Modern portfolio theory accords symmetrical treatment to all deviations from expected return, positive or negative. This assumption is vulnerable on both descriptive and behavioral grounds. Many of the predictive flaws in contemporary finance stem from mathematically elegant but empirically flawed Gaussian models. In reality, returns are skewed. The presumption that returns and volatility are symmetrical also defies human behavior. Losing hurts worse than winning feels good; investors do not react equally to upside gain and downside loss. Moreover, correlation tightening during bear markets, not offset by changes in correlation during bull markets, suggest that standard diversification strategies may erode upside returns without providing adequate protection during times of stress.
This article outlines mathematical tools for calculating volatility, variance, covariance, correlation, and beta, not merely across the entire spectrum of returns, but also on either side of mean returns. It pays special attention to beta. Beta is a composite measure that reflects changes in volatility and in correlation as returns move across either side of their expected value. Beta’s separate components address the distinct managerial concerns arising from loss aversion (or upside speculation) and from changes in correlation under different market conditions. Bifurcating beta in financial space describes both phenomena and anticipates the behavioral response to volatility and correlation in falling markets — problems appropriately described as sinking, fast and slow.