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Journal of Applied Mathematics and Physics, 2013, 1, 12-17
Published Online November 2013 (http://www.scirp.org/journal/jamp)
Open Access JAMP
Research on Credit Risk Measurement
Based on Uncertain KMV Model
Ni Zhan, Liang Lin, Ting Lou
College of Science, Guilin University of Technology, Guilin, China
Received May 25, 2013; revised June 25, 2013; accepted July 15, 2013
Copyright © 2013 Ni Zhan et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Regarding KMV model identification credit risk profile of small and medium-sized listed companies, at present, do-
mestic scholars has made some achievements in the process of the KMV model combined with China’s national condi-
tions. In this paper, we will amend the model by using uncertain in terest rate instead of fixed rate on the basis of exist-
ing research. Comparing the uncertain KMV model to traditional KMV model with ST-listed companies and non-
ST-listed companies in Shanghai and Shenzh en stock exchange, we find that it performs slightly be tter as a predictor in
uncertain KMV model and in out of sample forecasts.
Keywords: Credit Risks; KMV Model; Uncertain Interest Rate
With the improvement of the activeness in economic
activities and enhancement of national economic rela-
tionships between countries, people found that it’s im-
portant and necessary to be on guard and control the
various risks in economic activities. Barings Bank event
as well as a series of major financial institutions crisis
since the mid-1990 s of the 20th century, makes the Banks
increasingly concerned about the risk prevention, meas-
urement and management. Nowadays commercial banks
face two kinds of risk, one is non-systemic risk, such as
credit risk, settlement risk, and th e other is systemic risk,
such as market risk, interest rate risk, currency risk and
so on, and credit risk is the main risk in that commercial
banks face in the course of business. And how to guard
against and reduce credit risk is an urgent requirement to
the commercial banks.
W. Zhang  carried out a theoretical study of the
credit risk of listed companies measurement and ana-
lyzed the samples of China’s listed companies. Research
shows that at this stage, KMV model based on options
pricing breach theory could well apply to China’s credit
risk measurement of listed companies. Z. J. Zhang & X.
H. Chen  set two credit warning lines to monitor the
credit crisis of small and medium-sized listed companies
by KMV model. X. Y. Liang  made the empirical
analysis of KMV model widely used in foreign countries
combined with the actual situation in China, and they
amended parameters accordingly in order to enhance the
applicability of the model in the Chinese market. S. C.
Yang  used qualitative and demonstration method to
carry on the theoretical analysis and empirical research
on credit risk measure model, furthermore, he made
analysis on KMV model used in Chinese enterprise with
newest data o f corporat ion in sto ck market .
Most of the existing credit risk measurement methods
are “static” models by using risk-free rate to estimate the
uncertain risk. As we all know, interest rate in real life is
uncertain. In order to reflect and solve practical problems
better, we will introduce the uncertain variable into the
research of credit risk.
2. KMV Model Theory
2.1. The Basic Assumptions of the Model
KMV model basic idea is that viewed the company eq-
uity as a European call options, viewed market value of
assets as the subject, the nominal value of company’s
debt as exercise price. Compan y will choose to repay the
debt when market value of assets is greater than the value
of the debt, default while market value of assets is less
than the value of debt .
1) Satisfy the basic assumptions of the Merton model:
the company’s stock price is a random process, transac-
N. ZHAN ET AL. 13
tion is no friction, etc., and the change of enterprise’s
value obeys Ito Process.
2) Company will default to its creditors and share-
holders when the company’s asset value falls below a
3) The capital structure of the borrower only includes
owner’s equity, short-term debt, long-term debt and con-
vertible prefe rr ed st ock .
2.2. The KMV Calculated Steps
1) The Asset Value and Volatility
V represents market value of equity, A repre-
sents firm’s asset value, then the relationship between the
companies’ equity and asset value can be expressed as
where represents book value of debt which is due at
time , is the risk free interest rate,
·N is the
cumulative standard normal distribution function,
and are given by 1
Equity volatility and asset volatility relationship:
is equity volatility,
The asset value and volatility implied by the equity
value, equity volatility and liabilities are calculated by
solving the call price and volatili ty equ ation s, (1) and (4),
is asset volatility.
2) Default to Distance
The default point (DPT) is generally determined by the
short-term liabilities and long-term liabilities:
Distance to default (DD) is the relative distance be-
tween the future value of the assets and default point.
Once this numerical solution is obtained, the distance to
default can be calculated as
3) EDF (Expected Default Frequency)
The theoretical value of expected default frequency is
based on the assumption that market value of the assets
follows a normal distribution,
EDF DDN (7)
It still lacks large amounts of data resources to estab-
lish an effective database in China, so the empirical sec-
tion, we select the distance to default to measure credit
risk. The greater the distance to default, the smaller the
likelihood of default, the comp any’s credit status is better;
on the contrary, the smaller the distance to default, the
higher the risk of de faul t .
