UNIVERSITY Of ^ STACKS Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/discriminantanal447pinc Faculty Working Papers \ A DISCRIMINANT ANALYSIS OF ELECTRIC UTILITY BOND RATINGS George E, Pinches, J. Clay Singleton and All Jahankhani #447 College of Commerce and Business Administration University of Illinois at Urbana-Champaign FACULTY WORKING PAPERS College of Commerce and Business Administration University of Illinois at Urbana-Champaign November 4, 1977 A DISCRIMINANT ANALYSIS OF ELECTRIC UTILITY BOND RATINGS George E, Pinches, J. Clay Singleton and Ali Jahankhani /M47 A DISCRIMINANT ANALYSIS OF ELECTRIC UTILITY BOND RATINGS George E. Pinches, J. Clay Singleton and Ali Jahankhani September 1977 A DISCRIMINANT ANALYSIS OF ELECTRIC , UTILITY BOND RATINGS ABSTRACT In this study a six-variable multiple discriminant analysis model was developed that correctly predicted 70% of Moody's bond ratings, 76% of Standard & Poor's, and 81% where both agencies assigned the same bond rating. Fixed charge coverage was found to be 'the most important financial variable on a univariate basis for determining bond ratings, however, it did a relatively poor job in predicting bond ratings when employed by itself. AUTHORS George E. Pinches is Professor of Business Administration at the University of Kansas and 1977 Vice-President — Program for the Financial Management Association. He has done extensive research in recent years concerning bond ratings and recently testified on bond ratings in a public utility rate hearing. J. Clay Singleton is a doctoral candidate in finance at the Univer-i sity of Missouri-Columbia and has articles forthcoming in the Journal of Finance and the Journal of Financial and Quantitative Analysis . Ali Jahankhani is Assistant Professor of Finance at the University of Illinois , Urbana-Champaign and has published in the Journal of Financial and Quantitative Analysis . This research was completed while all three authors were at the University of Missouri-Columbia. -7 .r f rt '-^r ■ .'.w^ fe-iE^y A DISCRIMINANT ANALYSIS OF ELECTRIC UTILITY BOND RATINGS I. Introduction Bond ratings, which represent the judgment of Informed and sophisti- cated financial analysts concerning the credit risk of firms, have been the subject of numerous studies [9, 10, 14, 15, 18, 20] in recent years. By constructing statistical models of the bond rating process, insight has been gained concerning the type of information that analysts presumably employ in making their judgments about a firm's credit worthiness. For industrial firms variables related to size, profitability, financial leverage, fixed coverage, risk/earnings instability and subordination have been identified as important determinants of the bond ratings assigned by financial analysts. Using a variety of variables and statistical techniques, models have been developed that correctly classify between 55 and 75 percent of the industrial bonds into their assigned Moody's or Standard and Poor's rating categories. In addition, the information content of bond rating changes has also been examined [7, 8, 16]. In this study a discriminant analysis approach was employed to identify the variables for the electric utility industry that enabled us to best discriminate between bonds in different bond rating categories. The reasons for selecting the electric utility industry for examination were twofold. First, because the electric utility industry is viewed as being relatively homogeneous, financial variables may do a better job of discriminating between bonds in different categories for utilities than for the more heterogeneous industrial grouping. Second, with b^iii<- •aa -2- the general rise in Interest