The stochastic discount factor for the exponential-utility capital asset pricing model. Working Paper

In this paper we introduce the notion of a marginal stochastic dis-count factor for an asset. This is a function defined on the possible payoffs of the asset which can be used to value the asset and derivatives of it. We then show how one may derive the marginal stochastic dis-count factor for a common version of the Capital Asset Pricing Model (CAPM), namely the version derived under assumptions of exponen-tial utility and asset payoffs jointly normal with the market portfolio. The advantage of this is that it allows us to obtain equilibrium prices for assets which are derivatives of CAPM assets, even though these derivatives do not satisfy the CAPM. We then illustrate the practical application of this discount factor by showing how to value the liabilities of a firm with CAPM assets. Our derivation of the CAPM discount factor builds upon some papers published in the actuarial literature by Shaun Wang and Hans Bühlmann.