A methodology to estimate the optimal debt ratio when asset returns, and default probability follow stochastic processes

Amira Abid; Fathi Abid

doi:10.3934/jimo.2023017

Article Contents

2023, Volume 19, Issue 10: 7735-7752. Doi: 10.3934/jimo.2023017

This issue Previous Article A variable metric and nesterov extrapolated proximal DCA with backtracking for a composite DC program Next Article A variance-reduced stochastic gradient tracking algorithm for decentralized optimization with orthogonality constraints

A methodology to estimate the optimal debt ratio when asset returns, and default probability follow stochastic processes

Amira Abid^, and
Fathi Abid

University of Sfax, Tunisia Laboratory of probability, and Statistics Faculty of Business and Economic Sciences, Tunisia

^*Corresponding author: Amira Abid
^*Corresponding author: Amira Abid

Received: October 2022

Revised: January 2023

Early access: February 17, 2023

Published online: February 17, 2023

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Abstract

The purpose of this paper is to propose a methodology that allies both theoretical and empirical aspects to model and solve the problem of finding the optimal market-based corporate financial structure in a risky environment; the firm's asset value follows a geometric Brownian motion with a return adjusted by the probability of default which is stochastic and follows an Ornstein-Uhlenbeck process. We consider a decision manager to optimize the financial structure of his company, maximizes the expected utility of the shareholders' final wealth by solving a dynamic programming problem. The value function obeys a quadratic Hamilton-Jacobi-Bellman (HJB) equation and the explicit solution gives the optimal debt ratio. The Kalman filter approach is used to estimate the model parameters. We empirically examined the implications of the credit ratings on the optimal capital structure decision. The data used are daily, obtained from the "Data Stream" database and cover a sample of 16 U.S companies of various sectors and different rating categories over the period from January 01, 2015, to December 31, 2015. The estimated average optimal debt ratios are relatively high showing different patterns between and within investment and speculative-grade firms. In fact, the evolution of the optimal debt ratio according to the rating is influenced by a restrictive or an easy entry into the debt market and by the value of the estimated rate of return.

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Mathematics Subject Classification: 49J20, 49L12, 49L20, 49N05.

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Figure 1. Average CDS spread by rating classes during 2015

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Figure 2. Average CDS-implied default probability by rating classes during 2015

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Figure 3. Average optimal debt ratio by rating classes during 2015

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Table 1. Parameters estimates using the simple Kalman filter approach

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Table 2. Descriptive statistics of the average optimal debt ratio by rating class

