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The ASTIN Bulletin 1o (t979), THE THEORY OF INSURANCE RISK PREMIUMS -- A RE-EXAMINATION IN THE LIGHT OF RECENT DEVELOPMENTS IN CAPITAL MARKET THEORY YEHUDA KAHANE * 1. INTRODUCTION The premium

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The ASTIN Bulletin 1o (t979), THE THEORY OF INSURANCE RISK PREMIUMS -- A RE-EXAMINATION IN THE LIGHT OF RECENT DEVELOPMENTS IN CAPITAL MARKET THEORY YEHUDA KAHANE * 1. INTRODUCTION The premium calculation principle is one of the main objectives of study for actuaries. There seems to be full agreement among the leading theoreticians in the field that the insurance premium should reflect both the expected claims and certain loadings. This is true for policy, risk or portfolio. There are three types of positive loadings: a) a loading to cover commissions, administrative costs and claim-settlement expenses; b) a loading to cover some profit (a cost-plus approach) ; and c) a loading for the risk taken by the insurer when underwriting the policy. The administrative costs can be considered a part of expected gross claims . Thus, the insurer's ratemaking decision depends on his ability to estimate expected claims (including costs) and on the selection of a fair risk loading. The main concern in the literature is the appropriate measurement of the risk and the exact loading formula. BOI-rLSIANN [1970, ch. 5] and others identified four possible principles of risk loading, namely, the expected value principle, the standard deviation loading, the variance loading, and the loading according to the principle of constant utility. Various studies point to the advantages and disadvantages of these principles and also examine some additional loading forms--semi-variance, skewness, etc. (e.g., BOHLSIANN [1970], BENKTANDER [1971], BERGER [1972 ], I~URNESS [1972], BERLINER [1974], BERLINER and BENKTANDER [1976 ], BOHS{AN [t976], COOPER [1974], GERBER [1975] and others). Despite different pleferences in choosing the appropriate loading calculation principle, all seem to agree that the risk loading must be positive, since, otherwise, the firm would just have to wait for its ruin, that is bound to come sooner or later, according to risk theory. The purpose of this article is to re-examine the appropriate principle of premium calculation in light of the recent developments in the theory of finance and especially in the theory of capital market equilibrium. These developments may suggest a new point of view and raise a few questions regarding the loading rules. * Senior Lecturer at the Faculty of /XIanagement, Tel Aviv University, Israel, and Academic Director, Erhard Center for Higher Studies and Research in Insurance, Tel Avlv University, Israel. The author wishes to acknowledge the very helpful discussions with Dr. 13. Berliner on an earlier draft of this paper and the many remarks of the participants of the 14th ASTIN Colloquium in Taormina, October t978. 224 YEHUDA KAHANE The first question is related to whether or not, and how, investment income should be considered in premium calculation 1). Some insurers and insurance regulators tend to disregard investment income altogether. They misinterpret, perhaps, earlier models in risk theorv which concentrated on the insurance portfolio in isolation and disregarded the investments merely for the sake of simplicity. Other insurers, and especially in certain liues, deduct investment income through the calculation of the expected present value of the relevant cash flows (claims and expenses). This paper suggests that investment income should be considered in ratemaking, either through a present value calculation, or through a negative loading on expected claims. Another problem which can be solved with the use of financial theory is related to the appropriate measurement of risk for ratemaking purposes. It is suggested that the traditional measures of riskiness of an individual risk (standard deviation, variance, etc.) be replaced by the systematic element of the variance and that the risk loading be proportional to this element. It will be shown that, since the profit of the insurer is derived from both underwriting and investment incomes, the insurer might, under certain circumstances, even be willing to lose on his underwriting activities. The appropriate loading on the expected pure claims may therefore be negative, and this may offer a theoretical explanation for the willingness of some insurers to under-rate 2). The exact conditions for a negative loading will be studied later and an explicit expression for the profit (loss) will be presented. And finally, it is suggested that risk loadings should be determined by capital market equilibrium and must therefore be objective and uniform for all insurers. The main argument in the following analysis can be explained by viewing a very simple example: Assume an investment company which raises funds through the sale of bonds (debt) and invests its capital plus the external funds in an assets portfolio. The required return on