How to Apply Credit Risk Model Effectively: The Role of
Default Point
K. T. Wang＊
Department of Finance, National Sun Yat-sen University, 70 Lien-Hai Rd., Kaohsiung,
Taiwan (R.O.C). E-mail: tt9141217@gmail.com
Chau-Jung Kuo
Department of Finance, National Sun Yat-sen University, 70 Lien-Hai Rd., Kaohsiung,
Taiwan (R.O.C).
Chau-Hong Yu
Department of Finance, National Sun Yat-sen University, 70 Lien-Hai Rd., Kaohsiung,
Taiwan (R.O.C).
Abstract
Since the Basel Committee on Banking Supervision has declared the New Basel
Capital Accord on June 2004. The issue about credit risk becomes more significant
and the approaches used to measure credit risk have been complicated. Especially
when the sub-prime mortgage crisis has triggered a global financial crisis through
2007 and 2008; many people start to pay much attention to the issue on credit risk.
There are numerous models developed to measure the credit risk in the recent
years, and one of the well-known is Moody’s KMV credit risk model. Because it has
assembled large public and private company default and loss database in the world to
estimate the probability of default. Here, we propose a company is default since the
＊
Corresponding author.
firm’s value is below its default point which plays as one of the key role in estimating
the probability of default. So far as the Moody’s KMV credit risk model is concerned,
the default point is defined as current liabilities plus one-half of long-term liabilities,
and has been tested in many countries. However, we would like to observe whether
the default point in Moody’s KMV credit risk model is exactly suitable for different
countries whose regionalism, national condition, financial structure and industrial
characteristic may not be the same from each other. Obviously, there are numerous
papers published to examine the procedures and results. Besides, the historical
estimations do not account for variation over time.
Actual default probabilities are
not constant over time, but are conditional on the current state of the economy. In
other words, we should not apply the same definition of default point to such different
companies.
Thus, we will elaborate hence to renew the definition of default point base on the
Moody’s KMV credit risk model accompanied by the threshold regression, which can
observe whether it existed structural change or not. Combining this information with
assumptions by threshold regression, we believe the verified default point can, in
principle, be efficiently estimated and more appropriately tested Taiwan’s financial
market.
The main purpose of the article is to examine the appropriate default point
definition to apply to the financial market in Taiwan, and help to manage the credit
risk properly. By the modified definition of default point with threshold regression
model, we can calculate the probabilities of default of each company in Taiwan more
correctly. In our empirical results, we conclude that we have better accuracy and the
efficiency than original Moody’s KMV credit risk model after verifying the definition
of default point. This can help investors, lenders, and corporations understand and
adopt the verified methods to measure credit risk in one hand; and the financial
institutions and the regulatory authorities can mutually reinforce the credit
relationship in another hand. Simply speaking, they can prevent the investor, lenders
and corporations from bearing the default credit risk and loss by utilizing a proper
method to predict the probability of default.
1. Introduction
The definitions of credit risk are manifold. Narrowly definition of credit risk is
the debtor could not exercise the obligation in the period of time that he is asked to
pay until he did pay. In this period of time, the debtor may default by the reasons that
the limitation of liquidity and operating problem, and this will result in the loss of the
creditor. Broadly definition of credit risk is inclusive of all events which may damage
the creditors’ right. Consequently, no matter whether it is a listed company or not, it
might be suffered from this kind of credit risk or default risk. By the data of the
Financial Supervisory Commission show that the total market values of the listed
companies reach to 150% even to 200%, this reveals that the important role they play
in the economics system in Taiwan. Therefore, it will accompany bad influence as if
default was happened, so that we should develop a suitable credit risk model to
correctly evaluate the default problem in Taiwan as quickly as possible.
In fact, there are lots of models such as Credit Grades, Credit Metrics, Credit
Risk＋, Credit Portfolio Views and Moody’s KMV can used to measure the default risk
of companies. Because the market price of equity is available, the market value and
volatility of assets can be determined directly using an options pricing based approach,
which recognizes equity as a call option on the underlying assets of the firm,
addressed in Black and Scholes (1973) and Merton (1974). Accordingly, because of
the simplicity and accuracy of Moody’s KM credit risk model to measure the default
distance and default probability, it becomes popular and respect for industries and
academia. This credit risk model is developed based on the option pricing theory in
Black and Scholes (1973) and corporate debt pricing model in Merton (1974). This
model not only takes use of financial statement data, but also considers the equity
value of corporate. The significant advantage of this model is the instant information
in stock market to reflect the value of business and compute the bankruptcy
probability or default probability. For this reason, we can evaluate the credit risk of
business immediately without waiting for the rating announced by the rating agency.
In the viewpoint of regression model, non-linear or structural change is caused
by different time or different variable in the model. All the methods to solve these
kinds of problem at present are numerous, such as threshold regression and Quantile
regression. The former discussed in Hansen (1999) is the most popular and
affirmative, and the later is proposed in Koenker (1979). The central meaning of
quantile regression considered the different influence of independent variable as
dependent variable is on different position. The differences between traditional
regression and quantile regression are the coefficient we estimated. Instead of
estimating by the average samples, this model gives different weighted by the value of
error term. Accordingly, we can get different coefficient with the relative position. As
far as the threshold effect is concerned, it can be attributed to independent variable in
the threshold regression model, but the quantile regression model in Koenker (1979)
can be attributed to dependent variable.
