KMV model is one of the most important credit risk evaluation models in the world. It uses B-S option pricing and Morton formula based on the market value and volatility of the company’s equity, debt maturities, risk-free interest rates and the book value of liabilities to estimate the market value of the company’s assets and the volatility of the asset value. In this paper, based on the theory of KMV model, we can derive the listed company’s default rate, and assess credit risk. And the result is reasonable.
With the development of the world market economic globalization, financial globalization has swept through. Credit risk management draws more and more concern and attention of international financial institutions and other market participants. There are a lot of research about the identification, measurement, prediction and prevention of the credit risk internationally and the mature econometric models were established. Credit risk measurement methods are mainly two: the traditional measurement methods and modern measurement model. The traditional risk measurement approach focuses on qualitative analysis and modern metrics focus on quantitative analysis. With the post-crisis development of financial storm in 2008, credit risk is more complicated and the modern credit risk measurement model is more suitable for depicting the level of risk.
Chen Kai [
Yang [
Zeng and Wang [
Peng [
In this paper, at first, we respectively introduce the B-S optional pricing model, Merton model and KMV model. And then combining with the actual situation of China securities market, we conduct empirical study on credit risk of our listed company based on KMV model.
B-S model [
1) There is no friction in capital market, that is to say no transaction costs and taxes;
2) It has the risk-free interest rate;
3) The stock without dividends;
4) Share price follow normal distribution;
5) There is no risk-free arbitrage in the market;
Based on the above assumptions, there is
where S is stock price;
in which the expected value is 0 and the variance is 1.
B-S differential variance is derived by ITO lemma
Then we can get
where c is now option prices; S is the stock prices; r is the risk-free interest rate; T is the due time; t is the current time; X is executive price;
Merton [
First, it assume that the company’s assets value is follows geometric Brownian motion
The model presents the company’s equity value as the calls which set the company’s asset as subject matter (
Based on the above assumptions, T is the due date and D is the book value of debt. The call option value of the Debt maturity is
According to B-S model, at time t, the rights and interests of the company is
where
In the model,
The basic computing theory of the KMV model are on the base of Morton and B-S option pricing these two models and then according to the company’s share price information to analysis the company credit. Model assumes that when the market value of company assets is higher than the matured liability means not default behavior, on the contrary, if the asset value is lower than the amount of debt due payments, the company will choose to default. The main analysis tools of KMV model is expected default frequency (EDP), so it is also called EDP model.
1) Model is complying with the basic hypothesis of Merton option pricing model. That is to say, the stock prices of listed companies to obey random process and the trading is frictionless. And the value of listed companies change Process subject Ito Process.
2) It assumes that when the value of the assets of the listed company less than a certain level, the listed companies choose to default to its shareholders and creditors.
3) It assumes that the return on investment is fixed and known.
1) The market value of the assets and the value volatility
The value of the asset of company is assumed to subject to stochastic process:
The company’s equity value
where
The relationship between the company equity value and the market value of assets can be expressed as [
The relationship between the volatility of equity value and the volatility of asset market value is:
where
2) Distance to default DD
The default point is usually determined by short-term debt and 1/2 of long-term liabilities. Its mathematical definition is as follows:
It is known that the short-term debt of the company is STD and the long-term debt is LTD. So we define the default point as:
The calculation of the default distance can be expressed as
According to the standardized indicators of distance to default, different companies can be compared by the index, and the index can be used to measure the stand or fall of company’s credit situation. Therefore, distance to default can be used to as an indicator in the evaluation of company credit status.
Calculation of the default distance
3) Expected default frequency
The theoretical value of expected default frequency is concluded from the assumption that the market value of the assets follows the normal distribution. But it is unpractical obviously. So we observe some companies who have the same distance to default and record the actual number of defaults one year later. Expected default frequency is equal to the total number of companies who have the same distance to default divided by the number of companies who actual default. Formula is expressed as
The model estimation need to set a few parameters, most can directly get from the company’s financial statements or various trading software and a small number of parameters obtained by computing data. This article is based on data obtained from sina finance and economics website and everbright securities trading software.