the shareholder's investment reflects the risks of the investment portfolio and the financial leverage (debt) used. Notice that the shareholders derive an appropriate profit after the payment of a positive interest on the firm's debt. Now assume an insurer is silnilar to the investment company, except that it raises the additional funds as a by-product of the sale of insurance contracts, rather than through the use of regular debt instruments. According to QmRIN and WATERS [1975], this is analogous to a firm which charges a positive interest rate from its creditors, rather than paying them for the use of their money. A positive underwriting profit on the insurance portfolio would mean that the insurer x) This topic has attracted many econollllsts and actuaries. A discussion and references to some sources may be found in BIGER and I{AHANE [1978], PYLE [1971], QUIRIN and ~'VATERS [t975] or in a book by CooPI,:R [1974]. =) The traditmnal explanatmns for underrating are related to the attempt to preserve long-term connections with msureds, or to the lack of knowledge and experience (see BENKTANDER [197 t]). THE THEORY OF INSURANCE RISK PREMIUMS 225 makes a higher overall rate of profit than the investment company. Although the analogy is imperfect and very simplistic it may still demonstrate that consistent underwriting profits violate capital market equilibrium. Section 2 stumnarizes the developments in financial literature and the riskreturn relationships in capital market equilibrium. This will be used in Section 3 to analyze the treatment of investment income in ratemaking and the implications of the financial theory for the measurement of underwriting risks and tile loading factor to be used in ratemaking. Some reservations and a few concluding remarks are summarized in Section RISK RETURN RELATIONSHIPS AND CAPITAL MARKET EQUILIBRIUM Assume that the insurance company competes for investors' funds in the capital market. The firms' profits must therefore compensate the existing and potential shareholders for the risks they assume through their investment. The insurers' profitability is affected by the premium formula, and thus the relationship between the required expected return and the risk level on the insurer's shares may serve as a key to the ratemaking formula (BORCH [1974, ch. 22]). Fairly recent developments in financial theory suggest that exact relationships between the expected return and the risk must prevail in market's equilibrium. A brief summary of these developments follows prior to tile discussion of the implications for ratemaking. Risk and Diversification The basic idea in portfolio theory, which has been suggested by the pioneering work of Markowitz [1952], is imbedded ill the mathematical properties of the standard deviation. I.e., the standard deviation of a linear combination of stochastic variables is typically lower than the weighted sum of the individual standard deviations. Each individual risk is represented by a stochastic variable, which is assumed to be fully characterized by its expected value and standard deviation a). The expected value is taken as a measure of profitability, while the standard deviation is used as a measure of the risk. It can easily be seen that there would generally be some gain from holding diversified portfolios, since the standard deviation of the portfolio will be lower (i.e. less risky ) than that of an undiversified portfolio. This can be demonstrated by considenng two securities A and B (see fig. I). All portfolios obtained by holding these securities in varying proportions are represented by a curve APB. The nature of tlns curve depends on the correlation between the random variables A and B. In the extreme case, where the securities are perfectly positively correlated, there would be no gain from 3) See a short discussion in the concluding remarks. t5 226 YEHUDA KAHANE diversification (AQB in fig. 1). In the other extreme case, where all securities are perfectly negatively correlated, the investor would even be able to construct a portfolio with a positive expected leturn and zero standard deviation (i.e., a risk-free portfolio (R in fig. l)), although it is composed of individual risky securities. E Expected Rate of Return I i _ ~ B R ~ '~ A Standard Deviation of Rate of Return (=Risk) Fig. 1. The Effec~ of Diversification on the Po~folio's Expected Return and Risk. Efficiency Frontier Identifying the optimal portfolio is clearly not an easy task, since an infinite number of combinations of each pair of securities must be examined. The first step in the optimization is to calculate the efficient portfolio, which has the minimal standard deviation for a given level of expected value. This can be accomplished quite efficiently using the Quadratic Programming Technique (MARKOWlTZ [1952~). Repeating the same process for all levels of expected value creates the efficiency frontier which is the locus of all portfolios having the lowest standard deviation at each level of expected value (curve DEF in fig. 2). Knowing the efficiency