Relevant Distributions for Insurance Prices in an Arbitrage Free Equilibrium

Abstract

Relevant Distributions for Insurance Prices an Arbitrage Free Equilibrium The increased volatility of economic and financial-risk factors such as inflation, interest rates, investment returns, and exchange rates during the past decade have forced consideration of more financial factors along with underwriting risk factors insurance pricing models. Moreover, competition financial services markets have forced insurers to move more into the financial arena, for example, terms of products tied to investment performance, discounting of loss reserves, actuarial modeling of investment strategies, hedging interest-rate risks, and the internationalization of insurance operations. As a consequence, some of the research finance and risk management and insurance have started to converge as noted by Smith (1986), and Buhlmann (1987). Hence, more of the intertemporal models utilized insurance and actuarial applications involving financial linkages attempt to incorporate the financial concepts of market efficiency and the equilibrium notions underlying competitive market structures (e.g., Kraus and Ross 1982 and Cummins 1988). The same continuous time, stochastic process models are being used for insurance and asset pricing by scholars risk management and insurance and by researchers finance. One reason for this convergence is that insurers have most of their assets financial instruments (e.g. bonds, stocks and mortgages for life insurance companies and stocks and bonds for property casualty companies) and their liabilities consist of interest sensitive components, such as reserves both life-health and property-liability insurance which are discounted to a specific valuation date. Consequently, this article analyzes the probabilistic implications of efficiency and equilibrium from the perspective of potential stochastic models pertinent to actuarial calculations or insurance pricing involving financial transactions an efficient capital market equilibrium. Intuitively, an efficient capital market is the manifestation of a market system that works a cost-effective manner, and the study of efficient markets is a study of the (stochastic) process of price formation, or equivalently of the return generating stochastic process, and the market's adjustment to a sequence of relevant subsets. However, the primitive notion from finance that in equilibrium, price efficiency implies that prices reflect all relevant information is too general to have any practical quantitative applications for actuarial modeling of insurance products affected by financial prices. To quantitatively formalize and model this intuitive notion of an efficient market, scholars insurance, actuarial science and finance have developed several approaches to describing the stochastic process of prices. Two of these are the traditional independent increments or random walk model familiar to actuaries from risk theory, and the more general fair game or martingale model.(1) Some scholars, such as Cummins (1988), Boyle (1977), Black and Scholes (1972), and Boyle and Schwartz (1977), assume that rates of return, for example on stocks or bonds, follow a Brownian motion process. While there is some empirical support for the implied lognormality of the corresponding prices at any fixed point time, it would be desirable and preferable to complement this with an economically based theoretical argument showing why such continuous time probability models arise as a consequence of basic economic notions. Grossman and Shiller (1982, p. 197) also appeal for even a further basic economic rationale for the Brownian motion models which they use. The Brownian motion and stochastic calculus models referred to above are widely used insurance and actuarial research, for example see: Emanuel, Harrison and Taylor (1975), Boyle (1977), Martin-Lof (1986), Cummins (1988), and Sharp (1989). …