to improved models that could help practical men, ordinary insurance managers to get a better understanding of their company, their industry, their clients and their markets. This might even lead to better decisions and to better products at a lower price. The portfolio models presented in [23,35] form a good starting point but it is a question whether they are theoretically and operationally sound. The critisism with respect to mean-variance efficient model of portfolio selection are well known. Next to static nature of the model, there are the data requirements and the problems of estimating the paramet~.rs, particularly the variancecovariance matrix of portfolio returns. There are
some problems with the fact whether a portfolio is indeed adequately characterized by its mean and variance [ 16]. In [30] 'gross' underwriting profit is introduced as a criterion for profitability regarding the very large proportion of fixed cost in insurance; the concept of Direct Costing is introduced. Models should have operational relevance in the first place and it is an open question to the present author of this review whether the business of an insurer is any more stochastic than that in any other industry. While the underlying risk processes are undoubtly random it is an interesting question what consequences this does have for the company as a whole. I hope that nobody gets me wrong. I do not deny the stochastic nature of the insurance business but so is any other business in any other industry. What interests me is, whether deterministic models are appropriate approximations. The latter are so much easier to build, to operate and to comprehend. If this is not the case we might have to do with the beautiful and theoretically very interesting models of extremely limited scope and total non-operationality like Frisque's [28] dynamic model. One real prerequisite for further and fruitful expansion of operational modelling in insurance is a better understanding of the insurance market. Up till now there have only been published a limited number of studies by Borch [ 10,11,12,13,15] and Moffet [40] largely based on work by Arrow [3,41. The behavioural assumptions with respect to agents/brokers and client need to be tested and parameterized empirically. The role of government in price control, incomes policies and market regulation have to be analyzed. Market form and structure; competition and coalition as well as the relationship to money and capital markets have to be brought into the picture. Finally planning models are required for processing capacity; personnel and EDP planning systems while our models require the relevant inputs to provide us with adequate and timely management information for planning and control in a very important industry.
tl. van Gelder / Plamting and control in insurance
References [I] H.H. Agnew, R.A. Agnew, J. Rasmussen and K.R. Smith, An application of chance constrained programming to portfolio selection in a casualty insurance firm, Management Sci. 15 (10) (1969). [2] H. Ammeter, Experience rating a new application of the collective theory of risk, Astin Bull. II (II) (1962). [3] K.J. Arrow, The role of securities is the optimal allocation of risk bearing, Rev. Economic Studies (1964). [4] K.S. Arrow, Optimal insurance and generalized deductibles, Skand. Actuarial J. (1974). [5] A.L. Bailey, A generalized theory of credibility, Proc. Casualty Acturial Soc. 32 (1975). [6] A.L. Bailey, Credibility procedures, Laplace's generalization of Bayes rule and the combination of collateral knowledge with observed data, Proc. Casualty Acturial Soc. 37 ( i 950). [7] R.E. Beard, T. Pentikiiinen and E. Pesonen, Risk Theory (Methuen, London, I st ed., 1969). [8] K. Borch, An attempt to determine the optimum amount of stop loss reinsurance, Trans. 16th International Congress of Actuaries, Vol. 2. [91 K. Borch, The optimal reinsurance treaty, Astin Bull. V (II) (1969). [10] K. Borch, The utility concept applied to the theory of insurance, Astin Bull. I (1961). [11] K. Borch, Recent development in economic theory and their application to insurance, Astin Bull. II (II1) (1963). [12] K. Borch, The economic theory of insurance, Astin Bull. IV (III) (1967). [13] K. Borch, Problems in the economic theory of insurance, Astin Bull. 10 (I) (1978). [14] K. Borch, The safety loading of reinsurance premiums, Skand. Aktuartidskrift (1960). [151 K. Borch, Economic equilibrium under uncertainty, Internxt. Econom. Rev. (1968). II61 K. Borch, Portfolio theory is for risk lovers, J. Banking and Finance 2 (2) (1978). [171 K. Botch, The rescue of an insurance company after ruin, Astin Bull. V (II) (1969). [181 K. Borch, A short note on overall risk management in an insurance company, Astin Bull. VI (2) (1971). [191 K. Borch, Application of game theory to some problems in automobile insurance, Astin Bull. II (II) (1962). 120] H. B0hlmann, Experience rating and credibility, Astin Bull. V (II) (1969). [211 H. BUhlmann, Mathematical Methods in Risk Theory (Springer, Berlin, 1970). [22} DR. Cox and H.D. Miller, The Theory of Stochastic Processes (Chapman and Hail, London, 1965). [231 J. D,~,vid Cummins and David J. Nye, Portfolio optimization models for property-liability insurance companies: an analysis and some extensions, Management Sci. 27 (4)
