View FRM2009.E1.08

Category:FRM2009 -> Full1
Question:

E1.08 [source sample 2009 FRM Full Exam 1] You don’t have access to KMV’s data. Your boss wants you to tell him your estimate of the probability of default of a credit. To do so, you use the Merton Model because the credit you are considering has no systematic risk. In Merton’s Model, the distance to default (DD) and the expected default frequency (EDF) are:

a. positively and linearly related
b. negatively and linearly related
c. positively and non-linearly related
d. negatively and non-linearly related

my adds [some of these are tough]

E1.08e. Explain the Merton Model in a few brief sentences.
E1.08f. Cite two differences between Merton Model and Moody’s KMV.
E1.08g. Cite a few variables that would decrease the EDF?
E1.08h. Cite a disadvantage of this approach (i.e., equity-based model of default prediction).
E1.08i. [hard] The question implies that the Merton Model requires, or wants, an assumption that the credit has no systematic risk. Is this true?
E1.08j. [hard] As the relationship between DD & EDF is non-linear, can we be more specific: what is the distribution of DD? Reconcile this distribution with the lognormal property of stock prices.
E1.08k. The answer says the risk-neutral probability of default (PD) = 1 – N(d2). But de Servigny says PD = N(-d2). Which is correct?
E1.08l. If the distance to default (DD) is 2.0, what is Merton’s implied risk-neutral probability of default (PD)?
E1.08m. [hard] Under risk-neutral valuation, can we assume this PD applies in the real world?

Answers:

E1.08 [source] The risk neutral probability of default, EDF, in the Merton Model is 1‐ N(d2). The higher the distance to default, DD, the lower the risk neutral probability of default is. On the contrary, the lower DD, the higher EDF is. The relationship is non‐linear. When the DD is low, EDF, is high. If DD is imminent, EDF is high as well. Similarly, if DD is high, EDF is small and not imminent Reference: De Servigny and Renault, Measuring and Managing Credit Risk, Chapter 3.

E1.08e. Explain the Merton Model in a few brief sentences.

My short version: Merton model re-frames a (simple two-class) captial structure of an entire firm as a combination of two derivatives: shareholders are long a call option on the firm’s assets (with strike = face value of debt) and debtholders are short a put option on the firm’s assets. This implies that BSM OPM can be used to price the equity. Further, under the structural approach, a prediction of default is based on the probability that the future firm’s asset value will breach (fall below) the debt value; i.e., when asset value < debt value, then equity + debt < debt value, there is no equity cushion and default is predicted.

E1.08f. Cite two differences between Merton Model and Moody’s KMV.

1. Moody’s KMV does not use the total debt value for the default threshold (a.k.a., default point); they use (book) value of short-term debt plus some fraction of long-term debt.
2. Importantly, KMV does not parameterize the distance to default: - 3 standard deviations, for example, is not translated into EDF by a pure parametric distribution. Rather, it is emipircally based on default experience.

E1.08g. Cite a few variables that would decrease the EDF?

Future distribution (yes, that’s right, does not need to be normal or lognormal!)
Higher expected return in growth of firm’s assets;
Longer horizon;
Lower asset volatility;
Higher initial asset value and/or lower default point;

Although KMV has said that the critical variables are: asset value, future distribution, asset volatility, and level of default point; with expected growth and horizon length having “little discriminating power.”

You are understanding the Merton well to understand each of the above.

E1.08h. Cite a disadvantage of this approach (i.e., equity-based model of default prediction).

de Servigny cites their tendency for “false starts:” a fall in equity markets increases EDFs.
Further, de Servigny asserts the tendency for overreation makes the models highly pro-cyclical.

There can be other criticisms levied, based on Altman’s observations: “The EDF model has two fundamental differences with other approaches. First, it relies on the information in equity prices. Two, it does not try explicitly to be predictive. Whereas agency debt ratings are based upon trying to forecast future events, there are no real future forecasts in the EDF model. It simply looks at the current value of the ?rm relative to its default point and historical volatility. Thus, if it has predictive power, it is because the current value of the firm is a good prediction of future values. Since this value is derived from the ?rm’s equity market value, the EDF model is totally dependent on stock prices for its information content.”

E1.08i. [hard] The question implies that the Merton Model requires, or wants, an assumption that the credit has no systematic risk. Is this true?

No, the Merton does not require the absence of systematic risk; it does not requre a riskfree rate as an input. The phrase is misleading.
(the valuation of the equity takes riskfree rate as input, per the risk-neutral idea. But expected growth is used as an input in the default prediction).

E1.08j. [hard] As the relationship between DD & EDF is non-linear, can we be more specific: what is the distribution of DD? Reconcile this distribution with the lognormal property of stock prices.

The same properties are working here as in the Black-Scholes-Merton: the returns are normal and the firm (asset) levels are lognormal. The implied distribution of the distance to default—not it contains LN(firm value) and LN(default)!—is normal.

E1.08k. The answer says the risk-neutral probability of default (PD) = 1 – N(d2). But de Servigny says PD = N(-d2). Which is correct?
Both due to the symmetry of the normal: 1-N(d2) = N(-d2)

E1.08l. If the distance to default (DD) is 2.0, what is Merton’s implied risk-neutral probability of default (PD)?
=NORMSDIST(-2) = 2.275%; but this is the step KMV Moody’s does not perform. This, relying on “normal tails” understates the likely PD/EDF.
Therefore, Moody’s KMV EDF > Merton’s NORMSDIST(-2)

E1.08m. [hard] Under risk-neutral valuation, can we assume this PD applies in the real world?
No! Common mistake…The risk-neutral idea applies to the option value (i.e., the equity value), not to N(-d2) itself.

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