Document Type : Research Paper


Department of Statistics, Mathematics, and Computer Science, Allameh Tabataba'i University, Tehran, Iran.


One of the most critical aspects of credit risk management is determining the capital requirement to cover the credit risk in a bank loan portfolio. This paper discusses how the credit risk of a loan portfolio can be obtained by the stochastic recovery rate based on two approaches: beta distribution and short interest rates. The capital required to cover the credit risk is achieved through the Vasicek model. Also, the Black-Scholes Merton model for the European call option is utilized to quantify the Probability of Default (PD). Value at Risk (VaR) and Conditional Value at Risk (CVaR) are used as measures of risk to evaluate the level of risk obtained by the worst-case PD. A stochastic recovery rate calculates VaR related to the underlying intensity default. In addition, the intensity default process is assumed to be linear in the short-term interest rate, driven by a CIR process. The loan portfolio performance is evaluated by considering the relevant characteristics with the Data Envelopment Analysis (DEA) method. This study proposes the losses driven by the stochastic recovery rate and default probability. The empirical investigation uses the Black-Sholes-Merton model to measure the PD of eighth stocks from different industries of the Iran stock exchange market.


Main Subjects

[1]     Gordy, M. B. (2000). A comparative anatomy of credit risk models. Journal of banking and finance, 24(1–2), 119–149. DOI:10.1016/S0378-4266(99)00054-0
[2]     Gordy, M. B. (2003). A risk-factor model foundation for ratings-based bank capital rules. Journal of financial intermediation, 12(3), 199–232. DOI:10.1016/S1042-9573(03)00040-8
[3]     Andersson, F., Mausser, H., Rosen, D., & Uryasev, S. (2001). Credit risk optimization with conditional value-at-risk criterion. Mathematical programming, series b, 89(2), 273–291. DOI:10.1007/PL00011399
[4]     Alexander, S., Coleman, T. F., & Li, Y. (2003). Derivative portfolio hedging based on CVaR. New risk measures in investment and regulation, 321–336.
[5]     Fang, Y., Lai, K. K., & Wang, S. (2008). Fuzzy index tracking portfolio selection model. In Fuzzy portfolio optimization: theory and methods (pp. 155–161). Springer.
[6]     Vasicek, O. A. (2015). Finance, economics, and mathematics. John Wiley & Sons.
[7]     Di Clemente, A. (2020). Modeling portfolio credit risk taking into account the default correlations using a copula approach: implementation to an Italian loan portfolio. Journal of risk and financial management, 13(6), 129. DOI:10.3390/jrfm13060129
[8]     Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European journal of operational research, 2(6), 429–444.
[9]     Fallah, R., Kouchaki Tajani, M., Maranjory, M., & Alikhani, R. (2021). Comparison of banks and ranking of bank loans types on based of efficiency with DEA in Iran. Big data and computing visions, 1(1), 36–51.
[10]   Mirsadeghpour Zoghi, S. M., Sanei, M., Tohidi, G., Banihashemi, S., & Modarresi, N. (2023). Assets performance evaluation with the use of returns distribution characteristics. Journal of decisions and operations research, 8(3), 771-784. (In Persian).
[11]   Murthi, B. P. S., Choi, Y. K., & Desai, P. (1997). Efficiency of mutual funds and portfolio performance measurement: A non-parametric approach. European journal of operational research, 98(2), 408–418. DOI:10.1016/S0377-2217(96)00356-6
[12]   Basso, A., & Funari, S. (2001). A data envelopment analysis approach to measure the mutual fund performance. European journal of operational research, 135(3), 477–492.
[13]   Daraio, C., & Simar, L. (2006). A robust nonparametric approach to evaluate and explain the performance of mutual funds. European journal of operational research, 175(1), 516–542. DOI:10.1016/j.ejor.2005.06.010
[14]   Edalatpanah, S. A., Godarzi Karim, R., Khalilian, B., & Partouvi, S. (2020). Data envelopment analysis and efficiency of firms: a goal programing approach. Innovation management and operational strategies, 1(1), 1-16. (In Persian).
[15]   Rasoulzadeh, M., & Fallah, M. (2020). An overview of portfolio optimization using fuzzy data envelopment analysis models. Journal of fuzzy extension and applications, 1(3), 180–188.
[16]   Kavčáková, M., & Kočišová, K. (2020). Using data envelopment analysis in credit risk evaluation of ICT companies. Agris on-line papers in economics and informatics, 12(4), 47–60. DOI:10.7160/AOL.2020.120404
[17]   Grzybowska, U., & Karwański, M. (2015). Application of data envelopment analysis to calculating probability of default for high rated portfolio. Acta physica polonica A, 127(3), A66–A69. DOI:10.12693/APhysPolA.127.A-66
[18]   Portela, M. C. A. S., Thanassoulis, E., & Simpson, G. (2004). Negative data in DEA: A directional distance approach applied to bank branches. Journal of the operational research society, 55(10), 1111–1121. DOI:10.1057/palgrave.jors.2601768
[19]   Merton, R. C. (1974). On the pricing of corporate debt: the risk structure of interest rates. The journal of finance, 29(2), 449. DOI:10.2307/2978814
[20]   Dar, A. A., Anuradha, N., & Qadir, S. (2019). Estimating probabilities of default of different firms and the statistical tests. Journal of global entrepreneurship research, 9(1), 1–15. DOI:10.1186/s40497-019-0152-8
[21]   Jouanin, J. F., Riboulet, G., & Roncalli, T. (2004). Financial applications of copula functions. John Wiley & Sons.
[22]   Bakshi, G., Madan, D., & Zhang, F. (2001). Recovery in default risk modeling: theoretical foundations and empirical applications (Vol. 37). Divisions of Research & Statistics and Monetary Affairs, Federal Reserve Board.