An empirical assessment of the Merton model for prediction of the Probability

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model for prediction of the Probability

of Default

Georgia Skoufoglou

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Georgia Skoufoglou Student nr: 11088494

Email: gskoufoglou@gmail.com Date: August 15, 2018

Supervisor: Dr. L.(Lu) Yang Second reader: Dr L.J. van Gastel

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Statement of Originality

This document is written by Student Georgia Skoufoglou who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text

and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

The purpose of this study is to evaluate the performance of the Merton model on forecasting a default event. The model will be implemented on three companies that have already defaulted in the past in order to prove how many months in advance the Merton model can predict a default event. According to the main concept of the Merton model,our empirical methodology is based on the assumption that a default event occurs once a firm’s asset value have exceeded a certain amount. In order to estimate its probability, we will use the measure Probability of Default, that is the probability that a firm’s asset value exceeds the certain amount. In addition, the market value of the assets will be estimated and used in our case study, along with the assumption that the market value of the assets follow a logNormal distribution. At the end, we conclude that the Merton model can predict a default event on average 14 months in advance. It is a rather satisfactory performance given the fact that only publicly available data were used. However, an investor has to take into account the assumption used which limited the accuracy of the results.

Keywords Merton model, Probability of Default, KMV-Merton model, Distance-to-Default, CAPM, credit risk, Black-Scholes-Merton model

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Introduction

Credit risk modeling is one of the most crucial sections in Risk Management. Every single investor and financial institution wishes to be there a way to forecast the credit worthiness of a company or a financial instrument in which they intend to invest. In this way, it would be possible to evaluate the outcome of an investment or the fluctuation of a corporate loan portfolio value over time. It would be possible to control their exposure to the credit risk. It would be possible to convert the uncertainty into an expectation.

Although many researchers and financial analysts have developed several models of default prediction for a corporation, the main obstacle in their way is the absence of granular data. Since every corporation has its own structure of liabilities and assets, granular data are of great importance. However, there are some basic data publicly available, for instance the book values of liabilities (balance sheets)for daily prices of a firm’s stock. Therefore, many researchers focused on developing models of default prediction using publicly available data.

The first model developed for default prediction was the original Merton model,

Merton (1974). It focused only on publicly available data and its main idea is that a default event might occur once the total asset value is lower than the total value of liabilities, that is the equity value becomes negative. The equity value was assumed as a European call option on the asset value with strike price the value of the liabilities. In order to valuate the equity, the Black-Scholes-Merton model was employed. Later, this model was developed significantly by the corporation KMV,Crosbie and Bohn (2003). They adopted the same assumptions and methodology as those in the original Merton model but in addition they introduced the measure Distance-to-Default with which the Probability of Default could be estimated. This model is known as the Merton model.

The latter paper became the inspiration to our following research question :

“How well does the Merton model perform in forecasting the Probability of Default for a real company? ”

In order this case study to be accurate, we selected three companies that have already defaulted officially in the past, that is the Arch Coal Inc., the Vanguard Natural Resources and the Ford Motor Co. After obtaining all the necessary data, we estimate the Probability of Default per quarter and up to two years before the date of the default event. In this way, we can see how many months in advance the Merton model warns us of a possible default event in the near future. It is of great interest to see how well does this model perform beside the strict assumptions that will be adopted. This will increase the curiosity for further research and possible development of the model.

In chapter 2, we will review the definition of the credit risk, the financial instruments subject to this risk and the most important risk measures. Additionally, we will briefly introduce the regulatory frameworks which oblige the financial institutions to keep a buffer according to the value of their loan portfolios. In chapter 3, we present the

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2 Georgia Skoufoglou — Empirical assessment of the Merton model

Scholes-Merton model that will be used in the Merton model, the Merton model itself and the assumptions used. Moreover, we present the empirical methodology that we employed. In chapter 4, we introduce not only the three companies on which we will implement the Merton model, but also how the real data were obtained. Finally, in chapter 5 we present the results of the empirical implementation of the model and the answer to our research question.

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Credit risk

Credit risk was, is and will remain one of the most crucial sections in Risk Management. By definition, credit risk is the risk that losses might occur in a portfolio due to default or credit worthiness downgrade of counterparties or issuers in a financial contract. It consists of two risk subcategories, that is to say the default risk and the downgrade risk,McNeil et al.(2015). The former one indicates that the counterparty is not able to fulfill its contractual obligations. At this point, it is worth mentioning that according to each country’s bankruptcy codes, default is not always translated into bankruptcy. Furthermore, the latter one implies that the credit rating of a counterparty decreases. Particularly, the downgrade risk is related to rating agencies who provide a firm with a corresponding rate accompanied with probabilities of default as well as probabilities of migration to a higher or lower rate within one year.

2.1 Financial instruments subject to credit risk

Generally, there are many financial instruments that are subject to credit risk and worth mentioning in order for someone to be convinced of the importance of credit risk modeling. The most important ones are loans, bonds, OTC (that is to say Over-The-Counter) derivative contracts and credit derivatives.

Loans

There are many kinds of loans available in the market. However, they can be categorized by two main criteria, that is to say the type of the obligor and whether a collateral was offered up or not, McNeil et al. (2015). According to the first criteria, a financial institution may have a portfolio of retail, corporate, interbank and sovereign loans which have been offered to individuals or medium-size firms, large firms, banks and government respectively. According to the second criteria, a portfolio of secured and unsecured loans might be owned by a financial institution.

A portfolio of loans totally exposes the financial institution to the default risk of the obligor. Before the loan was granted to the borrower, the bank had estimated the borrower’s Probability of Default (that is PD) and defined the interest rate, the principal and the amortization schedule accordingly. The estimation was done by using the above-mentioned criteria as risk factors. In this way, the financial institution is able to evaluate the default risk that is taken over by granting such a loan. For instance, the principal of an unsecured retail loan will be significantly smaller than that of a secured corporate one. The reason is that the estimated Probability of Default of an individual without offering any collateral is rather higher than that of a firm with a collateral being offered up.

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Bonds

In general, there are two main types of bonds regarding the borrower, that is the corpo-rate and the treasury bonds issued by corpocorpo-rates and governments respectively,McNeil et al. (2015). When an investor buys a bond, he deposits the price of the bond to the issuer. Then, the issuer has the obligation to pay back the interest periodically and the principal at the maturity date.

As a result, bondholders like financial institutions face many risks. The most im-portant of them are the interest and the default risk. If the issuer cannot pay back the interest, the principal or even both of them, the financial consequences for the investor might be dramatic. Especially if the investor owns a portfolio of investments in bonds. Over-the-counter derivatives

Over-the-counter derivatives are private contracts written and signed by the contracting parties exclusively. Any exchanges or other intermediaries are absent. Generally, they are a useful risk-management tool for hedging or trading risks, such as credit risk. Firstly, they offer the possibility to small companies that cannot fulfill the exchange listing requirements to enter a derivative contract. Secondly, the terms of the over-the-counter derivatives can be negotiated between the two parties to both of their advantage.Thirdly, the administrative costs are lower than those of listed derivatives. Some examples of the most widely known over-the-counter derivatives are the swaptions, interest swap, longevity swap and credit default swap,McNeil et al. (2015).

