In an arbitrage-free economy, there will always exist a set of linear operators which map future contingent dividends of securities into their current prices. It happens that such operators will also form an “evolution semigroup” as a consequence of intertemporal analysis of the no-arbitrage restriction. This paper summarizes some of the major implications of the semigroup properties, but avoids almost all of the technical discussion which underlies them. Instead, several practical examples are presented. Some well-known continuous-time results are replicated by this alternative method, and certain new developments are explored.