Asset Value and Volatility Estimation for Corporate Credit Rating

1009611462 LUFEI Xiaoxin 1009611301 HE Yao

Abstract
The market-based credit models make use of market information such as equity values to estimate a firm’s credit risk. The Merton model and the Black-Cox model are two popular models that link asset value with equity value, based on the option pricing theories. Under these models, the distance to default can be derived and thus the default probability can be mapped to as long as a large database of companies is provided. The difficulty, however, is that some parameters, including asset values and asset volatilities, which are required in calculating the distance to default, are unobservable in market. Therefore, statistical methods need to be developed in order to estimate the unobservable parameters. In this project, our focus is on using KMV method, which has been widely used in the industry, to estimate corporates’ asset values and asset volatilities. We implemented two models and did numeric study by simulation, which shows that the KMV method gives generally accurate estimates under both models. We also analyzed the model risk under different circumstances. The barrier sensitivity analysis gives the result of how sensitive the Black-Cox model is in choosing different barriers, in the relation with asset volatility and debt level. Furthermore, the models are applied to real companies with different leverage ratios, which shows that structural models are more dynamic and accurate than traditional accounting based methods. Particularly, we applied the model to a recently defaulted company OGX Petr and the model shows a strong predicting capability in the default risk.

Table of Contents
1. 2. Introduction .................................................................................................................................... 3 The Models ................................................................................................................................... 11 2.1 The Merton Model ...................................................................................................................... 11 2.2 The Black-Cox Model................................................................................................................... 12 3. 4. The KMV Estimation Method ........................................................................................................ 14 Numeric Study............................................................................................................................... 16 4.1 Model Implementation (C++)...................................................................................................... 16 4.2 Numeric Result and Error Test .................................................................................................... 17 4.3 Model Risk Analysis..................................................................................................................... 20 4.4 Barrier Sensitivity Analysis .......................................................................................................... 25 5. Real Case Analysis ......................................................................................................................... 29 5.1 Difficulties in practice ................................................................................................................. 29 5.2 Distance-to-Default and Leverage Ratio ..................................................................................... 30 5.3 Model Comparison in Real Cases ................................................................................................ 34 5.4 Case Study ................................................................................................................................... 37 6. 7. Conclusion ..................................................................................................................................... 41 References .................................................................................................................................... 42

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1. Introduction
Credit risk refers to the potential that a borrower fail to make payments which it is obligated to do and thus leads to a default on any types of debt (Coen, 1999). There are different grades of such kind of risk. The most obvious and the worst case is that the counterparty actually default, which means that it ceases to make payments on its obligations. An intermediate risk occurs when the counterparty’s credit worthiness is downgrades, resulting in the loss of value in its obligations (Sooran, 2011). The effective management of credit risk is critical to any banking organisation, as it has been shown that the major cause of serious banking problems continues to be related to lax credit standards for borrowers, poor portfolio risk management, or a lack of attention to changes in economic or other circumstances that can cause the credit standing of a bank's counterparties to deteriorate (Coen, 1999). Additionally, a clear and impartial credit risk measurement provides guidelines for individual investors to more wisely choose their portfolios and better manage their risk exposure.

Over the last 20 years, there is a greater incentive in measuring credit risk in response to a number of secular forces. First, the number of bankruptcies structurally increased over the world. Second, there is a trend towards disintermediation by the highest quality and largest borrowers. Third, the value of real assets (and thus collateral) declines in many markets. Finally, a dramatic growth of off-balance sheet instruments with inherent default risk exposure, including credit risk derivatives, further motivates the development of credit risk measurement (Altman & Saunders, 1997).

The methods used to measure credit risk have undergone a dramatic change as well in the last 20 years. The starting point was expert systems and subjective analysis, in which financial institutions relied virtually exclusively on subjective analysis or so-called banker “expert” systems to assess the credit risk on corporate loans. Specifically, bankers base on so-called 4 “Cs” of credit, namely the borrower’s character, capital, capacity and collateral, to judge whether or not to grant credit. Later on, more systematic accounting based credit-scoring models were developed, where decision-makers of financial institutions compare various key accounting ratios of potential borrowers with industry or group norms (Altman & Saunders, 1997). Multivariate models are commonly used, in which the key accounting variables are combined and weighted to obtain either a credit risk score or a probability of default measure. One of the most popular multivariate models is the z-score model advance by Altman (1968)
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who took various accounting ratios, such as profit before tax (PBT) / current liabilities, current assets / total liabilities, into consideration and developed the z-score formula to calculate a corporate’s credit score (Agarwal & Taffler, 2008).

The accounting-based model, although based on a large sample of data to develop the measurement, suffers several limitations. First, since the ratios and their weightings are derived from sample analysis and the distribution of accounting ratios changes over time, the models need to be redeveloped periodically. Second, accounting statements reflect only past performance of a company and may not have a forward-looking perspective for the future. Third, book values recorded in the accounting statements may be very different from the true asset values due to conservatism and historical cost accounting (Agarwal & Taffler, 2008).

Regarding the drawbacks in accounting-based models, a more dynamic approach which takes a company’s equity value as source information that reflects the firm’s performance was put forward, from which market-based risk models were developed. Under the assumption of efficient market, stock prices reflects all information contained in accounting statements and also other information not including in accounting statement. In addition, despite the drawbacks of a backward-looking characteristic of accounting statement, market prices reflect future expected cash flows, and hence should be more appropriate for prediction purposes. Furthermore, output of market-based models is not time or sample dependent so that it does not need to be adjusted periodically (Agarwal & Taffler, 2008).

In consideration of the advantages of market-based models, numerous models of this kind have been developed, among which structural models are the most commonly recognized ones. In structural approach, it is assumed that the corporate’s asset value follow a geometric Brownian motion and the corporate’s capital structure is composed of a zero-coupon bond and common equity (Duan et al, 2005). Since all corporate securities are viewed as contingent claims on the assets (Lando, 2009), they can be modeled using option-pricing formulas.

In this research, we focus on the two commonly adopted structural models, namely the Merton model and the Black-Cox model. In the Merton model, default could happen only on the day of maturity and it defaults if the asset value is lower than the debt on that day (Lando, 2009). In this case, the corporate’s equity can be viewed as a call option. In the Black-Cox
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model, a default barrier is set so that default can be triggered at any time before maturity as long as the asset value hits the barrier. Therefore, a down-and-out barrier option is borrowed to model the Black-Cox world.