As in every financial contract, credit risk is involved in every over-the-counter deriva-tive. Although it is commonly referred to as counterparty risk, a default event of one of the contractual parties might occur causing a significant loss to the other party. Therefore, it is of crucial importance to be measured and to be integrated into the pricing and evaluation process of the derivative. Some commonly used techniques to reduce a party’s exposure to the counterparty risk is a netting agreement as well as a collateralization agreement,McNeil et al.(2015). The former technique is used in swaps where exchanges of fixed and floating cashflows take place among the contractual par-ties. Instead of operating both fixed and floating cashflows, only one transaction of their difference is being made. Regarding the latter technique, a collateral is pledged against a loan as a guarantee.

Credit default swap

As its name implies, a credit default swap is widely used mainly by banks, insurance companies and investment funds to hedge and trade the credit risk they are exposed to, for instance due to a loan or bond investment. Particularly, it is a contract between a protection seller and a protection buyer with regards to a reference entity, McNeil et al. (2015). Taking as an example a loan granted from a bank to a firm, the bank is exposed to the risk of loss if the firm defaults. Therefore, it is in need of buying a protection against this event. Assuming that a re-insurance company agrees on selling this protection to the bank, they negotiate a credit default swap between them. If the reference entity defaults before the maturity time of this contract, the protection seller has the obligation to make a deposit to the protection buyer with an amount interpreted as compensation for the bank’s loss due to the default event.This transaction is called default payment leg. On the other hand, the protection buyer has the obligation to pay a premium to the re-insurance company as long as the reference entity has not defaulted. This transaction is called premium payment leg.

Since the credit default swaps are over-the-counter derivatives, the counterparty credit risk exists. In addition to this, concentration as well as systematic risk among several default events exists. Assuming that the above-mentioned re-insurance company owns a portfolio of credit default swaps, default events occurring at the same period

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might cause absence of liquidity and apparently default events to other companies in-volved in such contracts might follow.

2.2 Risk measures

Credit risk modeling is important for the financial institutions not only by reason of protection against losses due to a possible default event of the counterparty, but also for reasons of competitive advantage in the market. It is easy for a financial institution to take over more risk as long as this will make it more competitive,since taking over more risk implies the ability to offer more attractive terms for the counterparty. However, this technique might cause liquidity problems and insolvency.

On the other hand, it is impossible for a financial institution to not take over any risk at all. Therefore, a trade-off decision has to be made regarding the exact amount of credit risk that a financial institution is willing and able to take over as well as the contract’s terms before engaging to a financial instrument. The risk measures that will lead to the most efficient decision are the Probability of Default, loss given default and exposure at default with their abbreviations PD, LGD and EAD respectively.

Probability of Default

By definition, the Probability of Default is the likelihood that a counterparty might default within a particular time horizon, usually one year,McNeil et al.(2015). It can be estimated using several techniques, for instance using historical default data, evaluating and comparing assets and liabilities or analyzing accounting data of the counterparty. Exposure at default

Exposure at default is actually the total exposure to the counterparty risk,McNeil et al.

(2015). For instance, in the case of a loan or bond it is the principal amount agreed. As one can easily assume, this particular risk measure depends on the time of default. As a consequence, high level of uncertainty is included. In the case of a loan, the exposure at default decreases as the time to maturity is getting closer. In the case of a bond or an over-the-counter derivative, the uncertainty of the future term structure of interests will make the exposure at default measure to fluctuate over time.

Loss given default

A loss given default measure represents the percentage of the total exposure to the risk that is going to be lost if a default event occurs,McNeil et al.(2015). As it is mentioned above, one of the most common techniques to decrease the exposure to the counterparty risk is the collateralization agreement. In particular, after a collateral asset being sold or a debt has been restructured, the final loss will be less for the financial institution. This consists the actual difference between the total exposure and the final loss at a default event and usually the first one is higher than the second one.

Furthermore, in some financial instruments, the loss depends on the exact time of default and not the whole year. For instance, we will consider an interest swap where a bank pays a fixed rate and receives a floating rate from the counterparty. We will also assume that the floating rates are higher and lower than the fixed ones at the beginning and at the end of the year respectively. A default event of the counterparty occurring at the beginning of the year will cause a significant loss to the bank since the floating amount to be deposited is larger than that with the fixed rate, that is to say the derivative has a negative value for the bank. However, because of the default, the bank will never receive these amounts. On the other hand, a default event occurring at the end of the year will cause a small or none loss to the bank, since the fixed rate becomes

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larger than that of the floating rate. Therefore, the bank has to deposit the difference of these amounts although the counterparty has defaulted.

2.3 The need of a regulatory framework

Many decades ago, credit risk has been acknowledged as being of paramount impor-tance for the financial sector. Initially, it was recognized by the financial institutions themselves who tried to model it internally. Within the last years, especially after the financial crisis of 2007-2008, financial institutions around the world faced the size of loss that is possible to occur as a consequence of a default event or inadequate credit risk modeling. As an additional outcome, people’s trust to the banks was lost. This fact led to liquidity problems as well as a significant increase in the Probability of Default since bank account holders withdrew their deposits under the fear of bank’s insolvency.

There is no doubt that financial institutions should implement advance risk manage-ment techniques in order to quantify expected as well as unexpected risks, accompanied with specific regulations. In particular, capital is needed to cover every possible loss that might occur. Its size will be different per enterprise since it depends on the risk that is included in their portfolio. Additionally, a committee or an organization outside the financial institutions should supervise the implementation of the specific regulations regarding the required capital. Supervision should be performed by one worldwide or-ganization or committee outside the financial institutions so that trust is built not only among the financial institutions but also between banks and account holders. Therefore, it is of crucial importance not only to model the risk in order to quantify it but also to perform outsourcing supervision .

Due to the above-mentioned reasons, the Basel Committee was established by the central bank Governance of the Group of Ten countries at the end of 1974, Goodhart

(2011). Its aim is to enhance financial stability in the banking sector by offering consis-tent and adequate banking supervision worldwide. Since then, the Basel Committee has enriched the banking regulation with three accords on capital adequacy, that is to say Basel I, Basel II and Basel III,Basel Committee on Banking Supervision(2018). All of them were followed by many supplements and revised versions throughout the years.

Basel I

The first regulatory framework was released by the Basel Committee in July 1988 to be implemented by the end of 1992. It focused on defining standard limitations on the credit risk taken by the banks.

In particular, claims were assigned to a category with a risk-weight (w) ranging between 0% and 100% based on the level of credit risk (e) they were exposed to. The risk-weight assigned to each category was calculated by the Basel Committee according to publicly available data on default probabilities and loan losses. Banks were not allowed to quantify it internally, that is to say using their own data. Additionally, credit ratings were not taken into consideration. Apparently, several claims assigned to a specific category had identical level of credit risk exposure independently of the counterparty’s credit rating.

