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Time reversal and last passage time of diffusions with applications to credit risk management

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Abstract

We study time reversal, last passage time and \(h\)-transform of linear diffusions. For general diffusions with killing, we obtain the probability density of the last passage time to an arbitrary level and analyse the distribution of the time left until killing after the last passage time. With these tools, we develop a new risk management framework for companies based on the leverage process (the ratio of a company asset process over its debt) and its corresponding alarming level. We also suggest how a company can determine the alarming level for the leverage process by constructing a relevant optimisation problem.

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Authors and Affiliations

  1. Graduate School of Economics, Kyoto University, Sakyo-Ku, Kyoto, 606-8501, Japan

    Masahiko Egami & Rusudan Kevkhishvili

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  1. Masahiko Egami
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  2. Rusudan Kevkhishvili
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Correspondence to Masahiko Egami.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author is supported by Grant-in-Aid for Scientific Research (C) No. 18K01683, Japan Society for the Promotion of Science.

The second author is a research fellow of the Japan Society for the Promotion of Science and is in part supported by JSPS KAKENHI Grant Number JP 17J06948.

Appendix

Appendix

We demonstrate the calculation procedure for \(q\) in (3.26) and (3.27) by using the end of December 2013 as a reference point. The values of \(A\), \(D\), \(E\) and “interest paid” are displayed in millions of U.S. dollars.

(1) The value of equity \(E\)\((=135.43)\) is calculated as the difference between the market value of assets \(A\)\((=292.98)\) (estimated by the method in Sect. 4.1.1) and the value of debt \(D\)\((=157.55)\) (used in the estimation of \(A\)). Using these estimates, the weights of the equity and debt are determined as \(w_{E}=\frac{E}{A}\) and \(w_{D}=\frac{D}{A}\), respectively.

(2) We set the daily risk-free rates equal to daily 1-year US treasury yield curve rates divided by 360. The company’s \(\beta \) is estimated by regressing (without intercept) the daily excess returns (of 2013) of the company stock on those of the NASDAQ composite price index. With this \(\beta \)\((=1.42)\), the annual NASDAQ return \(R_{m}\)\((=38.32\%)\) for 2013 and 1-year US treasury yield curve rate \(R_{f}\)\((=0.13\%)\) for the end of 2013, we calculate the cost of equity as \(C_{E}=R_{f}+\beta (R_{m}-R_{f})=54.31\%\).

(3) Dividing “interest paid” (\(=18.95\)) from the cash flow statement of the year 2013 by the average of the debt values of December 2012 (\(D_{2012}=117.05\)) and December 2013 (\(D_{2013}=157.55\)) gives us the cost of debt \(C_{D}=13.8\%\). The calculation of debt values is described in Sect. 4.1.1.

(4) Setting the corporate tax rate as \(35\%\), we obtain

$$ q=w_{E} C_{E}+w_{D} C_{D} (1-0.35)=29.9306\%. $$

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Egami, M., Kevkhishvili, R. Time reversal and last passage time of diffusions with applications to credit risk management. Finance Stoch 24, 795–825 (2020). https://doi.org/10.1007/s00780-020-00423-6

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