Good Deal Indices in Asset Pricing: Actuarial and Financial Implications

Abstract

We integrate into a single optimization problem a risk measure, beyond the variance, and either arbitrage free real market quotations or …nancial pricing rules generated by an arbitrage free stochastic pricing model. A sequence of investment strategies such that the couple (expected-return,risk) diverges to (+1; 1) will be called a good deal. The existence of such a sequence is equivalent to the existence of an alternative sequence of strategies such that the couple (risk,price) diverges to (1; 1). Moreover, by appropriately adding the riskless asset, every good deal may generate a new one only composed of strategies priced at one. We will see that good deals often exist in practice, and the main objective of this paper will be to measure the good deal size. The provided good deal indices will equal an optimal ratio between both risk and price, and there will exist alternative interpretations of these indices. They also provide the minimum relative (per dollar) price modi…cation that prevents the existence of good deals. Moreover, they will be a crucial instrument to detect those securities or marketed claims which are over-or underpriced. Many classical actuarial and …nancial optimization problems may generate wrong solutions if the used market quotations or stochastic pricing models do not prevent the existence of good deals. This fact is illustrated in the paper, and we point out how the provided good deal indices may be useful to overcome this caveat. Numerical experiments are included as well.