The main objective of the first Basel Capital Accord, that is to say Basel I, was to define an international measurement system for the capital required to secure bank’s solvency. As a consequence, the minimum Capital Requirement ( RC ) was introduced and established to be at least 8% of the risk-weighted assets. Moreover, the risk-weighted assets ( RWA ) were the product of the total exposure to a risk and its corresponding risk-weight, McNeil et al. (2015). This rule was supposed to be implemented on every

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category of risk-weighted claims mentioned above : RWA = w ∗ e

RC = 8% ∗ RWA.

In this way, banks were allowed to increase their risk-weighted assets as long as 8% of them is kept aside as a buffer.

Basel II

A revised version of Basel I was published in June 2004 named International Convergence of Capital Measurement and Capital Standards : a Revised Framework but also known as Basel II. In comparison to the Basel I framework, it has been significantly improved by using stronger risk management practices.

Initially, the Basel Committee introduced the new regulatory framework as a 3-pillar system, McNeil et al. (2015). The first pillar’s main subject is the quantification of the minimum capital requirement regarding the credit and operational risk. The second pillar is referring to the quantification of any residual risk not considered in the first pillar. As for the third pillar, guidelines for clear and transparent bank reports on the quantification under pillars 1 and 2 are provided.

Furthermore, the minimum Capital Requirement is established to be at least 8% of the risk-weighted assets, the same as in Basel I. However, the formula used in Basel II is more advanced. In particular, two ways of defining the risk-weight per claim are pro-vided, that is to say the Standardized and the internal-ratings-based approach,McNeil et al.(2015).

Under the Standardized approach, the risk-weights are defined by the Basel Com-mittee as in Basel I. However, in Basel II they range from 0% to 150%, credit rating is taken into consideration and in case the counterparty has not been assigned with any credit rating,then a fixed risk-weight is provided.

The internal-ratings-based approach, in short IRB, is divided into two subcategories. Under the first subcategory, named Foundation IRB approach, banks are given with the possibility to estimate the Probability of Default internally using their own data and methodology. However, the data used for estimating the PD must be of minimum 5 years and with a horizon of 1 year only. The risk measures exposure at default and loss given default are estimated by the Basel Committee and the systematic risk is taken into account. Under the second subcategory, named Advanced IRB approach, all the three risk measures mentioned previously can be estimated from the bank internally. At this point it is worth mentioning that the formula used for calculating the risk-weights is given by the Basel Committee and the risk measures estimated by the banks themselves are used only as inputs in it.

Basel III

After the financial crisis in 2007-2008, there is no surprise that the Basel Committee issued a new regulatory framework, known as Basel III. It was issued in December 2010 and is supposed to be implemented in 2019. There is no significant change with regards to the credit risk evaluation as in Basel II. Therefore, it will be left out of focus.

2.4 Firm-value and reduced-form models

At the time when a financial institution has to decide whether to grant a loan to a corporation or not, the first thing that is going to be analyzed is the official annual balance sheet. Information included in a balance sheet, for instance total assets, total liabilities and total shareholder’s equity, can provide a financial analyst with a general

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idea regarding the firm’s financial health as well as its Probability of Default. In partic-ular, a corporation finances itself by equity, for instance issuing shares, and by debts. Therefore, the total assets are the sum of its total liabilities and its total shareholder’s equity. Consequently, a negative shareholder’s equity works as a red flag for analysts since it means that the firm’s total assets are not enough to cover its total liabilities.

In a different perspective, the shareholder’s equity is treated as the firm’s net or book value. The former value consists of the market value of assets after subtracting the total liabilities while the second value is the residual of the book value of assets minus the book value of liabilities. In either case, a negative value of the shareholder’s equity does not enhance the corporation’s trustworthiness regarding the credit risk.

Furthermore, a more advanced method for evaluation of a firm’s financial health is required. Over the years, many researchers and financial analysts have created many models based on the above-mentioned concept of a firm’s value. Their aim was to create a model that can estimate the Probability of Default of either one firm or a portfolio of firms by using different perspectives.

Generally, the default models that have been developed can be divided into two main categories according to the mechanism by which default occurs, the structural or firm-value models and the reduced-form models, McNeil et al. (2015). In the former category, the models are supposed to be static. That is to say, the Probability of Default will be estimated using a pre-defined default point, a stochastic variable representing the asset value and a fixed time horizon. The default point is fixed throughout the time horizon and is estimated by the modeler or the firm’s executive committee. Usually, it lies somewhere between the short and the long liabilities of the firm. The structural models are built on the firm’s net or book value. Therefore, the default occurs at the time when the value of the asset stochastic variable is less than the default point,Crosbie and Bohn(2003).

In the reduced-form category, emphasis is given to the evolution of the credit risk in time, hence they are dynamic models. The perspective here differs from that in the structure models. The mechanism by which a default occurs is left unspecified and an analysis of the credit-risky securities is performed. Hence, the default time is assumed to be a non-negative random variable rather sensitive to economic variables, McNeil et al.(2015).

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The Merton model for default

prediction

It is of great importance for someone, especially for financial institutions, to be able to estimate a firm’s Probability of Default by using publicly available data. As a con-sequence, many financial analysts, academic researchers and financial institutions have tried to create an efficient model for default prediction over the years. However, a lot of obstacles were in their way, for instance an established way to price a derivative or a methodology to estimate the market value of the total assets.

It was in 1973 when the three economists named Fischer Black, Myron Scholes and Robert Merton introduced in their paper the well-known Black-Scholes-Merton formula for option pricing under specific assumptions, Black and Scholes (1973). The model became world widely used by many investors and option traders. After almost one year, 1974, the economist mentioned previously, that is Robert Merton, was the first who developed a structural model for default prediction, Merton (1974). The model, known as original Merton model, can predict whether a firm is likely to default or not at a specific point in time by assuming the firm’s equity as an European call option and then adopting the Black-Scholes formula in order to evaluate it. Until now, it is considered as a benchmark for all firm-value models. It is needless to say that many developments on the original model as well as extensions of it have followed until the present.

Later, the KMV Corporation was inspired by the original Merton model and man-aged to develop it further by using observed equity data and introducing the mea-sure called Distance-to-Default (DD),Crosbie and Bohn(2003). Since then, this KMV-Merton model is referred as KMV-Merton model. In 2002, the rating agency Moody’s bought the KMV Corporation and after a thorough research managed to introduce the well known Moody’s KMV model or EDFTMmodel. It exploits an enormous proprietary database of historical default events from all over the world in order to map empirically the DD and the Probability of Default. The exact methodology used in the EDFTMmodel is proprietary and it is being sold to the Moody’s Analytics subscribers.

Before we proceed with the introduction of the original Merton model, it is worth explaining the core elements upon which the original Merton model was built. Briefly, the original Merton model assumed the firm’s equity value as a European call option and tried to evaluate it using on of the most well-known option pricing techniques, that is the Black-Scholes-Merton formula. In order someone to better understand the implementation as well as the assumptions of this option pricing formula, a detailed explanation of the options, the risk-neutral world and the Black-Scholes-Merton formula itself will precedes the introduction of the original Merton model.

