Credit ratings are fundamental in assessing the credit risk of a security or debtor. The failure of the Collateralized Debt Obligation (CDO) ratings during the financial crisis of 2007-2008 and the massive undervaluation of corporate risk leading up to the crisis resulted in a review of rating approaches. Yet the fundamental metric that guides the construction of credit ratings has not changed. We study the inadequacies of the old metric in simple models of investment and in structured finance portfolio optimization tasks, and we propose a new methodology based on a buffered probability of exceedance. The new approach offers a conservative risk assessment, with substantial conceptual and computational benefits. We illustrate the new approach using several examples and report the results of a structuring step-up CDO case study, with details available in an online Supplement.

It has been argued that investors who optimize their portfolios with attention paid only to mean and standard deviation will all end up choosing some multiple of a certain master fund portfolio. Justification for the capital asset pricing model of classical portfolio theory, which relates individual assets to such a master fund, has come from this direction in particular. Attempts have been made to provide solid mathematical support by showing that the imputed behavior of investors is a consequence of price equilibrium in a market in which assets are traded subject to budget constraints, and optimization is carried out with respect to utility functions that depend only on mean and standard deviation. In recent years, reliance on standard deviation has come under increasing criticism because of inconsistencies in its effect on portfolio preferences. One response has been to introduce generalized measures of deviation which lead to alternative master funds. The market implications of such extensions of theory have hitherto been unclear, but in this paper the existence of equilibrium is established in circumstances where nonstandard deviations are admitted. Equilibrium is guaranteed even when different investors use different measures of deviation and thereby end up with portfolios scaled from different master funds. Whether they employ the same measure or not, they may impose caps on deviation, which likewise may be different.

A new class of asymmetric loss functions derived from the least absolute deviations or least squares loss with a constraint on the mean of one tail of the residual error distribution, is introduced for analyzing financial data. Motivated by risk management principles, the primary intent is to provide “cautious” forecasts under uncertainty. The net effect on fitted models is to shape the residuals so that on average only a prespecified proportion of predictions tend to fall above or below a desired threshold. The loss functions are reformulated as objective functions in the context of parameter estimation for linear regression models, and it is demonstrated how optimization can be implemented via linear programming. The method is a competitor of quantile regression, but is more flexible and broader in scope. An application is illustrated on prediction of NDX and SPX index returns data, while controlling the magnitude of a fraction of worst losses.