**6. Own approach to determine the probability of default**

One of the most criticized assumptions of Merton model, which is the starting model of structural models, is the assumption of a normal distribution of distance from defaults. This critique is also supported by empirical observations. It is for this reason that we have decided to leave the assumption of a normal distribution. The aim was to find a suitable functional relationship between the distance to default and its probability. An extensive statistical sample would be needed to identify such a functional relationship. Since, in pure region, it is not possible to obtain such a sample in view of the underdeveloped capital market, we have attempted to obtain a similar functional relationship that Moody's uses in its commercially successful KMV model. As we mentioned before, KMV is a commercial implementation of Merton's original model.

In 2003, an article was published in the CFO magazine, which contains hundreds of the largest bond issuers on the US capital market. In the article, the probability of default was quantified for companies based on Moody's internal methodology; asset volatility and the leverage of these companies were also presented [9]. Moody's predicts default probability with a 1-year horizon. In addition to this source, Moody's regularly publishes research on defaulted companies on their site in the research section and they have become another source of information for sample augmentation. Due to these studies, we also acquired companies with higher business risk.

From these collected data, we have subsequently calculated distance to default. In the case of risk-free interest rate, which is meant to express the growth of assets in rated company in a risk-neutral world, we have used US dollar yields with a maturity of up to 1 year for 2003 to match the sample of surveyed companies as much as possible. Based on this, it was then possible to assign the Moody's distance to default values. This relationship is shown in **Figure 13**.

Modeling Default Probability via Structural Models of Credit Risk in Context of Emerging… http://dx.doi.org/10.5772/intechopen.71021 123

**Figure 13.** Functionality between distance to default and Moody's probability of default.

company is traded. These data were inputs to calculate the market value of assets with the use of iterative procedures. An important source of information is also the analysis and collection

After analyzing the company's historical data, we have identified the input parameters of selected structural credit models and their subsequent quantification. In the process of quantification, not only the calculations but also their synthesis with the studies of credit rating agencies and the works of other authors played an important role. We used the procedure to determine the default barrier height, where we relied on Moody's approach to their commercial KMV model based on Black-Scholes equation. In the case of the Black-Cox model, based on the studies, we chose a default barrier discount rate of 7%. The values recommended by the technical document as well as by other authors have also been used in the Credit Grades

Finally, we went on to quantify credit risk by using the probability of default and credit spreads within each model. The results of the individual models differ in some cases. Credit Grades modeled in all cases different curves of default probability and credit spreads compared to other models based on the original Merton model. He alone worked with a stochastic

One of the most criticized assumptions of Merton model, which is the starting model of structural models, is the assumption of a normal distribution of distance from defaults. This critique is also supported by empirical observations. It is for this reason that we have decided to leave the assumption of a normal distribution. The aim was to find a suitable functional relationship between the distance to default and its probability. An extensive statistical sample would be needed to identify such a functional relationship. Since, in pure region, it is not possible to obtain such a sample in view of the underdeveloped capital market, we have attempted to obtain a similar functional relationship that Moody's uses in its commercially successful KMV model. As

we mentioned before, KMV is a commercial implementation of Merton's original model.

In 2003, an article was published in the CFO magazine, which contains hundreds of the largest bond issuers on the US capital market. In the article, the probability of default was quantified for companies based on Moody's internal methodology; asset volatility and the leverage of these companies were also presented [9]. Moody's predicts default probability with a 1-year horizon. In addition to this source, Moody's regularly publishes research on defaulted companies on their site in the research section and they have become another source of information for sample augmentation. Due to these studies, we also acquired companies with higher business risk.

From these collected data, we have subsequently calculated distance to default. In the case of risk-free interest rate, which is meant to express the growth of assets in rated company in a risk-neutral world, we have used US dollar yields with a maturity of up to 1 year for 2003 to match the sample of surveyed companies as much as possible. Based on this, it was then possible to assign the Moody's distance to default values. This relationship is shown in **Figure 13**.

of necessary input data from the annual reports of the surveyed companies.

122 Financial Management from an Emerging Market Perspective

model when we worked with the recommended yield rate and barrier volatility.

barrier, with its volatility having a significant effect on the calculated values.