We explore the appropriate default point and apply the Moody’s KMV credit risk
model to the listed companies in Taiwan, and find the threshold effect may exist in the
listed companies. However, the threshold effect is caused by the independent variable
(such as the corporate characteristic) or the dependent variable (default point) is an
essential issue to confirm. In order to clarify this question, we separately adopt the
threshold regression proposed in Hansen (1999) and quantile regression offered in
Koenker (1979) to examine the definition of default of listed company in Taiwan. We
can modify the original Moody’s KMV credit risk model and make up for the
shortcoming of linearity to properly measure the credit risk of listed companies in
Taiwan. To sum up, the main problems we want to solve are:
(1) Whether the structural change exists in the default point or not, we would like to
apply Moody’s KMV credit risk model to the listed companies in Taiwan? How to
define whether the structural change exists, and the new definition of default point?
(2) After redefining the definition of default point to modify the credit risk model,
whether it has better financial distress predictability of the listed companies in
Taiwan?
Except for the background and purpose above- mentioned, we review the relative
literature in the second section. Besides, we describe the methodology and the
empirical models we used in the third section. In regard to the empirical test and
analysis will be showed in the forth section. Finally, the conclusion and the suggestion
are appeared in the fifth section.
2. Literature
Since Altman created the Z-Score model in 1968, the subjects of Credit Risk
Management have started to be concerned. After the implementation of Basel I, these
subjects became more and more important, and how to take Credit Risk Management
into account is currently crucial to business. Many financial institutions and
businesses separate out Risk Management department in addition to putting academic
theory into practice. The subprime mortgage crisis further points out the importance
of Credit Risk to market investors.
For financial institutions, evaluating credit risk plays a crucial role in risk
management. In 1988, the Basel Committee on Banking Supervision issued the
International Convergence of Capital Standard (or Capital Accord of 1988), which
established regulations regarding the amount of capital banks should hold against
credit risk. Therefore, Basel I was drawn up, providing a standard approach to
measure credit risk and market risk to determine minimum capital requirements. Basel
II, implemented in 2006, also considered credit risk and market risk as in Basel I.
Moreover, this new accord addressed operational risk as well as focused on
improvements in measurement of credit risk and proposed the internal rating based
(IRB) approach, by which financial institutions will be allowed to use their internal
estimates of borrower creditworthiness.
At present, the listed companies in the Taiwan stock market raise capital by
issuing stocks or bonds. The most traditional way is to borrow required money from
financial institutions. However, the money borrowed from financial institutions is not
the same as small amount loans. Therefore, it brings a crucial issue for financial
institutions to evaluate the credit risk of public traded companies with their own credit
risk model.
In the past, the credit risk models, based on econometrics, such as the Z-Score
Model (Altman 1968), the Qualitative Model, Probit Model, the Logistic Model, the
Discriminate Model, and the Neural Network Analysis, have generally been used in
credit risk management. These models use the companies’ historical data as model
variables to predict the probable default in the future. Therefore, they are also called
Look Forward Analysis and belong in the category with actual models. However,
these models lack of the support of rigid theories and the past performance of
companies can’t represent their default in the future. Accordingly, they don’t have
abilities that precise and predictable. Since the 1990’s, the development of credit risk
models can be divided into two main categories: the structure-form model and the
reduced-form model. The basic difference between the two models is input variables.
Besides, these models use market information to predict probability of default in the
future. Hence, the scholars categorize the credit risk models since the 1990’s as Look
Forward Analysis.
Structural form models are based on the original framework developed by
Merton (1974), using the principles of option pricing in Black and Scholes. The input
variables in these models must be related to asset price and its volatility as well as
correlative capital structure parameters of the target company. Therefore, structure
form models have more instinct and economical meanings than reduced form models.
In addition, structure-form models adopt business individual information and use
non-linear estimation to determine the asset market value, volatility, and parameters
associated with capital structure. Then, the default probability of targeted companies
can be estimated, which have more institutional economic meaning. However, the
input variables of model are hard to observe directly. Accordingly, it’s difficult to
implement in quantification. Some representative articles are Black and Cox (1976),
Bernnan and Schwartz (1977; 1978; 1980), Geske (1977), Ingersoll (1976; 1977a;
1977b), Leland (1994), Leland and Tofe (1996), Longstaff and Schwartz (1995), and
Zhou (1997).
Reduced form models mainly use market information, such as credit spread,
credit rating conversion, or market price as input variables. This method ignores the
characteristic of each company and credit rating can’t reflect the public information
immediately. However, in practice, it simplifies the calculating process. Compared to
the structure form models, the reduced form models need not define bankrupt and
default. The following are representative studies: default-based approach (Jarrow and
Tumbull 1995), rating-transition approach (Jarrow et al. 1997; Das and Tufano 1996;
Lando 1998; Arvanitis, Gregory and Laurent, 1999), spread approach (Duffie and
Singleton 1997; Madan and Unal 2000), market-based risk neutral approach (Bierman
and Hass 1975; Yawitz 1977; Johnhart 1979; Lu and Kuo 2005).
Some cores or foundations of models mentioned above have been developed and
put into practice, for example Moody’s KMV, Credit Grades, Credit Metrics,
CreditRisk＋,and Credit Portfolio View. Among these models, Moody’s KMV credit
risk model is the most popular in credit risk management. It uses the concept of call
option, which considers the company market value of equity as the target of market
value of asset as well as the short-term liabilities plus half of long-term liabilities, also
called the default point, as the strike price. Therefore, according to the definition of
Moody’s KMV credit risk model, the default event happens while the market value of
asset is lower than the default point in a given period in the future. In terms of the
Moody’s KMV announcement, the definition of default point was determined through
an optimization process, making sure that the model power (accuracy ratio) is high.
