Distribution-Based Pricing Formulas Are Not Arbitrage-Free

A number of actuarial risk-pricing methods calculate risk-adjusted price from the probability distribution of future outcomes. Such methods implicitly assume that the probability distribution of outcomes contains enough information to determine an economically accurate risk adjustment. In this paper, it will be shown that distinct risks having identical distributions of outcomes generally have different arbitrage-free prices. This is true even when the outcomes are completely determined by the same underlying contingent events. Risk-load formulas that use only the risk’s outcome distribution cannot produce arbitrage-free prices and, in that sense, are not economically accurate for risks traded in markets where arbitrage is possible. In practice, most insurance underwriting risks are not traded in such markets. Distribution-based pricing usually does not carry a direct arbitrage penalty for insurance and can reflect an insurer’s risk preferences. A ratio is used to measure the implicit discount or surcharge for risk that is present in a price: the ratio of price density to discounted probability density. This ratio can be used to identify the qualitative nature of a risk as investment or insurance: a risk discount factor less than unity indicates investment, whereas a risk surcharge factor above unity indicates insurance.