The structural approach yields formulas for the default probability of the company which serves as an indicator of the company’s performance (Duan et al, 2005). Crosbie from Moody mentioned the default probability can be obtained from distance-to-default by establishing an empirical mapping from data on historical default and bankruptcy frequencies (Crosbie, 2003). Then a credit rating can be assigned based on the default probability. Moreover, default probability, additional to accounting reports, is also a piece of important information to investors. Since it is not included in the accounting reports, which only reflect historical and static performance of a company, default probability provides investors a dynamic, forward looking of the future.

The key to obtaining the default probability is distance-to-default, which is based on six parameters - current asset value, distribution of the asset value, asset volatility, book value of the company debt, the expected rate of growth in the asset value and time to maturity. Apart from the value of debt and time to maturity which is simplified by model assumption, thus, can be obtained with ease, asset value and volatility cannot be observed directly from the market. Therefore, estimating asset price and its volatility is the main purpose of our project. With estimated asset price and its volatility, it is straightforward to calculate the expected rate of growth in the asset value. Then we are able to calculate the distance-to-default and further obtain the default probability. However, as mentioned by Crosbie, the default probability can be obtained from an empirical mapping from a large database which consists of more than 25,000 company-years of data and over 4,700 incidents of default and bankruptcy (Crosbie, 2003). Since we do not possess such a large database, in this project, we are not able to perform the mapping.

In this project, we attempt to contribute from the following seven aspects:

1). Comparison analysis of two methods While structural models are conceptually elegant, they are entangled with implementation issues (Duan et al, 2005). Since asset values and its volatility is not directly available in the market, they impede the adoption of structural models. However, Moody’s KMV method
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which infers asset values and volatility from its equity value series, proved by Duan, renders us a way to implement the two models. According to the below Black-Scholes option pricing formula, it can be observed that if the asset volatility is known, asset value can be implied from a set of observed equity value by using bisection search algorithm. This holds because the pricing formula is monotone. The KMV estimation method arbitrarily selects a value as the asset volatility and iteratively calculates the volatility until convergence is achieved.

Since KMV method requires iterative computation of asset value and its volatility, in practice, people may prefer a relatively simple way if results do not deviate much. Here, since equity volatility can be obtained fairly easy from market, we directly take it as asset volatility and calculate the corresponding asset value and its volatility. We treat this method as Method 1 and name KMV method as Method 2. We simulate a set of asset prices and treat it as the ground truth. Then we use two methods to estimate the asset price and its volatility to compare with the ground truth to see which one is more accurate.

After conducting the comparison analysis, we see errors of Method 1, which directly takes equity volatility as asset volatility, unless both debt and equity volatility are kept at very low levels, could be very large. Equivalently, the equity volatility and asset volatility could be very different. Hence, taking equity volatility as asset volatility in credit risk estimation, though might be adopted in practice for simplicity, could lead to unsubstantiated results in calculating default probabilities. The Method 2, which is the KMV method, has been verified by our study, as the root mean square errors of Method 2 are acceptably small in all generated scenarios.

For the Merton Model, the error goes slowly up as the debt or real asset volatility increases, which implies that the model performs better under more stable or lower debt situations. In relatively extreme cases, however, the error could be slightly larger, but the overall error could still be controlled under 8% for cases where debt is as large as 90%. For the Black-Cox Model, the error appears to be more stable as to be less than 5% in all cases. Therefore, we conclude that the KMV method is a reliable approach in estimating the asset volatility for both Merton and Black-Cox model. Taking the equity volatility directly as the estimate, nevertheless, yields large errors and thus could lead to inaccuracy in default probability calculation.

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2). Barrier sensitivity analysis of the Black-Cox model The Black-Cox model, which allows defaults to occur at all times within the maturity of the company debt, is an extension of the Merton model. According to Giesecke, the Black-Cox model can be seen as a down-and-out call option of equity value on the company’s asset (Giesecke, 2004). When the underlying asset reaches a certain barrier, the company defaults and equity holders get nothing. Since in reality, the barrier level is also unobservable, and people, who intends to adopt the Black-Cox model, needs to specify the barrier themselves. Hence, whether the value of barrier determines the model performance is of vital importance.

In this project, a barrier sensitivity analysis is conducted to see the effects of different values of barrier on the model performance. We conclude that the value of barrier does affect the Black-Cox model performance when either debt or asset volatility is high. Under this situation, when estimated barrier deviates too much from the true barrier, which is unobservable from the market data, the model could yield unsatisfactory estimation of asset value and its volatility.

3). Model risk analysis Model risk is the risk of loss due to using models to make decisions, initially and frequently referring to valuing financial securities (Derman et al, 1995). In this project, we used two simplified models to fit the real world situation, which could be far more complicated. Hence, it is unknown that how accuracy of our estimation could be if the real world deviates from the model.

To test the model risk, instead of simulating real asset values and generating equity values with the assumed model, we take another model to do the simulation and make estimation with the original model. Specifically, first, the Merton model is used for simulation and the Black-Cox model is then adopted to estimate asset values and volatility using KMV method. Then, the positions of the two models are interchanged with simulation in the Black-Cox model and estimation in the Merton model.

We conclude, if only the two models are available, in the situation that both debt level and equity volatility are small, the model risk is negligible, as the two models show little difference regardless of which model is assumed. As either debt or equity volatility increases, however, the difference could be large, which indicates a higher model risk. The Merton
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model appears to be riskier as it produces more inaccurate results when the Black-Cox model is assumed. Both models show a deviation in the estimates when another model is assumed. While the Black-Cox model gives downward-biased estimates in the assumption of the Merton model, the Merton model tends to give larger results than the real value when the Black-Cox model is assumed.

4). Comparison of structural model and accounting ratio In finance, leverage is a general term for any technique to multiply gains and losses(Brigham & Houston, 2011). There are various leverage ratios used to quantitatively measure the leverage. One of the most frequently used ratio in literature is the financial-debt-to-asset ratio (Welch, 2011), which gives a simple indication on a firm’s capital structure. It is commonly assumed that a firm with more leverage has both higher-powered incentives and a higher probability of financial distress (Welch, 2011). In other words, a higher leverage in some way implies a higher risk.