Although Moody’s KMV credit risk model defines the default events happen as
those company’s asset market value is lower than its’ short-term liabilities plus half of
long-term liabilities, we think that the characteristics of company itself are the main
reasons affecting the probability of default. Ohlson (1980) used Logistic analysis to
create financial crisis prediction model. The empirical results showed that company
size, capital structure, return on asset, debt ratio and liquidity have significance
predictable power on financial crisis. Therefore, company size or capital structure
does affect company’s default risk in the future. Moreover, enterprise’s payment
ability may directly affect company’s default risk. Companies will face the risk of
bankrupt or shut down if they can’t afford payment. Beaver (1996) composed 79
crisis companies as samples and used financial variables as crisis indicators as well as
information on the financial statement to judge those companies. The results showed
that cash flow-to-total debt, and total debt-to-total asset had predictable power. Blum
(1974) used cash flow theory as well as indicators of liquidity, probability and
variability to create enterprise’s failure prediction model. Therefore, we consider that
the definition of default point should come from the related factors. In this way, the
estimated default probabilities can truly affect the operating characteristics of listed
companies in Taiwan stock market.
According to the researches mentioned above, the characteristics of business all
have significance relationship with enterprise’s default probability. Therefore, based
on these characteristics, we conclude the factors of affecting default probability, such
as operating ability, financial liquidity, business size, solvency, and financial leverage.
Traditional regression is often used to analysis the linear relationship between
dependable variables and independent variables. However, under the affection of time
difference, extraneous impaction, or variable characteristics, this linear relationship
between dependable variables and independent variables may change. In econometric,
this is so called structural change. Simple linear regression model no longer
completely interprets the relation between dependable variables and independent
variables in this situation. Although this problem can be controlled or interpreted by
presuming change point and using dummy variables, the way of presuming change
point is not obviously objective.
In econometric, the threshold regression is applied into empirical research of the
following characteristics, multiple equations of structural changes, non-linear and
time-series autoregression as well as dividing samples of continuous distribution.
Threshold auto-regression (TAR) was created by Tong (1978). Then, Tong and Lim
modified TAR, referring to discuss the level of autoregression of threshold variables
in different threshold intervals. Hansen (1996) used bootstrap method to examine
threshold effect. It solved the problems of non-standard distribution due to the
distribution of traditional test statistic. Besides, Hansen (1999, 2000) considered that
samples above or below threshold point should have different characteristics. Also, he
argued that the consistent estimators of threshold point can be determined by least
squares and then used likelihood ratio to determine the confidence interval of
threshold point. However, structural changes may exist in regression model. These
changes can result from the different levels of independent variables, the different
distribution of dependent variables or the level that the dependent variables affected
by independent variables. Therefore, we should survey the different levels of
structural changes caused by the degree that independent variables affect dependent
variables and concern that whether the structural changes, originated from the degree
that independent variables affect dependent variables, exist under different quantity
positions of dependent variables.
The quantile regression model considers both outliers and the concept of mean. The
coefficients of estimators, under different given quantile points, are estimated to
discuss how independent variables affect dependent variables under the different
distribution position of dependent variables. According to some descriptive statistics
in empirical studies related to financial topics, the distribution of return of financial
assets mostly belongs to non-symmetric distribution. The estimators estimated by
Ordinary Least Square will be biased. In addition, the down-side risk is more crucial
than the whole average risk in the issue of credit risk management. This is the reason
why this paper uses the threshold regression model created by Hansen (1999) to
estimate the default point in Moody’s KMV credit risk model as well as uses the
qunantile regression created by Koenker and Basset (1987) to avoid the sample
selection bias resulted from the threshold regression model and we try to study the
suitability for setting up the default points of listed companies in the Taiwan stock
market under different quantile positions of dependent variables.
3. Methodology
Moody’s KMV credit risk model estimates the market value of assets and its
volatility by market value of equity, standard deviation of equity return, debt value
and risk free interest rate. Then the expected default probability could be solved. The
following steps are the procedures used by Moody’s KMV:
Step 1: Estimate asset value and its volatility
With the concept of option, we recognize equity as a call option on the
underlying assets of the firm. Here, we suppose the creditor own the assets, and hold a
call option, and the shareholder have the right to buy the firm assets from them. In
other words, the seller is the creditors, the shareholders are the buyer, and the strike
price of the call option is equal to the book value of the company’s liabilities. As the
asset value is insufficient to meet the liabilities of the firm then the shareholders, the
owners of the call option, will not exercise the option and will leave the firm to its
creditors. On the other hand, when the asset value of firm is larger than the book value
of liabilities, the shareholder will exercise this option and pay back the debt to make
the firm exist. We can simultaneously solve the asset value and volatility implied from
the market value of equity:
VE  VA N  d1  Be  rT N  d 2 


VA
 E  V  A ,    N  d 2  
E

(1)
VE 、V A、B 、T and rf are separately represents the equity value of firm, the asset
value, book value of liabilities, the debt due at time T and risk free interest
rate. N  d1  and N  d2  show the cumulative probability functions in normal
distribution.
A
denotes
the
volatility
of
asset
value
of
firm
and
 V  
2  
d1  ln  A    rf  A   /   ， d2  d1    .
2  
  B 
Step 2: Calculate the distance-to-default
In Moody’s KMV credit risk model, the definition of default is the asset value of
firm is lower than the default point, and the default distance is measured by the
standard deviations. We can derive the default distance by measuring and
standardizing asset volatility. The larger the number means the farther the default
distance, in other words, the default probability of firm is lower. According to the
definition in the Moody’s KMV credit risk model, the default distance can be denoted
as follows:
ln
DD 
VA 
2 
    A t
Bt 
2 
A t
(2)
The default probability implies the probability of asset value reaching to the default
distance. Therefore, the default probability can be showed as:

VA 
 A2  
 ln    
t 
Bt 
2  

Pt  N 


A t




(3)
This is the probability in the Moody’s KMV credit risk model. However, in order
to calculate the default distance, we have to collect the information about the asset
value and asset volatility. In fact, we can not observe these two variables directly from
financial statement or in the stock market, so we can estimate the asset value and the
volatility simultaneously by non-linear approach with equation (1).
