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STRUCTURED FINANCE Special Report

December 13, 1996

The Binomial Expansion Method Applied to CBO/CLO

Analysis

AUTHOR:

Arturo Cifuentes, P h.D.

Senior Analyst(212) 553-1053

Gerard OConnor

Senior Analyst(212) 553-1494

CONTACTS:

Daniel Curry

M anaging D irector(212) 553-7250

J eremy G luck, P h.D.

M anaging D irector(212) 553-3698

Alicia J. Furma n

Investor R elations(212) 553-7941

CONTENTS:

Introduc tion

Why Use the BET

The BET Method

An Example of a BET Application

Conclusion

INTRODUCTIONM oodys ratings of collateralized bond obligations (C B O s) and collateralized loan

obligations (C LO s) are ultim ately based on the expected loss concept.Thus,the

need for an accurate m ethod of estim ating the expected loss for the notes to be

rated is of param ount im portance.

A num ber of m ethods can be used to estim ate the expected loss,ranging from

M onte C arlo sim ulation techniques (w hich are fairly accurate but cum bersom e to

im plem ent and com putationally expensive to run) to rather sim ple single-eventm odels (w hich are easy to im plem ent but m uch less accurate).

An alternative to simu lation or single-event models is the so-called Binom ial

Expansion Technique (B ET),w hich com bines the best of tw o w orlds: a high

degree of accuracy coupled w ith com putational friendliness (in term s of both speed

and im plem entation).This special report briefly describes the B ET m ethod in

M oodys analysis of C B O s and C LO s.

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WHY USE THE BET?The B ET is a straightforw ard approach to estim ating the total expected loss for a note in a

C B O /C LO structure.It offers a num ber of advantages in determ ining the appropriate rating:

It captures the effects of tail events,by accounting for all possible default scenarios.

Its im plem entation is m uch less com putationally intensive than that of a M onte C arlo

sim ulation.

It m akes the so-called stability (or sensitivity) analysis w hich is required for highly rated

investm ent grade notes fairly straightforw ard.(This topic w ill be covered in a future

Special R eport.)

THE BET METHODThe B ET m ethod is based on the diversity score concept. The idea is to use the diversity score

to build a hypothetical pool ofuncorrelated and homogeneous assets (bonds or loans) that w ill

m im ic the default behavior of the original pool.

Let D be the diversity score of the collateral portfolio.Then,the behavior of the original pool can

be m odeled using a fictitious portfolio consisting ofD bonds,each of w hich has the sam e par

value (total collateral par value divided by D).It is also assum ed that all these bonds have the

sam e probability of default (determ ined by the w eighted average probability of default of the

original pool).

Finally,as far as defaults are concerned,the behavior of this hom ogeneous pool ofD assets can

be fully described in term s ofD possible scenarios: one default,tw o defaults...up to D defaults.

The probability Pj that scenarioj (jdefaults) could happen can be com puted using the so-called

binom ial form ula:

Pj =D !

pj(1-p)D -jj!(D-j)!

w here p represents the w eighted average probability of default of the pool (stressed by the

appropriate factor).

Let Ej be the loss for the note to be rated under scenario j.(The loss,expressed as a percentage,

can be easily com puted by taking the present value of the cash flow s received by the note

holder,assum ing there are j defaults,and using the note coupon as the discount factor).

Finally,the total expected loss,considering all possible default scenarios,is calculated as follow s:

D

Expected Loss = PjEjj=1

AN EXAMPLE OF A BET APPLICATIONC onsider the sim ple tw o-tier structure depicted in Char t 1. Assum e that the collateral pool has a

diversity score of 20,an average probability of default of 25% (after factoring in the stressing

factor),a recovery rate of 30% ,a six-year tim e to m aturity,and pays an

average coupon of 11% .M oreover,assum e just for sim plicity that all

bonds are bullets,that there are no overcollateralization or interest rate trig-

gers,and that the excess cash is reinvested at 11% per year.And obviously,the senior piece has priority to receive the cash flow s from the collateral.

In principle,the senior note is supposed to receive the follow ing cash flow s

(on a sem iannual basis): {2.4,2.4,2.4,2.4,2.4,2.4,2.4,2.4,2.4,2.4,2.4,2.4 +

80}.As long as the num ber of defaults rem ains below ten,the senior note

w ill experience no losses.H ow ever,starting w ith ten defaults,the senior note

The Binomial Expansion Method Applied to CBO/CLO Analysis2

Chart 1

Hypothetical CBO Structure

$100

c=11%

p=25%

D=20

Rec rate=30%

Mat=6 years

$80

Senior Piece

c=6%

$20

c=12%

Equity

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w ill suffer increasing losses (see Table 1).The losses under each default

scenario are com puted sim ply by discounting,applying a 6% discount rate,

w hatever cash flow s the senior note receives and com paring that present

value w ith $80.