The vast majority of scholars simply assumed r is a
free-rate and did not give a reasonable explanation, but
we all know interest rate in real life is uncertain, we will
introduce the uncertain variable into the research of
3. Uncertainty KMV Model
Uncertainty theory was founded by Liu  in 2007 and
refined by Liu  in 2010, which is a branch of mathe-
matics based on normality, monotonicity, self-duality,
countable subadditivity, and product measure axioms.
3.1. Uncertain Measure and Uncertain Variable
be a nonempty set, and a
. Each element
is called an event. Uncertain
measure was introduced as a set function satisfying
the following five axioms :
Axiom 1. (Normality Axiom)
1for the uni-
Axiom 2. (Monotonicity Axiom)
Axiom 3. (Self-Duality Axiom)
for any event
Axiom 4. (Countable Subadditivity Axiom) For every
countable sequence of events
, we have
Axiom 5. (Product Measure Axiom) Let k be non-
empty sets on which k are uncertain measures
, respectively. Then the product uncertain
measure is an uncertain measure on the product
-algebra 12 n
Definition 3.1.1 (Liu ) Let be a nonempty set,
and be a
-algebra over , and an uncertain
measure. Then the triplet
,, is called an uncer-
Definition 3.1.2 (Liu ) An uncertain variable is a
measurable functio n from an uncertainty space
,, to the set of real numbers.
Open Access JAMP
N. ZHAN ET AL.
3.2. Uncertain Distribution and Inverse
Definition 3.2.1  The uncertainty distribution
an uncertain variable
is defined by
Definition 3.2.2 : An uncertain variable
called linear if it has a linear uncertainty distribution
denoted by where a and b are real numbers
with . Distribution shown in Figure 1.
Definition 3.2.3 : An uncertain variabe
normal if it has a normal uncertainty distribution
where e and
are real num-
bers with 0
. Distribution shown in Figure 2.
Definition 3.2.4 (Liu ) An uncertainty distribution
is call lognormal if
is a normal uncertain vari-
. In other words, a lognormal uncertain
variable has an uncertainty distribution
Figure 1. A linear uncertainty distribution.
xp 3, 0xx
, where and
numbers wit h 0
. Distribution shown in Fure 3.
Definition 3Liu )An uncertainty distributio
is said to be regular if its inverse function
sts and is unique for each
As we all know, the firm’slue is a asset vad
nction of interest r which is adjusted irregularly .
Let r be an uncertain variable with regular lognormal
.Then the inverse function
is called the inversuncertainty distribution ofr.
sy to verify that the inverse uncertainty distributi
of lognormal uncertain variable is:
4. Numerical Examples
nd *ST-listed companies,
In this paper, we choose 6 ST a
which are added ST and *ST in 2011, and 6 non-ST-
listed companies in Shanghai and Shenzhen stock ex-
change as empirical research samples, in which ST and
*ST companies are considered as default samples while
non-ST companies are considered as normal samples.
Data is from http://finance.sina.com.cn/.
After finishing the row data, we ge
4.1. Traditional KMV Model
% at the end of 2011,
own in Table 1. Compare the traditional V mo
which uses a fixed interest rate and uncertain KMV
model by using the above data.
We select deposit interest rate-3.5
into formulas (1)-(4), and obtain each
listempanmarket value of asset and asset volatil-
ity by the iterate program of MATLAB as shown in Ta-
Figure 3. A lognormal uncertainty distribution.
Figure 2. A normal uncertainty distribution.
Open Access JAMP
N. ZHAN ET AL.
Open Access JAMP
ble 2t to
art of the *ST and ST
4.2. Uncertain KMV Model
h normal uncertainty
imum interest rate?
ac o you think is the maximum interest rate?
ac o you think is a likely interest rate? , 0.75)
is the belief degree that the real interest rate
is pert’s experimental data (0.04, 0.8) is
ac is the belief degree that the real interest rate
. Put the data in Table 2 into (6) obtain defaulis less than 0.045?
distance, as shown in Table 3.
From Table 3, we find that pA5: 0.9. (an expert’s experimental data (0.045, 0.9) is
mpanies’ DD are bigger than non-ST companies which
may lead to an evaluation error. Where (0.02, 0.3), (0.035, 0.75), (0.04, 0.8), (0.04 5,
0.9), (0.05, 1) represents
Let r be an uncertain variable wit
distribution and can be obtain by Delphi method. The
consultation process is as follows:
Q1: What do you think is the minSo we obtain the inverse uncertainty distribution of
lognormal uncertain variable by using expert’s experi-
A1: 0.02. (an expert’s experimental data (0.02, 0.3) is
Q2: What d
A2: 0.05. (an expert’s experimental data (0.05, 1) is
Q3: What dWe get DD by using uncertainty interest rate shown in
A3: 0.03. (an expert’s experimental data (0.035
Q4: What isCompare DD1 and DD2 in the same drawing.
