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## John C. Hull, Izzy Nelken and Alan D. White

# Two predictions made by Merton’s model are:

❏ there should be a positive relationship with positive convexity between credit

spreads and at-the-money volatilities; and ❏ there should be a positive relationship between credit spreads and volatility

skews when the at-the-money volatility is high.

The first prediction is strongly supported by the data. The second is also supported by the data but somewhat less strongly than the first.

# APPENDIX

Kendall and Gibbons (1990) provide a great deal of information on the statistical properties of the Kendall and Spearman rank correlation measures. For n > 10, the probability distribution of Kendall’s rank order correlation, r_{k}, conditional on no rank order correlation between the variables is approximately normal with a mean of zero and a variance of [2(2n + 5)][9n(n – 1)]. The z-statistic for testing the null hypothesis that r_{k }is zero is therefore

3r_{k }n(n −1) 2(2n + 5)

F o r n > 3 0 t h e p r o b a b i l i t y d i s t r i b u t i o n o f S p e a r m a n ’ s r a n k o r d e r c o r r e l a t i o n , r s conditional on no rank order correlation is approximately normal with a mean of zero and a variance of 1 ⁄ (n – 1). The z-statistic for testing the null hypothesis that , r s i s z e r o i s t h e r e f o r e r s — — n – 1 — .

When the rank order correlation is non-zero, Kendall and Gibbons show that the standard deviation of the estimate of r_{k }depends on the true value of r_{k }and other unknown quantities concerned with the arrangement of the ranks in the p a r e n t p o p u l a t i o n . T h e s a m e i s t r u e o f r s . T h e e s t i m a t e d v a l u e o f r k c a n b assumed to be drawn from a normal distribution with a mean of ρ_{k }and a vari- e a n c e o f a t m o s t 2 ( 1 – ρ k 2 ) ⁄ n , w h e r e ρ k i s t h e t r u e K e n d a l l r a n k o r d e r c o r r e l a t i o n . T h e e s t i m a t e d v a l u e o f r s c a n b e a s s u m e d t o b e d r a w n f r o m a n o r m a l d i s t r i b u t i o n w i t h a m e a n o f ρ s a n d a v a r i a n c e o f a t m o s t 3 ( 1 – ρ s 2 ) ⁄ n , w h e r e ρ s i s t Spearman’s rank order correlation. In practice, ρ_{k }and ρ_{s }are set equal to the esti- h e t r u e m a t e s , r k a n d r s , i n t h e s e f o r m u l a s .

These results enable us to construct a conservative test of whether there is a significant difference between two rank order correlations. For example, suppose we observe a Spearman rank order correlation of r_{s,1 }from a sample of n_{1 }and a Spearman rank order correlation of r_{s,2 }from a sample of n_{2}. The z-statistic for testing whether they are significantly different is

r_{s,1 }− r_{s, }_{2 }

(A1)

r n r n s s , , 1 2 1 2 2 2 3 1 3 1 − ( ) + − ( )

## Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05