Based on a pooled time series and cross-sectional data of 200 Aa (high-quality) and Baa (medium-quality) bonds over the period 1961-1966, Joseph Cappelleri estimated the following bond rating prediction model.10
Yi = fa + fa X2/ + X3I + fa + fas Xsi + u, where Y, = 1 if the bond rating is Aa (Moody's rating) = 0 if the bond rating is Baa (Moody's rating) X2 = debt capitalization ratio, a measure of leverage dollar value of long-term debt
_ dollar value of total capitalization X3 = profit rate dollar value of after-tax income
dollar value of net total assets X4 = standard deviation of the profit rate, a measure of profit rate variability Xs = net total assets (thousands of dollars), a measure of size
A priori, fa and fa are expected to be negative (why?) and fa and fas are expected to be positive.
After correcting for heteroscedasticity and first-order autocorrelation, Cappelleri obtained the following results11:
Yi = 0.6860 - 0.0179X|(. + 0.0486X3, + 0.0S72X4, + 0.378(E-7)Xs
(0.1775) (0.0024) (0.0486) (0.0178) (0.039)(E-8) (15.3.1)
Note: 0.378 E-7 means 0.0000000378, etc.
All but the coefficient of X4 have the correct signs. It is left to finance students to rationalize why the profit rate variability coefficient has a positive sign, for one would expect that the greater the variability in profits, the less likely it is Moody's would give an Aa rating, other things remaining the same.
The interpretation of the regression is straightforward. For example, 0.0486 attached to X3 means that, other things being the same, a 1 percentage point increase in the profit rate will lead on average to about a 0.05 increase in the probability of a bond getting the Aa rating. Similarly, the higher the squared leveraged ratio, the lower by 0.02 is the probability of a bond being classified as an Aa bond per unit increase in this ratio.
10Joseph Cappelleri, "Predicting a Bond Rating,'' unpublished term paper, C.U.N.Y. The model used in the paper is a modification of the model used by Thomas F. Pogue and Robert M. Soldofsky, "What Is in a Bond Rating?'' Journal of Financial and Quantitative Analysis, June 1969, pp. 201-228.
"Some of the estimated probabilities before correcting for heteroscedasticity were negative and some were in excess of 1; in these cases they were assumed to be 0.01 and 0.99, respectively, to facilitate the computation of the weights wi.
CHAPTER FIFTEEN: QUALITATIVE RESPONSE REGRESSION MODELS 593
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