From Figure 4, we can find that, for *ST and ST com-
panies, calculations of uncertain KMV model are smaller
than traditional KMV model’s; for non-ST-companies,
calculations of uncertain KMV model are bigger than
traditional KMV model’s. By this example, it is verified
less than 0.04?
A4: 0.8. (an ex
Table 1. Equity value and equity volatility.
V (Ten thousand) Equity volatility
Debt D (Ten thousand)
*ST Yuancheng 94.10000 6 0.40924 47636.68
*ST Zhongda 302186.68 0.45036 2441
*ST Tianrun 123571.38 0.52767 27944.7
ST Guofa 178142.36 0.44814 8.76
Luodun 229618.29 0.58704 56683.21
ST Tianyi 103130.9 0.41686 41352.81
Jinma 240603 0.42448 63062.24
Kaile 364071.6 0.38353 181406.34
Batian 194009.26 0.53973 12566.91
Jiahua 1358366 0.3553 60442.38
henghua194814 0.47719 47517.91
ST Company (Ten thousand)
Table 2. Market value of asset and asset volatility.
Non-ST Company (Ten thousand)
*ST Yuancheng 148305.98 0.27410.333831 Jinma 304425.81 59
*ST Zhongda 549348.01 0.245261 Lianchuang
ST 1419 0.
*ST Tianrun 151854.66 0.427553 Kaile 547666.55 0.252987
ST Guofa 226851.81 0.350167 206727.72 0.5058
Luodun 287007.76 0.467563 Jiahua 537. 27 339649
ST Tianyi 144982.83 0.294551 242905.55 0.380962
N. ZHAN ET AL.
Taault to di
ST Company DDmpany DD1
ble 3. Defstance.
1 Non-ST Co
*2.8 2.1 ST Yuancheng 47616Jinma 37480
*ST Zhongda 2.265421 Lig
*ST Tianrun 1.90848 Kaile 2.643471
ST Guofa 2.249896 Batian 1.856881
ST Luodun 1.716353 Jiahua 2.818858
ST Tianyi 2.426656 2.111437
Tabllt to distance of uncertainV.
ST Company DD2
e 4. Defauty KM
DD2 Non-ST Company
*2.8 2.9 ST Yuancheng 19683Jinma 84075
*ST Zhongda 1.935192 Li
*ST Tianrun 1.678772 Kaile 3.087181
ST Guofa 2.010186 Bati an 2.352168
ST Luodun 1.48924 Jiahua 3.314398
ST Tianyi 2.163204 2.588359
Figure 4. Comparation of DD1 and DD2.
e model is flexib le and consistent with
f the two models above, we found
ained by Delphi method or assumed to
 W. Zhang, “A Study on Listed Company’s Credit Risk
th the actual situa-butions were obt
By the comparison o
that the uncertain KMV model can distinguish ST and
non-ST-companies better than traditional KMV model.
Interest rate changes such as the uncertainty factor in the
economic system, belong to the systemic risk which
cannot be dispersed.
In this paper, an uncertain variable method was pro-
posed to generate interest rate and the uncertainty distri-
be lognormal distribution. However, we will still con-
tinue to research on uncertainty interest rate.
Based on KMV Model,”
 Z. J. Zhang, X. H. Chen and F. Q. Wang, “Research on
Credit Risk of Small and Medium-Sized Listed Compa-
nies in China Based on KMV Model,” Journal of Finance
and Economics, Vol. 33 No. 1, 2007.
 X. Y. Liang, “A Study on Medium-Sized Listed Compa-
Open Access JAMP
N. ZHAN ET AL. 17
nies’ Credit Risk Measurement Based on KMV M
ethods and KMV Model,” 2010. J. Zhou, “Structure of Default Risk, Interest
odel,” for Modeling Human Uncertainty,” Springer-Verlag, Ber-
 S. Zeng and
 S. C. Yang, “Empirical Study on Credit Risk Measure-
 B. Liu, “Uncertainty Theory,” 2nd Edition, Springer-Ver-
lag, Berlin, 2007.
 B. Liu, “Uncertainty Theory: A Branch of Mathematics
Rate Fluctuations and Debt maturity—Interest Rate’s
Feedback Effects of Corporate Debt Financing,” Journal
of Jiangxi University of Finance and Ecomomics, No. 3,
Open Access JAMP