It is common knowledge that leverage can increase the potential returns of an investment, at the expense of increased risk. For a passive investor in the stock market, leverage can be achieved using margin debt or leveraged-ETFs. We perform bootstrapped Monte-Carlo simulations of leveraged (and unleveraged) mixed portfolios of stocks and bonds, based on past stock market data, and show that leverage can amplify the potential returns, without significantly increasing the risk for long-term investors.

We revisit mean-risk portfolio selection in a one-period financial market where risk is quantified by a positively homogeneous risk measure $\rho$. We first show that under mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical mean-variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation $\rho$-arbitrage, and prove that it cannot be excluded -- unless $\rho$ is as conservative as the worst-case risk measure.

After providing a primal characterisation of $\rho$-arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterisation of $\rho$-arbitrage. We show that the absence of $\rho$-arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual representation of $\rho$. A special case of our result shows that the market does not admit $\rho$-arbitrage for Expected Shortfall at level $\alpha$ if and only if there exists an EMM $\mathbb{Q} \approx \mathbb{P}$ such that $\Vert \frac{\text{d}\mathbb{Q}}{\text{d}\mathbb{P}} \Vert_\infty < \frac{1}{\alpha}$.

We propose a new algorithm for estimating treatment effects in contexts where the exogenous variation comes from aggregate time-series shocks. Our estimator combines data-driven unit-level weights with a time-series model. We use the unit weights to control for unobserved aggregate confounders and use the time-series model to extract the quasi-random variation from the observed shock. We examine our algorithm's performance in a realistic simulation based on Nakamura and Steinsson [2014]. We provide statistical guarantees for our estimator in a practically relevant regime, where both cross-sectional and time-series dimensions are large and show how to use it to conduct robust inference.

How to hedge factor risks without knowing the identities of the factors? We first prove a general theoretical result: even if the exact set of factors cannot be identified, any risky asset can use some portfolio of similar peer assets to hedge against its own factor exposures. A long position of a risky asset and a short position of a "replicate portfolio" of its peers represent that asset's factor residual risk. We coin the expected return of an asset's factor residual risk as its Statistical Arbitrage Risk Premium (SARP). The challenge in empirically estimating SARP is finding the peers for each asset and constructing the replicate portfolios. We use the elastic-net, a machine learning method, to project each stock's past returns onto that of every other stock. The resulting high-dimensional but sparse projection vector serves as investment weights in constructing the stocks' replicate portfolios. We say a stock has high (low) Statistical Arbitrage Risk (SAR) if it has low (high) R-squared with its peers. The key finding is that "unique" stocks have both a higher SARP and higher excess returns than "ubiquitous" stocks: in the cross-section, high SAR stocks have a monthly SARP (monthly excess returns) that is 1.101% (0.710%) greater than low SAR stocks. The average SAR across all stocks is countercyclical. Our results are robust to controlling for various known priced factors and characteristics.