Week 5: Default Probabilities – Part 2

Week 5: Default Probabilities – Part 2

Summaries

• Week 5: Default Probabilities - Part 2 > Lesson 1: Merton's Model P1 > Video Lesson
• Week 5: Default Probabilities - Part 2 > Lesson 2: Merton's Model P2 > Video Lesson
• Week 5: Default Probabilities - Part 2 > Lesson 3: The KMV Model > Video Lesson
• Week 5: Default Probabilities - Part 2 > Summary > Video
• Week 5: Default Probabilities - Part 2 > The Sofa > The Sofa Video

Week 5: Default Probabilities – Part 2 > Lesson 1: Merton’s Model P1 > Video Lesson

• Merton’s Model is the prototype of the class of structural models of default.
• Structural models are an important family of models, in which the default of a company happens as soon as a stochastic variable representing some asset value of the company falls below some given threshold, often representing liabilities.
• In Merton’s model, for example, we will consider the total value of the firm’s assets.
• Given the presence of the threshold, structural models are sometimes called threshold models.
• Merton’s model, the KMV model and many others all belong to this large family.
• Looking at the Basel framework, Merton’s model falls in the class of internal rating-based models.
• Many extensions of Merton’s model have been proposed in the literature, and it also represents the basis for many important industry solutions, such as the KMV and the CreditMetrics models.
• It is such an influential model that it is still used as a benchmark, even if some of the assumptions of the model are not really plausible.
• The model is strictly connected to Black-Scholes formula.
• In the field of credit risk it plays the same role of the Black-Scholes’ model in option pricing.
• For what concerns debt, we assume it is represented by one single debt obligation, such as a zero-coupon bond, with face value B and maturity T. Let St denote the value of equity at time t, while Bt represents debt at time t. Markets are assumed to be frictionless, therefore the value of the company’s assets at time t, Vt, for every t from 0 and T, is given by the sum of St and Bt. An important assumption of Merton’s model is that a company cannot pay dividends or issue new debt until time T. Default occurs if the firm is not able to pay debt holders, i.e. by missing a payment on debt.
• In the basic model this may only happen at time T, that is to say at maturity.
• As we will see, this is one of the points of weakness of the model.
• BT, the value of debt at maturity, what debt holders receive, is the minimum between VT and B. This can also be seen as B minus the maximum between B-VT and zero.
• If you are familiar with European options, all this implies that the value of the firm’s equity at time T corresponds to the payoff of a European call option on VT, while the value of the firm’s debt at maturity is equal to the nominal value of liabilities, B, minus the payoff of a European put option on VT, with exercise price equal to B. All this reminds us of Black-Scholes’ model.
• The Wikipedia page on Black-Scholes’ model can be a good starting point.

Week 5: Default Probabilities – Part 2 > Lesson 2: Merton’s Model P2 > Video Lesson

• We have seen that according to that model, the value of equity at maturity is equal to the payoff of a European call on VT, the market value of the company’s assets.
• At TU Delft, the complete derivation and analysis of Black-Scholes formula and Merton’s model is part of my master course in Financial Mathematics.
• According to Black-Scholes-Merton, the value of equity today is equal to V0 times the cumulative distribution function of a standard Gaussian, computed in d1, minus the actual value of B times, again, the cumulative distribution function of a standard Gaussian computed in d2.
• We have seen that, in Merton’s model, default happens, at maturity T, if the value of the assets of the company falls below the liabilities’ threshold B. Using some advanced maths, it can be shown that the probability of default is equal to the cumulative distribution function of a standard Gaussian computed in -d2.
• According to Merton’s model, the PD increases in the amount of debt B, decreases in V0 and it increases in the volatility of assets, for V0 greater than B. All this is clearly in accordance with the economic intuition that excessive debt, or the excessive riskiness of assets, make default more plausible.
• In particular, for what concerns the market value of the company, when dealing with real-life and actual data, it is not possible to exactly observe it.
• The market value of a company corresponds to the sum of the market values of its equity and its debt.
• While the market value of equities is known, if the company is publicly traded, for what concerns liabilities, only a relatively small part of firm’s debt is fully known.
• We may know issued bonds, but other types of debt are more difficult to observe If we assume that the company we are observing is publicly traded, S0, the value of equity today, is known.
• V0 and sigmaV can be found numerically, by minimizing the sum of the squares of functions F and G, which are two functions in V0 and sigmaV, which we obtain by rearranging equations 1 and 2.
• Question 1: What are the values of V0 and sigmaV? Question 2: What is the probability of default of the company in one year? To solve this exercise, we will make use of R. The code we are going to use is available in the course materials.
• We start by defining the quantities, whose value is given in the text of the exercise.
• These include, among the others, the value of equity at time zero, that is to say today, equal to 3 million euros; the risk-free rate, sigmaS, maturity equal to 1 year and liabilities equal to 10 million euros.
• We then define the function that we minimize in order to obtain V0, the value of the company’s assets today, and sigmaV, assets’ volatility.
• What we get are two values for V0 and sigmaV that are 12.39 and 0.2124 respectively.
• We now compute d1 and d2 explicitly, and thanks to pnorm(-d2) we find that PD in one year is 12.7%. What happens if the value of equity today increases to 5 million euros? To answer this question, we re-run the code, after changing S0 to 5 million euros.
• The first ones have an interest in the firm investing in risky projects, that increase the volatility of the underlying “security”, but may guarantee higher returns; while the second ones prefer a less volatile and less risky assets’ value.
• Regarding the points of weakness, Merton’s model essentially assumes we are in a Gaussian world, in which rare and extreme events, black swans in the terminology of Nicholas Taleb, are not taken into consideration.
• In Merton’s model, default can only happen at maturity, say in one year, and not within that period, say within one year.

