On Bonus and Bonus Prognoses in Life Insurance

Transcription

1 Scand. Actuarial J. 2001; 2: On Bonus and Bonus Prognoses in Life Insurance RAGNAR NORBERG ORIGINAL ARTICLE Norberg R. On bonus and bonus prognoses in life insurance. Scand. Actuarial J. 2001; 2: The surplus on a life insurance policy is de ned, at any time during the term of the contract, as the difference between the second order retrospective reserve and the rst order prospective reserve. General principles for redistribution of the systematic part of the surplus as bonus are formulated, and various special bonus schemes are discussed. Techniques for forecasting future bonuses are worked out in an extended model with stochastic experience basis. Numerical illustrations are provided. Key words: Technical basis, experience basis, safety margins, surplus, di idends, bonus, stochastic en ironment, bonus prognoses. 1. INTRODUCTION 1.1. Background The basic paradigm in life insurance mathematics is the principle of equivalence, which lays down that the expected present value of total bene ts less premium in respect of an individual insurance policy should equal 0 at the time of inception of the contract. The rationale of the principle is, roughly speaking, that the law of large numbers will make outgoes and incomes balance on the average in a large insurance portfolio. An implicit assumption underlying this consideration is that the experience basis, that is, the factual transition intensities, interest, and administration costs throughout the contract period, are known at the time of issue. In reality, however, the experience basis may undergo signi cant and unforeseeable changes within the time horizon of the contract, thus exposing the insurer to a risk that is indi ersi able, that is, can not he eliminated or mitigated by increasing the size of the portfolio. The risk stemming from the uncertain development of the interest rate can, under certain ideal market conditions, be eliminated by letting the contractual payments depend on the returns on the company s investments. Products of this type, known as unit-linked insurances, have been gaining increasing market shares ever since they emerged some few decades ago, and today they are also theoretically well understood, see Aase & Persson (1994), Moller (1998), and references therein. Unlike the unit-linked concept, a standard life insurance policy speci es contractual payments in nominal amounts, binding to both parties throughout the 2001 Taylor & Francis. ISSN

2 Scand. Actuarial J. 2 Bonus and bonus prognoses in life insurance 127 entire term of the contract. Thus, an adverse development of the experience basis can not be countered by raising premiums or reducing bene ts and also not by cancelling the contract (the right of withdrawal remains one-sidedly with the insured). The only way the insurer can prevent the indiversi able risk is to charge premiums to the safe side. In practice this is done by calculating premiums on a conservative so-called technical basis or rst order basis, which represents a provisional worst-case scenario for the future development of the experience basis. In the course of the contract period the insurer currently observes the experience basis or (rather assesses it by what is called) the second order basis. Upon identifying reserves of rst and second order, with the latter incorporating explicit safety contributions, one obtains an expression for the current contributions to the technical surplus showing how it emerges from safety margins in the individual technical elements. By statute the technical surplus belongs to the insured and is to be redistributed as bonus, which may be paid out either currently (cash bonus) or as a lump sum upon expiry of the policy (terminal bonus), or may be used as single premiums for current purchases of additional bene ts. Key references on this classical technique are the textbooks by Berger (1939) and Sverdrup (1969), which describe the basic principles in the context of a single-life policy. Width (1986), Ramlau-Hanse n (1991), and Linnemann (1993, 1994) carried the concept over to the multi-state policy Outline of contents The present study is a follow-up of a recent paper by the author (Norberg, 1999), which de nes the surplus at any time as the difference between the second order retrospective reserve (Norberg, 1991) and the rst order prospective reserve, investigates relationships between various notions of surplus de ned this way, and demonstrates how future contributions can be forecasted in an extended model where the empirical basis is driven by a stochastic process. We set out in Section 2 by describing the model framework, which is the traditional continuous time Markov chain model for the multi-state insurance policy extended with a stochastic environment, that is, the experience basis is taken to be stochastic. The rst order basis is taken to be deterministic. Section 3 presents the de nitions of the technical surplus and the policy s contribution to this surplus. In Section 4 we discuss some commonly used schemes for redistribution of the technical surplus, distinguishing between the di idends that are credited to the insured s account and the bonuses that are ultimately paid out. Section 5 shows how model-based prognoses of future bonuses can be worked out in the stochastic environment set-up. Numerical examples are supplied in Section 6. In Section 7 we indicate possible ways of incorporating administration costs in the analysis. In order to keep intermissions at a minimum, a number of comments and discussions are placed in the nal Section 8.