(1981). [24] R. Eisen, Zur Productionsfunktion der Versicherung, Z. Gesamte Versicherungswissenschaft ( 197 ! ) 407-419. [25] M. Etgar, An empirical analyses of motivations for the development of a centrally coordinated vertical marketing system: the case of the property casualty insurance in-
I 13
dustry, unpublished P h . D . Thesis, Graduate School of Business Administration, University of California, Berkeley (1974). [261 D. Farney, Produktions- und Kostentheorie der Versicherung, Karlsruhe (1965). [27] D. Farney 'Anslttz einer betriebswirtschaftlichen Theorie des Versicherungsunternehmens', The Geneva Papers on Risk and Insurance 5 (February 1977) 9-21. [28] A. Frisque, Dynamic model of insurance company's management, Astin Bull. VIII (1) (1974). [29l H.U. Gerber, Paret~,-optimai risk exchanges and related decision problems, Astin Bull. 10 (1978). [30] H. van Gelder and C. Schrauwers, Planning in theory and practice; with reference to insurance. Proc. 9th IFORS Conference on Operational Research, Hamburg ( 198 i ). [31] M.J. Goovaerts and FI. De Vylder, Premium calculations principles; some properties, Her Verzekeringsarchief 57 ( i ) (1980). I321 W.S. Jeweli, Regularity conditions for exact credibility, Astin Bull. VIII (3) (1975). [331 W.S. Jewell, The credibility distribution, Astin Bull. VII (3) (1974). [341 L. Joskow, Cartels, competition and regulation and the cost of capital in the insurance industry, J. Financial Quantitative Anal. (December 1971 ) 1283-1305. [351 Y. Kahane, Determination of the product mix and business policy of an insurance company--a portfolio approach, Management Sci. 23 (10) (June 1977). [36] Y. Kahane, Capital adequacy and the regulation of financial intermediaries, J. Banking and Finance I (2) (1977). [37] G.N.W. Leuven, Eigen behoud; enige kritische kanttekeningen bij optimaliserings-k~teria (Own retention: some critical remarks about optimality criteria) (Dutch) Her Verzekeringsarchief 56 (3) (1979). 1381 O. Lundberg, On Random Processes and Their Applications to Sickness and Accident Statistics (Aimquist and Wickselis, Uppsala, 2nd ed., 1964). [39] A.L. Mayerson, The uses of credibility in property insurance rate making, Giom. 1st. ltal. Attuari 24 (1964). [40] D. Moffet, Risk-bearing and consumption theory, Astin Bull. VIII (3) (1975). [41l P.M. Pechlivanides, Optimal reinsurance and dividend payement strategies, Astin Bull. 10 (1978). [42l T. Pentikliinen, On the solvency of insurance companies, Astin Bull. IV (III) (1967). [43] T. Pentikitinen, On the approximation of the total amount of claims, Astin Bull. IX (3) (1977). [44] T. Pentik~nen, The theory of risk and some applications, J. Risk and Insurance XLVIII (i) (March 1980). [45] E. Pesonen, On the calculation of the general Poisson function, Astin Bull. IV (I!,' (1967). 146] H.L. Seal, Stochastic Theory of a Risk Business (Wiley, New York, 1969). [471 H.L. Seal, Simulation of ruin potential of non-life insurance companies, Trans. Soc. Act. XXl, p. 562. [48] E. Straub, Application of reliability theory to insurance, Astin Bull. VI (2) (1971). [49] K.H, Wolff, Methoden der Unternehemensforschung im Versicherungswesen (Springer, Berlin, 1966).
Invited Review
Planning and control in insurance H a n s van G E L D E R Delta Llovd. Amsterdam. Netherlands
Recef red July 1981
1. Introduction
Practitioners in insurance face the same problem many practical men do, they do not share in recent theoretical developments. Although they have extensive data processing facilities and vast amounts of data at their disposal they do extract relatively little information from it as they generally do not know what questions to ask. Insurance is a very traditional industry to which planning, budgetting and control have come only lately [30]. A survey of the 'traditional' OR/MS-literature (Management Science, Interfaces, Journal" of the O.R. Soc. or Operations Research) confirms the critical opinion stated by the present author as most journals carry none or at most just a few articles denoted to insurance. This is all the more surprising to the uninitiated as one realizes that 'arithmetic' is the tool of the trade in insurance. Actuarial mathematics originated towards the end of the 17th century, when E. Halley's and Johan de Wit's mortality tables permitted the mathematical treatment and calculation of annuity values for the first time. The underlying model expressed in mathematical language at that time has been so closely adherent to in life insurance-the classical field of application for actuarial mathematics--that actuarial theory in this classical sense has often been characterized as closed upon itself. In recent decades a stimulus has come from general (non-life), property, casualty and accident insurance. This development has been made possiNot~,h-Hoiland Publishing Company European Journal of Operational Research 9 (1982) 105-113