Step 3: Re-define the default point
After estimating the parameters of assets value and the volatility of defaulted
companies by non-linear approach simultaneously, we take use of the assets value of
defaulted companies as the dependent variable and the short-term liabilities and
long-term liabilities as the independent variable in the threshold regression. With the
five factors affected above-mentioned as our threshold variables to evaluate the
coefficient value of short-term liabilities and long-term liabilities which have
influence on assets value of default companies separately by threshold regression and
quantile regression. Consequently, we will take these coefficients as the default point
to measure the default probability which is appropriate for the listed companies in
Taiwan. The threshold regression model could be showed as follow:

αˆ ti  ˆ11SDi  ˆ12 LDi  ei ,
DPi  
ˆ
ˆ

αˆ ti   21SDi   22 LDi  ei ,
q
q 
(4)
The DPit , SDi and LDi are separately denotes as the assets value, short-term
liabilities and long-term liabilities of the ith default company. Let q be the threshold
variable of the five business characteristics, and  represent the threshold point in
the threshold regression. In order to examine if it exists the threshold effect or not, we
firstly establish the null hypothesis and alternative hypothesis as follow:
H0： ˆ11  ˆ 21 , ˆ12  ˆ 22
(5)
H1： ˆ11  ˆ 21 , ˆ12  ˆ 22
(6)
The test value is F1 
RS 0  RS 1 ˆ 
1
1
, in which ˆ 2  eˆ'eˆ  RS  ˆ  . As RS0
2
n
n
ˆ
 
denotes the mean square deviation value of all samples, RS1 k
is the mean square
deviation value under the threshold value.
To avoid the difficulty by non-linear approach, Hansen (1999) developed a
method which can apply non-dynamic panel data to solve this problem by setting,
estimating and testing in the threshold model. We will take us of two stage least
squares to estimate the threshold value. At first, we set a threshold value separately by
least square method to find the sum of squared errors, SSE. Then we can get the
estimated value of threshold through SSE, and use this value to calculate the
coefficient of regression to analyze and observe the consistency of the result.
Although the original samples will divided into different subsample according to
different threshold values in the threshold regression model, the dependent variable
may bring out different influences with different independent variable. On the other
hand, the estimated results may be different with different threshold values. Besides,
as we divide the sample into subsample in this model, we may make a mistake of
sample selection because of setting different threshold values. Therefore, we still
apply quantile regression to analyze our results.
Suppose yi shows default points of different companies, and denotes the
variable affect the default point, such as the sort-term liabilities and long-term
liabilities of companies. Here,  i presents the position of assets value of ith company
in all samples. ui is the error term under different quantile point. Thus, we set the
quantile regression model as follows:
yi  xi'  u i


Quant  yi xi   inf y : Fi  y x   xi'
(7)
Quant  u i xi   0
m i
n   yi  xi' 
n
i 1
 u
  u   
  1u
i fu 
i uf
(8)
0
0
 is a coefficient given a quantile point  , Quant  yi xi  denotes the
estimated equation of quantile regression given a quantile point  . Fi  y x  is a
conditional distribution given x . If the error term ui in equation (7) is positive, it
presents the position of assets value of ith firm is above the conditional quantile value
of dependent variable. Instead, if the error term ui in equation (7) is negative, it
presents the position of assets value of ith firm is under the conditional quantile value
of dependent variable. We take the quantile point which is between zero and one as
the weighted coefficient in the quantile regression to distinguish the position in the
dependent variable. In this article, it denotes different default point of different
companies.
Therefore, we only have to take equation (4) into xi in equation (7) or equation
(8); we can get the coefficient of short-term liabilities and long-term liabilities of
companies. Accordingly, this becomes another new definition of default point which
different from the threshold regression model to measure the default probability of the
listed companies in Taiwan.
4. Empirical results
4.1 Data and Variables
The subjects of the research are all listed companies in Taiwan excluding
financial companies, and the sample period is during January 1, 1998 to December 31,
2007. The data are obtained in the database of Taiwan Economics Journal (TEJ). In
fact the definitions of default companies from TEJ are many kinds. For instance,
check bounce and running on a bank, closing down and bankruptcy, doubts about
continuing operations, restructure, bailing out and asking for support, taking over,
delisting of stocks requiring full delivery, financial distress and stoppage, and net
value is negative .
There were 161 defaulted companies and 1392 normal companies (total 1553)
our sample period of time. Here, we divided them by industries, all sample companies
were distributed across 27 industries. Among them, the number of default building
material and construction companies was on the top of list, total 19 companies;
followed by electronic component industry, 17 companies, food industry and
opto-electronic industry, 13 companies. The food industry occupied the largest share
in the total samples by 37.14%; followed by glass and ceramic industry 30% and
building material and construction industry 26.03%.
If we analyze it by the of time, from Table 1, we could find that the number of
default companies in 2005 occupied the largest share by 29 companies (18.24%),
followed by 28 companies (17.61%) in 2001 and 23 companies (14.47%) in 2004. In
2007, 1998 and 1999, there were only two companies (1.26%), three companies
(1.89%) and nine companies (5.66%) respectively.