To check whether the result of leverage ratio is consistent with that of our measure, we randomly selected some companies that are in different leverage levels. Also, we categorize the corporates into three leverage levels: Low Leverage with (debt/asset) ≲ 30%, Medium Leverage with 30% < (debt/asset) ≲ 60% and High Leverage with (debt/asset) > 60%. For each level, we selected three companies and calculated their asset values, asset volatilities, as well as the distance to default.

From our analysis, we conclude that, although the leverage ratio gives a quick overview of a firm’s capital structure and may have a weak implication of the firm’s credit risk, it generally considers too few aspects of the characteristics of the firm’s overall conditions to yield reasonable results. In our study, some companies with relatively low leverage actually are closer to default and thus are riskier. Hence, using only leverage ratios to estimate corporates’ financial risks may lead to unconvincing results.

5). Model comparison in real cases In the “Model risk analysis” section, a study is conducted to examine the risk of loss due to using the Merton model and the Black-Cox model based on computer simulated data. However, in real cases, values of model parameters such as equity volatility and company

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debt can be less drastic than simulated values. Hence, it is interesting to know under real situations, the performance difference of the two models.

A sample which consists of 30 Hong Kong companies is used to perform the model comparison task. Besides, in order to better depict the debt level for different companies, “leverage” is introduced. During sample selection phase, we try to diversify companies with different leverage level.

We discover that there is no direct relationship between the model difference and either of company’s leverage or equity volatility, while under certain proportion of them, we can see an upward trend showing the Merton model differs further from the Black-cox one when the sum of 0.7*Equity volatility and 0.3* leverage increases.

6). Case study: OGX Petr OGX Petróleo e Gás Participações S.A. is a Brazilian publicly listed oil and gas company belonging to the EBX Group. It was once Brazil's second largest oil company by market value after Petrobras before it went bankruptcy a few days ago on 30/10/2013(Blount, 2012). To see how our models could predict default in real cases, we applied the Merton model to estimate the firm’s asset value and its volatility. In this study of a default case, our model works well in estimating the company’s default probability and predicting its risk status. At about one year ago, it could be already implied from the prediction that OGX is almost surely to default. By adopting the model, the debt holders could have forced the company to default earlier so that more assets could have been recovered.

7). Excel add-in In structural approach, asset value and its volatility can be inferred from the equity price series. People who would like to adopt the approach need to work with equity prices which are available from stock exchange websites or financial news and research websites such as Yahoo! Finance. Since downloaded equity prices always appear in excel format, it could facilitate people’s practice if an excel add-in which estimates the asset price and volatility by using the two studied models is available. Therefore, we have developed the corresponding add-in which is easy to use.
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The report proceeds as follows: the next section describes two different structural models the Merton model and the Black-Cox model and their respective model assumptions and formulation. In Section 3, the Moody’s KMV estimation method is presented in order to address the implementation problem of structural models. Numeric study on the two models is conducted in Section 4. Since the two models are initially implemented in C++, a detailed program flow is explained, followed by numeric result and error test. A model risk analysis and barrier sensitivity analysis are then presented. In Section 5, real case analysis is developed. We discuss the difficulties in adopting the models in real practice and then check whether the result of leverage ratio is consistent with that of our models by selecting firms with different leverage ratios and performing a comparison. A real case study of OGX Petr is given to demonstrate the effectiveness of our model. Finally, conclusions are drawn in Section 6.

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2. The Models
2.1 The Merton Model
In Merton model, a corporate’s equity is viewed as a call option on its asset, the reason for which is that “the equity holders are residual claimants on the corporate’s assets after all other obligations have been met” (Vassalou & Xing, 2004). In other words, equity holders get nothing if the company defaults, while have the claim on the remaining assets after the debt being paid back if default does not occur. This characteristic can be expressed in the following equation: ET = max(AT – D, 0) where ET is the equity value at maturity, AT is the asset value at maturity, and D is the debt that the company has borrowed.

Clearly, the equation gives the same form as the payoff of a European call option (Hull, 2010). The Black-Scholes formula gives a closed-form equation for the price of a European option, which we can borrow to be used here and the relationship of a firm’s equity value and assets can be written as:
( )
( )

(

)

where
( ) ( √( √( ) )( ) )

and Et is the equity value at time t, At is the asset value at time t, r is the risk free interest rate, T is the time of maturity.

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Therefore, with Black-Scholes formula, the unobservable asset value and asset volatility is possible to be estimated with the firm’s equity provided.

2.2 The Black-Cox Model
The major limitation in Merton model concerns with its assumption that default only happens at the maturity date of company’s debt. However, in reality, debt holders could force the company into bankruptcy liquidation when the company’s asset value is far below a level which could pay for its debt. Therefore, default is possible to occur prior to the maturity of the debt (Lando, 2009). Mathematically, default will happen when the asset value hits a lower boundary. It can be expressed as
( )

Where Mt is the historical low value of a firm’s asset at time t B is the barrier D is the debt It can be seen that the equity is a down-and-out call option on the company’s asset (Giesecke, 2004). When the underlying asset reaches a certain barrier, equity holders get nothing. In this report, we discuss a deterministic barrier only which does not change over time. Besides, the barrier is below the initial asset price and company’s debt. Giesecke gives the rationale about this. If D >= B, when the asset value never touch the barrier B over T, then debt holders receive the full amount of debt D and the equity holders take the remaining At - D. However, if the asset value falls below the barrier D at some point within T, then the company defaults. The debt holders take over its assets and leave nothing to equity holders. Then we discover that debt holders are not taking any risks: they receive at least the debt value D upon default. However, this situation does not occur if D < B because debt holders are also exposed to default risk.

In the Black-Scholes setting, we find the relationship between equity value and assets expressed below:

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( (

)

)

(

)

(

)

(

)

(

)

(

)

Where C is the vanilla call value and where
( )( √ ) ( )

Therefore, with a given barrier, asset value and volatility can be estimated by using the above formula.