**6. Own approach to determine the probability of default**

**Figure 13** shows two areas of anomalies. The first is the area around the probability of defaults at 20%. It is obvious that the negative dependence of distance from defaults and their probability is not ensured. The following conclusions can also be drawn from **Figure 13**:


We used these findings to calibrate our own approach to determine the probability of default. In the case of companies that have lower market value of assets than the default point or if their distance to defaults is negative, we can assume default in the near future. Such companies therefore should not be able to meet their obligations and the owners of the capital would thus not apply the purchase option that was written out on the assets of such company. We will associate these companies with the probability of default at 35%. On the contrary, those with a very good financial position, financial stability with a low degree of creditor risk assigned probability at 0.02%. The probability of default for other companies moves between these two extremes of function. Depending on the distance to default based on Moody's (*EDFM*) data, the probability of default is described by the following exponential function:

$$EDF\_M = 0.1797 \, e^{-0.652 \times \text{DD}} \tag{1}$$

The selected functional relationship showed the determination index at 0.9683. The synthesis of the acquired knowledge has determined the resulting function describing the relationship between *EDFM* as follows:

$$EDF\_M = \begin{cases} 35\% & pre\ DD \le 0\\ 0.1797 \, e^{-0.632 \times DD} & pre\ DD > 0 \end{cases} \tag{2}$$

At the same time, *EDFM* ∈〈0.02%; 35%〉. *EDFM* expressed in this way, defines the probability of default in a risk-neutral environment. **Figure 14** compares the probability of a default calculated on the base of normal distribution with the probability of default on the base of Eq. (2).

Based on **Figure 14**, we can say that the probability of default—*EDFM* calculated with the use of relationship differs from EDF with normal distribution. This is particularly the case for companies with a very low distance to default values. In this extreme, *EDFM* values are systematically overestimated. With the increase in distance to default, the opposite situation occurs, and thus the systematic underestimation of probability of default for the normal distribution. For companies with high values of distance to default, these differences are negligible. EDF with normal distribution is asymptotically approaching zero and *EDFM* has a minimum of 2%.

Quantification of the probability of default with the use of KMV model takes several forms [10]. Basic and generally known works with a normal distribution of distance to default. However, empirical studies show that this distance varies depending on the probability of default, taking into account several different factors like business sector, its geographical situation as well as other relevant factors. The calibrated relationship between these two variables offered different results in comparison with other models presented in this paper as we can see in **Figure 15**:

To offer better overview, we offer only comparison of default probability for 2 years. From the results, we can see that modeled probabilities of default have an expected progression. During these two curves, there should be two intersections at specific points as we can see in **Figure 15**.

In practice, credit ratings established by renowned agencies are the most commonly used, because they offer sufficient reporting ability in terms of the financial stability of the analyzed companies. However, most of these ratings are in the form of paid service. Moody's officially

**Figure 14.** Comparison of functionalities between distance to default and probability of default for normal and calculated distribution.

**Figure 15.** Comparison of probabilities of default with normal and calculated distribution for ČEZ, a.s.

*EDFM* <sup>=</sup> {

124 Financial Management from an Emerging Market Perspective

distribution.

35% *pre DD* ≤ 0

At the same time, *EDFM* ∈〈0.02%; 35%〉. *EDFM* expressed in this way, defines the probability of default in a risk-neutral environment. **Figure 14** compares the probability of a default calculated on the base of normal distribution with the probability of default on the base of Eq. (2).

Based on **Figure 14**, we can say that the probability of default—*EDFM* calculated with the use of relationship differs from EDF with normal distribution. This is particularly the case for companies with a very low distance to default values. In this extreme, *EDFM* values are systematically overestimated. With the increase in distance to default, the opposite situation occurs, and thus the systematic underestimation of probability of default for the normal distribution. For companies with high values of distance to default, these differences are negligible. EDF with normal distribution is asymptotically approaching zero and *EDFM* has a minimum of 2%. Quantification of the probability of default with the use of KMV model takes several forms [10]. Basic and generally known works with a normal distribution of distance to default. However, empirical studies show that this distance varies depending on the probability of default, taking into account several different factors like business sector, its geographical situation as well as other relevant factors. The calibrated relationship between these two variables offered different results in comparison with other models presented in this paper as we can see in **Figure 15**:

To offer better overview, we offer only comparison of default probability for 2 years. From the results, we can see that modeled probabilities of default have an expected progression. During these two curves, there should be two intersections at specific points as we can see in **Figure 15**. In practice, credit ratings established by renowned agencies are the most commonly used, because they offer sufficient reporting ability in terms of the financial stability of the analyzed companies. However, most of these ratings are in the form of paid service. Moody's officially

**Figure 14.** Comparison of functionalities between distance to default and probability of default for normal and calculated

0.1797 *<sup>e</sup>* −0.652∗*DD pre DD* <sup>&</sup>gt; <sup>0</sup> } (2)

assigns ČEZ, a.s. rating at Baa1 level. The 1-year probability of default level is 1.18%. This value is in the group also assigned by RMA study to this rating as we can see in **Table 2**. Therefore, we can say that our approach to probability of default is relatively close to that one of Moody's agency, also more we would need more data to verify this even further.


**Table 2.** Relationship between *EDFM* and Moody's rating scale.
