Covariance appears throughout investment management, e.g., in risk reporting and control, portfolio construction, risk parity, smart beta, algorithmic trading, and hedging. It is usually represented via multi-factor model. The form’s fewer parameters and structure—comovement through...

Historical VaR, CVaR and ES (Expected Shortfall) to LIQUIDATION Software is a model characterized by its straightforwardness, allowing regulators measure risk using a standard database of primitive factors and portfolio positions only, leaving little error margin in comparing market risk for...

Asset allocation strategies which utilize stop-loss and stop-gain rules may dramatically decrease risk and even increase long-term return relative to passive investing. I introduce an asset allocation strategy which shifts portfolio weights based on simplistic stop rules. The two-asset (S&P...

We consider an investor who faces parameter uncertainty in a continuous-time financial market. We model the investor's preference by a power utility function leading to constant relative risk aversion. We show that the loss in expected utility is large when using a simple plug-in strategy for...

Risk decomposition is a standard tool for analyzing investment portfolio risk. The portfolio is divided into notional parts—e.g., individual securities, holdings by sector or region, factor exposures—whose contributions to net risk are estimated and reported. Convention regards the...

We study Aumann and Serrano's (2008) risk index for sums of gambles that are not dependent. If the dependent parts are similarly ordered, then the risk index of the sum is always larger than the minimum of the risk indices of the two gambles. For negative dependence, the risk index of the sum is...

This paper analyses and develops insights to systematic risk and diversification when random, imperfectly dependent, losses are aggregated. Systematic risk and diversification are shown to vary across layers of component losses according to local dependence and volatility structures. Systematic...