1

Benchmarking Modelof Default Probabilities of

Listed Companies

Cho-Hoi Hui, Research Department, HKMA

Tak-Chuen Wong, Banking Policy Department, HKMA

Chi-Fai Lo, Institute of Theoretical Physics and Department of Physics, CUHK

Ming-Xi Huang, Physics Department, CUHK

10 November 2005

*The conclusion herein do not represent the views of the Hong Kong Monetary Authority

2

Introduction

• Basel II allows banks to choose among several approaches to determine their capital requirements to cover credit risk.

• The foundation IRB approach for corporate, sovereign, and bank exposures allows banks to provide estimates of probability of default (PD) but requires banks to use supervisory estimates of loss given default (LGD), exposure at default (EAD), and maturity. The advanced IRB approach for such exposures allows banks to provide estimates of all these risk characteristics.

• As credit risk measures are estimated by banks, systematic underestimation of such measures and the corresponding regulatory capital in a bank (or a number of banks) will increase the bank’s vulnerability to adverse changes in market conditions, in particular during a financial or banking crisis.

• The validation methodologies of IRB systems have emerged as one of the important issues of the implementation of Basel II.

3

Introduction (2)

• This paper proposes a benchmarking model for the purpose of IRB validation of listed companies, which is developed upon using a credit risk model and a simple mapping process. The credit risk model is based on the recent studies of the predictive capability of structural models.

• Leland (2002) finds that PDs generated from the Longstaff and Schwartz (1995) model fit actual default rates provided by Moody’s (1998) for longer time horizons quite well for reasonable parameters with proper calibrations.4 However, the default boundary in the Longstaff and Schwartz model should be specified as a certain fraction of the principal bond value. This specification imposes a constraint to some of the calibrations for the model (for example the asset volatility), that may not be empirically reasonable, in order to obtain consistent results.

4

Structural model of term structures of PDs

• The structural model employed for generating term structures of PDs follows the model proposed by Hui et al. (2005).

T h e r i s k - a d j u s t e d d y n a m i c o f t h e l e v e r a g e r a t i o L i s m o d e l l e d b y t h e f o l l o w i n gs t o c h a s t i c d i f f e r e n t i a l e q u a t i o n :

LL LdztLdttdL , ( 1 )

w h e r e t a n d tL a r e t h e d r i f t a n d t h e v o l a t i l i t y o f L r e s p e c t i v e l y a n d a r e t i m e

d e p e n d e n t . T h e d r i f t t i s e f f e c t i v e l y t a k e n a s z e r o i n t h e p a p e r . T h ec o n t i n u o u s s t o c h a s t i c m o v e m e n t o f t h e i n t e r e s t r a t e r f o l l o w s :

r rd r t t r d t t d Z ( 2 )

w h e r e tr i s t h e i n s t a n t a n e o u s v o l a t i l i t y . T h e s h o r t - t e r m i n t e r e s t r a t e r i s m e a n -

r e v e r t i n g t o l o n g - r u n m e a n t a t s p e e d t . T h e W i e n e r p r o c e s s e s LdZ a n d

rdZ a r e c o r r e l a t e d w i t h

dttdZdZ rL .

5

2 2 22 2 2

2 2

1 1

2 2L r L r

P P P Pt L t t t t L

t L r L rP P

t L t t r r PL r

. ( 3 )

02

22 1

01

2 10

1

l n, 1 e x p 4 l n 1 6

2

l n 8

2

d e f

Lb t

LLP L t N b t b t

Lb t

Lb t b t

LN

b t

( 4 )

w h e r e N ( . ) i s t h e c u m u l a t i v e n o r m a l d i s t r i b u t i o n f u n c t i o n , i s a r e a l n u m b e rp a r a m e t e r , a n d b 1 ( t ) a n d b 2 ( t ) a r e d e f i n e d a s f o l l o w s :

t

L dtttb0

21 ''

2

1 ,

t

dtttb02 '' ,

ttatattttt LrL2

12 2

1exp ,

t

dttta01 '' ,

t

dttata0 12 ''exp .

6

PD term structures

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Tenor (Year)

Def

ault

Pro

babi

litie

s (%

)

ModelS&P' s dataB model PDaveB S&P' sBB model PDaveBB S&P' sBBB model PDaveBBB S&P' s

CCC

B

BB

BBB

PD term structures generated from the structural model and actual cumulative defaultrates reported by S&P’s (2002) of ratings of CCC, B, BB and BBB. The leverage ratiosof ratings CCC, B, BB and BBB are 0.732, 0.538, 0.495 and 0.315 respectively. Theleverage volatilities L of ratings CCC, B, BB and BBB are 0.299, 0.27, 0.241 and 0.213respectively.

7

Benchmarking process of benchmark PD estimation

Input market parameter:Leverage ratio and its volatility of a listed company

Model Engine Generate the PD term structure of the company

Mapping with S&P’s default ratesMap the model PD term structure of the company to S&P’s default-rate term structures of different ratings (static pools cumulative average default rates)

Assigning benchmark “S&P’s” rating Based on the mapping result, a rating is assigned to the company.

Implied one-year benchmark PD of the companyIt is based on the actual one-year average default rate of the benchmark rating.

Compare the one-year benchmark PD with the bank’s one-year PD of the company based on its IRB system. Based on comparisons for a number of companies, the results will indicate any inconsistencies / systematic underestimation in the bank’s PD estimates.

