In the same fassion thresholds for all 22 credit ratings can be found for all 3 companies. Note that there is no need to find a threshold for AAA rating, since every log-return above the AA+ threshold will lead to the upgrade of a company to AAA. Asset return thresholds for all companies can be found in Tables 13,14,15. Now in order to perform Monte-Carlo simulations of possible porfolio values, it was needed to define joint behavoir of firms' credit migration. To model that behavoir, appropriate Copula function must be fitted.
3.2 Fitting Copula
Since the margins of Copula functions must be uniform on the unit interval, CDFs of respective t distributions were used to apply Probability integral transform to our log-returns so that they follow uniform distributions on [0,1]. With margins defined, the next step was to construct trivariate distribtuion by means of pair-copula cunstruction method discussed above. "CDVineCondFit" function of "CDVineCopulaConditional" package in R was exploited in order to fit bivariate copula to each pair according to the algorithm explained above by conditioning Nornickel returns. The function is implemented in such a way that it fits all 40 copulas (list of implemented copulas can be found in R documentation), finds maximum likelihood estimators of parameters for each of them, and then chooses the best fit by BIC crteria. Table 4 shows the structure of pair-copula construction.