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3.1 The Black-Scholes-Merton model

3.1.1 Risk-neutral world

It was difficult to create a widely common model for option evaluation since we should take into account the level of risk aversion by every investor. The risk preferences of an investor are totally connected to the risk premium required. That is to say, the higher the level of risk aversion by an investor, the higher the risk premium required for taking over the risk of a given stock while a less risk-averse investor might require lower risk premium for the same stock.

The above-mentioned economists, Fischer Black, Myron Scholes and Robert Merton, managed to successfully introduce their model for option evaluation in a risk-neutral world, that is a model independent of any risk preference,Hull and Basu(2016). In this way, they established a common method to evaluate an option world widely, making it more easy and more accurate to compare the value of two or more options. Therefore, two assumptions are adopted in a risk-neutral world :

• the risk-free rate is used as the expected return on a stock

• the risk-free rate is used as the discount rate for the expected payoff of a financial instrument.

3.1.2 Options

At this point, it is worth mentioning the definition of a European call option. Generally, options are contracts traded on exchanges as well as in the over-the-counter market. In each option, there are some specifications defined at the time of purchase, that is the underlying asset, the strike price, the expiration date and the type of the option. There are two main types of options available in the market, the call and the put options,Hull and Basu(2016). A call-option holder has the right to buy while the put-option holder has the right to sell the underlying asset at the strike price on the expiration date. In addition, another two subcategories of options exist, the European and the American options. The difference between them is the possible time of exercising an option. The European options can be exercised only at the expiration date, while the American options can be exercised at any time up and including the expiration date.

In this paper, we are going to use the European call option on a stock (St) with a strike price (K) and an expiration date (T ). A holder of such an option has the right but not the obligation to buy the stock at the time T at the price K. Therefore, the payoff for the option-holder at the expiration date will be

max(ST − K, 0). (3.1)

3.1.3 Stock price as a stochastic process

In the Black-Scholes-Merton model, the stock prices are assumed to follow a logNormal distribution under the real world probability measure, Hull and Basu(2016). In partic-ular, the stock price is assumed as a stochastic process St which follows a Geometric Brownian Motion, that is a continuous time stochastic process where the logarithm of the stock price follows a Generalized Brownian Motion. Therefore, the following stochas-tic differential equation is being satisfied by the stock price :

dSt= µStdt + σStdWt. (3.2) At the latter formula, Wt follows a Brownian Motion, that is Wt− Wt−∆t=

√ ∆tZ where Z ∼ N(0, 1), µ is the expected rate of stock return per year and σ the per annum

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volatility of the stock prices. Both µ and σ are constant.As a solution to the equation (3.2), we are given that :

ST = S0e(µ−σ 2_{/2)T +σW} T ST ∼ logN(S0eµT, S02e2µT(eσ 2_T − 1)). (3.3)

In the latter formula, ST and S0 represents the stock price at time T and at current time (zero) respectively.

Additionally, by using the Ito’s Lemma, we can derive the process that the ln ST follows: d ln St= (µ − σ2/2)dt + σdWt

ln ST ∼ N (ln S0+ (µ − σ2/2)T, σ2T ).

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3.1.4 Assumptions adopted by the Black-Scholes-Merton model In order for the Black-Scholes-Merton model to be accurate, some assumptions have been made, Hull and Basu(2016).

Firstly, the two main assumptions adopted are the evaluation in a risk-neutral world and that the stock prices are logNormally distributed. The model does not take into account the risk preferences of each investor, that is the expected rate of return µ. Therefore, it replaces the drift rate µ under the logNormal distribution of the stock prices with the risk-free interest rate r. Additionally, both the risk-free interest rate and the volatility of the stock prices are assumed to be constant.

Secondly, it is assumed to be there no transaction costs as well as no tax. Thirdly, the Black-Scholes-Merton model can evaluate only European options with no dividends paid during its life. Lastly, there is no arbitrage opportunity.

3.1.5 The Black-Scholes-Merton model

As an investor, you want to be able to compare the values of two options in order to come to a decision of purchasing one of them. One optimal way to do so is to compare the current prices of each call option. Since the payoff (3.1) is known, one would guess that only a discounting of its payoff to the present suffices. However, the crucial question is which discount rate you are going to use.

The Black-Scholes-Merton model is capable of determining the current price of a Eu-ropean call option (Ct) on a stock (St) by adopting the above-mentioned assumptions,

Hull and Basu(2016). Under the risk-neutral valuation property, the formula discounts the expected payoff at the maturity with the risk-free interest rate. The latter, in com-bination with the logNormal assumption for stock prices, gives us the following well known model: C0 = S0Φ(d1) − Ke−rTΦ(d2) d1 = ln(S0/K) + (r + σ2/2)(T ) σ√T d2 = d1− σ √ T . (3.5)

In the latter formula, we assume that the current time is t = 0 and the time to maturity T. Not only the strike price (K) but also the continuously compounded risk-free interest rate (r) as well as the stock price volatility (σ) are assumed to be constant and known. The function Φ(*) is the cumulative distribution function for a standardized normal distribution, that is a normal distribution with mean equal to zero and volatility equal to one. Lastly, the S0and C0 represent the stock price and the option price at the current time t = 0, respectively.

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3.2 The Merton model

3.2.1 The original Merton model

It was Robert Merton who developed the first firm-value model for default prediction,

Merton(1974). As this model-category implies, the original Merton model was developed on the fact that the assets of a firm are the sum of its equity and its liabilities. Therefore, a default event might occur once the value of the assets are smaller than that of the liabilities, that is to say equity has a negative value, Crosbie and Bohn (2003). As a conclusion, the equity value of a firm Etis assumed to be a European call option on the firm’s asset value At with maturity time T and strike price the firm’s liability value L. Moreover, it is assumed that there are no transaction costs or taxes. The asset trading is continuous over time and no dividends will be paid until the option matures at time T . As for the firm’s liabilities, they assumed to be one zero-coupon bond with principal L and maturity time the same as that of the option. In this spirit, one class of equity is assumed as well. Additionally, according to the definition of the European-category option, default might occur only at the time of maturity and not earlier.

As one would expect, the Black-Scholes-Merton model is used in the original Merton model in order to evaluate the firm’s equity value, that is to say the European call option on the firm’s assets. As a consequence, the two main assumptions adopted in the former model will be adopted to the second one as well, McNeil et al. (2015), Hull and Basu

(2016). Firstly, under the physical probability measure P, the value of the firm’s assets is assumed as a stochastic process At which follows a Geometric Brownian Motion, that is equation (3.2) after replacing stock price St with the asset value At. Accordingly, the symbols µ, σA and Wt represent the expected asset value return, the per annum volatility of the asset value and the Brownian Motion respectively.