rates, the cost of debt capital has increased for electric utility firms. This has led to greater consideration of debt costs by all parties concerned with regulatory proceedings. Recent testimony in at least two electric utility rate cases (Public Utilities Control Authority of Connecticut Dockets Nos. 760604 and 760605, and Public Utility Commission of Texas Docket No. 178) stressed the need for substantial revenue increases in order to preseirve or increase a firm's bond rating. Specifically, it was suggested that if sufficient rate increases were not granted, the firm's fixed coverage ratio would decrease causing the firm' s bonds to be downgraded leading to an increase in the effective interest rate; consequently, the company and its consumers would have to pay the increased interest costs. While fixed coverage may be important in the rating process, we are unaware of research suggesting that any single variable captures virtually all of the information informed financial analysts presumably employ when rating public utility bonds. Previous research on electric utility bond ratings [2] does not address this question; in addition, this previous research appears to suffer from certain methodological problems. The purposes of this study were: (1) to develop a model to predict (or discriminate between) electric utility bonds in different bond rating classifications for both Moody's and Standard & Poor's; (2) to examine the relative importance of the financial variables employed in these models; and (3) to compare the predictive ability of the multiple discriminant analysis model with a univariate model employing the fixed coverage ratio. -:>- II. Methodology A. Sample Firms Data for 1970-1975 were gathered on ninety-seven electric utility firms listed on the COMPUSTAT data tapes as of December 31, 1975 that had first mortgage bonds outstanding rated in the top four bond rating classifications by both Moody's and Standard & Poor's. (A list of the ninety-seven firms is available from the authors. Virtually all electric utility firms had their first mortgage bonds rated in the top four categories.) For Moody's the top four categories were Aaa, Aa» A and Baa while for Standard & Poor's they were AAA, AA, A and BBB. Recently, Standard & Poor's further subdivided the AA, A and BBB groups by appending pluses and minus to these ratings; for our purposes we disregard these subdivisions. The number of firms in the top four Moody's categories were Aaa — 5, Aa — 41, A — 32 and Baa — 19. For Standard & Poor's there were 3 AAA rated firms, 32 rated AA, 45 A and 17 BBB rated firms. B. Variables Based on a thorough review of previous bond rating studies and related work on the predictive ability and interrelationships between financial variables [1, 3, 17, 19], the nineteen variables listed in Exhibit 1 were considered for inclusion in the multiple discriminant model. Seventeen of these variables were financial variables while two, X- (Regulatory Climate) and X. , (Geographical Area) attempted to measure certain non-financial factors that might influence the bond rating process, and consequently, bond ratings. X. was based A.) EXHIBIT 1 VARIABLE NUMBER AM) DESCRIPTION VARIABLE NUMBER X, ^6 X. X, X X "10 15 17 X. ^18 19 NAME Regulatory Climate* Total Assets Total Operating Revenue Long-Term Debt/Invested Capital Debt & Preferred Stock/Total Assets Net Income/Total Assets Earnings Before Taxes/Total Operating Revenue Cash Flow**/Fixed Charges Earnings Before Interest & Taxes/Fixed Charges Cash Flow**/Total Assets Residential Electric Sales/Total Electric Sales AFUDC***/Net Income Construction Expenses/Total Assets Geographical Area**** Dividend Payout