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3. The KMV Estimation Method
In the previous section, structural modeling approach is discussed. It can be seen that the approach is conceptually elegant but as pointed out in Jarrow and Turnbull (2000), the company’s asset values and volatility are unobservable. Hence, it becomes the major drawback of the structural credit risk models. However, Moody’s KMV method provides an intuitive and numerically efficient way to obtain estimates of asset value and volatility under the Merton model (Duan et al, 2005). It adopts an iterative algorithm which derives the company’s unobserved asset value, expected return and volatility from the company’s equity price. The method is validated by Duan (2005). Since the three sets of derived values are essential to calculate credit spread and default probability, the KMV estimation method has gained popularity in financial institutions. In academia, it is also adopted in the study of equity returns (Vassalou and Xing, 2004).

In this section, the KMV estimation method is discussed under the Merton model. The equity values are observed from time 0 to T, and we denote the set of T+1 observations as {E0, E1h, E2h , ..., Enh} where h is the length of time between two observations. Here we observe a sample on daily basis over one-year period. Correspondingly, h = 1/250 and n = 250.

According to the below Black-Scholes option pricing formula, it can be observed that if the asset volatility is known, asset value can be implied from a set of observed equity value. This holds because the pricing formula is monotone. Let function f(.;σ) denote the option pricing formula. It is obvious to have Et = f(At; σ). Since function f is invertible, we can express A t = f-1(Et; σ). It can be easily done numerically using bisection search algorithm.
( )
( )

(

)

where
( ) ( √( √( ) )( ) )

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The KMV estimation method is a two-step iterative algorithm which arbitrarily selects a value as the asset volatility and repeats the two steps until convergence is achieved. The two steps from the i-th to (i + 1)-th iteration are described as following: Step 1: Calculate the set of asset value by At(σ( i )) = f-1(Et; σ( i ) ) with given equity value set {E0, E1, E2 , ..., En}. Step 2: Calculate the asset returns {R1(i) , R2(i),... , Rn(i)} where Rt(i) = ( ln(At(σ(i)) / At-1(σ(i))). Update the asset volatility and drift as follows:

(

(

)

)
)

(∑ ( ∑

(

()

∑ (
( )

( )

)

(

( )

)

)

By recursively performing the aforementioned two steps until convergence is achieved, asset value and volatility can be obtained.

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4. Numeric Study
4.1 Model Implementation (C++)
To do the numeric study and sensitivity analysis, a C++ implementation of both models is used to simulate a large number of scenarios and to obtain corresponding asset values, asset volatility and asset drift. The class diagram shown in Figure 1 gives an overview of the program.

Figure 1

The class EquityValueGenerator is able to simulate a sequence of asset values based on the log-normal distribution of asset values. The equity values under this set of asset values are then calculated using option-pricing formulas. The AssetValueCalculator, then with given generated equity values and a specified asset volatility, is able to calculate the asset values using Bisection Search Method under the option-pricing formulas. A set of asset values is obtained by directly taking the equity values’ volatility as the parameter of AssetValueCalculator, which we call it Method 1. For Method 2, the RecursiveCalculator is used to recursively call AssetValueCalculator, which takes the asset values’ volatility of the current iteration as the volatility parameter of the next iteration, until the volatility converges. Here, we take 1*10^-6 as our convergence criteria. We repeat this procedure and a sample of a large amount of scenarios could be simulated and asset values, volatility and drift under each scenario are obtained.

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4.2 Numeric Result and Error Test
To test the models, numerous scenarios are generated under different sets of parameters. For simplicity, the initial asset value S0, the time interval h, total time period T, the risk-free rate r and the asset drift μ are fixed when simulating as S0 = 100, h = 1/250, T = 250 days, r = 3%, μ= 10%.Different combinations of asset volatility σ and debt D are set to see their effects on model performances. Under each combination of the parameters, 250 random steps are taken to simulate the asset values of one year and we repeat it for N = 500 times for different random paths. The obtained 500 estimations of asset volatilities are compared with the real sigma σ and the root mean square error is then calculated for these 500 estimates. With similar process, errors under various sets of parameters for both models are obtained. Note that for Black-Cox Model, we set Barrier B = 0.9 * D for two reasons. First, to eliminate the arbitrary opportunity, the default barrier should be set below the debt (Giesecke, 2004). Second, in order to make the difference of the Merton Model and the Black-Cox Model prominent, barrier should be set relatively larger since the smaller the barrier, the closer a down-and-out call option price is to a vanilla call option price with all other parameters set the same.

The errors of both models are shown in the following tables.

The Merton Model
With Fixed D r 0.03 μ 0.1 D 50 σ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Error - Method 1 89.76% 97.34% 116.06% 138.45% 138.44% 138.39% 127.78% 116.92% 101.64% Error - Method 2 4.47% 4.47% 4.56% 4.75% 5.09% 5.45% 5.69% 5.76% 5.65%

Table 1

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With Fixed σ r 0.03 μ 0.1 D 10 20 30 40 50 60 70 80 90 σ 0.5 Error - Method 1 13.26% 33.23% 62.96% 101.46% 138.44% 176.80% 216.27% 235.00% 274.57% Error - Method 2 4.47% 4.47% 4.55% 4.70% 5.09% 5.69% 6.26% 6.41% 7.60%

Table 2

The Black-Cox Model
With Fixed D r 0.03 μ 0.1 D 50 B 45 σ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Error - Method 1 89.76% 97.35% 116.88% 141.69% 143.63% 142.72% 141.96% 135.78% 115.46% Error - Method 2 4.47% 4.47% 4.54% 4.58% 4.62% 4.55% 4.49% 4.50% 4.55%

Table 3

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With Fixed σ r 0.03 μ 0.1 D 10 20 30 40 50 60 70 80 90 B 9 18 27 36 45 54 63 72 81 σ 0.5 Error - Method 1 13.26% 33.26% 65.49% 105.98% 143.63% 176.20% 259.63% 278.14% 357.73% Error - Method 2 4.47% 4.47% 4.52% 4.55% 4.62% 4.56% 4.51% 4.59% 4.65%

Table 4

As can be seen from the tables, errors of Method 1, which directly takes equity volatility as asset volatility, unless both D and σ are kept at very low levels, could be very large. Equivalently, the equity volatility and asset volatility could be very different. Hence, taking equity volatility as asset volatility in credit risk estimation, though might be adopted in practice for simplicity, could lead to unsubstantiated results in calculating default probabilities.