Secondly, the valuation will be performed in the risk-neutral world ignoring any risk preference and using the risk-free interest rate instead of the drift rate of the asset value. By definition, the original Merton model implies that the option’s payoff to the equity-holders is ET = max(0, AT − L). This information, in combination with the logNormal distribution of the firm’s asset value under the risk-neutral world, results to the current value of the firm’s equity. That is equation (3.5) after replacing the current option price C0, the current stock price S0 , the stock price volatility σ and the strike price K with the corresponding current equity value E0, the current asset value A0, asset value volatility σA and the liability value at the time of maturity L.

E0= A0Φ(d1) − Le−rTΦ(d2) d1= ln(A0/L) + (r + σ2A/2)T σA √ T d2= d1− σA √ T . (3.6)

As a result, the Probability of Default under the risk-neutral world was calculated as the probability that the asset value at the time of maturity is less or equal than the liability value. By taking into account the dynamics in equation (3.2) after replacing the drift rate with the risk-free interest rate in the Black-Scholes-Merton model due to the risk-neutral world assumption, one ends up with the following Probability of Default under the risk-neutral world:

P (AT ≤ L) = P (ln AT ≤ ln L) = Φ ln(L/A₀) − (r − 1/2σ2 A)T σA √ T = Φ(−d2)

3.2.2 The Merton model

The original Merton model motivated many academic researchers as well as financial analysts in order to end up with a further developed model for default prediction of

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corporations. As it was mentioned above, the KMV Corporation managed to develop the original Merton model significantly. This new model is also known as the KMV-Merton model. In this paper, it will be called as the KMV-Merton model.

The Merton model adopts all the assumptions used in the original Merton model with one more that the model is to be implemented to publicly traded companies,

McNeil et al.(2015). In addition, it uses the market value of a firm’s assets and equity,

Bharath and Shumway(2004). The current market value of equity is easily observable for a publicly traded firm since it is the multiplication of the share price with the number of the outstanding shares. However, the market value of assets is not always possible to be observed since some of the assets might not be available for trading. Therefore no market price exists for them. Thus, we end up with two unknown variables, that is the market value of the total assets Atand their volatility σA.

In order to estimate the market value of a firm’s assets along with their volatility, the Merton model uses two equations. The first equation was obtained in the original Merton model under the assumption that the equity value is a European call option on the firm’s assets in a risk-neutral world, that is (3.6). The second equation defines the relation between the equity volatility σE and the asset volatility σA. Under the Merton’s model assumptions, we get the equation σE = (A_Et_t)∂E_∂AσAdirectly from the Ito’s Lemma. By taking into account that under the Black-Scholes model ∂E_∂A = Φ(d1) holds as well, we end up with the following equation used in the Merton model :

σE = A_t

Et

Φ(d1)σA. (3.7)

Then, by solving equations (3.6) and (3.7) simultaneously, we obtain the market value of a firm’s assets as well as their volatility, Bharath and Shumway(2004).

Moreover, in the Merton model, the KMV Corporation introduced a new variable named Distance-to-Default (in abbreviation DD) under the real world measure. After obtaining the market value of the firm’s assets along with their volatility as described above, the Merton model implements these values in the estimation of the Probability of Default under the real world probability measure. Therefore, under the assumption that the logarithm of the asset value follows a Normal distribution, (3.4), the DD variable is defined as follows: DD = ln A0− ln L + (µ − σ 2 A/2)(T ) σA √ T . (3.8)

It incorporates all the variables calculated so far except that of the firm’s equity and the equity volatility. Some analysts translate it into the number of standard deviations a firm’s asset value is away from its value of liabilities.

As a result, the Probability of Default is given by the following formula:

P D = P (AT ≤ L) = P (ln AT ≤ ln L) = = Φ ln(L/A₀) − (µ − 1/2σ2 A)T σA √ T = Φ(−DD). (3.9)

As one would notice, the variable DD is rather similar to the variable d2 after replacing the risk-free interest rate r with the drift rate µ. The main idea behind is that the DD measures the Probability of Default under the real world measure while the d2 measures the Probability of Default under the risk-neutral world. Of course, the former one is of great interest to the investment world since it takes the level of risk aversion into account. If you would compare the two formulas, that is to say Φ(−DD) and Φ(−d2), we would end up with the following equation:

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Φ(−d2) = Φ(Φ−1(−DD) + (µ − r) √

T /σA). (3.10)

In the latter equation, the part (µ − r)/σA is actually the Sharpe ratio of the asset value where the numerator represents the excess asset return, that is to say the expected return minus the risk-free rate. By definition, the Sharpe ratio measures the asset return on top of the risk-free rate after a per unit change of its volatility. Therefore, investments with high sharpe ratio are relatively attractive to investors with low level of risk aversion. Equation (3.10) is used sometimes in the Merton model concept to convert a Prob-ability of Default under the risk-free world measure into a ProbProb-ability of Default under the real world measure and vice versa. However, we need to keep in mind that it provides poor results when the assets do not follow a logNormal distribution.

3.3 Capital asset pricing model

Another variable that needs to be estimated in order to obtain the Probability of Default under the real world probability measure (3.9), is the drift rate µ of the log asset value changes. Under the logNormal assumption for the asset values, µ − σ_A2/2 is the expected change in the log asset values per year and eµ the expected change in the asset values per year. As a consequence, the expected asset return has to be estimated. Generally, there are many methods in order to define the expected asset return. For instance, you could assume it equal to the risk-free rate, define it as a fixed number according to the investor’s level of risk aversion or you could estimate it using the well-known CAPM (which stands for Capital Asset Pricing Model), Hull and Basu (2016), L¨oeffler and Posch (2011).

The general idea behind the CAPM is to estimate the expected asset return by taking into account the systematic risk. In general, a portfolio of assets caries two kind of risks, the systematic and the idiosyncratic risk. The former one is referring to the market risk and it is common for all the securities. It cannot be diversified so every investor takes it over. On the other hand, idiosyncratic risks can be diversified.

As one would assume, an investor is expecting a compensation for taking over the market risk, on top of the risk-free rate of return. This kind of compensation is known as the market risk premium E(RM− r), where RM is the return on the market portfolio and r is the risk-free rate. In order this model to provide the most accurate results, one should choose a well diversified index as a representative of the market portfolio, for instance the S&P 100 or S&P 500.

Additionally, the CAPM introduces the systematic risk measure beta β. It indicates how risky an asset is in comparison to the overall market risk. It can be estimated by running a linear regression on the excess returns on assets against the excess returns on market, that is RAt− r = β(RM − r), where RAt stands for the asset value return.

Then, the value of the beta β is the second coefficient.

Summarizing, according to the CAPM, the expected asset return (E[RAt]) is the

sum of the risk-free rate of return and the risk premium :

E[RAt] = r + β(E[RM] − r). (3.11)

Finally, the drift rate of the assets is the logarithm of the expected asset returns, due to the logNormal assumption mentioned above :

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3.4 Empirical methodology

3.4.1 Assumptions

In order to empirically apply the Merton model on companies using real data, we need to define the assumptions that will be adopted.

