CreditMetrics - Bivariate Normal Distribution Probabilities
Hi! Here is one small clarification I am seeking regarding the Portfolio probability distribution (where no of Obligors are 2). It is the classical example as given in the CreditMetrics document. BOND "BBB" Possible Migrations ?? AAA?? ?? AA ?? ??? A ? ? ?? BBB?????? BB????? B ? ??? CCC????? Default Probabilities(%) ? ? ???? 0.02 ? ? 0.33 ??? 5.95 ? ?? 86.93 ? ? 5.30 ?? 1.17 ??? 0.12? ???? 0.18 ????????? Year End values*?????? 109.37? 109.19? 108.66? 107.55?? 105.02? 98.10? 83.64??? 51.13 Mean(BBB)? =? 107.09 (= mu1) sd(BBB)???? =?????? 2.99 (= sd1) * These are the values taken by BBB bond after migrating to respective ratings after 1 year. Similarly for Bond "A" Possible Migrations ?? AAA?? ?? AA ?? ??? A ? ? ?? BBB????? BB?????? B ? ??? CCC????? Default Probabilities(%) ? ? ???? 0.09 ?? ? 2.27 ?? 91.05? ???? 5.52 ??? 0.74 ??? 0.26?? ? ? 0.01? ???? 0.06 ????????? Year End values*?????? 106.59? 106.49? 106.30? 105.64?? 103.15? 101.39??? 78.71??? 51.13 Mean(A)?? =?? 106.55 (= mu2) sd(A)?? =?? 1.49 ? ? ??? (= sd2) #### Case 1 :- The correlation between BBB and A is 0. So Probability that after 1 year, BBB will remain BBB and A will remain A is 0.8693*0.9105 = 0.7915 #### Case 2 : Let the correlation between BBB and A is 0.30 (= r ) So Probability that after 1 year, BBB will remain BBB and A will remain A = 0.7969 I have tried to arrive at this value but somehow I am missing something. I have used the following formula (assuming the normalized returns on two assets BBB and A) p(BBB, A) = 1/[2*pi*sd1*sd2*sqrt(1-r^2)] * [exp (-z / 2*(1 - r^2))] where z = ((x1-mu1)/sd1)^2 + ((x2 - mu2)/sd2)^2 - 2 * r * (x1-m1)/sd1 * (x2-mu2) / sd2 (formula available at http://mathworld.wolfram.com/BivariateNormalDistribution.html) I have taken x1 = 107.55 and x2 = 106.30 and the probability value I arrive at using above formula = 0.03612 The actual answer is 79.69% i.e. 0.7969 as given in the Creditmetrics technical document (page 26 - table 1.7). I am held up at this particular point. I will be grateful if someone can guides me. Also I am not sure I will be able to attach the excel file with this mail, but nevertheless I am trying the same. I sincerely apologize for writing such a long mail. Regards Milano -------------- next part -------------- An HTML attachment was scrubbed... URL: <https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20100607/2a48ea9b/attachment.html> -------------- next part -------------- A non-text attachment was scrubbed... Name: biv_prob.xls Type: application/vnd.ms-excel Size: 32256 bytes Desc: not available URL: <https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20100607/2a48ea9b/attachment.xls>