[R-SIG-Finance] CreditMetrics - Bivariate Normal Distribution Probabilities
Amy Milano
milano_amy at yahoo.com
Mon Jun 7 14:24:13 CEST 2010
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
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