# [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

-------------- 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>
```