8

Assignment of ordinal numbers to S&P’s ratings and numbers of sample companies with

S&P’s ratingsS&P’s ratings Ordinal numbers Numbers of sample

companies

A- and above 1 942BBB+ 2 422BBB 3 618BBB- 4 430BB+ 5 338BB 6 368BB- 7 457B+ 8 207B 9 91B- 10 46

CCC 11 24

9

Data and empirical results

I n o r d e r t o o b t a i n t h e t e r m s t r u c t u r e s o f P D s s p e c i f i e d i n e q u a t i o n ( 4 ) , i t i s

n e c e s s a r y t o l i n k t h e l e v e r a g e v o l a t i l i t y L t o t h e e q u i t y v o l a t i l i t y S . T h e v a l u e s

o f L a r e a s s u m e d t o f a l l c l o s e t o t h e a s s e t v o l a t i l i t i e s o f c o m p a n i e s . T h i s m e a n s

t h a t t h e v o l a t i l i t y o f a c o m p a n y ’ s l i a b i l i t y i s a s s u m e d t o b e i m m a t e r i a l . T h e d a i l y

s t a n d a r d d e v i a t i o n o f e q u i t y r e t u r n s )( tS i s c a l c u l a t e d b a s e d o n a w i n d o w o f 1 , 0 0 0

d a y s , w h e r e t i s t h e o b s e r v a t i o n d a t e . T h e e s t i m a t e o f t h e d a i l y a s s e t v o l a t i l i t y

)( tL a t t i m e t i s o b t a i n e d b y a p p l y i n g a g e a r i n g r a t i o t o )( t

S a s :

DS

StS

tL

)( ( 5 )

w h e r e S d e n o t e s t h e c o m p a n y ’ s e q u i t y p r i c e a t t i m e t .

10

Mismatch statistics of benchmark ratings versus S&P’s ratings of 3,943 sample companies

Mismatch statistics

Differences Sample companies % Cumulative figures

0 21.7% 21.7% 25.9% 47.6% 21.2% 68.8% 12.8% 81.7% 8.8% 90.5% 5.7% 96.1% 2.2% 98.4%

11

Mismatch distribution of benchmark ratings versus S&P’s ratings of 3,943 sample companies

0.0% 0.0% 0.0% 0.0% 0.2%0.8%

1.4%

2.7%

4.7%

21.7%

16.6%16.5%

10.1%

7.4%

4.9%

2.1%

0.8%0.2% 0.6%

0.1%

9.3%

0%

5%

10%

15%

20%

25%

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Mismatches

Per

cen

tage

of

the

Sam

ple

12

Discriminatory powerT h e h i t r a t e H R ( P D ) i s d e f i n e d a s

NIN

PDHPDHR

)()( , ( 6 )

w h e r e H ( P D ) i s t h e n u m b e r o f c o m p a n i e s a s s i g n e d c o r r e c t l y a s n o n - i n v e s t m e n tr a t e d c o m p a n i e s b a s e d o n t h e b e n c h m a r k i n g m o d e l , a n d N N I i s t h e t o t a l n u m b e r o fn o n - i n v e s t m e n t r a t e d c o m p a n i e s i n t h e s a m p l e s .

T h e f a l s e a l a r m r a t e F A R ( P D ) i s d e f i n e d a s

IN

PDFPDFAR

)()( , ( 7 )

w h e r e F ( P D ) i s t h e n u m b e r o f f a l s e a l a r m s , i . e . t h e n u m b e r o f i n v e s t m e n t r a t e dc o m p a n i e s t h a t a r e c l a s s i f i e d i n c o r r e c t l y a s n o n - i n v e s t m e n t r a t e d c o m p a n i e s b a s e do n t h e b e n c h m a r k i n g m o d e l . T h e t o t a l n u m b e r o f i n v e s t m e n t r a t e d c o m p a n i e s i nt h e s a m p l e s i s d e n o t e d b y N I .

121

0 FARdFARHRAR . ( 8 )

13

Receiver operating characteristic curve and accuracy ratio of the benchmarking model

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0%

False Alarm Rate (%)

Hit

Rat

e (%

)

Accuracy ratio = 0.7104

Benchmarking model

Random model

14

Degree of association between benchmark ratings versus S&P’s ratings of 3,943 sample companies

Degree of Association

Type of statistics Estimates Asymptoticstandard error

p-value of noassociation

Kendall's tau (b) 0.5241 0.0090 0.0000Stuart's tau (c) 0.5070 0.0088 0.0000Gamma () 0.5873 0.0101 0.0000

15

Summary

• Credit risk measures of listed companies can be obtained from the structural model developed by Hui et al. (2005) without any specific calibration in this paper.

• The benchmarking model assigns benchmark ratings and one-year PDs to companies by mapping the term structures of PDs of the companies generated by the chosen structural model to the term structures of default rates reported by S&P’s.

• The empirical results show that the benchmark ratings could broadly track the S&P’s ratings of the US sample companies. The association between them is statistically significant. The results demonstrate that the benchmarking model has adequate discriminatory power of ranking credit risk of the sample companies in terms of differentiating investment rated and non-investment rated companies.

• Benchmark PDs obtained from the model could thus be used as alternative external and independent PD estimates for comparison with banks’ internal PD estimates of listed companies. Significant deviations from this benchmark provide a reason to review the banks’ internal estimates and their credit rating processes.