Ratio 1970-1975 Growth Rate in Cash Flow** 1970-1975 Growth Rate in Net Earnings Standard Deviation of 1970-1975 Cash Flows** Fuel Expenses/Total Electric Sales * As determined by T-Jhite, Weld & Co. (1 = most favorable, 4 = least favorable) ** Cash flow = operating income after taxes + income taxes + de- preciation + interest charges - AFUDC *** AFUDC = Allowance for Funds Used During Construction **** Federal Power Commission regional breakdown where company supplied a major portion of its electricity. -4- on a major brokerage firm's assessment [6] of the regulatory climate in the state where the majority o£ the firm's revenue originated from, while X-, indicated the general geographic region in which the firm derived the majority of its sales. (While both of these variables were classifactory variables, available evidence [12] suggests discrimi- nant analysis is robust In such situations.) Variables X, (Total Assets) and X^ (Total Operating Revenue) measure size; variables related to size have been found to be Important in previous bond rating studies. X. (Long-Term Debt/Invested Capital) and X- (Debt & Preferred Stock/Total Assets) measure financial leverage which has also been found to be Important in previous work. Variables X, (Net Income/Total Assets) , X- (Earnings Before Taxes/Total Operating Revenue) , and X. ^ (Cash Flow/Total Assets) measure various aspects of profitability, while Xg (Cash Flow/Fixed Charges) and X- (Earnings Before Interest & Taxes/Fixed Charges) measure fixed charge coverage; similar variables have also been Important in earlier studies. Variables X,, (Residential Electric Sales/Total Electric Sales), X 2 (Allowance for Funds Used During Construction/Net Income), X.- (Construction Expenses/Total Assets) and X-g (Fuel Expanses/Total Electric Sales) are all unique to the electric utility industry. X. , (Dividend Payout) was Included to reflect relative differences in dividend policy, while variables X^^ (1970-1975 Growth Rate in Cash Flow), X^^ (1970-1975 Growth Rate in Net Earnings) end X^^g (Standard Deviation of 1970-1975 Cash Flows) measure various aspects of growth and stability for electric utility firms. In Appendix A the means, standard deviations and univariate F ratios (testing for differences between the means') for all 19 variables -5- are presented, by bond rating group, for both Moody's and Standard & Poor's. The correlation matrix between all of the variables Is presented in Appendix B. C. Discriminant Analysis Multiple discriminant analysis Is a multivariate statistical technique that allows observations (firms In this study) to be classified into appropriate a priori groups (bond ratings) on the basis of a set of Independent or predictor variables. While all 19 variables could be employed, this would result In a great deal of "noise" In the discriminant model as Lachenbruch has noted [12, 75]. Complete stepwise procedures were employed to reduce the original 19 variables to a six variable model employing the same variables for both Moody's and Standard & Poor's. (The six variable model was selected after ~ examining a number of models for both Moody's and Standard & Poor's including from four to ten variables. Six variables appeared reasonable based on the relative Independence of the variables and the very small incremental increases In discriminatory ability when more than six variables were employed. Slightly better models were obtained for either Moody's or Standard & Poor's; however the reported six variable model was the best for both groups simultaneously.) Due to the small number of Moody's Aaa bonds (5) and Standard & Poor's AAA bonds (3) it was Impossible (employing normal discriminant analysis techniques) to develop a four-group model. Since the number of variables exceeded the number of cases, the dispersion (variance- covariance) matrices for the Aaa and AAA groups were singular. Hence, ■t dp LW'-J9?3 - ~ ' J . rf Kilii J1.'J.J- ^^.Lcni . :J',: -6- we excluded the five Aaa-ratad flras for Moody's leaviug 92 bonds for analysis. For Standard & Poor's, ti.