frontier, the main problem is to select the optimal portfolio on that frontier. The traditional economic solution is based on the introduction of a set of indifference curves which represent the subjective trade-off between risk (standard deviation) and profitability (expected return). The optimal portfolio would be obtained at the tangency point between the THE THEORY OF INSURANCE RISK PREMIUMS 227 E Expected Rate of Return JA.. F E C Standard Deviation of Rate of Return Fig. 2. Efhciency Frontier and the Optimal Portfolio. r (=Risk) highest possible indifference curve and the efficiency frontier (point E in fig. 2). This solution depends on the individual's subjective attitude toward risk reflected by the indifference curves and assumes a full knowledge of individual utilities. The Capital Asscls Pricing Model (CAPM) The CAPM offers a new solution which does not depend on the individual's preferences and which is uniform for all investors. Its main assumption is the existence of a perfect capital market (i.e., there is a uniform interest rate at which each investor can borrow or lend any amount of money with no other transactions costs). The introduction of this interest rate, which is a risk-free security (Ri), causes dramatic changes in the efficiency frontier; combining a risky security, or portfolio, A with the risk-free security R I will generate portfolios on the straight line RIA (see fig. 3). The best combinations will lie on the ray RIM which is tangent to the original efficiency frontier at M. Being on the section RiM means that the investor lends part of his initial capital (purchases risk-free bonds). A portfolio represented by a point on ray RIM but to the right of M is obtained by borrowing money at the risk-free rate and investing tile capital and the borrowed funds in the risky 228 YEHUDA KAHANE portfolio M (i.e., by using financial leverage ). The optimal portfolio is selected in two isolated stages. The first consists of finding the portfolio M of risky securities. In the second stage the desired mix of this portfolio with the risk-flee asset is selected according to the tangency of RiM to the indifference curves. Expected Rate of Return! / jj x [ ~ l j ~ B Rf Fig. 3. Capital Asstes Pricillg Model Standard Deviation of Rate of Return (=Risk) r The next step in the development of the CAPM is based on the assumption that all investors have the same expectations concerning the means, standard deviations and covariances between all securities. Under a model of full agreement, all investors must hold the same portfolio composition of risky securities (point M). This portfolio is composed of all the risky ventures and is called the market line portfolio. The combinations of this portfolio with the risk-free interest rate, lie on a straight line called the market line which represents the risk-return relationship for al! portfolios in the market. It is impossible to create a portfolio with a better performance which would be represented by a point above this capital market line. Any portfolio below this line would be inferior. The equation :for the capital market line is Em- Rf (t) E~ = R.f + ~, THE THEORY OF INSURANCE RISK PREMIUMS 229 where E and e denote expected value and standard deviation, respectively, and the subscripts p and m denote a portfolio and the market portfolio, respectively (SuARPE [1964], LINTNER [1965], MOSSlN [1966]). Equation (1) represents the objective risk-return relationship for a portfolio in market equilibrium and can be interpreted as follows: The expected return on any investment portfolio equals the risk-free rate of interest plus a risk loading which is proportional to the standard deviation of the porttolio. Under the CAPM, the appropriate risk measure for a portfolio of securities is the standard deviation and not its variance. This result stems from the basic assumption of the model and therefore cannot be used as an argument against the use of a risk loading proportional to the variance, which is recommended by some of the leading authorities in the Collective Risk Theory (BfUHLMANN [197o ], BERLINER [1974], BOHMAN [1976 ], etc.). Risk-Relur~ Relalionship for an Individual Risk The capital market line is obtained through the holding of a combination of securities which are typically below it (like points A, B, C, in fig. 3). What is the appropriate risk-return relationship for the iudividual security? Further analysis of the CAPM showed that the expected return of each individual investment under equilibrium must satisfy the following equation E,,- Rf (2) E, = R.r + ~ ~, (~?