Fair Premium Rating Methods and the Relations between Them

Abstract

Capital Asset Pricing Model, Advisory Filing of the Massachusetts Automobile and Accident Prevention Bureau for 1982 Rates, August 1981. [Appears in J. D. Cummins and S. E. Harrington, eds., 1987, Fair Rates of Return in Property-Liability Insurance (Dordrecht: Kluwer Nijhoff Publishing).] Sharpe, W. F., 1964, Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance, 19: 425-442. Greg Taylor is Actuary responsible for property and casualty insurance at GIO Australia, Sydney. The majority of the research leading to the present article was carried out at the Wharton School of the University of Pennsylvania. It received major financial support from NRMA Insurance Ltd, Sydney, Australia. Introduction The operation of an insurance business requires capital to provide security that claims will be met. This article is concerned with the case in which this capital is provided by shareholders (as opposed to the case of a mutual insurer). Shareholders' capital is placed at risk by the insurance process and so must attract a return commensurate with the risk to which it is subject. To the extent that the return exceeds that which would be generated by the mere holding of the capital base in investments, the excess must be drawn from premiums paid by policyholders. Hence, these premiums must be based on a pricing formula of the form premium = risk premium + expense loading + margin to service capital. (1) The margin appearing as the final premium component in equation (1) will be referred to as the profit margin. Fair premium rating methods are those which quantify the profit margin in such a way as to provide shareholders with a fair return on capital but no more. Details differ from one method to another, but all have this concept of fairness at their base. Several fair rating methods have been developed, and the major ones are summarized below. The methods yield different results when applied to a specific numerical case, even though each has been developed more or less rigorously from a certain theoretical base. Thus, the methods are not equivalent in general. Nevertheless, their derivations indicate that they have a good deal in common. It is therefore useful to render manifest those equivalences which do exist by identifying circumstances in which different fair rating methods yield identical results. The obverse of the establishment of these equivalences will be the identification of the major differences between methods. The identification of these equivalences and differences is the purpose of this article, and, in this respect, it forms a sequel to Cummins (1990), which compared the Myers and Cohn and internal rate of return methods of rating. Notation, Terminology, and Conventions Consider a portfolio of insurance contracts. The parties financially involved in this portfolio comprise policyholders, shareholders, issuers of investments to the insurer, tax authorities, and providers of goods and services to the insurer. To anticipate the more detailed treatment given below, the cash flows between these parties comprise premiums, claims and expenses, investment income, taxes, and payments to and from shareholders. Subsequent sections will consider the subset of this portfolio consisting of policies underwritten in a particular period. This subset will be referred to as the cohort. It will be traced from its origins in the underwriting period, through subsequent periods, until it generates no further transactions. Let the various periods be denoted by t = 0, 1, ..., T, with t = 0 denoting the underwriting period, and t = T the period in which the final transaction occurs. All periods t are of equal length, but the length is unspecified; it may be years, quarters, months, etc. The premium underwritten in period zero provides coverage during periods 0, 1, ..., k, where k is a fixed but arbitrary integer less than or equal to T. …