ble by advances in probability theory and mathematical statistics. This all has lead to what is usually called (collective) Risk Theory. In the preface of his excellent book Btihlmann [21] characterizes this branch of actuarial mathematics an undertaking "... to solve the technical problems of all branches of insurance and that it concerns itself particularly with the operational problems of the insurance enterprise". This claim sounds very familiar to any knowledgeable student of O R / M S and is--to the present authormeven more preposterous than the statements made in this respect by members of the latter fraternity. As Btihlmann writes "... to solve the technical problem..." he is undoubtedly right that modern actuarial mathematics have made considerable theoretical contributions towards modeling important problems in insurance. Btihlmann made the following remark in the preface of his earlier mentioned book: "Characteristic of the present stage of development, however, is the fact that the current profusion of scientific publications in the field of actuarial mathematics deals above all with detached individual problems". Is it this detachment that actuarial attention is focused on the kisk Theoretical problems and that made them turn their back to various Managerial problems outside the narrowly defined confines of the stochastic model of a risk business? Many important management decisions have to be made in various stages of planning and control as well as in the day to day operations of an insurance company that are not addressed at all by Risk Theory. This all may sound like a harsh condemnation of the actuaries. It is not, because equally right would be the statement that OR/MS, Marketing, Production, Personnel or Finance have
i
037%2217/82/0000-0000/$02.75 © 1982 North-Holland
tl. van Gelder / Plmming and control in insurance
106
not addressed the problem in a coherent manner. This is not to say that neither Risk Theorists nor students of the other disciplines have not contributed, but that very few if any integrated analytical studies on the insurance industry have been published. May be the problem lies somewhere else? With the practitioners of insurance? As aid before insurance is a traditional industry where business folklore is held in high regard. Few of the senior-managers have quantitative-analytical background while those that have are generally employed in life insurance. Many of the seniormanagers have had a training in the legal discipline, while in some countries the road from the bottom to the top, all through the ranks still seems the most appropriate. Anyhow one thing seems clear; the worlds of theorists and practitioners are in insurance at least as far apart as in any other industry. The Risk Theorists most certainly have drawn up an impressive and elegant theory of the insurance cartier as a 'Risk Business" based on F ( x ) , the distribution function of the total amount of claims. If P~ and (7, are the premium earned and the sum of claim amount incurred in the time interval (0, t) respectively than only the difference of the two defining functions is essential for the mathematical analyses of the risk. Risk Theory does just that and generally--but not necessarily--in the study of risk processes Pt is assumed to be deterministic while Ct is a stochastic function. Every risk is characterized by a claims number process, generally assumed to follow a Poisson distribution function, and a claim amount process. The joint outcome of these two functions is oc
(I) k=O
the generalized Poisson function, where Pk = Prob(v = k) when v is a random variable denoting the number of claims and C~.(x) is the conditional probability that when the number of claims is exactly k the sum of these k claims is <~x. If it is assumed that the claim amounts are mutually independent, the function C~.(x) is the k th convolution of the distribution function C(x). Among the standard text books of Risk Theory
are tile works of Biihlmann [21], Seal [46] and Beard, Pentikiiinen and Pesonen [7]. The generalized Poisson function is basic to all Risk Theoretical work. One of the problems with " this function in practice is that it's usefulness for numerical computations is restricted; specific assumptions with regard to the distribution function of the claim amount of one claim, or a very small number for n are required. Finding approximations and computational methods for practical applications has been one of the main endeavours of the theory [43,45]. From the basic--generalized Poisson--function the Risk Theorists develop their theory and it's quantitative-analytical and computational methods for such important insurance processes as: Premium calculation, Reinsurance, Reserve calculation, Solvency requirements. One of the distinct weaknesses of their theory is that almost all of them restrict themselves almost always to the risk process without considering other and equally important processes in insurance that have a direct bearing on the parameters of the risk process. One of such ancillary processes is the financial intermediary function of insurance.