The Method 2, which is the KMV method, has been proved to be an EM algorithm and gives unbiased estimates of asset volatility (Duan et al, 2005). Our study verifies the result, as the root mean square errors of Method 2 are acceptably small in all generated scenarios. For the Merton Model, the error, as shown in table 1 and table 2, goes slowly up as the debt or real sigma increases, which implies that the model performs better under more stable or lower-leverage situations. In relatively extreme cases, however, the error could be slightly larger, but the overall error could still be controlled under 8% for cases where leverage is as large as 90%. For the Black-Cox Model, the error appears to be more stable as to be less than 5% in all cases.

Therefore, we conclude that the KMV method is a reliable approach in estimating the asset volatility for both Merton and Black-Cox model. Taking the equity volatility directly as the estimate, nevertheless, yields large errors and thus could lead to inaccuracy in default probability calculation.

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4.3 Model Risk Analysis
Model risk is the risk of loss due to using models to make decisions, initially and frequently referring to valuing financial securities (Derman et al, 1995). Here, we used two simplified models to mimic the real world, which could be far more complicated. Thus it is unknown that how accuracy our estimation could be if the real situation deviates from the model.

To test the model risk, instead of simulating real asset values and generating equity values with the assumed model, we take another model to do the simulation and make estimation with the original model. Specifically, first, the Merton model is used for simulation and the Black-Cox model is then adopted to estimate asset values and volatility using KMV method. Then, the positions of the two models are interchanged with simulation in the Black-Cox model and estimation in the Merton model.

For the purpose of comparison, estimation is done under both models for a model assumption. Similar to the error test, different combinations of σ and D are set to track their influences on the model risk and for each combination, 500 scenarios are generated. Again, barrier is set to be 90% of debt. Three aspects of estimated asset volatilities are compared for two models, namely the root mean square error, the average, and the mean difference. Note that the difference is relative, that is for each scenario we calculate the difference as “ | σ of alternative model – σ of original model| / σ of original model” and average is then taken for 500 scenarios. The results are shown below. In general, the result shows that if both σ and D are kept at a low level, the difference could be very small between the two models no matter which model we assumed in advance. Also, if either σ or D is kept at very low level, even though another is high, the difference could be small as well, which can be seen from the first row of each table. While σ and D increase, error of the alternative model could rise sharply with the mean of the estimation deviating from the real σ, which results in large difference between the two models. The increasing rates are different in various scenarios and we draw them in line charts to better illustrate.

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Model Difference Comparison

Figure 2

Figure 3

Figure 4

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Figure 5

As shown in the charts, both σ and D have significant influences on the model risk. The effect of one parameter could be magnified when another parameter is kept at a relatively high level. When D is fixed at 30, the differences of both models are kept at a low level with less than 10% even though σ is high. Thus we conclude that in low-leverage scenarios, the model risk is relatively small and regardless of what the real situation is, both models give an acceptably accurate estimate. When D is fixed at a high-level, which we set as 90, however, the difference increases rapidly at first. Nevertheless, it shows a convergence trend as σ keeps increasing. If the Merton model is assumed, the difference is kept below 25%, while if the Black-Cox model is assumed, the difference could be as large as 100%. It indicates that the model risk could be very large in such scenarios, especially using the Merton model to estimate under the Black-Cox Model. As σ is fixed, only when the Merton model is assumed can the model difference be controlled under 25%. In other scenarios, the differences could all be larger than 40% as the debt level goes up. Again, the risk of the Merton model is larger since the model differences grow more rapidly when the Black-Cox model is assumed. Note that unlike the charts with fixed D, lines with fixed σ shows an upward trend as debt increases, which implies that the difference could keep rising as D further goes up. In practice, however, the leverage of normal companies would not be higher than 100%. Therefore, it could be concluded that the model risk of the Black-Cox model is relatively small, and it is able to give a reasonably accurate estimate even the real situation follows the Merton model. Nevertheless, it is not the case for the Merton model as it produces large difference from the estimate using the original model.

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Model Mean Comparison

Figure 6

Figure 7

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Figure 8

Figure 9

To further explore the performance of a model in the situation that actually follows another model, the average of the estimates is calculated to see if there is a systematic bias for the alternative model. As can be seen in the charts above, the estimates calculated using the assumed model shows no bias as the blue lines show, which verifies the conclusion drawn by Duan et al (2004). For the alternative model, however, as D becomes larger, the red lines deviate from the real σ. When the Merton’s model is assumed, the estimates using the Black24

Cox model shows a downward trend, which indicates that it gives smaller asset volatility than the true value. While if the Black-Cox model is assumed, the estimates using the Merton’s model is upward biased. Again, the drift is more obvious for the Merton model under the assumption of the Black-Cox model, which is consistent with the result in the difference comparison. To conclude, if only the two models are available, in the situation that both D level and σ are small, the model risk is negligible, as the two models show little difference regardless of which model is assumed. As D or σ increases, however, the difference could be large, which indicates a higher model risk. The Merton model appears to be riskier as it produces more inaccurate results when the Black-Cox model is assumed. Both models show a deviation in the estimates when another model is assumed. While the Black-Cox model gives downwardbiased estimates in the assumption of the Merton model, the Merton model tends to give larger results than the real value when the Black-Cox model is assumed.

4.4 Barrier Sensitivity Analysis
While the Black-Cox model is less restrictive than the Merton model by allowing defaults to occur prior to the maturity of the debt, it introduces the concept of “barrier” which cannot be observed directly from the available market data. When people adopt the Black-Cox model, they need to specify the barrier by themselves. Hence, whether the value of barrier determines the model performance becomes a concern.