& 3 A*A-rated firns were excluded leaving 94 bonds rated AA, A or BB3 for analysis. In Appendix C an alternative approach was employed (based on assuming the dispersion matrix for the Aaa[AAA] group was equal to the dispersion laatrix for the Aa[AA] group) that allov/ed the four group model to be estimated. Tests for the equality of the dispersion tatrlces between the three bond rating groups resulted in the rejection of the null rypothesis of equal dispersion matrices for both Moody's (.C'';5 significance level) and Standard & Poor's (.005 sigrificr.tice IjvcI); h£.ic3, qr.adratic as opposed to linear classification rules ivere caployad. Also, because quadratic classification procedurss vera enplcyed the typical discriminant functions (two in this case) and their coefficients were not reported. (Discriminant functioni are not particularly T'.eaniugful when quadratic classification rules are employed.) Finally, v& employed equ^l prior probabilities for classification purpoge^. Ideally the prior probabilities should reflect the dictributr.on of ^ords in the population; lio-rever, in recent years the n-jiiber of bonda ia different rating categories has been undergoing ccnsiderabls change. Given thi-s instability in the popi'lation prior probabilities, we believed e.d prior probabilities were more appropriate r.ad provide results that were more consistent and generalizable. Tho specific cccputer program er-ployed for this analysis is described in [4j. (For further cli-bcratioa on the mathe- matical assumptions rid difficulties encourtered in employing multiple discriminant analysis, see [4, 12, 13]). •7- III. Empirical Findings A. Analysis of the variables in the MDA model The six variables selected by the complete stepwise procedure were: X (Regulatory Climate), X^ (Total Assets), X- (Net Income/Total Assets), Xg (Earnings Before Interest & Taxes/Fixed Charges), X.. (Construction Expenses/Total Assets) and X _ (1970-1975 Growth Rate in Net Earnings). An examination of Appendix A indicated that, as expected, the more favorable the Regulatory Climate, X., the higher t]' ■ bond rating. The results indicate that except for the Baa(BBB) group, the larger firms (in terms of X-, Total Assets) tended to have higher bond ratings. The large average size for the Baa(BBB) group was caused by the presence of several large firms including Consolidated Edison Co. of New York and Detroit Edison Co. The higher-rated firms . tended to be more profitable as seen by X- (Net Income/ Total Assets), o and had higher fixed coverage levels, X- (Earnings Before Interest & Taxes/Fixed Charges), than lower-rated firms. In addition, the higher-rated firms tended to have a higher ratio of Construction Expenses to Total Assets (X,-), while they had lower Growth Rates in Net Earnings (X^-,) than lower-rated firms during the 1970-1975 time period. The higher construction expenses for higher-rated firms may be due to the fact that firms in the Aa(AA) group tend to cluster in the Midwest and Southern regions of the country — areas where the demand for electrical energy \ras growing faster than the national average. The seeming inconsistency in the lower Growth Rates in Net Earnings (X^y) for higher-rated firms may be due, in part, to the accounting treatment for two separate items. First, the