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

For the Kendall rank order correlation measure, r_{k}, we look at the n(n – 1) ⁄ 2 alternative pairs of observations. If the rankings of variables for a particular pair of observations are in the same order we score +1, and if they are in inverse order we score –1. For example, if for a particular pair of observations the rankings of the first variable were 5 and 10, respectively, and the rankings of the second variable were 6 and 8, respectively, we would score +1. If the rankings of the first variable were 5 and 10, respectively, and the rankings of the second variable were 8 and 6, respectively, we would score –1. The rank order correlation is the sum of the scores for all pairs divided by n(n – 1) ⁄ 2.

F o r t h e 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 , w e c a l c u l a t e f o r e a c h o b s e r v a tion i the difference, d_{i}, between the rank order of the first variable and the rank order of the second variable. The correlation measure is -

r d s = − n^{3 }− n ∑ 1 6 n i=1

2 i

Kendall and Gibbons (1990) provide a great deal of information on the properties of the two correlation measures. The properties we will use and the statistical tests they give rise to are outlined in the appendix.

The formal statistical tests based on the rank order correlation between implied spreads and observed CDS spreads are reported in Table 3. The first panel in Table 3 shows the results from pooling all the data. The middle panel presents a firm-by-firm analysis. Rank order correlations were calculated for every company that had 30 or more observations, as in Table 2. The final panel in Table 3 tests whether Merton’s model can be used to rank the relative credit quality of different firms at a point in time. Rank correlations were computed for every day on which data were available for 30 or more firms, as in Table 2.

The broad conclusions from the results in Table 3 are similar to those from the results in Tables 1 and 2. The z-statistics show that we can reject the null hypoth- esis that the rank order correlations are zero with a very high degree of confi- dence in all cases for both implementations of Merton’s model. The rank order correlations within firms across time are always higher than those measured within a day across firms.^{14 }This indicates that both implementations do better at tracking a single firm over time than they do at distinguishing between firms at a point in time.

The correlations from the ImpVol implementation of Merton’s model are always higher than for the Trad implementation. In the case of the firm-by-firm analysis they are significantly higher at the 1% level. This is in contrast to the results in Table 2, where the R^{2 }for Trad is slightly higher than the R^{2 }for ImpVol in the firm-by-firm case. A close inspection revealed that the R^{2 }for the firm-by- case case in Table 2 was greatly influenced by a few outliers. Outliers have far less effect on rank order correlations.

14 Although the calculations are not shown, in every case the within-firm rank order correlation is significantly larger than the corresponding within-day rank order correlation at the 1% level.

# Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05