2. A macro view of an insurance company
Discussing problems in OR/MS, or planning and control--terms that to the present author have a very broad meaning--one generally has the choice of following an analytical or a synthetic approach. That is to say: either start from the whole and divide that into it's most important constitutant parts, or start with the parts and construct the system which they make up. Topdown versus bottom-up. An insurance company is a financial intermediary that offers it's liabilities to the public and invests the money received into suitable assets'. A t = L, + Kt;
(2)
assets equals liabilities plus equity (capital and surplus). In banking L, is formed by client deposits and credit balances in current account. In insurance the liabilities are created by underwriting insurance business [23,30,35]. Investment income is as essential to an insurance company as any other variable in the
tt. van Gelder / Planning and control m insurance
business. To ignore this could easily lead to wrong conclusions with regard to other Plements in the process or their interaction. The ratio L , / K ,
(3)
would be called the leverage, solvency or capital adequacy. Not so in insurance; there we generally use the ratio
(4)
e,/r,
as the the solvency ratio; premium (Pt) over capital (K,). But as
(s)
gP=L
premium times a 'fund generating factor' equals liabilities [23,30,35]. As premiums are paid in advance and expenses and claims are, paid gradually later on, in subsequent periods, the insurer holds some funds to invest. If we assume that expenses in any period are paid out of current premium income than we can write the following formula where P, is the net premium earned in period ¢ after dedueth.,fl of expenses: t
L,=
2 ¢=t--i
t
P,-
2
2
ity of ruin in Risk Theoretical terms). To the contrary each of the constraints by itself could worsen the situation, but used in combination may do the trick. In [23,30,35] a portfolio approach is used. The objective in [35] is to derive an efficient frontier in mean-variance space:
Min
VAR(Pp)
a,jP,
(6)
T=t--i j=T
where Y.~=~apyP, is the total amount of claims incurred on the premium of period ~- and where ~=,a~j is the socalled 'claims ratio' of period and the a~j are the fractions of the total claims in "r being settled and payed in periodj (j~> ~); a~j = 0 for j --, oo. (6) is a retrospective view of the funds generating process. A prospective view formulates the liabilities as the net present value of all claims payments to be made from premium received up to time t plus premium received in advance at t but not yet earned. In most countries financial intermediaries, and thus insurers, operate under a number of restrictions and guidelines set out by the supervisory authorities. Kahane [36] studied the implications of the customary restrictions on leverage ( L / K ) as well of constraints upon the balance sheet composition. Analyzing side effects on the efficient frontier in mean-variance space he concludes that neither restriction on its own realizes the objective of lowering the probability of bankruptcy (probabil-
(7)
.) .j = I ..... m
where
6
=
--- = r,,, =
wj = j j
= =
w,, =
return on equity; ~ / k o, return on either classes of insurance business written or on risky investments, tildes denote stochastic variables, return on riskfree investment, Pj /ko; insurance leverage or Aj/ko; investment leverage, l ..... n; lines of insurance written, n + l, .... m - l; risky investments, A,,,/ko; riskfree investment,
subject to E ( Pp) : E ( Pp) *,
(8)
wj~>O,
(9)
I!
Y, gjwj+l=
t
107
j= I
rlt|
E wj.
(10)
j=n+ I
(10) is essentially (2) divided by k; the balance sheet equation. Kahane [35] employs Sharp's Single Index Technique and does not suggest rules for selecting an operating point on the efficient frontier. In [23] such rules are suggested by introducing Utility Theory--originally introduced by Borch [10,11,12,13,14,15] into Risk Theory--and ruin probability [7,21,22], a classical measure of solvency in Risk Theory. In general Risk Theory distinguishes between probability of ruin during a finite and during an infinite period. Assuming that the insurer's capital funds amounts at to,tl,t 2..... t,, to ko,kl,k 2..... k,,. Then all of the k t (except for k 0, which is known with certainty) are random variables. The function k, is the random walk of the company's capital. The probability sought is
~,,(ko)=e{K~ ~<0forsome
l <~k<~n]ko}.
(11)
Monte Carlo simulation has been employed by Seal [47] for the calculation of ~k,,(ko). This method makes is possible to change F(x) --(1) the distribution function of period results--
H. can Gelder / Plamaing and control in insurance
108
over successive periods if the law of change is known or can be estimated. In [7] a case is presented where trend, short and long term, oscillations in the basic probabilities are operative. Straub [48] introduced reliability theory to the ruin problem, from which the notion of 'failure rate' is intuitively appealing in the context of ruin probability. The probability of ruin in an infinite time interval is an application of Gambler's Ruin to insurance where the company plays against the portfolio. Then it can be shown that 6,,ko <~e -Rr°
(12)
or
~/,,(ko) =C(ko) e -RK°
(13)
and C( k I) ~- 1. When the distribution function of one claim is Poisson we can solve R. i + (i + x)
dS(c).
(14)
Then R = - - - 2 ~ - m / a ~ where A is the so called safety loading in the premium, m the expected value of one claim and a, the variance around m. Ruin probability as an analytical method to the stability and solvency problem of an inrurance company incorporates an unrealistic property. Profits are continuously accumulated--no dividend payments--and capital stock grows indefinitely with time. The dividend criterion as an alternative has some definitely more realistic characteristics. What is the optimum value of k 0 (k~)--or what are the optimum values of K,(K*)--in order to maximize the expected net present value (ENPV) of future dividend payments, which are made whenever K t > Kg' (or K*), then a dividend (d, = K*, d>~O) is paid, before ruin (K, ~<0) sets in. K** is an alternative absorbing barrier. (K, <~K**) is a more general formulation of K t ~<0. Borch has written very interesting papers on the dividend criterion for stability of an insurer [17,18]. • A third and more general criterion for evaluating objectives is the utility criterion. [10,11, 12,13,14,15]. In itself not a new criterion but only a different way of measuring and discounting effects. Using the ruin probability as an objective, which value would be more suitable? Is ( = 10-4 for one
year preferable to ( = 10 -3 for 10 years or to ( = 10-2 for infinity? Concerning dividend. What is to be preferred a stable dividend stream with a low ENPV; an erratic dividend with a higher ENPV or first for a number of years no dividend at all and then a very high dividend with a very high ENPV. Utility theory can be of help here in incorporating the risk attitudes in this problem. The link between Utility theory and ruin probability is employed in [23]. A critique on all these methods is implicitly aired by Pentikiiinen in [44] and more explicitly in [30]. The objectives of management are more diversified and wider then ruin probabilities below a very small number or dividends to share holders who play only a minor part in most managerial thinking. In [30] it is argued that organizational theory of the last three decades has made it clear that organizations are made up of individuals with at least some (but not more) goals in common. Organizational objectives are the product of the objectives of the dominant participants, constrained by the objectives of the other participants in the organization or the ,mvironment. Continuity is generally one of the main goals of most organizations and therefore a favourable balance of output over input is a necessity. Pentikiiinen [44] model is attractive that it takes many aspects of an insurance company into account. It allows for variations in the basic probabilities; introduces reflecting barrier, upper capital level, the excess being paid out in dividends, an absorbing (winding up) barrier at which special measures have to be taken to continue the company. He introduces investment income, promotional expenses, differences between expected and actual cost. His model can be used for various purposes; solvency testing, determination of fluctuation reserves, reinsurance decisions, rate making, business planning in general and finally as a teaching tool. It is obvious that capital funds are not only required to absorb the risk resulting from the randomness in the Risk Theoretical model, but also to form a buffer stock when unexpected and unforseeable losses occur because interest rates rise faster than expected or premiums can not rize with inflation and rising costs of claims' acquisition and administration [42].