1) The setting of the parameter E standing for the market value of Equity
The estimation method of the company equity value set by Traditional KMV model is not applicable to the listed company of our country. In view of the experience of forefathers’ research, this paper set the value of the company equity as the sum of the market value of the circulation stock and non-tradable shares. That is to say: E = E1 + E2.
The market value of circulation stock (E1) = the number of shares of circulation stock × the annual average closing price.
The market value of non-tradable shares (E2) = the number of shares of non-tradable shares × Net assets per share.
2) The setting of the parameter D standing for the Book value of debt
The Book value of debt is the sum of the Current Liabilities and the noncurrent liability.
3) The setting of the parameters standing for the volatility of equity value
There is more mature method to calculate the stock price volatility At home and abroad. This article uses the
The relationship between EDF and DD
historical volatility method to conduct calculation:
where
after day trading;
days of stock market in a year;
4) The setting of the parameters r standing for interest rate
Traditional model mainly choose the interest rates whose default risk is low, such as the one-year deposit rate and Treasury rates. Western countries usually use Treasury rates to replace the risk-free interest rate. But The Treasury bond market is not perfect in our country and the interest rate market is just open. Therefore in this section, we set the one-year deposit interest rate by the end of 2012 year as the risk-free interest rate, so r = 3%.
5) The setting of the parameters T standing for the debt maturity
In this section, we select a year for debt maturity.
6) The setting of the parameters DPT standing for default point
The classic formula of default point is: DPT = Current Liabilities + 0.5 × Long-term Liabilities. But the market value of the assets of our country is not consistent with them. The domestic most of the credit risk researchers on the selection of parameters are directly quoted all parameters of the original model, it reduce application accuracy of the KMV model in our country’s listed companies to a certain extent. we will set 10 different default points in This section and calculated to obtain the default point of KMV risk measurement model which accord with the situation of listed companies in our country. So the default point can be expressed as
where STD are current liabilities and STD are long-term liabilities. And
This paper randomly select real estate, biopharmaceutical, ceramic industry, food industry, hotel tourism, agriculture & farming, coal profession, transportation, cement industry and automobile production, etc., based on the sina finance and economics stocks in Shanghai and Shenzhen. We select a non-ST company and a ST company every industry. There are 20 companies in all. (http://finance.sina.com.cn/stock/. Description: ST company operate deficit two consecutive years and *ST company operate deficit three consecutive years). We collect capital stock, long-term liabilities, short-term liabilities, net assets per share and other financial data from different listed company. And after data processing, we calculate the market value of the liabilities, equity market value and equity value volatility of the following 20 listed companies. As shown in
We can see from the
We get the market of debt and equity and the equity annual volatility. Then we put them into the formulas (10)-(12). We can calculate each listed company’s asset market value and the volatility of asset value by using the MATLAB software. It was expressed as
We can see from the
. The liabilities, equity value and equity annual volatility of listed companies
Industry | Stock code | Listed company | The market value of the liabilities (yuan) | Equity value (yuan) | Equity annual volatility |
---|---|---|---|---|---|
Real estate | 000024 | Merchants Property | 79,758,000,000 | 30,071,264,223 | 0.3434 |
000056 | *ST Guoshang | 2,411,513,000 | 3,217,479,672 | 0.4067 | |
Biopharmaceutical | 000538 | Yunnan Baiyao | 3,636,674,000 | 39,734,773,517 | 0.2692 |
