Multi-Period Discounted Cash Flow Rate-making Models in Property-Liability Insurance

Abstract

Multi-Period Discounted Cash Flow Rate-making Models in Property-Liability Insurance Introduction Property-liability insurance contracts are characterized by a time lag between the premium payment and loss settlement dates. During this time lag, the insurer earns investment income on the unexpended component of the premium. Given this timing difference, it is surprising that the recognition of investment income in ratemaking is a relatively recent phenomenon. Prior to the late 1960s, property-liability ratemaking formulas included as a profit margin a flat percentage of the gross premium (usually 5 percent). Timing differences between premiums and claims and the resulting investment income were ignored in formal ratemaking procedures.(1) During the late 1960s, rising claim costs and higher interest rates began to motivate regulators to scrutinize ratemaking formulas more carefully. The result was that states such as New Jersey and Texas began to require insurers to give explicit consideration to investment income in ratemaking. Since property-liability insurance involves cash flows at different points, the models developed in corporate finance would seem to have been the logical starting point for the recognition of investment income in insurance rates. However, property-liability insurance is deeply mired in statutory accounting. As a result, the earliest models were based on accounting concepts. The accounting models have been extremely influential and are still proposed in some jurisdictions by insurers and their rating bureaus such as the Insurance Services Office (ISO). Descriptions and analyses of accounting models can be found in NAIC (1983), Cummins and Chang (1983), and Williams (1983). The most serious defects of the accounting models are that they are retrospective rather than prospective and they use embedded yields to measure the rate of return on policyholder funds. Both of these characteristics are contrary to well-known principles of corporate finance.(2) To estimate the cash flows that will result from any given insurance policy, the accounting models look backward instead of forward. They typically measure policyholder funds as a proportion of unearned premiums and loss reserves. Policyholder funds are multiplied by the rate of return on the company's investment portfolio (the embedded yield) to obtain the investment income credit. Reserves are an imperfect proxy, at best, for the amount and timing of future cash flows. (See Cummins and Chang (1983) for an analysis of this problem.) Reserves represent sunk costs which should be irrelevant in setting rates for policies issued in the future. They do contain some information on the time lag between the premium and loss payment dates, but this information can be easily extracted for use in a correct ratemaking formula. The embedded yield is also irrelevant for ratemaking. The correct rate of investment return is the estimated rate that will be earned on the funds received under any given policy or policy cohort.(3) This has nothing to do with past investment yields. When the insurer receives the premium under a newly issued policy, these funds (net of expenses) will be invested at current market rates, not at the embedded yield. Hence, ratemaking should always reflect the best possible estimate of the yields that will be attainable when the cash flows are received.(4) The recognition of these and other defects has led to the development of more appropriate financial pricing models for property-liability insurance contracts. An excellent review of these models is presented in D'Arcy and Doherty (1988). Among the methods proposed for insurance pricing are the capital asset pricing model (CAPM) [e.g., Biger and Kahane (1978) and Fairley (1979)], the option pricing model (OPM) [e.g., Doherty and Garven (1986)], arbitrage pricing theory [Kraus and Ross (1982)], and more general models of continuous time finance [Cummins (1988a)]. …