For exam ple,if there are 10 defaults,the senior note receives {2.4,2.4,2.4,

2.4,2.4,2.4,2.4,2.4,2.4,2.4,2.4,78.86} yielding a loss of 3.1% .In this case,it

has been assum ed that the defaults are front-loaded that is,50% occur

at the end of year one and 10% at the end of each year for the five subse-

quent years.Table 1sum m arizes the results of the B ET com putation.The first colum n

show s the num ber of defaults under the scenario; the second colum n

show s the probability that that scenario w ill occur; and the third colum n

show s the loss under that scenario.

The total expected loss is as follow s:

Expected Loss = 0.3171% x 0.00% + ...

...+ 0.00% x 45.1629% = 0.067%

Table 2show s the expected loss for different ratings and different m aturities

based on M oodys idealized historic data.A ccording to this table (see six-

year colum n),the senior note w ould be rated Aa3 (the cut-off value is

0.10065% for the Aa3).

It is also interesting to see the variation of the expected loss (and hence,the

rating) as a function of the diversity,D,assum ing all the rem aining variables

are kept constant.Chart 2 depicts such a graph.C learly,a variation in the value ofD has a

m ajor im pact for low -diversity pools; for higher values ofD,the expected loss tends to be m uch

m ore stable.

CONCLUSIONThis exam ple dem onstrates the application of the

B ET to the analysis of a C LO /C B O.O f course,addi-tional m odeling com plexities arise in real situations,

w hich m ust address am ortization,reinvestm ent

criteria,overcollateralization tests,m anagem ent

fees,sw aps,caps,different priority of paym ents,

and the like.Also,m ore nonhom ogeneous portfo-

lios (for exam ple,portfolios in w hich a few bonds

account for a large portion of the collateral portfolio)

m ight require som e special m odifications of the

B ET m ethod.These extrem e cases m ust be exam -

ined carefully.

The Binomial Expansion Method Applied to CBO/CLO Analysis 3

Table 1

Summary of BET Calculation

Probability# of Def. Scenario(% ) Loss(% )

0 0.3171 0.0000

1 2.1141 0.0000

2 6.6948 0.0000

3 13.3896 0.0000

4 18.9685 0.0000

5 20.2331 0.0000

6 16.8609 0.0000

7 11.2406 0.0000

8 6.0887 0.0000

9 2.7061 0.0000

10 0.9922 3.1026

11 0.3007 7.8958

12 0.0752 12.6890

13 0.0154 17.4822

14 0.0026 22.2754

15 0.0003 27.0686

16 0.0000 31.5621

17 0.0000 34.7819

18 0.0000 38.0531

19 0.0000 41.6080

20 0.0000 45.1629

Chart 2

Expected Loss versus Diversity

J

J

J

J

J

J

J

J

J

5 8 10 12 17 20 25 35 50

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

Expected

Loss

Diversity

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The Binomial Expansion Method Applied to CBO/CLO Analysis

C op yright 1996 by M oo dys Investors Service, Inc., 99 C hurch S treet, N ew York, N ew York 100 07.All rights reserved. ALL IN FO RM ATIO N C O N TAIN ED H ER EIN IS C O PY R IG H TED IN TH E N AM E O F M O O D YS IN VES TO RS SE RV IC E, IN C . (M O O D YS), AN D N O N E O F SU C H IN FO R M ATIO N M AY B E C O PIED O ROTH ERW ISE REPR OD UC ED, REPAC KAG ED, FURTHER TRANSM ITTED, TRANSFERRED , DISSEM INATED, REDISTRIBUTED O R R ESOLD, OR STOR ED FO R SU BSEQ UEN T USE FO R A NY SUC H PUR PO SE, INW H O LE O R IN P AR T, IN A N Y FO RM O R M AN N ER O R B Y AN Y M EAN S W H ATSO EVER , BY A N Y PER SO N W ITH O UT M O O DYS PR IO R W RITTEN C O N SEN T. All inform ation contained herein is obtained byM O O D YS from sources b elieved by it to b e accurate and reliable. Because o f the possibility of hum an or m echanical error as w ell as o ther factors, how ever, such inform ation is provided as isw ithout w arranty of anykind and M O O D YS, in particular, m akes no representation or w arranty, express or im plied, as to the accuracy, tim eliness, com pleteness, m erchantability or fitness for any particular purpo se of any such inform ation.U nder no circum stances shall M O O D YS have any liability to any person or entity for (a) any loss or dam age in w hole or in part caused by, resulting from , or relating to, any error (negligent or otherw ise) or other circum -stance or contingency w ithin or outside the control of M O O D YS or any of its directors, officers, em ployees or agents in connection w ith the procurem ent, collection, com pilation, analysis, interpretation, com m unica-tion, pub lication or delivery of any such inform ation, or (b) any direct, ind irect, special, consequential, com pensatory or incidental dam ages w hatsoever (includ ing w ithout lim itation, lost profits), even if M O O D YS isadvised in advance of the possibility of such dam ages, resulting from the use of or inability to use, any such inform ation. The credit rating s, if any, constituting part of the inform ation contained herein are, and m ust beconstrued solely as, statem ents of opinion and not statem ents of fact or recom m endations to purchase, sell or hold any securities. N O W ARR AN TY , EXPRES S O R IM PLIED , AS TO TH E A C C U RAC Y, TIM ELIN ES S,CO M PLETENESS, M ERCH ANTABILITY OR FITNESS FOR ANY P ARTICU LAR PUR PO SE O F ANY SU CH RATING O R O THER O PINIO N O R INFO RM ATIO N IS G IVEN O R M ADE B Y M O OD YS IN A NY FO RM O RM AN N ER W H ATS O EV ER .E ach rating or other opinion m ust be w eighed solely as one factor in any investm ent decision m ade by or on behalf of any user of the inform ation contained herein, and each such userm ust acco rdingly m ake its ow n study and evaluation of each security and of each issuer and g uarantor of, and each provider of credit sup port for, each security that it m ay consider purchasing, holding orselling. Pursuant to S ection 17(b) of the S ecurities A ct of 19 33, M O O D YS hereby discloses that m ost issuers of debt securities (including corpo rate and m unicipal bo nds, debentures, notes and com m ercialpaper) and p referred stock rated by M O O D YS have, prior to assignm ent of any rating, agreed to pay to M O O D YS for app raisal and rating services rendered by it fees ranging from $1,000 to $3 50,000.

