  We construct a deep portfolio theory. By building on Markowitz's classic
risk-return trade-off, we develop a self-contained four-step routine of encode,
calibrate, validate and verify to formulate an automated and general portfolio
selection process. At the heart of our algorithm are deep hierarchical
compositions of portfolios constructed in the encoding step. The calibration
step then provides multivariate payouts in the form of deep hierarchical
portfolios that are designed to target a variety of objective functions. The
validate step trades-off the amount of regularization used in the encode and
calibrate steps. The verification step uses a cross validation approach to
trace out an ex post deep portfolio efficient frontier. We demonstrate all four
steps of our portfolio theory numerically.