!!. van Gelder / Plamting and control m insurance
3. Parameters and decision variables in insurance After the macro view we now embark on a course which will lead us along the subsystems of our insurance model; an image of an insurance company in real life.
3.1. Premium calculation Below eq. (1) we have briefly touched upon the claims process. P = (1 + y ) P , ,
(15)
where P
Pr
= gross premium = risk premium = proportional loading to cover cost of acquisition, administration, overheads, miscellaneous expenses and profit.
Premium calculation is based on the assumption that a contingent claims experience can be compensated by fixed payements, called premiums. Premium then is calculated on the basic of an 'equivalence principle'; net premium equals the expected value of claims [21]. So called premium calculation principles have been introduced by Btihlmann [211 and a good discussion on it can be found in [31 ]. On the premium a contingency, or safety loading is required which is either proportional to the expected value of claims {(1 + h ) E ( S ) }, proportional to standard deviation {ao(S)} or to the variance {fla2(S) } of claims. There is a relationship between 'ris~< premium' and 'collective premium' but a difference as well. The risk premium is essentially unknown as the collective premium is calculated from the claims statistics. Only when the collective is strictly homogeneous we know the (true) risk premium. If we observe a risk long enough we may approximate the (true) risk premium through experience rating also called the Theory of Credibility [5,6,38,39,201. In general the original (collective) premium is (1 + h ) P and actual amount of claims in a year is ~, then a bonus or 'profit sharing' can be agreed upon:
B=k{(i +X)P-¢}.
(16)
109
In general B~>0, although in the Bonus-Malus systems of automobile insurance B < 0 is not uncommon. In case of (16) the cost of the insurance--the 'net' premium--is then (! + h ) P - B; the credibility premium. The credibility premium is stepwize approximation of the risk premium by a function dependent on risk experience [2,21,32,33]. Experience rating is a very powerful instrument that has important marketing implications. An insurance company that does not engage in selective marketing and does not discriminate in pricing between good and bad risks may end up loosing its good customers while charging a premium rate that is inadequate to compensate for the risks of customers of dubious quality. "Bad risks drive out good risks," when the same premium is charged. Finding 'correct' premium rates for all sub groups in the market is an important problem for practical insurance management. Due to the stochastic nature of insurance a larger portfolio leads--ceteris paribus--to reduction in the variance of outcomes and thus to a reduction in the safety loading. That correct or attractive premiums are not always identical to actuarial fair premiums and that the principle of equivalence can be ignored in that respect is shown in a attractive study by Borch [19], a prolific writer who has introduced many concepts familiar to O R / M S students to Risk Theory and insurance.
3. 2. Market volume and market share Little has been published on the subject, particularly in Europe. For the U.S. Etgar [25] and Joskov [34] published on the subject. In [30] the present author has stated on the basis of empirical research among a great number of our independent agents and brokers that the market form (at least in Holland) is that of monopolistic competition where a quick and generous claim handling, speed and quality of administrative procedures, the willingness to accept 'somewhat more difficult risks' as well and the quality of staff are all considered to be generally more important than price. On the other hand there is a growing market share for Direct Writers, those companies that sell directly to the public without the intermediary of agents or brokers. They aim at
tl. van Gelder / Planning and control in insurance
I i0
price buyers, while the traditional companies stress convenience and quality. The marketing strategies of the future have to be based on: marketing; discrimination between distribution channels and risk groups; experience rating; Improving the quality of staff and procedures to speed up and guarantee the quality of the administrative process. Organizational structure correlated with mission; - High quality of claims handling and underwriting by commercially sensitive staff that communicate well with agents and insured and that can take operational decisions. -
S
e
l
e
c
t
i
v
e
-
3.3. Reinsurance We have before given a description of the risk process as a combination of a random claims number process and a random claims amount process. Then
where Pk is the probability that the numbers of claims is exactly k, while the conditional probability that the total amount of claims ~
The former cases are called proportional reinsurance. Non-proportional reinsurance is:
- Exces~of loss: Zo--e z if Zo > e . , Zr=
r
where Z n -- net claim retained by the original insurer, Z o = original amount, Z r = reinsurer's part of the claim. If Zr/Z o = constant we have a 'quota share' reinsurance contract Where a surplus contract is used, the allocation of sum insured (Q) between cedant ( M ) and reinsurer ( Q - M) is fixed in advance and in the event of a claim
Zr:{QZo
for Q > M, for Q~
0
ifZ0 ~
where e~ is the maximum loss retained by the • original insurer.