heavier capital expenditures -8- (as evidenced by variable X^^^) experienced by higher-rated finos indicated that relatively more generating capacity was being placed into service by these firms. This would cause depreciation expenses to be greater for the higher-rated firms, thus resulting in lower reported earnings and lower growth rates. Second, an examination of variable X. ^ indicated that the Allowance for Funds Used During Construction (AFUDC) was a larger percent of net earnings for lower-rated finns. Hence, a second reason for the higher growth rates in net income for lower-rated firmt, may be due to the relatively large amounts of AFUDC (as a percent of net income) for lower-rated firms during this time period. In such situations, total reported earnings may be growing faster for lower- rated firms, but financial analysts rating electric utility bonds recognize, the lower "quality" of earnings growth when it was due to the inclusion of larger amounts of AFUDC. In order to test the null hypothesis that the difference in the six group means (centroids) when considered simultaneously was zero between the thrse bond rating grcvps (for both Moody's and Standard & Poor's), the F test based on Wilks lamba was employed. The null hypothesis of no difference was rejected for both Moody's and Standard . & Poor's at the .001 significance level; hence, we inferred there was a significant difference between the group centroids for the three bond rating groups when the six variables were considered in a multivariate context. The next step was to examine the ability of the models to predict which bonds should be assigned to each specific bond rating category. -9- B. Classification Results To test the discriminatory power of the aodei, every saaple firm was classified into one of the three bond rating groups on the basis of the closeness of the firms' observation values to the respective group centroids. The classification matrix (Exhibit 2) shows that 70.65% (65/92) of the firms were classified correctly into their Moody's bond rating category and 76.60% (72/94) ware classified according to their Standard & Poor's classification. (The total number correctly classified was determined by summing the main upper left-lower right diagonal element of the classification table.) The six-variable model did slightly better, In total, for Standard & Poor's than for Moody's suggesting that Standard & Poor's bond ratings more "''.ossly followed these six variables than did Moody's bond ratings. For both Moody's and Standard & Poor's, the model did very well for Baa(BBB)-rated firms, and did the poorest for the A~rated firms. In addition, the model did slightly better for Standard & Poor's top two categories examined, AA and A, than for Moody's (Aa avA A). While these results T;ere lmpreGci\e, they suffer an upward bias since the same firms were reclassified that were employed to develop the model. In order to validate the model, the Lachenbruch jackknife procedure [11] was employed. The essence of this procedure was to estimate the model on all but one of the observations (firms) and then classify the omitted observ-ation. This was repeated sequentially until all observations had been classified en thn basis of a model determined by the rest of the observations. The results of this validation procedure (Exhibit 3) indicated that 54.35% (50/92) were correctly EXHIBIT 2 CLASSIFICATION RESULTS Moody ' 6 Actual Bond Rating Aa A Baa Predicted Bond Rating Percent Aa A Baa Correct 30 10 1 73.17 7 18 7 56.25 2 17 89.47 Standard & Poor's Actual Predicted Bond Rating Percent Bond Rating AA A BBB Correct AA 27 5 84.38 A 8 30 7 66.67 BBB 2 15 88.24 --^^ ;*:i* M»»i3r*a ^^r^fe^1-r*^^^^^^tttS^^i^: co-,;_-:--e-.5nu-,>; s: u. 1 1} 12 2 10 u 37.SO f f«cc• o 0) 0) O »-l M •o u a cd to 4J 1 C/3 >^ 4J • « to a C -H eO a V . s ^'-N /-s ^— \ ^^ *— V .