Conditional Beta Pricing and the CAPM

Introduction In this chapter we discuss the counterparts in the multidate setting of the results of Chapter 18 deriving beta pricing and of Chapter 19 deriving the Capital Asset Pricing Model, each in the two-date setting. The counterpart of the beta pricing relation of Chapter 18 is the conditional beta pricing relation. The derivation of conditional beta pricing is based on the observation that each nonterminal event and its immediate successor events are formally indistinguishable from the two-date model. Accordingly, the pricing relation can be derived in the same way in the multidate case as in the two-date case. In the derivation of the Conditional CAPM we restrict our attention to the case with quadratic utilities. Two-Date Security Markets at a Date- t Event We want to construct two-date security markets associated with nonterminal event ξ t by viewing variables at ξ t and the immediate successor events of ξ t as the analogues of the corresponding variables at date 0 and date 1, respectively, of the two-date model. The first step is to note that some terms that have a clear meaning in the two-date model have several possible distinct analogues in the multidate model. For example, consider portfolio payoffs. In the two-date model the payoff of a portfolio h is xh . In Chapter 21 we defined the multidate payoff in event ξ t +1 of a portfolio strategy h as [ p (ξ t +1 )+ x (ξ t +1 )] h (ξ t )– p (ξ t +1 ) h (ξ t +1 ).