/t where the a,m represents the covariance between the return on security i and the return on the market portfolio (The proof for these relationships is given by SVlARPE [1970, pp ]). Equation (2) means that the expected return on the individual security equals the return on the risk-free asset plus a proportional risk loading. Unlike the relationship for a portfolio (equation (I)), the risk for an individual security is measured by **m, the covariance of the return on the security and the market portfolio. This suggests a new measure for the risk level of an individual security--the systematic risk clement. A variation of this term, namely, a,m/%2n, is often used in financial literature for the same purpose and is called the beta coefficient. The risk for an individual security, unlike the measure of risk for a portfolio (collective risk), is not measured by its standard deviation or variance. The full variance of the return on each security is split into two components: the systematic risk (representing the correlation with the market portfolio), and a non-systematic element (representing random fluctuations or noise). This is demonstrated by fig. 4, which shows the return on a hypothetical security i and the return on the market portfolio. The dots on this graph represent individual observations (periodic observations). The systematic element is captured by the slope of the regression line. The vertical deviations 23 YEHUDA KAIIANE Noise Rate of ri Return on Security i Fig. 4. Systematic and Non-SystenlaticRisk. Rate of Return m on Market Portfolio r of the observed return from its conditional expected value represent a random noise. The non-systematic element (the noise ) is excluded from the measurement of risk because it can be diversified away and eliminated to a great extent by holding appropriately diversified portfolios 4). T15s results from the assumption that the random fluctuations of securities i and j are uncorrelated. The return on securities fluctuates. Despite these fluctuations some securities may be regarded as risk-free where their rates of return have no consistent relationships with those of the market portfolio. In such a case their expected return must equal the risk-free rate of interest. Such securities are represented by lines with zero slope in fig. 5. Other securities may be represented by a slope of unity. Holding such securities has an effect similar to the holding of the market portfolio itself (despite their higher variance caused by random noise). Securities having slopes steeper than unity are aggressive , i.e., they augment the fluctuation of the market and are therefore more risky than the market portfolio. Some securities may even have negative slopes, which means that they behave counter to the market portfolio. The expected return on these securities would be lower than the risk-free rate of interest since they have a risk reduction effect in a portfolio context. 4) See a quite similar idea in BERLINER [1974]. THE THEORY OF INSURANCE RISK PREMIUMS 23I Expected Return on Security 'i E m I R I I I I 0 i Systematic Risk l:ig 5. The frisk-return llelationship for Individual Risks IMPLICATIONS FOR INSURANCE RATEMAKING The CAPM is obviously an over-simplified representation of financial markets in the real world. Tile model rests on the assumption that a security is completely described by a stationary probability distribution and that only the first two moments of the distribution are relevant. Ii1 addition, the model assumes uniform information among investors, identical investment planning horizons, and perfect capital markets with a risk-free rate of interest. Despite the over-simplifications, the model seems to capture some of the essential elements in real situations and has demonstrated a fairly good explanatory power in empirical tests 5). Unfortunately, this model has hardly received the attention it deserves in actuarial literature. Among the few exceptions are the works by BOl~CH [1974, ch. 9, 21, 22] and by QUIRIN ET AL. [1974]. 5) There are a great nulriber o~ elnpirical tests for the vahdity of the CA.PM. A review of some of the tests cart be found m ~,{ODIGLIANI arid POGUE [1974]. 232 YEHUDA I{AHANE The potential of the CAPM for the analysis of the ratemaking issue is quite obvious. According to the CAPM, there should be an objective market price per unit of risk. This may suggest that the insurance risk loadings must be determined objectively, rather than through subjective considerations of the insurance company. It means that the loading should not depend on management attitude toward risk (i.e., its utility function). Moreover, the CAPM may be used to find the exact parameters for the risk loading. The profit of the insurer is derived from two sources' its underwriting profits and its investment income. Thus, the ratemaking problem should be analyzed by considering the two income sources simultaneously. It will be shown that previous studies which simplified the analysis by examining the insurance portfolio in isolation (e.g., BENKTANDER [1971], BOHI.MANN [1970]) offered only a partial solution for the ratemaking problem. Assume that the firm has m insurance activities (policies or lines). The firm collects SX, in premiums for contract i and expects to make an underwriting loss (profit) of X,h dollars, h is a stochastic variable representing the rate of underwriting loss in this line (a negative value will denote profit). The stochastic variables arc cl

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