Ta

ble2

MoodysIdealizedCumulativeEx

pectedLossRates(%)

Year

Ra

ting

1

2

3

4

5

6

7

8

9

10

A

aa

0.0

00028

0.0

0011

0.0

0039

0.0

0099

0.0

0160

0.0

0220

0.0

0286

0.0

0363

0.0

0451

0.0

0550

A

a1

0.0

00314

0.0

0165

0.0

0550

0.0

1155

0.0

1705

0.0

2310

0.0

2970

0.0

3685

0.0

4510

0.0

5500

A

a2

0.0

00748

0.0

0440

0.0

1430

0.0

2585

0.0

3740

0.0

4895

0.0

6105

0.0

7425

0.0

9020

0.1

1000

A

a3

0.0

01661

0.0

1045

0.0

3245

0.0

5555

0.0

7810

0.1

0065

0.1

2485

0.1

4960

0.1

7985

0.2

2000

A

1

0.0

03196

0.0

2035

0.0

6435

0.1

0395

0.1

4355

0.1

8150

0.2

2330

0.2

6400

0.3

1515

0.3

8500

A

2

0.0

05979

0.0

3850

0.1

2210

0.1

8975

0.2

5685

0.3

2065

0.3

9050

0.4

5595

0.5

4010

0.6

6000

A

3

0.0

21368

0.0

8250

0.1

9800

0.2

9700

0.4

0150

0.5

0050

0.6

1050

0.7

1500

0.8

3600

0.9

9000

B

aa1

0.0

49500

0.1

5400

0.3

0800

0.4

5650

0.6

0500

0.7

5350

0.9

1850

1.0

8350

1.2

4850

1.4

3000

B

aa2

0.0

93500

0.2

5850

0.4

5650

0.6

6000

0.8

6900

1.0

8350

1.3

2550

1.5

6750

1.7

8200

1.9

8000

B

aa3

0.2

31000

0.5

7750

0.9

4050

1.3

0900

1.6

7750

2.0

3500

2.3

8150

2.7

3350

3.0

6350

3.3

5500

B

a1

0.4

78500

1.1

1100

1.7

2150

2.3

1000

2.9

0400

3.4

3750

3.8

8300

4.3

3950

4.7

7950

5.1

7000

B

a2

0.8

58000

1.9

0850

2.8

4900

3.7

4000

4.6

2550

5.3

7350

5.8

8500

6.4

1300

6.9

5750

7.4

2500

B

a3

1.5

45500

3.0

3050

4.3

2850

5.3

8450

6.5

2300

7.4

1950

8.0

4100

8.6

4050

9.1

9050

9.7

1300

B

1

2.5

74000

4.6

0900

6.3

6900

7.6

1750

8.8

6600

9.8

3950

10.5

2150

11.1

2650

11.6

8200

12.2

1000

B

2

3.9

38000

6.4

1850

8.5

5250

9.9

7150

11.3

9050

12.4

5750

13.2

0550

13.8

3250

14.4

2100

14.9

6000

B

3

6.3

91000

9.1

3550

11.5

6650

13.2

2200

14.8

7750

16.0

6000

17.0

5000

17.9

1900

18.5

7900

19.1

9500

Caa

14.3

00000

17.8

7500

21.4

5000

24.1

3400

26.8

1250

28.6

0000

30.3

8750

32.1

7500

33.9

6250

35.7

5000

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