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References

[1]	A. Abid, F. Abid and B. Kaffel, CDS-based implied probability of default estimation, Journal of Risk Finance, 21 (2020), 399-422. doi: 10.1108/JRF-05-2019-0079.
[2]	B. Around, M. Todd and S. Anjolein, Credit ratings as coordination mechanisms, Review of Financial Studies, 19 (2006), 81-118. doi: 10.1093/rfs/hhj009.
[3]	S. Das, R. Sundaram and S. M. Sundaresan, A simple model for pricing securities with equity, interest-rate and default risk, AFA 2004 San Diego Meetings, (2004), 28pp. doi: 10.2139/ssrn.472680.
[4]	B. David, Counterparty Credit Risk, Working Paper, 2013. doi: 10.1002/9781118266403.ch20.
[5]	M. Davis and F. Lischka, Convertible bonds with market risk and credit risk, AMS IP Studies in Advanced Mathematics, 26 (2002), 45-58. doi: 10.1090/amsip/026/05.
[6]	D. Dwyer, Z. Li, S. Qu, H. Russell and J. Zhang, CDS-implied EDF TM credit measures and Fair-value spreads, Moody's Analytics, Modeling Methodology, From Moody's KMV, (2010), 1-43.
[7]	M. Faulkender and M. Petersen, The society for financial studies does the source of capital affect capital structure?, The Review of Financial Studies, 19 (2006), 45-79. doi: 10.1093/rfs/hhj003.
[8]	M. Flannery, J. Houston and F. Partnoy, Credit default swap spreads as viable substitutes for credit ratings, University of Pennsylvania Law Review, 158 (2010), 2085-2123.
[9]	B. Frank and M. Usha, The determinants of capital structure choice: A survey of european firms, Financial Management, 33 (2004), 103-132. doi: 10.2139/ssrn.299172.
[10]	C. Godlewski, Study of the coherence of the ratings of banks with the probability of failure banking in the emergent countries, Ssrn, (2004), 38pp. doi: 10.2139/ssrn.588161.
[11]	J. Graham and C. Harvey, 9o Artigo 5 - The theory and practice of corporate finance, Journal of Financial Economics, 60 (2001), 187-243. doi: 10.1016/S0304-405X(01)00044-7.
[12]	J. Hilscher and M. Wilson, Credit ratings and credit risk: Is one measure enough?, Management Science, 63 (2017), 3414-3437. doi: 10.1287/mnsc.2016.2514.
[13]	J. Hull, Options, Futures and other Derivatives, 2014, (Pearson, Ed.).
[14]	J. Hull and A. White, Valuing credit default swaps Ⅱ: Modeling default correlations. The Journal of Derivatives, 416 (2000).
[15]	C.-H. Hung, A. Banerjee and Q. Meng, Corporate financing and anticipated credit rating changes, Review of Quantitative Finance and Accounting, 48 (2017), 893-915. doi: 10.1007/s11156-016-0571-3.
[16]	M. Jacobs Jr, A Karagozoglu and N. Layish Dina, Credit risk signals in CDS market vs agency ratings, The Journal of Risk Finance, (2016).
[17]	A. Jan and C. Marc, Modelling European Credit Spreads, (Prinsstraat, Ed.), Universiteit Antwerpen (13th ed.), 2000.
[18]	C. Janko, Are credit rating announcements contagious? Evidence on the transmission of information across industries in credit default swap markets, The Journal of Fixed Income, 23 (2013), 27-60.
[19]	R. Jarrow, D. Lundo and S. Turnbull, A markov model for the term structure of credit risk spreads, The Review of Financial Studies, 10 (1997), 481-523.
[20]	Z. Jin, Optimal debt ratio and consumption strategies in financial crisis, Journal of Optimization Theory and Applications, 166 (2015), 1029-1050. doi: 10.1007/s10957-014-0629-0.
[21]	Z. Jin, H. Yang and G. Yin, Optimal debt ratio and dividend payment strategies with reinsurance, Insurance: Mathematics and Economics, 64 (2015), 351-363. doi: 10.1016/j.insmatheco.2015.07.005.
[22]	Y. Kabanov and S. Pergamenshchikov, On ruin probabilities with investments in a risky asset with a regime-switching price, Finance and Stochastics, 26 (2022), 877-897. doi: 10.1007/s00780-022-00483-w.
[23]	G. Kara, A. Özmen and G.-W. Weber, Stability advances in robust portfolio optimization under parallelepiped uncertaintty, Central European Journal of Operations Research, 27 (2019), 241-261. doi: 10.1007/s10100-017-0508-5.
[24]	A. Khadimallah and F. Abid, Stochastic optimal control with Contingent Convertible Bond in banking industry, Journal of Mathematics and Modeling in Finance, (2022).
[25]	A. Leif and B. Dan, Calibration and implementation of convertible bond models, Available at SSRN, (2002), 355308.
[26]	A. Lindquist, On feedback control of linear stochastic systems, SIAM Journal on Control, 11 (1973), 323-343. doi: 10.1137/0311025.
[27]	W.-H. Liu and Z. Jin, Analysis of optimal debt ratio in a markov regime-switching model, In conference, (2014), (pp. 1-37). Australia: Early Career Research Grant and Faculty Research Grant by The University of Melbourne. Retrieved from http://sfm.finance.nsysu.edu.tw/php/Papers/CompletePaper/006-1764937268.pdf
[28]	K. Maritato and S. Uryasev, Derivative of reduced cumulative distribution function and applications, License CC BY 4.0, (November 2022). doi: 10.21203/rs.3.rs-2223961/v1.
[29]	R. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.
[30]	R. Merton, On the pricing of corporate debt: The risk structure of interest rates, The Journal of Finance, 29 (1974), 449-470.
[31]	M. Norton, V. Khokhlov and S. Uryasev, Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation, Annals of Operations Research, 299 (2021), 1281-1315. doi: 10.1007/s10479-019-03373-1.
[32]	K. Pavlikov and S. Uryasev, CVaR norm and applications in optimization, Optimization letters, 8 (2014), 1999-2020. doi: 10.1007/s11590-013-0713-7.
[33]	I. Rodríguez, K. Dandapani and E. Lawrence, Measuring sovereign risk: Are CDS spreads better than sovereign credit ratings?, Financial Management, 48 (2019), 229-256.
[34]	F. Sajjad and M. Zakaria, Credit rating as a mechanism for capital structure optimization: Empirical evidence from panel data analysis, International Journal of Financial Studies, 6 (2018), 13. doi: 10.3390/ijfs6010013.
[35]	R. Samaniego-Medina and F. di Pietro, Rating and capital structure: How do the signs affect the speed of adjustment?, Journal of International Financial Management & Accounting, 30 (2019), 188-202. doi: 10.1111/jifm.12106.
[36]	E. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973. doi: 10.1111/j.1540-6261.1997.tb02721.x.
[37]	N. Shammyla, Financial Structure of UK Firms: The Influence of Credit Rating, PhD thesis. Glasgow: University of Glasgow, 2012.
[38]	A. Shi, X. Li and Z. Li, Optimal portfolio selection with life insurance under subjective survival belief and habit formation, Journal of Industrial and Management Optimization, 19 (2023), 2464-2484. doi: 10.3934/jimo.2022051.
[39]	J. Stein, Stochastic Optimal Control and the U.S. Financial Debt Crisis, Springer, New York, 2012. doi: 10.1007/978-1-4614-3079-7.