600421 | *ST Guoyao | 220,412,000 | 913,533,500 | 0.3965 | |
Ceramic industry | 300089 | Saatchi & Saatchi PLC | 252,823,000 | 1,169,612,479 | 0.5774 |
002162 | *ST Shangkong | 1,345,741,900 | 2,918,366,186 | 0.5585 | |
Food industry | 000895 | Shuanghui Development | 4,141,016,000 | 43,368,603,527 | 0.2593 |
000972 | *ST Zhongji | 3,508,537,600 | 1,688,277,765 | 0.4241 | |
Hotel tourism | 601888 | CITS | 2,736,546,490 | 24,367,274,576 | 0.3095 |
600358 | *ST Lianhe | 709,132,000 | 1,767,704,564 | 0.3239 | |
Agriculture & Farming | 600251 | Guannon fruit | 1,629,518,000 | 6,881,060,577 | 0.3557 |
600265 | *ST Jinggu | 456,412,900 | 910,666,983 | 0.4450 | |
Coal profession | 601088 | China Shenhua Energy | 150,460,000,000 | 479,712,161,052 | 0.2265 |
600381 | *ST Xiancheng | 1,899,580,000 | 7,165,719,403 | 0.7436 | |
Transportation | 600009 | ShangHai airport | 3,356,014,000 | 20,397,263,346 | 0.1550 |
600087 | *ST Changyou | 15,861,330,000 | 5,306,963,483 | 0.6455 | |
Cement industry | 600585 | Anhui Conch Cement | 36,347,400,000 | 85,584,174,626 | 0.3412 |
600539 | ST Shitou | 125,449,950 | 965,149,215 | 0.3835 | |
Automobile production | 601633 | Great Wall Motors | 20,926,060,000 | 33,667,439,375 | 0.3433 |
600760 | *ST Heibao | 2,316,713,000 | 2,100,841,298 | 0.4588 |
. The market value of the asset and its volatility
Industry | Stock code | Listed company | The market value of asset (yuan) | Volatility of asset value (yuan) |
---|---|---|---|---|
Real estate | 000024 | Merchants Property | 110,400,000,000 | 0.0961 |
000056 | *ST Guoshang | 5,646,200,000 | 0.2355 | |
Biopharmaceutical | 000538 | Yunnan Baiyao | 43,397,000,000 | 0.2472 |
600421 | *ST Guoyao | 1,135,500,000 | 0.3213 | |
Ceramic industry | 300089 | Saatchi & Saatchi PLC | 1,424,300,000 | 0.4773 |
002162 | *ST Shangkong | 4,275,900,000 | 0.3861 | |
Food industry | 000895 | Shuanghui Development | 47,539,000,000 | 0.2373 |
000972 | *ST Zhongji | 5,226,200,000 | 0.1407 | |
Hotel tourism | 601888 | CITS | 27,123,000,000 | 0.2791 |
600358 | *ST Lianhe | 2,481,800,000 | 0.2331 | |
Agriculture & Farming | 600251 | Guannon fruit | 8,522,000,000 | 0.2892 |
600265 | *ST Jinggu | 1,370,300,000 | 0.2994 | |
Coal profession | 601088 | China Shenhua Energy | 631,230,000,000 | 0.1736 |
600381 | *ST Xiancheng | 9,091,300,000 | 0.5925 | |
Transportation | 600009 | ShangHai airport | 23,777,000,000 | 0.1337 |
600087 | *ST Changyou | 21,659,000,000 | 0.1688 | |
Cement industry | 600585 | Anhui Conch Cement | 122,190,000,000 | 0.2416 |
600539 | ST Shitou | 1,091,500,000 | 0.3405 | |
Automobile production | 601633 | Great Wall Motors | 54,740,000,000 | 0.2141 |
600760 | *ST Heibao | 4,437,100,000 | 0.2218 |
We put the formulas from the table above to calculate default distance and default probability of ST and non-ST companies in all industry. We choose
From the graph 3 we can see that the distance to default of each listed company falls in range (1, 7). Default distance is smaller, the listed company is closer to the default point, namely, the greater the probability of default is, and credit is bad. It generally accords with the calculation results of KMV model.
We first make transverse comparison. The default distance when a = 0.8 is less than the default distance when a = 0.5 to all listed companies. For the same industry, the change range of default distance of ST and non-ST companies is different. Most of the change range of non-ST companies’ default distance is smaller than ST companies’. This can well distinguish default companies and non-default and is more sensitive. When a = 0.8, the identifiability of the KMV model is stronger.
Then, we make vertical comparison. Non-ST companies’ default distance is mainly bigger than the ST in the same industry. But the default distance of ceramic industry and hotel tourism’s non-ST companies is smaller than the ST companies in the same industry when a = 0.5. That shows a = 0.5 has some limitations to the calculation of default distance. The default distance of the two industries’ ST and non-ST companies are revised when a = 0.8, CITS’s default distance is bigger than *ST Lianhe and the default distance of Saatchi & Saatchi PLC still slightly smaller than *ST Shangkong.