- Stop loss: z r = { Zr-e"r
ifZr°>e:r'ifZr<~e,r
where e~r is an upper limit for the total (accumulated) loss in a period. Reinsurance arrangements are certain functions which define the portion of claims experience to be retained by the original insurer. The ultimate goal of reinsurance is quite the same as of any other insurance contract; the only difference being that the insurer who concludes a reinsurance contract is much better informed than his client. Whether he is the rationally calculating homo stochasticus (a close relative to homo economicus) may be open to discussion; there is quite some folklore to be found in the insurance industry. "The quality of the lunch with a reinsurer is as important as his price and conditions", has been said [3C]. As all reinsurance is conducted at a price there is of course a lively interest for 'optimal reinsurance'. As with all problems of 'optimally' there is the requirement of objectives and constraints. Objectives can be formulated as related to: - cost of reinsurance, the variance of the original insures net retention, the probability of ruin, - the dividend policy, the utility, etc. Reinsurance is only one variable that effect the latter three objectives. We can conclude that the reinsurance policy is not an independent variable, but has to be chosen in relation to other decisions, as well as to objectiv¢~ and constraints. The constraints are formed by the initial state in which the company is with respect of its funding, the size and composition of its portfolio, as well as with respect of its expectations concerning the claims distribution functions of its portfolio, -
Zn~--Zo--Z
{
-
-
-
ll. van Gelder / Piamling and coJltrol m insurance
Finally there are the types of reinsurance contracts and their premium in the market. In theory it can be shown that stop loss reinsurance is the optimal solution from all reinsurance methods in the sense that it gives for a ilxed net premium P (without or with a safety |.~ading proportional to expected claims), the smallest variance V,,(a) of the companies net retention. In practice however heavy safety loadings in stop loss premiums (proportional to the variance of claims) are common as this form of reinsurance gives maximum relative variance to the reinsurer. Authors that have studied this subject extensively are among others, Borch [8,9] and Gerber [29]. Pechlivanides [41] presents a normative model for the sequential reinsurance and dividendpayment problem of an insurance company. Optimal strategies are found in closed form for a class of utility functions. The model is formulated as a discrete time dynamic program. Leuven [37] critisises the usual optimality criteria (see preceding pages) as non-operational. He proposes the criterion of profit stabilization. Maximize G as a function of the retention (t) whereby
a(ti
= e(w(t))
where E(w(t)) = expected profit when retention is t, a(w(t)) = standard deviation of profit when retention is t, a = the risk attitude parameter. He demonstrates that the risk attitude is one of the determining functions in choosing the optimum level of the retention while the retention has the tendency to increase with the size of the reinsurance premium's loading which both are intuitively rather obvious. A good treatment of the subject can also be found by Wolff [49] from which it appears that the conditions of the reinsurance market are generally not in accordance with the conditions assumed under which optimal reinsurance strategies are devised. 3. 4. Cost structure of non.life insurance companies Cost of insurance companies are the cost of claims handling, acquisition, underwriting, transaction-processing, administration, housing, super-
III
vision and management. For companies that write insurance through agents we have two sets of costs: (la) Commissions payable to brokers and agents; these costs are about linearly proportional to premium volume. (b) Cost of specific services to and financing of agents; these costs are generally more or less linearly proportional to premium volume or relatively small. (2) Cost of: (a) Acquisition: field and internal staff, Marketing department, advertising and promotion; these costs are program costs and as such invariant to premium volume (at least over a certain range). (b) Underwriting, transaction processing and claims handling. These activities involve variable (to volume) cost of directly productive labor and fixed cost of indirectly productive labor, supervision and management. (c) Housing, data processing and ancillary services. For a very minor part variable with volume and for the major part fixed; at least within a reasonable range insensitive to premium volume. A very high percentage of the cost under point two are personnel costs. Insurance is a relatively labor intensive service industry. Next to salaries, pension premiums and other benefits, the cost of maintaining an adequate labor force are important. Selection, training, placement and renumeration are key variables to succes. So is the organization structure and the quality and levels of communication. Recalling the determinants of the agent's choice demonstrates that quality and motivation of staff and their communication with the intermediaries is the key factor of succes. We see within the industry a growing interest for personnel planning and management development. Particularly so as the recruitement of today determines the composition and quality of staff for the next 20 to 30 years. Literature on production and cost functions in insurance is scarce. Eisen [24] and Farney [26,27] are the only authors mentioned by Botch [13]. It is obvious that the rapid development of electronic data processing and communications techniques will offer a number of problems to insurers in the not to distant future. (i) The planning component in management
i 12
H. can Gelder
/
Planning and control in insurance
will increase in volume and importance. (2) Unless a considerable increase in business volume can be realized companies will be faced with serious social/labor problems as a result from strongly increased productivity; lay offs. (3) New insurance companies that do not have a large, generally older and strongly established labor force are in a position to introduce micro electronics rapidly, have much lower pr';.ces and will undercut traditional companies with 'older' and unfavorable cost structures.