To see the effects of different values of barrier on the model performance, a barrier sensitivity analysis is conducted. Here, we let the initial asset value S0 = 100, the asset drift μ= 10%, the time interval h = 1/250, total time period T = 250 days and the risk-free rate r = 3%. We assume the true barrier is 70% of debt level and study the mean error of the model under the estimated barriers (50%, 60%, 70%, 80%, 90% of debt level). Moreover, due to the unknown roles played by debt D and asset volatility σ, we classify them into high and low levels and test the combination of them separately. By following the similar process discussed in “Numeric result and error test” section, under each combination of the parameters, 250 random steps are taken to simulate the asset values and we repeat it for N = 500 times for different random paths. The only difference here is when simulating the true equity values,
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we set Barrier as B = 0.7 * D, while estimating asset values, we set B = 0.5 * D, 0.6 *D, 0.7 * D, 0.8 * D and 0.9 * D respectively. The obtained 500 estimations of asset volatility are compared with the real one, and the root mean square error is calculated.

The following table shows the root mean square errors under different combinations of the parameters.

High level debt with high level asset volatility Real Barrier Debt Sigma Risk-free Rate Real Mu Esti-barrier Mean Error 0.7 * D 90 0.9 0.03 0.1 0.5 * D 0.6 * D 0.7 * D 0.8 * D 0.9 * D
Table 5

0.140417 0.0801731 0.0490526 0.0792695 0.13092

High level debt with low level asset volatility Real Barrier Debt Sigma Risk-free Rate Real Mu Esti-barrier Mean Error 0.7 * D 90 0.3 0.03 0.1 0.5 * D 0.6 * D 0.7 * D 0.8 * D 0.9 * D
Table 6

0.0742966 0.0681166 0.0615739 0.0689574 0.124741

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Low level debt with high level asset volatility Real Barrier Debt Sigma Risk-free Rate Real Mu Esti-barrier Mean Error 0.7 * D 30 0.9 0.03 0.1 0.5 * D 0.6 * D 0.7 * D 0.8 * D 0.9 * D
Table 7

0.0601528 0.0516423 0.0478433 0.0516158 0.0634346

Low level debt with low level asset volatility Real Barrier Debt Sigma Risk-free Rate Real Mu Esti-barrier Mean Error 0.7 * D 30 0.3 0.03 0.1 0.5 * D 0.6 * D 0.7 * D 0.8 * D 0.9 * D
Table 8

0.0446789 0.0446789 0.0446789 0.0446789 0.0446788

From the tables above, it can be seen that as long as either debt or asset volatility remains at high level, the value of barrier does affect the model performance. In particular, when both of them are high, the root mean square error is low only when the estimated barrier coincides with the true one. Other estimated barriers drive the model to reach an unacceptable level of error. When only one of the debt and asset volatility is high, the errors still deviate from the one with true barrier, although the range is narrowed. When both the debt and asset volatility is kept at low level, shown by the last table, the model performance is stable. Under different estimated barrier, the root mean square errors are almost the same.

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Therefore, we conclude that the value of barrier does affect the Black-Cox model performance when either debt or asset volatility is high. Under this situation, when estimated barrier deviates too much from the true barrier, which is unobservable from the market data, the model could yield unsatisfactory estimation of asset value and its volatility.

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5. Real Case Analysis 5.1 Difficulties in practice
As we move from theories to real world, there are several problems that we need to handle. The first problem is that the debt in a corporate’s balance sheet consists of several components, including short-term liabilities and long-term liabilities, each of which may have different influences on the corporate’s default probability. Understandably, short-term liabilities have more significant effects on a firm’s credit risk, since the debt has to be paid off in the near future. To reflect this difference, instead of simply add all liabilities together, a weighting ratio should be applied to different kinds of liabilities. How to set an appropriate ratio, however, is difficult to decide in practice. In our study, we follow the method adopted by Vassalou and Xing (2004), in which the total debt is calculated as:

Total Debt = (Short-term Debt) + 0.5 (Long-term Debt) As mentioned by Vassalou and Xing (2004), the Moody’s uses the same ratio in the KMV method, for which they argue that it catches adequately the financing constraints of firms.

The second problem is that the barrier is usually not be clearly defined. Since the KMV method is not be able to estimate the barrier, it has to be arbitrarily defined. Nevertheless, as shown in our study of barrier sensitivity, with asset volatility and debt level kept relatively low (σ = 30%, D = 30), the barrier is not sensitive for it gives almost the same results even though the set barrier is about 30% deviate from the true barrier. In reality, this is roughly an average level for normal companies, while the extreme cases as we studied with σ = 90% or D = 90 are rare. As shown by Brockman and Turtle (2003), in their 7,787 observations, the mean and median of asset volatilities are 29.04% and 22.86%, and the mean and median of debt ratio are both approximately 45%. Therefore, it can be concluded that in practice, an arbitrarily chosen barrier could be with small sensitivity and thus would not have significant effects on the result. The result of Brockman and Turtle (2003) shows that the average implied barrier-to-debt ratio is 0.6920 with a corresponding standard deviation of 0.2259. Following their result, here we set the barrier as 0.7 of debt for simplicity. The third difficulty is that both Merton and Black-Cox model requires an input of a firm’s lifespan, representing the maturity of the options. It is unobservable in market neither and has
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to be arbitrarily specified. As suggested by Brockman and Turtle, a relatively long lifespan is more suitable in the consideration of the standard going concern assumption. Also, they argue that the barrier estimation is not sensitive to the selecting of the lifespan. Here we choose the same lifespan in their research as 10 years.

5.2 Distance-to-Default and Leverage Ratio
In finance, leverage is a general term for any technique to multiply gains and losses (Brigham & Houston, 2011). There are various leverage ratios used to quantitatively measure the leverage. One of the most frequently used ratios in literature is the financial-debt-to-asset ratio (Welch, 2011): (D/A) = Total Debt / Total Asset The financial-debt-to-asset ratio gives a simple indication on a firm’s capital structure. It is commonly assumed that a firm with more leverage has both higher-powered incentives and a higher probability of financial distress (Welch, 2011). In other words, a higher leverage in some way implies a higher risk.

To check whether the result of leverage ratio is consistent with that of our measure, we randomly selected some companies that are in different leverage levels. Note that here the total asset is calculated as (total liabilities + total equity), since total asset is unknown before estimation. Also, we categorize the corporates into three leverage levels: Low Leverage with (D/A) ≤ 30%, Medium Leverage with 30% < (D/A) ≤ 60% and High Leverage with (D/A) > 60%. For each level, we selected three companies and calculated their asset values, asset volatilities, as well as the distance to default. The result is shown in the chart below, with green lines representing the companies with low leverage, yellow lines representing the companies with medium leverage, and red lines for high-leverage companies.