Table 1: The Number of Default Companies from 1988 to 2007
1998
Number of Default
Companies
3
1999
9
5.59%
2000
22
13.66%
2001
28
17.39%
2002
10
6.21%
2003
18
11.18%
2004
23
14.29%
2005
29
18.01%
2006
15
9.32%
2007
4
2.48%
Total
161
100%
Year
Percentage
1.86%
Source：Reorganized and Compiled by the Author
4.2 The adjusted definition of default point in threshold regression model
According to Hansen (1999, 2000), we find five indices such as operating
capability, financial liquidity, business size, solvency, financial leverage as the
threshold point separately by the minimum MSE in regression. First, we get the asset
value, short-term liability and long-term liability of firms which is defaulted in
Taiwan, and sorted by the seventeen threshold variables from the five indices. Second,
we will divide all samples into two subsamples by each of the threshold point. In this
way, we have the asset values of defaulted firms as independent variable, and the
short-term liabilities and long-term liabilities as dependent variable. Finally, we can
estimate the coefficient of short-term liabilities and long-term liabilities in each
subsample. Here, we take it as the new definition of default in our research.
We find that only account receivable turnover rate, current ratio, ratio of
short-term liabilities to size (1), ratio of short-term liabilities to size (2), business size
(1), business size (2), time interest earned, financial leverage, debt ratio and
short-term liabilities exist threshold effect significantly. The adjusted definition of
default point show as follow.
Table 2: The adjusted definition of default by threshold regression
Threshold variable
Account Receivable
Turnover Rate
Current ratio
Quick ratio
Short-term
liabilities/sizes(1)
Short-term
liabilities/ sizes (2)
Business sizes (1)
Business sizes (2)
Cash flow ratio
Time interest
earned
separation
Formula of default point
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
DP＝1.885(short-term liabilities)＋1.068
(long-term liabilities)
DP＝1.4136(short-term liabilities)＋0.9495
(long-term liabilities)
DP＝1.4594(short-term liabilities)＋0.8987
(long-term liabilities)
DP＝1.7714(short-term liabilities)＋1.9972
(long-term liabilities)
DP＝1.4435(short-term liabilities)＋1.0364
(long-term liabilities)
DP＝2.8754(short-term liabilities)＋
(-0.7538) (long-term liabilities)
DP＝2.2153(short-term liabilities)＋1.1594
(long-term liabilities)
DP＝1.5329(short-term liabilities)＋0.988
(long-term liabilities)
Threshold
value
5.77%
149.65%
51.36%
0.3139%
DP＝2.3706(short-term liabilities)
0.3084%
DP＝1,5274(short-term liabilities)＋0.9764
(long-term liabilities)
DP＝1.0734(short-term liabilities)＋2.9721
(long-term liabilities)
17.95hundre
DP＝1.4288(short-term liabilities)＋0.7808 d million
(long-term liabilities)
DP＝1.2603(short-term liabilities)＋2.288
20.93
(long-term liabilities)
hundred
DP＝1.436(short-term liabilities)＋0.8218
million
(long-term liabilities)
DP＝2.9228(short-term liabilities)＋
(-0.9589) (long-term liabilities)
-7.51%
DP＝1.4101(short-term liabilities)＋1.0537
(long-term liabilities)
DP＝1.8477(short-term liabilities)＋1.0865
(long-term liabilities)
-2.73 倍
DP＝1.3504(short-term liabilities)＋1.0315
(long-term liabilities)
Financial leverage
Debt ratio
Short-term
liabilities
Total debt
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
Lower than the
threshold
Higher than the
threshold
DP＝1.3862(short-term liabilities)＋0.8096
(long-term liabilities)
22.92%
DP＝1.5686(short-term liabilities)＋0.9483
(long-term liabilities)
DP＝2.3204(short-term liabilities)＋2.4158
(long-term liabilities)
47.15%
DP＝1.4819(short-term liabilities)＋0.932
(long-term liabilities)
DP＝1.9387(short-term liabilities)＋1.4134
(long-term liabilities)
1.21 hundred
million
DP＝1.5365(short-term liabilities)＋0.9726
(long-term liabilities)
DP＝1.3993(short-term liabilities)＋1.9914
18.55
(long-term liabilities)
hundred
million
DP＝1.2849(short-term liabilities)
Source：Reorganized and Compiled by the Author
4.3 The adjusted definition of default point in quantile regression model
We adopt the quantile regression model in Koenker and Baset (1987) to estimate
default point in order to avoid the selection bias which may be caused by the
threshold regression model. We investigate the fitness default point of the listed
companies in Taiwan through different quantile position of dependent variable.
According to the empirical results of quantile regression, we find the definition of
default point exists structural change only when the asset value of firm is higher than
11.5 hundred million. As far as all firms are concerned, this threshold is much lower.