4. In conclusion
We opened this paper with a statement that the 'traditional' O R / M S literature showed a scarcity of publications on insurance. If we had chosen for a survey on decision making under uncertainty instead we would have fared much better and in that sub-discipline lies an attractive avenue to improved decision making for planning and control in insurance. Now we had to turn to the actuarial journals. Although we criticized the Risk Theorists for an number of omissions in their models and a too strict adherence on their side to the risk process proper with the consequential neglect of other, and related problems in insurance we have to give a lot of credit to quite a number of them for the excellent treatment of many of the important issue's in insurance. A closer link between the 'traditional' actuaries and the as 'traditional' M S / O R practitioners with a good dose of Analytical Economics, Marketing, Finance and Accounting could lead to improved models that could help practical men, ordinary insurance managers to get a better understanding of their company, their industry, their clients and their markets. This might even lead to better decisions and to better products at a lower price. The portfolio models presented in [23,35] form a good starting point but it is a question whether they are theoretically and operationally sound. The critisism with respect to mean-variance efficient model of portfolio selection are well known. Next to static nature of the model, there are the data requirements and the problems of estimating the paramet~.rs, particularly the variancecovariance matrix of portfolio returns. There are
some problems with the fact whether a portfolio is indeed adequately characterized by its mean and variance [ 16]. In [30] 'gross' underwriting profit is introduced as a criterion for profitability regarding the very large proportion of fixed cost in insurance; the concept of Direct Costing is introduced. Models should have operational relevance in the first place and it is an open question to the present author of this review whether the business of an insurer is any more stochastic than that in any other industry. While the underlying risk processes are undoubtly random it is an interesting question what consequences this does have for the company as a whole. I hope that nobody gets me wrong. I do not deny the stochastic nature of the insurance business but so is any other business in any other industry. What interests me is, whether deterministic models are appropriate approximations. The latter are so much easier to build, to operate and to comprehend. If this is not the case we might have to do with the beautiful and theoretically very interesting models of extremely limited scope and total non-operationality like Frisque's [28] dynamic model. One real prerequisite for further and fruitful expansion of operational modelling in insurance is a better understanding of the insurance market. Up till now there have only been published a limited number of studies by Borch [ 10,11,12,13,15] and Moffet [40] largely based on work by Arrow [3,41. The behavioural assumptions with respect to agents/brokers and client need to be tested and parameterized empirically. The role of government in price control, incomes policies and market regulation have to be analyzed. Market form and structure; competition and coalition as well as the relationship to money and capital markets have to be brought into the picture. Finally planning models are required for processing capacity; personnel and EDP planning systems while our models require the relevant inputs to provide us with adequate and timely management information for planning and control in a very important industry.
