In evaluating credit risk models, it is common to use metrics such as power curves and their associated statistics. However, power curves are not necessarily easily linked intuitively to common lending practices. Bankers often request a specific rule for defining a cut-off above which credit will be granted and below which it will be denied. In this paper we provide some quantitative insight into how such cut-offs can be developed. This framework accommodates real-world complications (e.g., “relationship” clients). We show that the simple cut-off approach can be extended to a more complete pricing approach that is more flexible and more profitable. We demonstrate that in general more powerful models are more profitable than weaker ones and we provide a simulation example. We also report results of another study that conservatively concludes a mid-sized bank might generate additional profits on the order of about $4.8 million per year after adopting a moderately more powerful model.

We study the stock price distributions that arise when prices follow a diffusion process with a stochastically varying volatility parameter. We use analytic techniques to derive an explicit closed-form solution for the case where volatility is driven by an arithmetic Ornstein–Uhlenbeck (or AR1) process. We then apply our results to two related problems in the finance literature: (i) options pricing in a world of stochastic volatility, and (ii) the relationship between stochastic volatility and the nature of “fat tails” in stock price distributions.

[We study the stock price distributions that arise when prices follow a diffusion process with a stochastically varying volatility parameter. We use analytic techniques to derive an explicit closed-form solution for the case where volatility is driven by an arithmetic Ornstein-Uhlenbeck (or AR1) process. We then apply our results to two related problems in the finance literature: (i) options pricing in a world of stochastic volatility, and (ii) the relationship between stochastic volatility and the nature of "fat tails" in stock price distributions.]

It is often the case in default modeling that the need arises to calibrate a model to some prior probability of default. In many situations, a researcher may not know the true prior default rate for the population because the data set at hand is itself incomplete, either with respect to default identification (hidden defaults) or default under reporting. In situations where a researcher has access to two incomplete default data sets, for example in the case of two banks that have merged, it is possible to infer the number of “missing” defaults, which we demonstrate in this short note. We discuss an approach to estimating this quantity and show an example in which we infer the number of missing defaults in the combined legacy databases of the former Moody’s Risk Management Services and the former KMV Corporation. While calibration is one application of this approach, the method is a general one that can be applied in other settings as well.

Traditional financing sources such as private and public equity may not be ideal for investment projects with low probabilities of success, long time horizons, and large capital requirements. Nevertheless, such projects, if not too highly correlated, may yield attractive risk-adjusted returns when combined into a single portfolio. Such “megafund” portfolios may be too large to finance through private or public equity alone. But with sufficient diversification and risk analytics, debt financing via securitization may be feasible. Credit enhancements (i.e., derivatives and government guarantees) can also improve megafund economics. We present an analytical framework and illustrative empirical examples involving cancer research.