Arbitrage and Growth Rate for Riskless Investments in a Stationary Economy

A sequential investment is a vector of payments over time, (a0, a1, . . . , an), where a payment is made to or by the investor according as ai is positive or negative. Given a collection of such investments it may be possible to assemble a portfolio from which an investor can get “something for nothing,” meaning that without investing any money of his own he can receive a positive return after some finite number of time periods. Cantor and Lipmann (1995) have given a simple necessary and sufficient condition for a set of investments to have this property. We present a short proof of this result. If arbitrage is not possible, our result leads to a simple derivation of the expression for the long-run growth rate of the set of investments in terms of its “internal rate of return.”