To begin with, three publicly traded companies were selected to be used for this study. All three of them are companies with limited liability. That is to say, the share-holders in such a company are responsible up to the amount they have invested in it and not for its total debts.

Default is the situation where the total asset value (At) of a company falls below a specified default point. In our case studies of limited liability companies, we will define the default point as the total book value of the firm’s liabilities (Lt),L¨oeffler and Posch (2011). Therefore, once the asset value of a firm becomes less than the value of liabilities, the equity value (Et) becomes negative meaning that the firm is worth nothing. Then, the equity holders have to decide whether they will keep the firm or they will give it to the creditors at no cost. As a consequence, we will assume that the equity value of a firm is a European call option on the firm’s asset value with strike price the total value of liabilities and maturity time T .

We will estimate the Probability of Default for three companies on a one-year hori-zon, that is to say T = 1. In order for the rest of the data to be aligned with this assumption, we will also assume the firm’s liabilities to be a zero-coupon bond with a maturity time of one year and principal L, that is the total liabilities as mentioned on the firm’s balance sheets. Although in real world, companies have many liabilities with several maturity dates, in this paper we will assume that the firms have only liabilities which mature in one year. The reasons for taking such a decision is firstly for the sake of convenience and secondly because the data needed for that case are not publicly avail-able. However, we do keep in mind that this decision will slightly decrease the reliability of the outcome.

Furthermore, we will adopt the same assumptions that were used in the Black-Scholes-Merton model in order to estimate the equity value, Bharath and Shumway

(2004), Afik et al. (2016) and Barsotti and Del Viva (2015). That is to say, there will be no transaction costs, no taxes, no arbitrage opportunity and no dividend payments. Additionally, we will assume a risk-neutral world as well as that the underlying asset value follows a logNormal distribution under real world probability measure, that is the logarithm of the asset value follows a Normal distribution.

Under the risk-neutral-world assumption, a risk-free interest rate is being used. Though, it is of common knowledge that in the real world there is no risk-free in-terest rate since every single investment carries at least a tiny amount of risk. In our case study, we will use the one-year US treasury rate as the risk-free interest rate. Al-though the three-month US treasury rate is most commonly used as a risk-free rate for firms that are based in the United States of America, we chose the corresponding rate of one-year maturity in order to be aligned with our above-mentioned assumption of one-year horizon.

Under the Normal distribution for the logarithm of a firm’s asset value in the real-world probability measure, we come to the decision that equation (3.4) holds after replacing the stock price ST with the asset value AT and the σ2 with σA2 which will be the symbol of the per annum variance of the log asset value changes. Then, µ will remain as the symbol for the drift parameter. Therefore, µ − σ_A2/2 is expected change in the log asset values, including the assumption of one-year horizon.

In addition, after adopting all the Black-Scholes-Merton model’s assumptions men-tioned above, the current equity value of a company as a European call option can be estimated using the equation (3.6).

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event might occur only at the maturity, that is T = 1 as long as t = 0 symbolizes the present. As a consequence, we are interested about the equity,asset and liability value of a firm at time T .

As a last assumption, we take into granted that the balance sheet data are genuine and not manipulated. Since the only source of the liability value of a firm is its balance sheet, this assumption is of crucial importance.

3.4.2 Estimation of the equity, asset value and volatility

In order to estimate the Probability of Default for a firm, we need to obtain all the necessary variables. Some of them are directly observable, some others are not. Addi-tionally, in order to robust the performance of the Merton model, we will use the market value of the firm’s equity and assets. Consequently, the volatility of the assets will be calculated using the market values of the assets.

Since the three companies used in our paper are publicly traded, the daily market value of the equity for a period of one year can easily be observed. However, the market value of the assets and their volatility are unknown. Generally, it is difficult and some-times even impossible to obtain the market value of a firm’s assets. The reason is that some of the assets might not be tradable in the market at the time of the valuation. For instance, real estate and businesses.

In order to tackle the problem of obtaining the market value of the company’s assets, we will apply the iterative estimation approach, L¨oeffler and Posch (2011), instead of the method introduced in the Merton model. This alternative approach produces rather similar results to those from the simultaneous solution of the equations (3.6) and (3.7),

Bharath and Shumway(2008).

According to the initial step of the iterative estimation approach, we obtain the mar-ket value of the equity by multiplying the stock price with the number of the outstanding shares per trading date. The book value of liabilities and the risk-free interest rate have already been obtained as it was described above. In this way, we end up with the fol-lowing sets of observations per variable Et−α, Lt−α and rt−α, where α = 0, 1, 2, ..., m represents the trading dates per one-year period to be studied, m represents the total trading dates per period and t represents any time between the present t = 0 and the maturity T .

In order to perform the iterations, we rearrange the equation (3.6) while we keep the formulas of the d1 and d2 the same:

At= [Et+ Lte−rtΦ(d2)]/Φ(d1) In this way we end up with a system of m + 1 equations:

At= [Et+ Lte−rΦ(d2)]/Φ(d1)

At−1= [Et−1+ Lt−1e−rt−1Φ(d2)]/Φ(d1) . . .

At−m= [Et−m+ Lt−me−rt−mΦ(d2)]/Φ(d1).

(3.13)

At this point it is worth mentioning that d1 and d2 change over time since the variables included in them are changing over time. However, for reasons of convenience we do not use the corresponding notation. Though, we should keep in mind that they differ per equation in the above system, (3.13).

As initial values for the assets, we define the sum of the market value of the equity and the book value of the liabilities for all the trading days of the prior year until the present time t. That is to say, we end up with the initial system of m + 1 equations where At−α = Et−α+ Lt−α for every α = 0, 1, 2, ..., m. The initial value of the asset volatility is obtained by taking the standard deviation of the log returns of the initial

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asset values we just calculated. Then, in order to convert it into a per annum volatility, we multiply it with the square root of the total trading dates m.

Then, we perform the 1st iteration. We compute the d1 and d2 per trading date using the initial values of the assets and their volatility. We apply the d1 and d2 values in the system of equations (3.13) and we obtain new values for the assets At−α where α = 0, 1, 2, ..., m. We calculate the per annum volatility of these new asset values with the same way as we calculated it for the initial asset values.

We proceed to the next iteration until the sum of the squared differences between the asset values obtained in the current iteration and those obtained in the previous one is smaller than 10−10. Our aim is to obtain asset values and their volatility such that the iteration procedure converges. For convenience, we symbolize the market values of assets obtained from the iteration approach as At−α where α = 0, 1, 2, ..., m and their volatility σAfor the rest of the case study.

3.4.3 Estimation of the drift rate

The last variable that needs to be estimated in order to obtain the Probability of Default under the real world probability measure is the drift rate µ of the log asset value changes. We will estimate it by applying the CAPM, (3.11).

In our case study, we take the S&P 500 index return as a representative of the return on the market portfolio. The index S&P 500 is based on the market capitalization of the 500 largest companies in America. Therefore, it is well diversified and it will represent a market portfolio successfully.

