# Buffered Probability of Exceedance (bPOE) Ratings for Synthetic Instruments

Credit ratings are widely used by investors to assess the credit risk of a security. The financial crisis of 2008 showed that credit ratings might not measure the risk appropriately for the synthetic instruments such as Collateralized Debt Obligations (CDOs).

Recently, Buffered Probability of Exceedance (bPOE) emerged as a new topic in risk management. This paper presents a new rating assignment model based bPOE, that is an improvement over the current Probability of Exceedance (POE) based methodology. We demonstrate how bPOE ratings can be designed and used in portfolio optimization and structuring synthetic financial instruments. The structuring optimization problems are reduced to convex and linear optimization. The case study results are posted at a website.

POE is frequently used to measure uncertainties in outcomes. For instance, POE is used to estimate probability that assets of a company fall below liabilities. POE measures only the frequency of outcomes and ignores magnitude of outcomes. POE counts outcomes exceeding the threshold, and it “does not worry” about the amount by which each outcome exceeds the threshold. POE is lumping together all threshold exceedance events, potentially “hiding” quite large and very troublesome outcomes. Moreover, POE has poor mathematical properties when used to characterize discrete distributions of random values (e.g., when distributions are defined by observed historical data). POE for discrete distributions is a discontinuous function of control variables, making it difficult to analyze and optimize. POE is used for defining financial ratings of companies and financial derivative instruments.

This presentation discusses a new bPOE probabilistic measure. With bPOE, it is possible to count outcomes close to a threshold value, rather than only outcomes exceeding the threshold. To be more precise, bPOE counts tail outcomes averaging to some specific threshold value. For instance, 4% of land-falling hurricanes in US have cumulative damage exceeding $50 billion (i.e., POE = 0.04 for threshold=$50 billion). It is estimated, that the average damage from the worst 10% of hurricanes is $50 billion. In terms of bPOE, we say bPOE=0.1 for the threshold=$50 billion. bPOE shows that the largest damages having magnitude around $50 billion have frequency 10%. bPOE can be considered as an important supplement to POE. We think that bPOE should be routinely calculated together with POE. This example shows that bPOE exceeds POE, which is why it is called Buffered Probability of Exceedance. The positive difference, bPOE-POE, can be interpreted as some “buffer.” bPOE is an inverse function of Conditional Value-at-Risk (CVaR); it inherits a majority of exceptional mathematical properties of CVaR (which is a so called “coherent measure of risk”). Similar to CVaR, minimization of bPOE can be reduced to Convex and Linear Programming.