It is because the stock price of default firms dropped immediately to reflect the default
event, the asset values of firms become lower consequently. Here, we especially
examine the influences of short-term liabilities, long-term liabilities of firms
individually because total debt which is inclusive of short-term liabilities, long-term
liabilities and other debt. Therefore, we show the estimation of quantile regression in
Table 3:
Table 3: the definition of default point under different levels in quantile regression
Variables in quantile
regression
Long-term Short-term
Constant
liabilities liabilities
coefficient 0.02
1.00
1.01
p-value
0.99
0.99
0.29
coefficient 0.04
1.05
1.02
p-value
0.97
0.83
0.28
coefficient 0.04
1.09
0.99
P-value
0.97
0.74
0.31
coefficient 0.03
1.10
1.06
p-value
0.98
0.68
0.24
coefficient 0.04
1.10
1.24
p-value
0.97
0.68
0.12
coefficient 0.03
1.15
1.22
p-value
0.97
0.54
0.13
coefficient 0.05
1.17
1.21
p-value
0.96
0.50
0.13
coefficient 0.04
1.19
1.17
p-value
0.97
0.31
0.16
coefficient 0.09
1.17
1.16
p-value
0.93
0.28
0.16
coefficient 0.11
1.17
1.14
p-value
0.91
0.23
0.17
Quantile
statistics
level
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Variables in quantile
regression
DW
Long-term Short-term
Constant
liabilities liabilities
coefficient 0.06
1.26
1.06
0.55
1.97
p-value
0.95
0.18
0.23
coefficient 0.11
1.36
1.06
0.60
1.97
p-value
0.91
0.14
0.23
coefficient 0.20
1.48
1.15
0.65
1.98
p-value
0.84
0.21
0.16
coefficient 0.33
1.47
1.20
0.70
1.98
p-value
0.74
0.09*
0.13
coefficient 0.40
1.56
0.98
0.75
1.98
p-value
0.68
0.09*
0.30
coefficient 0.63
1.62
1.06
0.80
1.98
p-value
0.52
0.06*
0.23
coefficient 1.02
1.67
0.95
0.85
1.98
p-value
0.30
0.04**
0.34
coefficient 0.56
1.97
1.47
0.90
1.90
p-value
0.58 <0.01*** 0.04**
coefficient 1.82
2.37
1.23
0.95
1.72
p-value
0.10 <0.01***
0.17
notes：***、**and*are separately denotes the
coefficientthat is different from 10%、5%以及 1%
Quantile
DW
statistics
level
1.89
1.90
1.91
1.92
1.93
1.94
1.94
1.95
1.95
1.96
Source：Reorganized and Compiled by the Author
From Table 3, we find when the asset values of firm is higher (the quantile point
is higher than 0.85), the coefficient of the short-term liabilities to default point is
significantly different from the original one in Moody’s KMV credit risk model.
Obviously, there exists some improvement space to set the default point in the original
Moody’s KMV credit risk model. In order to define the formula of default point under
different asset values level exactly. Besides, we consider the differences between
independent variable and dependent variable under different position of dependent
variable.
We sorted all defaulted firms by the asset values from small to large, and take
use of imputation method to get asset value which corresponding to the 85% in the
sequence. Finally, we get the threshold point which is 11.5 hundred million, and
divided the defaulted firms into two subsamples according to the asset values. Then,
we use the traditional regression model to estimate this coefficient of the two
subsamples individually, and redefine the default point.
Table 4: the coefficient of long-term and short-term liabilities to defaulted point under different
asset value
Low asset value
High asset value
short-term liabilities
long-term liabilitied
1.2248
0.043**
1.4584
<0.0001***
coefficient
P-value
R2
Adjusted R2
0.8519
0.8497
short-term liabilities
long-term liabilitied
1.2919
0.036**
0.7097
0.1375
coefficient
P-value
R2
Adjusted R2
0.8322
0.8155
notes：***, **and*are separately denotes the coefficient that is different from 10%, 5%, and1%.
The coefficient of debt is 0.5.
Source：Reorganized and Compiled by the Author
After getting the coefficient of long-term liabilities and short0term liabilities, we
redefined the new default point. Put the new definition of default point into Moody’s
KMV credit risk model, we can get the default probability of all firms which is
inclusive of default and non-default. Besides, we compare this new definition to the
original ones (coefficient of short-term liabilities is 1, and the coefficient of long-term
liabilities is 0.5) in the Moody’s KMV credit risk model and the default probability
which is estimated by the threshold model regression. Here, we use cumulative
accuracy profiles curve (hereafter is CAP curve) and receiver operating characteristic
curve (hereafter is ROC curve) to analyze.
4.4 Efficiency and predictability of the adjusted default point
We have to understand the theorem of ROC curve before verifying by ROC
curve. It takes the relationship of type Ⅰ error and type Ⅱerror to verify the efficiency
of the model. The type Ⅰ error means the firms actually defaulted but are considered
the normal firms from the estimation of model. However, The typeⅡ error is defined
the firms are normal but determined as a defaulted ones.
The application of CAP curve is to verify the predictability. Here we take the
proportion of the normal and default firms to draw the CAP curve and therefore
distinguish the effectiveness. After we match the whole samples by the model, we can
get a default probability and will sort this from large to small. Firstly, suppose we
calculate the proportion that defaulted firms exist in the select the preceding 5% of all
samples. Secondly, we select the preceding 10% of all samples and repeat the above
steps. Finally, we draw the percentage on the cross axle and also draw the proportion
of the defaulted firms to the selected samples on the vertical axle. Thus, we can get
the CAP curve and calculate the area under the CAP curve.
