Plots of ACF of residuals and PACF of squared residuals can be found in Figures 4,5,6. Furthermore, Ljung-Box test for serial independence, which was performed on the residuals of ARIMA-GARCH models, resulted in p-values 0.9652 for Lukoil, 0.9944 for Gazprom and 0.4609 for Norilsk Nickel, meaning that at 1% significance level residuals are serially independent.
3.2 Finding asset return thresholds
Another huge assumption made by Credit Metrics approach is that company's asset returns are normally distributed and when portfolio consists of bonds issued by multiple companies, joint asset returns are multivariate normally distributed. In order to check the validity of that assumption, Normal distribution and Student t distributions were fitted to residuals obtained earlier. By comparing AIC and BIC criteria of both fits, it was found that t distribution describes returns data slightly better than Normal distribution for all 3 companies. Results of those fits and their corresponding parameters can be found in Tables 11,12. Then, using CDFs of these t distributions, asset return thresholds were calculated. As an illustration, let us consider thresholds of Lukoil. Let R denote the stock returns of Lukoil. From Table 1 it is evident that Lukoil is currently rated as BBB. To visualize the problem, consider the following Figure.