Speaker: Yosi Rinott, Hebrew University of Jerusalem
Yosef Rinott, Joint ongoing work with Sergiu Hart
Suppose there is a set of states that are relevant to the value of an asset, and the market has a prior distribution on the state space. Consider signals having distribution depending on the true state. The price of an asset depends on the market’s posterior probability that the asset is in a good state, given a signal. Before the signal is given, the market considers the expectation of this (random) posterior probability, and more generally, its distribution. It is well known that the expected posterior equals the prior, but the actual posterior probability may be higher or lower than the prior depending on the signal. Suppose an insider has a different prior than the market. We study the question of what kind of priors will make the insider expect that the asset value following the signal will be higher than the market’s value (suggesting to buy, or to be optimistic). Since the posterior is a random variable, this leads to stochastic ordering, and in fact, we obtain a very strong ordering, namely likelihood ratio order. Given such a setup, if signals are generated independently according to a given state, then Doob’s Bayesian Consistency Theorem says that the posterior probability of this state converges to one. We study some monotonicity properties of this convergence, and some (perhaps) surprising lack of monotonicity.