Initially, we calculate the excess returns on the daily S&P 500 index , that is RM−r, and on the market value of assets, that is RAt − r. Thus, we subtracting the risk free

rate from both the return of the S&P 500 index and the market value of assets per trading date.

Then, in order to estimate the risk measure beta β for market value of the assets, we run a linear regression on the excess returns on assets against the excess returns on market, that is RAt− r = β(RM − r). In this way, the value of the beta β is the second

coefficient.

The next step is to calculate the expected asset return as the CAPM defines. Firstly, we assume as the market risk premium the fixed value of 4%. Secondly, the expected asset returns are obtained from the equation (3.11)

Finally, the drift rate of the assets is obtained from the equation (3.12).

Then, the mostly wanted Probability of Default can be obtained from the equation (3.9)

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Chapter 4

Data and results

The primary objective of this paper is to estimate the Probability of Default for three selected firms using the Merton model, under specified assumptions. All the three se-lected firms have defaulted in the past. This fact gives us the advantage of comparing the estimated probabilities of default up to two years before the default date. In this way, we can evaluate the performance of the Merton model and get an indication of how many months in advance the model warns us about a possible default event.

4.1 Companies

Three firms were selected to be used for this study, the Arch Coal Inc., the Vanguard Natural Resources LLC and the Ford Motor Co.

All three of them are companies with limited liability. This property, in combination with the assumption in the Merton model that the equity is treated as an European-call option, gives the equity holders the possibility to exercise their walk-away option and give the company to the creditors at no cost, in case the liabilities are larger than the assets. In mathematical terms, the payoff of the equity to the shareholders at the time of maturity can be either positive and equal to AT − LT or zero, that is to say max(AT − LT, 0). It cannot be negative.

Arch Coal Inc.

Arch Coal was formed in 1997 when two coal companies merged, the publicly traded Ashland Coal, Inc. and the privately held Arch Mineral Corporation. It is a leading coal producer for power generation and steel-making not only in the United States of America, where it is originated, but also worldwide.

The company was listed on the New York Stock Exchange with the ticker symbol ACI until and including the 8th of January 2016. The reason for stop being listed is that it filed for the chapter 11 bankruptcy protection on the 11th of January 2016,

Missouri Eastern Bankruptcy Court(2016). The first indication of its financial troubles became publicly available in the 2015’s fourth quarter balance sheet. The shareholder’s equity became negative, meaning that the total liabilities exceeded its total assets. At this point, it is worth mentioning that a negative shareholders’ equity is presented in the balance sheets as positive under the name shareholders’ deficit.

The company managed to emerged from the chapter 11 protection on the 5th of July 2016, Wikimedia Foundation (2018). Since then, it has been listed on the New York Stock Exchange with the ticker symbol ARCH.

Vanguard Natural Resources LLC

Vanguard Natural Resources LLC was founded in 2006 and it is a natural gas and oil company originated in the United States of America. The company was listed on the

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New York Stock Exchange with the ticker symbol VNR.

The company filed for the chapter 11 bankruptcy protection on the 2nd of February 2017, Vazza and Kraemer (2018). According to its balance sheets, the firm’s liabilities exceeded its assets for the first time on the 31st of December 2015. However, the firm managed to emerged from the chapter 11 protection on the 1st of August 2017 with the new name Vanguard Natural Resources Inc.,Vanguard Natural Resources(2017). Ford Motor Co

Ford Motor Co is the worldwide famous automaker company founded in 1903 by Mr. Henry Ford. It has become one of the largest automaker companies in the whole world. Only in 2017, the company produced 6.6 million vehicles.

Ford Motor Co has been listed on the New York Stock Exchange with the ticker symbol F. Additionally, it is a component of the S&P 500 American stock market index which is based on the market capitalization of the 500 large companies in America.

According to the annual global corporate default study and rating transitions pub-lished by the S&P Global,Vazza and Kraemer (2018), Ford Motor Co defaulted on its payments in 2009. However, there is no official announcement that the firm entered the chapter 11 bankruptcy protection. The reason is that it managed to erase some of its debts after negotiations with its creditors.

4.2 Data

Arch Coal Inc.

Since the default date of the Arch Coal Inc. was on the 11th of January 2016, we are going to estimate the Probability of Default per quarter and within a period of two years. Therefore, the older one-year Probability of Default will be as of the 31st of December 2013 and the latest one will be as of the 31stof December 2015. For the former case, we will need data for the period [31/12/2012, 31/12/2013] while for the latter case we will need data for the period [31/12/2014, 31/12/2015]. Corresponding data will be needed for the dates in between.

At this point, it is worth mentioning that the default probabilities will be estimated per quarter by using data of the whole last year and not by converting the one-year default probability into four quarters. For instance, the Probability of Default as of 31st of March 2015 will be estimated using data of the period [31/03/2014, 31/03/2015]. The reason of doing so is to not decrease the reliability of the outcome since data of a whole year will give a better impression of the actual credit situation of the firm.

Initially, we obtain the daily stock prices under the ticker symbol ACI for every trading date of the period [31/12/2012, 31/12/2015]. Then, we get the number of the outstanding shares in the market at the latest trading date of each quarter for the same period of time. The amount of the outstanding shares will remain the same throughout the corresponding quarter of the year. For instance, the number of the outstanding shares on the 31st of March 2015 will be the same for all the trading dates up to 30th of June 2015. The reason for doing so is that the Merton model assumes that there will be no changes of either the equity or the liability structure. Both the stock prices and the number of the outstanding shares were obtained from the CRSP (which stands for Center for Research in Security Prices) database through the Wharton Research Data Services from the Wharton School of the University of Pennsylvania, Wharton School

(1993b).

Additionally, from all the quarterly balance sheets of the firm for the period [31/12/2012, 31/12/2015], we collect the book value of the liabilities Lt under the section ”Total Lia-bilities”. The same assumption as the one for the number of outstanding shares, applies here. Therefore, the book value of the liabilities will remain constant throughout the

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corresponding quarter of the year. This information was collected from the Campustat database through the Wharton Research Data Services from the Wharton School of the University of Pennsylvania, Wharton School(1993a).

Moreover, we collect the one-year US treasury rate at the latest trading date of each quarter for the period [31/12/2012, 31/12/2015], www.macrotrends.net (2018). As it was applied above, the one-year US treasury rate will remain constant throughout the corresponding quarter of the year.

Another two variables are needed in order to estimate the asset drift rate, the S&P500 index and the one-year US treasury rate. Both of them must be daily observa-tions for the period [31/12/2012, 31/12/2015]. The information for the former variable was obtained from the CRSP database through the Wharton Research Data Services from the Wharton School of the University of Pennsylvania,Wharton School(1993b). Vanguard Natural Resources LLC

Almost the same kind of data will be needed for this firm as the ones needed for the Arch Coal Inc. In particular, the Vanguard Natural Resources LLC defaulted on the 2nd of February 2017. Therefore, the one-year Probabilities of Default will be obtained at the last trading date of each quarter between the 31st of December 2014 and the 31st of December 2016.