Week 5: Default Probabilities – Part 2 > Lesson 3: The KMV Model > Video Lesson

• Today, we will consider the KMV model, which is a very important industry model, derived from Merton’s one.
• The KMV model has been introduced in the late 80’s by KMV, a research driven company that soon became a leading provider of quantitative credit analysis tools.
• From a theoretical point of view, the KMV model is not that different from Merton’s one.
• As we will see, the KMV model essentially tries to overcome some of the flaws of Merton’s model, by making an extensive use of empirical data.
• The model, and the data set on which it relies, is now maintained and developed by Moody’s Analytics, which acquired KMV in 2002.
• A fundamental quantity in the KMV model is the so called Expected Default Frequency, or EDF, if we use the acronym, a registered trademark.
• In order to understand the way in which KMV obtains the EDF, we can use Merton’s model.
• As you can imagine, some specific parameters and implementations of the model are not completely known to the public, and we have to rely on what the company allows us to know.
• The difference of log(V0) and log(B) is approximately equal to V0-B over V0. If we substitute B with B tilde, which is the threshold used by Moody’s KMV, we have that our term A can be approximated by the DD! The next step, in the KMV approach, is to substitute, as we have anticipated, Phi bar with an empirical function, which we can call F bar KMV. This decreasing function is obtained using historical data.
• KMV has in fact estimated, for virtually every meaningful time horizon, and for many small intervals of DD values that we call “cells”, the proportion of firms that in each cell have defaulted within a given time horizon.
• The use of empirical data allows Moody’s KMV to better represent extreme events, and defaults are extreme events.
• All in all, the EDF is given by this empirical F computed in the DD. An important assumption of the KMV model is that companies having the same DD have the same probability to default, i.e. the same EDF.
• For completeness, we have to say that Moody’s KMV uses a proprietary algorithm to obtain V0 and sigmaV.
• The KMV model is hence able to overcome two problems of Merton’s one: Gaussianity and the fact that default only happens in T. In fact, now, thanks to B tilde, the possibility of defaulting within T is taken into consideration.
• Ok, let’s now see some little graphics, to better understand the KMV model as a whole.
• We now want to transform this model into the KMV one.
• V0 and sigmaV are estimated using a special algorithm developed by KMV. And this is it.

Week 5: Default Probabilities – Part 2 > Summary > Video

• Ok, let’s start our review from Merton’s model.
• What we have seen is that Merton’s model is a structural model of default.
• In Merton’s model, we have that default happens when the market value of the assets of the company, at maturity, is smaller than the value of liabilities.
• Always on Merton’s model, we have seen that the probability of default is expressed in terms of a cumulative distribution function of a standard Gaussian.
• The KMV model is a derivation of Merton’s model.
• In the KMV model, default can happen at any time before T. So…not only at maturity.
• The cumulative distribution function of the standard Gaussian, which is one of the main characteristics of Merton’s model, here is substituted with an empirical function, which is estimated by Moody’s KMV using a huge proprietary data set.

Week 5: Default Probabilities – Part 2 > The Sofa > The Sofa Video

• Welcome everybody… This week, we have studied two important structural models of default: Merton’s model and the KMV one.
• For what concerns the KMV model, we have said that this model can be seen as a derivation of Merton’s model.
• Now, a question we can ask, when dealing with this type of models, is the following: are these models effective in studying, in predicting the probability of default of a company? In other words, are these models useful in practice? Now, for what concerns Merton’s model, we can say that this model is rarely used in practice.
• It is surely one of the most influential theoretical models in the literature, it is still a benchmark for scholars and practitioners – from a theoretical point of view, but its practical use is limited, is constrained by its own flaws.
• Merton’s model does not take into account the dependence among defaults.
• Finally, there is a criticism which is in common with all structural models of default.
• Scholars and practitioners argue that, probably, the threshold mechanism, which is behind Merton’s model, and in general structural models of default, may be too naive, too simple, to describe, to capture, the complexity, which is behind an event like a default.
• Now, if we consider the KMV model, we can surely say that this model improves Merton’s model, for example by substituting the liability structure, and assuming a more plausible, a more realistic structure; or by substituting the Gaussian distribution with an empirically estimated distribution that should guarantee a better treatment of extremes and real-life events.
• Now, without entering into the discussion about risk-neutral probabilities, without giving too much details, what we can say is that the KMV model relies on some assumptions that are perfectly ok, from a purely financial mathematics’ point of view, but they may be not the best assumptions from a real-life point of view.
• If we consider empirical studies about the efficacy of the KMV model, what we find is that these studies support the idea that the KMV model is able to satisfactorily predict the PD companies, using the EDF, for what concerns the 1-year time horizon.