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Figure 10

As shown in the chart, the financial-debt-to-asset ratio and the distance to default, overall, gives similar estimation to a firm’s risk. As can be seen, the red lines, which are for companies with lower leverage, generally lie at the bottom of the chart, which indicates that they have smaller distance to default and thus lower risk. In contrast, the yellow lines are basically in the middle and the green lines are at the top, indicating that medium-leverage companies have average risk while high-leverage companies are more dangerous. Hence, in general, the leverage ratio gives a rough estimation for a corporate’s credit risk.

However, as we more closely check individual companies, the leverage ratio does not always yield the same result as the distance to default. Taking the company 8026.HK (Prosten Technology Holdings Ltd.) as an example, which is represented by the green line lying between the red lines. The company has a relatively low leverage as the debt-to-asset is 23%, while the distance to default of which is smaller than the company with leverage as high as 65% (8003.HK: Great World Company Holdings Ltd). To further examine the company, we found that its equity volatility is as high as 96%, which then leads to a high asset volatility,
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which is about 93% calculated in the Merton model. As shown in the chart below illustrating the company’s 1-year stock, there are often large gaps in the stock prices, which results in the large volatility of equity values.

Figure 11

Large volatility in equity value then leads to the large volatility of estimated asset values, which intuitively indicates that Prosten Technology is in an unstable status and thus is more likely to default. It is consistent with the fact that the stock is in the Growth Enterprise Market (GEM) and the newly started companies are usually less steady than the mature companies. We then formerly check the distance-to-default formula, which is
( ) ( √( ) )( )

As can be seen from the equation, the debt-to-asset ratio takes into consideration only the first term of the equation, while other parameters μ, σ, and time to maturity are dismissed. To further check the company mentioned above, although for Prosten Technology, the term (ln(A/D)) is larger, the second term of the numerator is small and the denominator is large with relatively high σ, and overall it produces a small DD. In other words, despite of more assets relative to debt that Prosten Technology possesses the default point of which is high because of the large volatility. And in this case, the latter factor dominates and thus the

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overall risk, indicated by the smaller distance to default, is higher for Prosten Technology than other companies with low debt-to-asset ratio.

Another example is China Telecom with company code 0728.HK, which in the chart, is the yellow line lying in between the red lines. Although its debt-to-asset ratio is at medium level, which is 52%, and the equity volatility is adequately small (22%) as well, its distance-todefault is as small as high-leverage companies. The reason is that the estimated asset drift is small and actually negative (-0.02), which indicates a downward trend in the assets. Clearly, a decrease in asset value implies that the company is undertaking more risk, and thus a smaller distance to default. This is also consistent with the DD equation, since the second term in the numerator would be smaller with smaller μ. It can also be implied from a downward tendency in the company’s stock prices.

Figure 12

Therefore, although the leverage ratio gives a quick overview of a firm’s capital structure and may have a weak implication of the firm’s credit risk, it generally considers too few aspects of the characteristics of the firm’s overall conditions to yield reasonable results. In our study, some companies with relatively low leverage actually are closer to default and thus are riskier. Thus, using only leverage ratios to estimate corporates’ financial risks may lead to big mistakes.

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5.3 Model Comparison in Real Cases
In “Model Risk Analysis” section, a study is conducted to examine the risk of loss due to using the Merton model and the Black-Cox model based on computer simulated data. However, in real cases, values of model parameters such as equity volatility and company debt can be less drastic than simulated values. Hence, it is interesting to know under real situations, the performance difference of the two models.

A sample which consists of 30 Hong Kong companies is used to perform the model comparison task. Besides, in order to better depict the debt level for different companies, “leverage” is introduced as the criteria and calculated by the formula below. During sample selection phase, we try to diversify both leverage level and industry type. The sample covers seven industries, including jewelry, semiconductor, coal, food and beverage, mining, household goods and automobile. It also supports companies of high, medium and low leverage ratio.

Leverage = Total liability / (Total liability + Equity Price) In this study, the company’s daily stock price between 07/11/2012 and 04/11/2013 is observed from Yahoo! Finance, where the corresponding share outstanding can also be obtained. Since for some companies, debt information is outdated on Yahoo Finance, we obtain it from ET Net. Other model parameters are set as risk free rate r = 2%, h = 1 / 250, debt D= short term debt + 0.5 * long term debt, barrier B = 0.7 * D.

Figure 13 shows the model difference with respect to leverage. There is no pattern indicates how the model difference changes under different leverage level. While Figure 14 shows the difference with respect to equity volatility. Although in general, when equity volatility becomes large, the model difference increases, the pattern is still unclear as too many points are jammed together. Therefore, we guess that the model difference could be associated with both leverage and equity volatility, instead of either one of them.

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Figure 13

Figure 14

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Figure 15

Hence, we tried different portion of equity volatility and leverage to see if there is any pattern associated with model difference. Under 0.7*Equity volatility and (1-0.7)* leverage, there is an upward trend indicating that the model difference increases when the sum of 0.7*Equity and (1-0.7) leverage becomes larger.

To conclude, we intend to compare the Merton model and Black-Cox model by using real cases in this section. Since computer simulated data can be far different from real scenarios, it worths effort to study the model difference under real world. However, it is discovered that there is no direct relationship between the model difference and either of company’s leverage or equity volatility, while under certain proportion of them, we can see an upward trend showing the Merton model differs further from the Black-cox one when the sum of 0.7*Equity volatility and 0.3* leverage increases.

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5.4 Case Study

OGX Petr

OGX Petróleo e Gás Participações S.A. is a Brazilian publicly listed oil and gas company belonging to the EBX Group. It was once Brazil's second largest oil company by market value after Petrobras before it went bankruptcy a few days ago on 30/10/2013 (Blount, 2012). The bankruptcy protection request, filed in a Rio de Janeiro court, came after OGX failed to reach an agreement with creditors to renegotiate part of its $5.1 billion debt load (Lorenzi & Blount, 2013).