^■^x /«s <^ ^> ^N /^ #*■• v-x sr ^-' ( * * ro v-" \^ >*• ^^ ^i" H 128 (136 rH I--. >* sr * ?5 * * * * « •K ■K a * •K * * ■K * * « H m CM m 00 St o> r«» CM ON t-< CM CM sr r-t "^ t^ CO m CO VO VO CM -* ON VO O Ov rH m in CO CM r-~ O VO CO <• CO m VO sj- CO sr rH CO CM VO 00 CM CTv in CO y^\ CM CN ON vO 00 • 00 r^ CM o vr • m 00 O 00 sr rH r^ o m o m o o o CM O CO VO cjN sr rH O CO o CM H rH O c^ o ro -^ 'O o CM CM N-^ v--* \^ N-^ CO »-' CM ^^ • • • • • • CO CO P3 -^ r^ o m m S.X CQ r^ CN VO «3- VO CO « rH M rH >._• ^ N— ' \^y g p ^— N *^N y^x ^-N /-s y*^ ^-N /^N y^^ ^-N /•"v /-s ^^ •■^s. P4 C^ CTk CM lO O CO CO CO VO m sr rH LO vD r^ 00 O r~ CO CM rH 00 CM r^ rH ■>3- CM CM fc^ Si- r^ O vO t^ CO m o m o o o OJ O OV 00 «3- m rH O CO o CM 1-4 rH O 00 CM ^-N Cvl ^-' cj^ m CM r^ v^ v-^ V— ' s^ CO v-' CO ^— ' ' * o N-X >— ' • • • * CO CN ^ in VO cr. m CM v-' s < <■ H o C) CO ^ N-^ tH rH v-/ H ^•■v t^^ ^^ /-*N ^—s •*^ •-\ /-^ ^— \ (*^ /-N M vO CM VO ^ cr\ rH CM CO CM in sr rH VO in VO rH VO C3N CO CM 00 VO H O CM ■^ si- in v-/ ^z O O r~- rH in o> CO H m CO v^ rH rH vO O CO /-S rH rH CO rH tn ON O CM VO CO o o . \o Ov 00 <. v->* O CO CM ^-' * « * * « * •K « * « O -K H< HC H« * •K * M r^ CO Ch r^ VO m OV cr> CM r>- 00 sr CO o r- >o m rH o vO 00 m VD sr sr 1^ VO CM in sr CN sr CO a\ CM rH sr rH CM • O CM CO r~» CO Ov sr CO r^ sr CO rH CM VO rH CM c^ O CM CM in r^ ON 00 V o 00 •0- 00 CM VO m o m o o o CM O CO VO 00 «a- rH O CO o CM rH rH O o o cd ^-N CO "^^ 00 o^ 00 r^ v<^ *^x V— ' Vw* CO •—' tM N-' >-» s^ v—* v^ St CO (0 ^ in sr rH •vT >3- t-~ t "-^ c^.' r^ CO m v-' s^ v-' Vw' CO ^-' CO >-' ^ v-^ ^^ v^ CO eg CO 00 CO CO sr \^ >w O rH CO CO W rH r-^ v—^ ■• •—^ g U O ^-N •-N •-V /— ^ ^'"N /->. ^-N •*\ ^-N /--x /-\ ^-N >•-% •-S 2 CM si- CO r-- m CO CM CO ■^ m sr rH VO VO CN in \D rH CO CM CTv r^ CM CO CM -J >-{ rH S ^5 CM a\ CO 00 CO CO m o m O o o CM O sr o 1^ l~» rH O CM o CM rH <-l O r». <3N CM -"^ <• LO O CO Vw^ v-^ v_y ^^ •a- rH CO -w • • * • CO r-i >.->• o o CO CO rH rH CM m v-^ ' v--" ^^ ^^ ^X *^% ^-v ^^ rH ON 00 r^ sr rH VD ON /-N ITV CO o o o m rH u-1 CM O 0^ CM CO sr -3- rH cjv m O v£> rH ^ iH o sr CO rH O rH O CM CO vf m m rH 00 VO sr o m o o o CM o • • • • • . . • a • • . • * /— V • . • . • • • . . . . . « . m -^ sr "^ ^^ »-' V^ v-/ sr rH 03 m r-i ^-' O CO CM CM \^y V-' v^ v-/ >^ g ^-^ o -* o tn < 4 h-i rH CM CO X X X X X X X X X X X X X X ir> fo en .-too o o m 00 o o tN CM « * CM CM o l-l o o m CM o o .H in <• CM T-t 1-1 o o v-^ 1 ^-• >—» CM CM *—' o o o o o m o r-l o o ON CTi * NO CM VO cn «r> r-i iO ■* CM CM cn O O CO o o CO CM o o CM CO O O 00 •* o o On in CM CO CM CM O O «— ' CO ON f^) c^ H « r-i CO •* NO CO iH f-~ i-l r-i • 0) V 1-1 • o o • • o • O • CO • CM • o o • • • I-l <0 > rH 1 ^x' N.^* CM CM O CM **• • CO m > r-l H O iH i-H o o « r^ O o o * ^^ /-N y-N ^-\ .^-s n] o • 4J CTn •* CO in fH C 1 •rl iH C/3 •K •K j-H >r X M t-4 c?^ CO * * * * * ■K * 01 4) in T-i r'. Xi a\ « 1-4 •H 1^ * m (A > p c •H a> (<< 0) w >. Q) 4J C •H •H i-l Z •H ^ )~l C3 0) > u o Ti u CO u 4J u c <0 0) r^ •H M O •H ft u-< 01 U-l > (U 0) o to o >» a u o < X ON 00 X « X^ en CM X X o o o o o o o o o o O o CM CO o CO o> •* eo 00 o O CM O CM iH I-l • • • • • • • I-l 1 1 1 1 1 o 00 \o r-- CO CO m -£) «3- •* O «n O o rH o O O CO CO CO •* • • • • « • • • • r-i 1 1 1 1 1 r-l CM CO ^ m VO 1^ 00 ON X X X X X X X X X o o CO CO 00 CM iH ON CM r-l O O •* 1^ 1^ ""^ O iH CO O tH 1-5 r r r * O • 00 00 NO O CM o o H o CN* sr r-l I 11 i I O r-i CM O ON 00 O CM O CO CM I-l 1-1 r^ ' { I ' C I* ovooor^ONOOCMt^ OCOOO«*'OfNj'^ 1-5 * r * r * * * OCvJU-iiHrHOOOONO Or-liHOOOOOO r4 * ' r r ' * •* * OvOI^O«*N£«^SCr;!IJ oor^f^oc>ir>cvif-ii-< r^ r r r i' >* * r r i (ri^rHCOr-iUOOOvO>OrJ t>-i-53oOCMCOeMiHf>l III I III oocMcocoincoNor«;0 OOHr>-rHOrHsrCOr-|CM r r \ ' \ ' I* •' •' oomooovotoi-j'n^ON Si-HOCOCMHOiHOCO i i ' ' r r * •' '* vor<»iniAONOorv«^cM OOOi-lOiTiOfOOO * t* I* II II • ' oco«*Nou-ii-iooor^CM II III uirHCOCJNCMCvJOr^OON COOCMOr-tCOrHOOO .