Finally, we compare the non-ST and ST companies. We sort the default distance in two columns and find that part of the non-ST companies’ default distance is smaller than the other industries’ ST companies’, such as, the default distance of Saatchi & Saatchi PLC is smaller than *ST Guoyao, *ST Jinggu companies. And the default distance of *ST Lianhe is bigger than Anhui Conch Cement and Guannon fruit.
We put the default distance when default point is a = 0.5 into the default probability formula to calculate. The result is shown in
Distance to default is inversely proportional to default frequency. The smaller the default distance is, the
. The comparison of the two default distance
Industry | Stock code | Listed company | DD (a = 0.5) | DD (a = 0.8) |
---|---|---|---|---|
Real estate | 000024 | Merchants Property | 3.57195573 | 3.1616919 |
000056 | *ST Guoshang | 3.105545462 | 2.7018266 | |
Biopharmaceutical | 000538 | Yunnan Baiyao | 3.714754827 | 3.709688 |
600421 | *ST Guoyao | 2.509312742 | 2.5086549 | |
Ceramic industry | 300089 | Saatchi & Saatchi PLC | 1.723222521 | 1.7232221 |
002162 | *ST Shangkong | 1.800229609 | 1.7850068 | |
Food industry | 000895 | Shuanghui Development | 3.85827 | 3.8515058 |
000972 | *ST Zhongji | 2.380985292 | 2.3539452 | |
Hotel tourism | 601888 | CITS | 3.221639017 | 3.2215242 |
600358 | *ST Lianhe | 3.392640076 | 3.1955813 | |
Agriculture & Farming | 600251 | Guannon fruit | 2.853072821 | 2.8192103 |
600265 | *ST Jinggu | 2.252185048 | 2.2373967 | |
Coal profession | 601088 | China Shenhua Energy | 4.592211016 | 4.4692806 |
600381 | *ST Xiancheng | 1.37685396 | 1.3518101 | |
Transportation | 600009 | ShangHai airport | 6.816059743 | 6.5806705 |
600087 | *ST Changyou | 2.966850027 | 2.1382074 | |
Cement industry | 600585 | Anhui Conch Cement | 3.277507847 | 3.0557062 |
600539 | ST Shitou | 2.603262629 | 2.6008935 | |
Automobile production | 601633 | Great Wall Motors | 2.953742348 | 2.9126105 |
600760 | *ST Heibao | 2.297882745 | 2.2118765 |
. Default probability of the listed companies
Industry | Stock code | Listed company | Non-ST company’s EDP | Stock code | Listed company | ST company’s EDP |
---|---|---|---|---|---|---|
Real estate | 000024 | Merchants Property | 0.0784% | 000056 | *ST Guoshang | 0.3400% |
Biopharmaceutical | 000538 | Yunnan Baiyao | 0.0104% | 600421 | *ST Guoyao | 0.6100% |
Ceramic industry | 300089 | Saatchi & Saatchi PLC | 4.2400% | 002162 | *ST Shangkong | 4.3200% |
Food industry | 000895 | Shuanghui Development | 0.0059% | 000972 | *ST Zhongji | 0.9300% |
Hotel tourism | 601888 | CITS | 0.0638% | 600358 | *ST Lianhe | 0.0698% |
Agriculture & Farming | 600251 | Guannon fruit | 0.2400% | 600265 | *ST Jinggu | 1.2600% |
Coal profession | 601088 | China Shenhua Energy | 0.0004% | 600381 | *ST Xiancheng | 8.8200% |
Transportation | 600009 | ShangHai airport | 0.0000% | 600087 | *ST Changyou | 1.6200% |
Cement industry | 600585 | Anhui Conch Cement | 0.1100% | 600539 | ST Shitou | 0.4600% |
Automobile production | 601633 | Great Wall Motors | 0.1800% | 600760 | *ST Heibao | 1.3500% |
greater the default frequency is. The worse the listed company’s credit is, the higher the level of risk is. We can see from the
This research was supported by the National Natural Science Foundation of China (71361002).