Research on Credit Risk Measurement Based on Uncertain KMV Model

Regarding KMV model identification credit risk profile of small and medium-sized listed companies, at present, domestic scholars has made some achievements in the process of the KMV model combined with China’s national conditions. In this paper, we will amend the model by using uncertain interest rate instead of fixed rate on the basis of existing research. Comparing the uncertain KMV model to traditional KMV model with ST-listed companies and nonST-listed companies in Shanghai and Shenzhen stock exchange, we find that it performs slightly better as a predictor in uncertain KMV model and in out of sample forecasts.


Introduction
With the improvement of the activeness in economic activities and enhancement of national economic relationships between countries, people found that it's important 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-1990s of the 20 th century, makes the Banks increasingly concerned about the risk prevention, measurement and management.Nowadays commercial banks face two kinds of risk, one is non-systemic risk, such as credit risk, settlement risk, and the 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 [1] carried out a theoretical study of the credit risk of listed companies measurement and analyzed 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 [2] set two credit warning lines to monitor the credit crisis of small and medium-sized listed companies by KMV model.X. Y. Liang [3] 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 [4] 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 of corporation in stock 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.

The Basic Assumptions of the Model
KMV model basic idea is that viewed the company equity as a European call options, viewed market value of assets as the subject, the nominal value of company's debt as exercise price.Company 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-tion is no friction, etc., and the change of enterprise's value obeys Ito Process.
2) Company will default to its creditors and shareholders when the company's asset value falls below a certain level.
3) The capital structure of the borrower only includes owner's equity, short-term debt, long-term debt and convertible preferred stock.

1) The Asset Value and Volatility If E
V represents market value of equity, A represents 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, is the cumulative standard normal distribution function, and are given by Equity volatility and asset volatility relationship: where The asset value and volatility implied by the equity value, equity volatility and liabilities are calculated by solving the call price and volatility equations, (1) and ( 4), simultaneously.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 between 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, It still lacks large amounts of data resources to establish an effective database in China, so the empirical section, we select the distance to default to measure credit risk.The greater the distance to default, the smaller the likelihood of default, the company's credit status is better; on the contrary, the smaller the distance to default, the higher the risk of default.
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 credit risk.

Uncertainty KMV Model
Uncertainty theory was founded by Liu [5] in 2007 and refined by Liu [6] in 2010, which is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.

Uncertain Measure and Uncertain Variable
Let  be a nonempty set, and a   -algebra over  .Each element    is called an event.Uncertain measure was introduced as a set function satisfying the following five axioms [5]: Axiom 4. (Countable Subadditivity Axiom) For every countable sequence of events   i  , we have   Axiom 5. (Product Measure Axiom) Let k be nonempty sets on which k are uncertain measures , respectively.Then the product uncertain measure is an uncertain measure on the product where , 1,2, ,

 
, ,    to the set of real numbers.

Uncertain Distribution and Inverse Uncertain Distribution
Definition 3.2.1 [6] The uncertainty distribution  of an uncertain variable  is defined by
From Table 3, we find that p A5: 0.9.(an expert's experimental data (0.045, 0.9) is acquired)).mpanies' DD are bigger than non-ST companies which may lead to an evaluation error.

Co
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 proposed to generate interest rate and the uncertainty distri-be lognormal distribution.However, we will still continue to research on uncertainty interest rate.
is called an uncer- tainty space.Definition 3.1.2(Liu[5]) An uncertain variable is a measurable function from an uncertainty space

Definition 3 . 2 . 3 [ 6 ]
: An uncertain variabe  is called normal if it has a normal uncertainty distribution

 2 .Definition 3 . 2 . 4 (
where e and  are real num- bers with 0   .Distribution shown in Figure Liu [6]) An uncertainty distribution is call lognormal if  ln  is a normal uncertain variwords, a lognormal uncertain variable has an uncertainty distribution

Table 1 . Equity value and equity volatility.
respectively.