We open up the black box behind Deep Learning for portfolio optimization and prove that a sufficiently wide and arbitrarily deep neural network (DNN) trained to maximize the Sharpe ratio of the Stochastic Discount Factor (SDF) is equivalent to a large factor model (LFM): A linear factor pricing model that uses many non-linear characteristics. The nature of these characteristics depends on the architecture of the DNN in an explicit, tractable fashion. This makes it possible to derive end-to-end trained DNN-based SDFs in closed form for the first time. We evaluate LFMs empirically and show how various architectural choices impact SDF performance. We document the virtue of depth complexity: With enough data, the out-of-sample performance of DNN-SDF is increasing in the NN depth, saturating at huge depths of around 100 hidden layers.

We propose a novel and easy-to-implement framework for forecasting correlation risks based on a large set of salient realized correlation features and the sparsity-encouraging LASSO technique. Considering the universe of S&P 500 stocks, we find that the new approach manifests in statistically superior out-of-sample forecasts compared to commonly used procedures. We further demonstrate how the forecasts translate into significant economic gains in the form of higher pairs trading profits, better equity premium predictions, more accurate portfolio risk targeting, and superior overall risk control and minimization.

I develop a method to identify the strongest determinants of expected returns. Instead of sorting stocks on characteristics, I sort stocks into portfolios based on their realized returns---the variable of interest---at each month in the past and find the average of each characteristic among assets in each portfolio. Then I create out-of-sample portfolios such that they are as similar as possible to the returns-sorted portfolios regarding 178 characteristics. This approach separates low-mean stocks from the high-mean ones so that a long-short portfolio gains an out-of-sample monthly alpha of 1.74% (t = 13.78). Characteristics that differ between low- and high-mean stocks drive the dispersion in expected returns. I find price-based characteristics are the strongest predictors.