Table 5: the ROC comparison of the adjusted model and original Moody’s KMV
year
Moody’s KMV
Account
Receivable
turnover rate
Thres Current ratio
hold Short-term
regres liabilities/sizes(1)
sion Short-term
model liabilities/sizes(2)
Business sizes (1)
Business sizes (2)
Time interest
1998
0.4235
1999
0.4302
2000
0.4637
2001
0.4297
2002
0.4066
2003
0.4404
2004
0.4537
2005
0.5656
2006
2007
0.5441 0.4682
average
0.4626
0.3444
0.7061
0.4513
0.4709
0.4097
0.5880
0.4875
0.6776
0.6932 0.8818
0.5711
0.3370
0.6924
0.4751
0.4585
0.4088
0.4604
0.3728
0.6340
0.6554 0.5888
0.5083
0.5679
0.6364
0.4610
0.4556
0.5393
0.5078
0.5521
0.6938
0.6349 0.8637
0.5913
0.5759
0.6068
0.4653
0.4472
0.5480
0.4972
0.5523
0.6835
0.6465 0.8606
0.5883
0.3765
0.3691
0.3605
0.3710
0.3668
0.3636
0.4913
0.4828
0.4747
0.4821
0.4727
0.4734
0.4495
0.4925
0.4713
0.4349
0.4749
0.5410
0.4415
0.4242
0.4257
0.6152
0.6326
0.6958
0.5803 0.6304
0.6733 0.6319
0.7360 0.8615
0.4873
0.5021
0.5404
earned
Financial
leverage
Short-term
liabilities
Debt ratio
Quantile regression
model
0.5426
0.7907
0.5176
0.5257
0.4238
0.5043
0.4033
0.6573
0.7017 0.8453
0.5912
0.5537
0.6734
0.4737
0.4601
0.4664
0.5435
0.5473
0.7033
0.6573 0.8777
0.5956
0.2827
0.5677
0.4459
0.4827
0.2817
0.3935
0.3108
0.5902
0.6322 0.7264
0.4714
0.3889
0.3816
0.4924
0.4388
0.4169
0.4436
0.3928
0.6132
0.6593 0.8779
0.5105
Source：Reorganized and Compiled by the Author
From the examination of ROC in table 5, the accuracy ratio of the most of the
variables which is adjusted are larger than the original Moody’s KMV credit risk
model. But the debt ratio is an exception which means we can not find significantly
that the predictability and efficiency are better than the original Moody’s KMV credit
risk model under the ROC examination. However, as far as the great majority is
concerned, the predictability and efficiency are superior to the original Moody’s KMV
credit risk model in estimating default probability. In other words, no matter what we
adopt the adjusted definition of defaulted point by Hansen (1992) or Koenker and
Basset (1978), the result would be constant.
But if we only consider the predictability of the credit risk model with adjusted
default point, we can not conclude that the new Moody’s KMV credit risk model is
better than the original ones. Therefore, we continuously compare the efficiency by
CAP curve, and the results are showed in Table 6:
Table 6: the CAP comparison of the adjusted model and original Moody’s KMV
year
1998
Moody’s KMV
Account
Receivable
Thresh turnover rate
old
Current ratio
regressi
Short-term
on
liabilities/sizes(1)
model
Short-term
liabilities/sizes(2)
1999
2000
2001
2002
2003
2004
2005
2006
averag
e
0.7015 0.5081
0.5842
0.8749
2007
0.3750
0.5750
0.4083
0.3231
0.4813
0.4517
0.4063
0.6303
0.7288
0.4000
0.6750
0.4717
0.4212
0.4563
0.6283
0.4875
0.7039
0.7231
0.3750 0.7250
0.4767
0.4019
0.5000
0.5017
0.3875
0.6645
0.6673
0.4750
0.6750
0.4933
0.4462
0.5625
0.5450
0.5500
0.7184
0.6769
0.7756 0.5475
0.6018
0.8753
0.5000
0.6250
0.4933
0.4423
0.5688
0.5417
0.5750
0.7250
0.6769
0.8509
0.5999
Business sizes (1)
Business sizes (2)
Time interest
earned
Financial leverage
Short-term
liabilities
Debt ratio
Quantile regression model
0.3000
0.3750
0.4750
0.6250
0.4950
0.4967
0.4500
0.4346
0.5063
0.5250
0.5150
0.5350
0.4625
0.4750
0.6776
0.6855
0.6750
0.6846
0.8082 0.5365
0.8031 0.5640
0.6057
0.8506
0.4250
0.7250
0.5067
0.4115
0.5438
0.5850
0.5000
0.7382
0.7712
0.4250
0.7750
0.5067
0.4519
0.4813
0.5583
0.4688
0.6816
0.6923
0.4500
0.6750
0.5067
0.4385
0.5438
0.5683
0.5313
0.7197
0.6846
0.8439 0.5885
0.5993
0.8750
0.3000
0.3750
0.5750
0.6250
0.4467
0.4633
0.4731
0.3885
0.3375
0.4875
0.4350
0.5150
0.3063
0.4375
0.5776
0.6592
0.6173
0.6981
0.7138 0.4782
0.9085 0.5558
Source：Reorganized and Compiled by the Author
We take the listed companies in Taiwan as our samples, and we both use the
night variables as the threshold variable, and quartile regression to modify the original
Moody's KMV credit risk model. According to the empirical results, there are better
improvement in predictability and efficiency when we estimate the default probability
with the two models. We conclude that mat because the development of accounting
system and legal rules are getting mature, and the financial report becomes clear to
reflect the real condition of firms. Therefore, the stock price can correctly respond the
public information, so that stock market becomes an efficient one in the recent year.
These developments can help us to increase the predictability when utilize the
financial data and market price to estimate the Moody’s KMV credit risk model.
Eventually, in order to compare the predictability and efficiency in different models,
we compare the ROC, CAP of threshold regression, quantile regression and original
Moody’s KMV credit risk model to analyze.
We find the ROC and the CAP of threshold regression model are more accurate
than quantile regression and Moody’s KMV credit risk model. In a word, we conclude
that the predictability of threshold regression model is more powerful than quantile
regression when we measure the credit risk of the listed companies in Taiwan.
4.5 The statistic inference of default probability after modifying the model
In most of the relative literatures, they estimate the default probability by only
correcting the definition of default point without statistic inference of default
probability. Thus, we adopt the method established in Hansen and Schuermann (2006)
to calculate the confidence level of default probability. Besides, we also examine
whether the default probability is significantly different from zero in our sample
period?