tl. van Gelder / Plamting and control in insurance
References [I] H.H. Agnew, R.A. Agnew, J. Rasmussen and K.R. Smith, An application of chance constrained programming to portfolio selection in a casualty insurance firm, Management Sci. 15 (10) (1969). [2] H. Ammeter, Experience rating a new application of the collective theory of risk, Astin Bull. II (II) (1962). [3] K.J. Arrow, The role of securities is the optimal allocation of risk bearing, Rev. Economic Studies (1964). [4] K.S. Arrow, Optimal insurance and generalized deductibles, Skand. Actuarial J. (1974). [5] A.L. Bailey, A generalized theory of credibility, Proc. Casualty Acturial Soc. 32 (1975). [6] A.L. Bailey, Credibility procedures, Laplace's generalization of Bayes rule and the combination of collateral knowledge with observed data, Proc. Casualty Acturial Soc. 37 ( i 950). [7] R.E. Beard, T. Pentikiiinen and E. Pesonen, Risk Theory (Methuen, London, I st ed., 1969). [8] K. Borch, An attempt to determine the optimum amount of stop loss reinsurance, Trans. 16th International Congress of Actuaries, Vol. 2. [91 K. Borch, The optimal reinsurance treaty, Astin Bull. V (II) (1969). [10] K. Borch, The utility concept applied to the theory of insurance, Astin Bull. I (1961). [11] K. Borch, Recent development in economic theory and their application to insurance, Astin Bull. II (II1) (1963). [12] K. Borch, The economic theory of insurance, Astin Bull. IV (III) (1967). [13] K. Borch, Problems in the economic theory of insurance, Astin Bull. 10 (I) (1978). [14] K. Borch, The safety loading of reinsurance premiums, Skand. Aktuartidskrift (1960). [151 K. Borch, Economic equilibrium under uncertainty, Internxt. Econom. Rev. (1968). II61 K. Borch, Portfolio theory is for risk lovers, J. Banking and Finance 2 (2) (1978). [171 K. Botch, The rescue of an insurance company after ruin, Astin Bull. V (II) (1969). [181 K. Borch, A short note on overall risk management in an insurance company, Astin Bull. VI (2) (1971). [191 K. Borch, Application of game theory to some problems in automobile insurance, Astin Bull. II (II) (1962). 120] H. B0hlmann, Experience rating and credibility, Astin Bull. V (II) (1969). [211 H. BUhlmann, Mathematical Methods in Risk Theory (Springer, Berlin, 1970). [22} DR. Cox and H.D. Miller, The Theory of Stochastic Processes (Chapman and Hail, London, 1965). [231 J. D,~,vid Cummins and David J. Nye, Portfolio optimization models for property-liability insurance companies: an analysis and some extensions, Management Sci. 27 (4)
(1981). [24] R. Eisen, Zur Productionsfunktion der Versicherung, Z. Gesamte Versicherungswissenschaft ( 197 ! ) 407-419. [25] M. Etgar, An empirical analyses of motivations for the development of a centrally coordinated vertical marketing system: the case of the property casualty insurance in-
I 13
dustry, unpublished P h . D . Thesis, Graduate School of Business Administration, University of California, Berkeley (1974). [261 D. Farney, Produktions- und Kostentheorie der Versicherung, Karlsruhe (1965). [27] D. Farney 'Anslttz einer betriebswirtschaftlichen Theorie des Versicherungsunternehmens', The Geneva Papers on Risk and Insurance 5 (February 1977) 9-21. [28] A. Frisque, Dynamic model of insurance company's management, Astin Bull. VIII (1) (1974). [29l H.U. Gerber, Paret~,-optimai risk exchanges and related decision problems, Astin Bull. 10 (1978). [30] H. van Gelder and C. Schrauwers, Planning in theory and practice; with reference to insurance. Proc. 9th IFORS Conference on Operational Research, Hamburg ( 198 i ). [31] M.J. Goovaerts and FI. De Vylder, Premium calculations principles; some properties, Her Verzekeringsarchief 57 ( i ) (1980). I321 W.S. Jeweli, Regularity conditions for exact credibility, Astin Bull. VIII (3) (1975). [331 W.S. Jewell, The credibility distribution, Astin Bull. VII (3) (1974). [341 L. Joskow, Cartels, competition and regulation and the cost of capital in the insurance industry, J. Financial Quantitative Anal. (December 1971 ) 1283-1305. [351 Y. Kahane, Determination of the product mix and business policy of an insurance company--a portfolio approach, Management Sci. 23 (10) (June 1977). [36] Y. Kahane, Capital adequacy and the regulation of financial intermediaries, J. Banking and Finance I (2) (1977). [37] G.N.W. Leuven, Eigen behoud; enige kritische kanttekeningen bij optimaliserings-k~teria (Own retention: some critical remarks about optimality criteria) (Dutch) Her Verzekeringsarchief 56 (3) (1979). 1381 O. Lundberg, On Random Processes and Their Applications to Sickness and Accident Statistics (Aimquist and Wickselis, Uppsala, 2nd ed., 1964). [39] A.L. Mayerson, The uses of credibility in property insurance rate making, Giom. 1st. ltal. Attuari 24 (1964). [40] D. Moffet, Risk-bearing and consumption theory, Astin Bull. VIII (3) (1975). [41l P.M. Pechlivanides, Optimal reinsurance and dividend payement strategies, Astin Bull. 10 (1978). [42l T. Pentikliinen, On the solvency of insurance companies, Astin Bull. IV (III) (1967). [43] T. Pentikitinen, On the approximation of the total amount of claims, Astin Bull. IX (3) (1977). [44] T. Pentik~nen, The theory of risk and some applications, J. Risk and Insurance XLVIII (i) (March 1980). [45] E. Pesonen, On the calculation of the general Poisson function, Astin Bull. IV (I!,' (1967). 146] H.L. Seal, Stochastic Theory of a Risk Business (Wiley, New York, 1969). [471 H.L. Seal, Simulation of ruin potential of non-life insurance companies, Trans. Soc. Act. XXl, p. 562. [48] E. Straub, Application of reliability theory to insurance, Astin Bull. VI (2) (1971). [49] K.H, Wolff, Methoden der Unternehemensforschung im Versicherungswesen (Springer, Berlin, 1966).