We argue that arbitrage-pricing theories (APT) imply the existence of a low-dimensional nonnegative nonlinear pricing kernel. Loose Definition of Arbitrage: A trading strategy that earns something from nothing, no matter how the market evolves in the future.Arbitrage pricing theory (APT) is an alternative to the capital asset pricing model (CAPM) for explaining returns of assets or portfolios. No arbitrage means "create profit without any risk". Arbitrage-free valuation is the value of an asset or financial instrument based solely on the real performance or cash flows that it generates. No arbitrage opportunity exists among well-diversified portfolios. I will use an old joke about economists.