Merton’s model, credit risk and volatility skews

Suppose that r_{k, }_{j }and r_{s, }_{j }are the Kendall rank order correlation and Spearman rank order correlation for company j, that n_{j }is the number of observations for company j, and that there are N companies. Under the null hypothesis that there is no correlation between the two variables, each of the r_{k,j }is normally distributed with mean zero and variance

V j =

2(2n_{j }+ 5) (9n_{j }(n_{j }−1))

The mean value of the r_{k, }_{j }variance

is then normally distributed with mean zero and

1 N^{2 }

### N

V j ∑

j =1

and the z-statistic for testing whether the mean is significantly different from zero is

# ∑

N j =1

r_{k, j }

z_{k }=

∑

N j =1

2(2n_{j }+ 5) (9n_{j }(n_{j }−1))

An upper bound for the standard error of each of the r_{k, }_{j }standard error of the mean value of the r_{k, }_{j }is

i s 2 ( 1 – r 2 k , j ) ⁄ n j a n d t h e

1 N

∑

N j =1

2 1 2 r n k j j − ( ) ,

Analogously, the z-statistic for the mean value of the r_{s, }_{j }is

z_{s }=

∑ ∑ N j =1 N j =1

r_{s , j }

1 n_{j }−1

and an upper bound for the standard error of the mean value of the r_{s, }_{j }is

1 N

∑

N j =1

3 1 2 r n s j j − ( ) ,

The expressions for the z-statistics and standard error for daily means in the day- by-day analysis are similar to those in the company-by-company analysis. In this case, j counts days rather than companies and N is the number of days for which we are able to calculate rank order correlations.

## Research papers

www.journalofcreditrisk.com

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