To see how our models could predict default in real cases, we applied the Merton model to estimate the firm’s asset value, asset volatility and asset drift. With the estimation, the distance-to default as well as the default probability is then calculated to measure the firm’s credit risk. In order to examine the prediction performance of the model, we take positions at different time before the bankruptcy and apply the models pretending that we have only available information prior to that time. Specifically, we examine the situations at 1 month back, 2 months back, until 12 months back and a period of one-year stock prices is taken into consideration. That is, we will apply the model with equity values from 30/9/2013 to 30/9/2012, from 8/30/2013 to 8/30/2012, etc. The result is shown in the table and charts below.

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Time 1 month back 2 month back 3 month back 4 month back 5 month back 6 month back 7 month back 8 month back 9 month back 10 month back 11 month back 12 month back
Table 9

Est - σ 110.51% 98.17% 89.98% 71.88% 106.29% 131.53% 142.06% 65.15% 62.10% 61.95% 62.81% 63.14%

Est - μ -240.98% -204.38% -160.59% -132.65% -51.69% 2.65% 30.55% -141.35% -122.41% -91.21% -96.75% -90.21%

DD -9.120346721 -8.486732042 -7.285372706 -7.098433483 -3.165811714 -1.917422388 -1.444450266 -7.395718311 -6.580198174 -4.991721058 -5.233085434 -4.855583523

Default probability 100.00% 100.00% 100.00% 100.00% 99.92% 97.24% 92.57% 100.00% 100.00% 100.00% 100.00% 100.00%

Figure 16

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Figure 17

Figure 18

As can be seen from the table, the default probability of OGX has been as high as very close to 100% even 1 year ago. Although it recovered a little between 7 to 5 months back, the overall default probability is above 90% for the whole time period. As shown in the chart of distance-to-default, after a sudden boom in 6 months back, it keeps dropping until it reached about -9 in the previous month. Tracing back to the estimated parameters, σ is very large with the extreme point to 142%, which is distant from the normal companies with an average asset volatility of approximately 30%. For the asset drift μ, it is generally below zero, which indicates that the assets are generally in decrease. Only in 6 month and 7 month ago, μ is positive, which is consistent with the fact OGX pumped its first oil in January 2012 (Lorenzi & Blount, 2013). However, by mid-year, it became clear that the field would not produce near expectations and the asset drift kept dropping after then.
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In this study of a default case, our model works well in estimating the company’s default probability and predicting its risk status. At about one year ago, it could be already implied from the prediction that the company is almost surely to default. By adopting the model, the debt holders could have forced the company to default earlier so that more assets could have been recovered.

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6. Conclusion
In conclusion, both the Merton model and the Black-Cox model work well in estimating the parameters in the simulated situation. The KMV method is then proved to be a reliable approach in estimating the asset volatility for both Merton and Black-Cox model. In model risk analysis, we see if only the two models are available, in the situation that both debt level and equity volatility are low, the model risk is negligible. While both models show a deviation in the estimates when another model is assumed, the Black-Cox model gives downward-biased estimates in the assumption of the Merton model, the Merton model tends to give larger results than the real value when the Black-Cox model is assumed. In barrier sensitivity analysis, we discover that the value of barrier does affect the Black-Cox model performance when either debt or asset volatility is high. When considering leverage ratio, although it gives a quick overview of a firm’s capital structure, it generally considers too few aspects of the characteristics of the firm’s overall conditions. Hence, using only leverage ratios to estimate corporates’ financial risks may lead to big mistakes. In real case analysis of the two models, it is discovered that there is no direct relationship between the model difference and either of company’s leverage or equity volatility, while under certain proportion of them, we can see an upward trend showing the Merton model differs further from the Black-cox model.

It is also interesting to classify companies into different industries and apply the two structural models to study the asset volatility of individual industry. Moreover, the Merton and Black-Cox model simplify company debt structure, while in reality, it may be far more complicated. More sophisticated models which consider complicated debt structure require further research.

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7. References
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Altman, E. I., & Saunders, A. (1997). Credit risk measurement: Developments over the last 20 years. Journal of Banking & Finance, 21(11), 1721-1742.

Blount, J. (2012). OGX to buy part of Maersk Brazil oil stakes. In Reuters. Retrieved November 23, 2013 from, Website: http://uk.reuters.com/article/2012/03/16/ogx-maerskidUSL2E8EG8IE20120316

Brigham, E. F., & Houston, J. F. (2011). Fundamentals of financial management. CengageBrain. com.

Brockman, P., & Turtle, H. J. (2003). A barrier option framework for corporate security valuation. Journal of Financial Economics, 67(3), 511-529.

Coen, W. (1999). Principles for the management of credit risk - Consultative document. Retrieved November 23, 2013 from Bank for International Settlements, Website: http://www.bis.org/publ/bcbs54.htm

Derman, E., Kani, I., Ergener, D., & Bardhan, I. (1995). Quantitative Strategies Research Notes. The Volatility Smile and Its Implied Tree.

Duan, J. C., Gauthier, G., & Simonato, J. G. (2005). On the equivalence of the KMV and maximum likelihood methods for structural credit risk models. Groupe d'études et de recherche en analyse des décisions.

Giesecke, K. (2004). Credit risk modeling and valuation: An introduction.Available at SSRN 479323.

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Hull, J. (2010). Options, Futures, and Other Derivatives, 7/e (With CD). Pearson Education India.

Jarrow, R. A., & Turnbull, S. M. (2000). The intersection of market and credit risk. Journal of Banking & Finance, 24(1), 271-299.

Lando, D. (2009). Credit risk modeling: theory and applications. Princeton University Press.

Lorenzi, S. & Blount, J. (2013). Batista's OGX files for bankruptcy protection in Brazil. In Reuters. Retrieved November 23, 2013 from, Web site: http://www.reuters.com/article/2013/10/30/us-brazil-batista-idUSBRE99T19620131030

Sooran, C. (2011). Credit risk: Definition and management. Retrieved November 23, 2013 from http://www.finpipe.com/credit-risk/

Vassalou, M., & Xing, Y. (2004). Default risk in equity returns. The Journal of Finance, 59(2), 831-868.

Welch, I. (2011). Two Common Problems in Capital Structure Research: The Financial‐Debt‐ To‐Asset Ratio and Issuing Activity Versus Leverage Changes. International Review of Finance, 11(1), 1-17.

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