In order to proceed with the statistic inference, we apply bootstrap method to
calculate the original Moody’s KMV credit risk model, threshold regression model
and quantile regression model by selecting 1000 random samples, and repeat the step
10,000 times. By means of bootstrapping, we can get the statistic inference such as
average value, standard deviation of default probability, and then measure the 95%
confidence interval. The results are showed in table 7 to 9:
Table 7: the statistic inference of default probability in Moody’s KMV credit risk model
Non-Parameteric
Paramteric
Moody’s KMV
Standard
Standard
Average
Z-value
Average
Z-value
deviation
deviation
1998
0.0141
0.1173
4.7279
0.0141
0.0892
6.2305
1999
0.0168
0.1282
5.1780
0.0168
0.1013
6.5580
2000
0.0208
0.1424
5.7667
0.0208
0.1111
7.3952
2001
0.0422
0.2007
8.2899
0.0422
0.1590
10.4735
2002
0.0408
0.1974
8.1440
0.0408
0.1594
10.0936
2003
0.0424
0.2012
8.3085
0.0424
0.1555
10.7536
2004
0.0221
0.1468
5.9508
0.0221
0.1128
7.7389
2005
0.0214
0.1443
5.8426
0.0214
0.1022
8.2400
2006
0.0189
0.1359
5.4914
0.0189
0.1020
7.2975
2007
0.0220
0.1463
5.9270
0.0220
0.1102
7.8605
Source：Reorganized and Compiled by the Author
Table 8: the statistic inference of default probability in threshold regression model
Non-Parameteric
Paramteric
Moody’s KMV
Standard
Standard
Average
Z-value
Average
Z-value
deviation
deviation
1998
0.0385
0.1921
7.9011
0.0385
0.1348
11.2604
1999
0.0525
0.2227
9.2915
0.0525
0.1592
13.0060
2000
0.0788
0.2693
11.5492
0.0788
0.1954
15.9240
2001
0.1450
0.3519
16.2538
0.1451
0.2650
21.6101
2002
0.1494
0.3564
16.5430
0.1495
0.2602
22.6756
2003
0.1624
0.3688
17.3827
0.1626
0.2595
24.7239
2004
0.1259
0.3316
14.9788
0.1260
0.2257
22.0341
2005
0.1534
0.3603
16.8010
0.1535
0.2415
25.0751
2006
0.1626
0.3689
17.3896
0.1627
0.2534
25.3357
2007
0.1335
0.3400
15.4926
0.1336
0.2319
22.7346
Source：Reorganized and Compiled by the Author
Table 9: the statistic inference of default probability in quantile regression model
Non-Parameteric
Paramteric
Moody’s KMV
Standard
Standard
Average
Z-value
Average
Z-value
deviation
deviation
1998
0.0350
0.1835
7.5222
0.0350
0.1393
9.9106
1999
0.0464
0.2101
8.7132
0.0466
0.1625
11.3070
2000
0.0683
0.2521
10.6904
0.0683
0.1953
13.8011
2001
0.1287
0.3347
15.1696
0.1288
0.2696
18.8485
2002
0.1306
0.3368
15.2977
0.1306
0.2613
19.7205
2003
0.1404
0.3473
15.9553
0.1402
0.2594
21.3238
2004
0.1032
0.3041
13.3903
0.1031
0.2221
18.3127
2005
0.1241
0.3295
14.8557
0.1241
0.2367
20.6843
2006
0.1321
0.3384
15.3978
0.1321
0.2497
20.8689
2007
0.1069
0.3088
13.6575
0.1071
0.2266
18.6443
Source：Reorganized and Compiled by the Author
Although the default probability we estimated in the modified model or the
Moody’s KMV credit risk model is very small, the average default probability each
year is significantly different from zero in 1998 to 2007 from the results. In other
words, this outcome appears that the listed companies in Taiwan will still default and
it reveals the importance of credit risk management.
5. Conclusion
Our empirical analysis is based on 161 listed companies in the Taiwan stock
market, which are qualified for the definition of default in TEJ database. The sample
period was between 1998 and 2007. This paper uses five major factors (operating
ability, financial liquidity, business size, solvency, and financial leverage) as proxy
variables of the companies’ characteristics. Meanwhile, we also use total 17 threshold
variables to modified Moody’s KMV credit risk model and calculate the default
probability of 1551 companies in the Taiwan stock market. Then, we compare the
result obtained from our modified model to from the original Moody’s KMV model.
Through the threshold regression model, we find that there are ten threshold
variables (the account receivable turnover ratio, the current ratio, the short-term
debt-to-size (1), the short-term debt-to-size (2), the Business size (1), the Business
size (2), the time interest earned, the financial leverage, the short-term liabiliti and the
debt ratio) exist significant threshold effect on the default point of listed companies in
the Taiwan stock market. In another word, the definition of default point is not linear
relationship as described in Moody’s KMV credit risk model. Accordingly, we have
sufficient evidence that the original Moody’s KMV credit risk model is not suitable
for the listed companies in the Taiwan stock market.
We use the modified definition of default point obtained by the empirical results
of threshold regression model to estimate the “modified” default probability.
Meanwhile, from the point of quantile regression analysis, we use short-term
liability and long-term liability to analyze the asset value and then find that the
coefficients of long-term liability and short-term liability, as well as the company’s
market asset value are not totally linear relation. From the results, we find that under
the 5% of significant level, we have sufficient evidence to believe that the coefficient
of short-term liability is not equal to 1, which proves what this paper mentions.
Companies’ characteristics and asset market value lead to structural changes in the
definition of default point in Moody’s KMV credit risk model.
At the end, we also use CAP curve and ROC curve to examine the accuracy and
the efficiency of the modified Moody’s KMV default probability. It appears that the
modified default point performs better than the original Moody’s KMV credit risk
model. Compared the threshold regression model and the quantile regression model,
the threshold regression model has better predictability of credit risk, when we test the
listed companies in Taiwan stock market.
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