Next, we obtain the daily stock prices under the ticker symbol VNR as well as the number of the outstanding shares in the market at the latest trading date of each quarter for the period [31/12/2013, 31/12/2016]. Both of them were obtained from the CRSP database through the Wharton Research Data Services from the Wharton School of the University of Pennsylvania, Wharton School(1993b).

Additionally, from all the quarterly balance sheets of the firm for the above-mentioned period, we collect the book value of the liabilities Lt under the section ”Total Liabili-ties”,Securities and Commission (1984). Also, we collect the one-year US treasury rate at the latest trading date of each quarter for the same period, www.macrotrends.net

(2018).

Another two variables are needed in order to estimate the asset drift rate, the S&P500 index and the one-year US treasury rate. Both of them must be daily observa-tions for the period [31/12/2013, 31/12/2016]. The information for the former variable was obtained from the CRSP database through the Wharton Research Data Services from the Wharton School of the University of Pennsylvania,Wharton School(1993b).

Once more, the values of the liabilities, the number of the outstanding shares and the one-year US treasury rate will remain constant throughout the quarter of the year with the exception of the CAPM application where daily observations are required. Ford Motor Co

Following the same pattern, almost the same kind of data will be needed for this firm as the ones needed for the previous companies. In particular, the Ford Motor CO defaulted 2009. Therefore, the older one-year Probability of Default will be as of the 29th of December 2006 while the latest one will be as of the 31st of December 2008.

We obtain the daily stock prices under the ticker symbol F and the number of the outstanding shares in the market at the latest trading date of each quarter of the period [29/12/2006, 31/12/2008]. Both were obtained from the CRSP database through the Wharton Research Data Services from the Wharton School of the University of Pennsylvania, Wharton School(1993b).

Additionally, we obtained the book value of liabilities Lt, the one-year US treasury rate and the S&P500 index for the same time period and with the same frequency as in the previous companies. The sources of information were Campustat database through the Wharton Research Data Services from the Wharton School of the University of

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Pennsylvania, Wharton School (1993a), www.macrotrends.net (2018) and the CRSP database through the Wharton Research Data Services from the Wharton School of the University of Pennsylvania, Wharton School(1993b) respectively.

4.3 Results

The main objective of this paper was to estimate the Probability of Default for three specific companies in order to evaluate the empirical performance of the Merton model using real data. All three companies are based in the United States of America, they are from different industries and they have defaulted in the past. After obtaining all the necessary data, we implemented the methodology described above using the program-ming language R, R Development Core Team (2012). In particular, we ran the model for every company and for 0, 3, 6, 9, . . . , 21 and 24 months before the default date.That is to say, we end up with 9 Probabilities of Default. Every single Probability of Default was calculated by using data of the whole one past year and not by just converting an annual Probability into four quarters.The reason for doing so is that the Merton model is designed in such a way that only data from a whole past year can provide accurate results. In this way, we end up with a clear picture of how many months in advance is the model able to warn us for a possible default event.

4.3.1 Arch Coal Inc.

The table below shows the results obtained through the programming language R.

Date PD 31/12/2013 4.4% 31/03/2014 2.33% 30/06/2014 4.21% 30/09/2014 13.75% 31/12/2014 20.46% 31/03/2015 39.66% 30/06/2015 68.33% 30/09/2015 90.09% 31/12/2015 99.88%

One can see that the Probability of Default is low for the period between 24 and 18 months before the actual default date, that is the 11th of January 2016, with numbers between 2.3% and 4.21%. However, it increases from then onward. In particular, from a default probability of 13.8% 15 months prior to the default date, it goes to 68.3% 6 months prior to default and ends up to 99.9% some days before the company filed for the chapter 11 bankruptcy protection.

If we would have taken into account only the book values of the shareholder’s equity given in the firm’s quarterly balance sheets, our estimation of default would be rather different. The reason is that the book value of the shareholder’s equity becomes negative on the September 2015, that is 3 months before the company officially defaults.

As a conclusion, for the Arch Coal Inc. company, the Merton model gives us the first indication of a possible future default event 15 months in advance.

4.3.2 Vanguard Natural Resources LLC

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22 Georgia Skoufoglou — Empirical assessment of the Merton model Date PD 31/12/2014 6.27% 31/03/2015 8.36% 30/06/2015 7% 30/09/2015 19.09% 31/12/2015 46.58% 31/03/2016 78.06% 30/06/2016 78.28% 30/09/2016 82.26% 31/12/2016 76.45%

The same pattern is followed for the Vanguard Natural Resources LLC as the one followed in the Arch Coal Inc. That is, the Probability of Default is low for the period between 24 and 18 months before the company officially defaults on the 2nd of February 2017. However, the probability ranges between 6.27% and 8.36%. These amounts are neither very low so that investors would not expect a default event, nor very high so that a default event would be assured.

However, it is from the end of September 2015 , that is 15 months prior default, when the probability rises to 19.09% and the model gives us the first indication that the company might default in the near future. This indication is being assured by the default probabilities estimated for some months before the company filed for the chapter 11 bankruptcy protection. For instance, 2 months before the event the default probability increases to 76.45%

As a conclusion, for the Vanguard Natural Resources LLC company, the Merton model gives us the first indication of a possible future default event 15 months in ad-vance.

4.3.3 Ford Motor Co

The table below shows the results obtained through the programming language R.

Date PD 29/12/2006 3.88% 30/03/2007 37.98% 29/06/2007 30.22% 28/09/2007 38.22% 31/12/2007 9.78% 31/03/2008 18.03% 30/06/2008 14.61% 30/09/2008 34.35% 31/12/2008 73.42%

At this point, it is worth mentioning that the company faced several financial prob-lems within the year of 2007. This is clearly reflected in the above estimated Probabilities of Default as well as company’s quarterly balance sheets where the equity became nega-tive. For instance, the probability that the company will default within the near future was 3.88% in December 2006 and 38.22% in September 2007. Then, company restruc-tured its liability structure and its profit allocation efficiently. Therefore, its profit rose and the equity value became positive again. Consequently, the corresponding default probability dropped to 9.78%.

However, the financial crisis followed in 2009 before the company manage to ob-tain its financial strength back. The profits dropped dramatically and the equity value became negative once more. As one would expect, the probabilities of default start to rise as well from 18.03% on March 2008 to 73.42% in December 2008. As it was men-tioned previously, the company never filed for the chapter 11 bankruptcy protection, However, it defaulted in 2009 by not paying its obligations to the debtor on time, as it

An empirical assessment of the Merton model for prediction of the Probability

model for prediction of the Probability

of Default

Georgia Skoufoglou

Contents

Introduction

Credit risk

2.1

Financial instruments subject to credit risk

2.2

Risk measures

2.3

The need of a regulatory framework

2.4

Firm-value and reduced-form models

The Merton model for default

prediction

3.1

The Black-Scholes-Merton model

3.2

The Merton model

3.3

Capital asset pricing model

3.4

Empirical methodology

Chapter 4

Data and results

4.1

Companies

4.2

Data

4.3

Results