Research article

Resolutions to flip-over credit risk and beyond-least squares estimates and maximum likelihood estimates with monotonic constraints

  • Received: 03 December 2018 Accepted: 18 March 2019 Published: 22 March 2019
  • Given a risk outcome y over a rating system $\{R_i\}^k_{i=1}$ for a portfolio, we show in this paper that the maximum likelihood estimates with monotonic constraints, when y is binary (the Bernoulli likelihood) or takes values in the interval 0≤y≤1 (the quasi-Bernoulli likelihood), are each given by the average of the observed outcomes for some consecutive rating indexes. These estimates are in average equal to the sample average risk over the portfolio and coincide with the estimates by least squares with the same monotonic constraints. These results are the exact solution of the corresponding constrained optimization. A non-parametric algorithm for the exact solution is proposed. For the least squares estimates, this algorithm is compared with “pool adjacent violators” algorithm for isotonic regression. The proposed approaches provide a resolution to flip-over credit risk and a tool to determine the fair risk scales over a rating system.

    Citation: Bill Huajian Yang. Resolutions to flip-over credit risk and beyond-least squares estimates and maximum likelihood estimates with monotonic constraints[J]. Big Data and Information Analytics, 2018, 3(2): 54-67. doi: 10.3934/bdia.2018007

    Related Papers:

  • Given a risk outcome y over a rating system $\{R_i\}^k_{i=1}$ for a portfolio, we show in this paper that the maximum likelihood estimates with monotonic constraints, when y is binary (the Bernoulli likelihood) or takes values in the interval 0≤y≤1 (the quasi-Bernoulli likelihood), are each given by the average of the observed outcomes for some consecutive rating indexes. These estimates are in average equal to the sample average risk over the portfolio and coincide with the estimates by least squares with the same monotonic constraints. These results are the exact solution of the corresponding constrained optimization. A non-parametric algorithm for the exact solution is proposed. For the least squares estimates, this algorithm is compared with “pool adjacent violators” algorithm for isotonic regression. The proposed approaches provide a resolution to flip-over credit risk and a tool to determine the fair risk scales over a rating system.


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  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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