3 Scand. Actuarial J. 2 Bonus and bonus prognoses in life insurance 145 shall not pursue this idea any further here, but note, by way of warning, that prognostication in this kind of in ation model will present problems in addition to those solved in Section DISCUSSIONS 8.1. The principle of equi alence This principle, as formulated in (2.3), is basic in life insurance. The expected value represents averaging over a large (really in nite) portfolio of policies, the philosophy being that, even if the individual policy creates a (possibly large) loss or gain, there will be balance on the average between outgoes and incomes in the portfolio as a whole if the premiums are set by equivalence. The deviation from perfect balance, which is inevitable in a nite world with nite portfolios, represents pro t or loss on the part of the insurer and has to be settled by an adjustment of the equity capital. (The possibility of loss, about as likely and about as large as the possible pro t, might seem unacceptable to an industry that needs to attract investors, but it should be kept in mind that salaries to employees and dividends to owners are accounted as part of the expenses discussed in Section 7.) 8.2. On the notion of second order basis The de nition of the second order basis as the true one is slightly at variance with practical usage (which is not uniform anyway). The various amendments made to our idealized de nition in practice are due to administrative and procedural bottlenecks: The factual development of interest, mortality, etc. has to be veri ed by the insurer and then approved by the supervisory authority. Since this can not be a continuous operation, any regulatory de nition of the second order basis must to some extent involve realistic, still typically conservative, short term forecasts of the future development. However, our de nition can certainly be agreed upon as the intended one The role of the stochastic en ironment model The equivalence requirement (4.1) is exercised conditionally, given the second order basis. Therefore, it does not depend on our choice of marginal distribution of the second order elements. The same goes for the contributions and the dividends. Thus, the distribution placed on the second order plays a role only in the prognostication of bonus. Subsidiary as it is, this role is still an important part of the play; although a prognosis does not commit the insurer to pay the forecasted amounts, it should as much as possible be a reliable piece of information to the insured. Therefore, the distribution placed on the second order elements should set a reasonable scenario for the course of events, but it need not be perfectly true. In particular, the Markov chain model used here can certainly serve well.

4 146 R. Norberg Scand. Actuarial J. 2 Table 3. Expected alue (E) and standard de iation (SD) of present alues of a term life insurance (TI) with sum 1 and a life annuity (LA) with le el intensity 1 per year, with interest d¾e id À and mortality m¾e mm À for various choices of e i and e m TI em : 1.5 LA ei: 0.5 E: SD: E: SD: E: SD: A digression: Which is more important, interest or mortality? Actuarial wisdom says it is interest. This is, of course, an empirical statement based on the fact that, in the era of contemporary insurance, mortality rates have been smaller and more stable than interest rates. Our model can add some other kind of insight. We shall again be content with a simple illustration related to the single life described in Section 6. Table 3 displays expected values and standard deviations of the present values at time 0 of a term life insurance and a life annuity under various scenarios with xed interest and mortality throughout the term of the policy. The impact of interest variation is seen by reading column-wise, and the impact of mortality variation is seen by reading row-wise. The overall impression is that mortality is the more important element by term insurance, whereas interest is the (by far) more important by life annuity insurance. ACKNOWLEDGMENT This paper is dedicated to the memory of Professor Erling Sverdrup, whose didactics shaped my understanding of the issue of bonus. REFERENCES Aase, K. K. & Persson, S. A. (1994). Pricing of unit-linked life insurance policies. Scand. Actuarial J. 1994, Berger, A. (1939). Mathematik der Lebens ersicherung. Verlag von Julius Springer. Hoem, J. M. (1969). Markov chain models in life insurance. Blätter der Deutschen Gesellschaft für Versicherungsmathematik 9, Linnemann, P. (1993). On the application of Thiele s diverential equation in life insurance. Insurance: Math. & Econ. 13, Linnemann, P. (1994). Bonus, salary increases and real value of pensions. Scand. Actuarial J. 1994, Møller, T. (1998). Risk minimizing hedging strategies for unit-linked life insurance contracts. ASTIN Bull. 28,

5 Scand. Actuarial J. 2 Bonus and bonus prognoses in life insurance 147 Norberg, R. (1991). Reserves in life and pension insurance. Scand. Actuarial J. 1991, Norberg, R. (1995). A time-continuous Markov chain interest model with applications to insurance. J. Appl. Stoch. Models and Data Analysis 11, Norberg, R. (1999). A theory of bonus in life insurance. Finance and Stochastics 3, Ramlau-Hansen, H. (1991). Distribution of surplus in life insurance. ASTIN Bull. 21, Sverdrup, E. (1969). Noen forsikringsmatematiske emner. Stat. Memo. No. 1, Inst. of Math., Univ. of Oslo. (In Norwegian.) Width, E. (1986). A note on bonus theory. Scand. Actuarial J. 1986, Address for correspondence: Ragnar Norberg Department of Statistics London School of Economics Houghton Street LondonWC2A 2AE United Kingdom R.Norberg@lse.ac.uk.