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# Merton’s model, credit risk and volatility skews - page 23 / 26

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Mertons model, credit risk and volatility skews

Suppose that rk, j and rs, j are the Kendall rank order correlation and Spearman rank order correlation for company j, that nj 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 rk,j is normally distributed with mean zero and variance

V j =

2(2nj + 5) (9nj (nj 1))

The mean value of the rk, j variance

is then normally distributed with mean zero and

1 N2

### N

V j

j =1

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

# ∑

N j =1

rk, j

zk =

N j =1

2(2nj + 5) (9nj (nj 1))

An upper bound for the standard error of each of the rk, j standard error of the mean value of the rk, 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 rs, j is

zs =

N j =1 N j =1

rs , j

1 nj 1

and an upper bound for the standard error of the mean value of the rs, 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

25

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