* I* * ' I I ' ' OrHCMCO-3-inNDr-.COC x^ x-" x-^ x-^ x-" x'' x-^ x^ x-^ X APPENDIX C A Four-Group Multiple Discriminant Model Because of the small sample size of the Aaa group for Moody's (5) and the AAA group for Standard & Poor's (3), and the necessity to calculate separate dispersion matrices (because of the inequality of the dispersion matrices) , a four-group multiple discriminant model could not be estimated by the normal procedure. Two alternative approaches might be employed in attempting to develop a four-group model. First, we could assume that the dispersion matrices for all four groups were equal so that a pooled dispersion matrix could be estimated over the 97 bonds. This approach would allow a linear four-group model to be estimated in the usual manner, but does not take account of the inequality in the dispersion matrices for the Aa(AA), A, and Baa(BBB) groups. This inequality in the group dispersion matrices indicates that an assumption of equal dispersion matrices is not valid; hence, quadratic as opposed to linear classification procedures should be employed. A second approach to the problem involves the estimation of the Aaa (AAA) group dispersion matrix so that quadratic classification procedures can still be employed. This subject has not been examined widely and, we believe, requires some elaboration. The crux of the problem is how to obtain a "reasonable" estimate of the dispersion matrix for the Aaa(AAA) group that cannot be estimated because of the small sample size. One possible approach would be to estimate this unobservable dispersion matrix as being the same (or equal to) -li- the dispersion matrix for the next closest observable group. Thus, following this procedure we would estimate the Aaa(AAA) dispersion matrix as being equal to the dispersion matrix for the Aa(AA) group. Once this assumption is made, we have separate estimates of all four dispersion matrices (even though txjo of them are equal) — hence the quadratic classification can be completed. Four-group classification results from employing this latter procedure are reported in EXHIBIT CI for the 97 cases for Moody's and Standard & Poor's, and the 72 cases where both agencies rated the bonds the same. For all three models the classification results did. not change for the A and Baa(BBB) groups. However, the number of Aa(AA)-rated bonds classified correctly decreases for all of the models because some of the bonds formally estimated as Aa(AA), were now placed in the Aaa (AAA) group. The overall (percentage) classifitory ability of the multiple discriminant models decreased in all three cases from the three group results; in addition, the Moody's four- group model also suffered an absolute decrease of one less bond being correctly classified than for the three-group model. T^Jhile this procedure does allow estimation of a four-group quadratic discriminant model, it suffers one drawback — the reasonableness of the assumption that the unobservable Aaa (AAA) group dispersion matric being equal to that of the Aa(AA) group cannot be ascertained. o •a o 04 3 O u o I u 0) 4-1 V4 O 3 (0 g a o •H •H CO CO QJ U O •H •a u CO 3 4-1 O <: C •H 4-t 4J 4-1 a V M u 4 04 ^ CO o o CO ^ CM r- P^ rH ^ ^ CM CTN r^ CO r-H to H ct) C3 < rt H