In this final instalment of the series we’re going to take a look at Robert Merton’s 1974 ‘Distance-to-Default’ method for predicting financial distress. Merton is the often forgotten third player in the Black-Scholes formula (properly called the Black-Scholes-Merton model) and won the 1997 Nobel Prize in Economics accordingly. His work in expanding the mathematical understanding of the options price model inspired to him to apply the technique to other financial problems including the prediction of company defaults. It is one of the most popular approaches to default probability estimation and has the advantage of being insensitive to the leverage ratio. It can therefore be applied more readily to banks and other highly leveraged firms where the other models can’t. For that reason alone we thought it was worth looking at in more depth.
Having developed the Black-Scholes model for pricing options at a time when option trading was in its infancy, Merton cast around for other purposes the basic approach could be used for. He found that by characterising a company’s equity as an option on its assets (we’ll try and explain this a bit later) he could accurately assess the credit risk of a company. Put –call parity (the relationship between the price to long or short the stock) was then used to value the long option which is treated as a proxy for the firm’s credit risk. To try and simplify this a bit we’ll go through it step by step below. For any newcomers to the field of options and option pricing, you can find a good summary here.
- Firstly, the model assumes that a company has an amount of debt that will become payable at a fixed time in the future T (calculated through a combination of short-term and long-term debt obligations to assume all the debt matures at the same time);
- A company is deemed to have defaulted if the value of its assets falls below the face value of the debt promised at time T;
- By introducing the options pricing formula, we can calculate the expected asset value at time T and compare it to the debt value. The equity of the company is characterised as a European call option on the assets (a long position on the theoretical right to buy the equity at a price equal to the face value of the debt at time T) and should the option suggest a value for the assets lower than the company’s debt obligations, it is classed as being likely to default.
This process can be expressed through the following formula which looks a bit daunting at first but which we will break down afterwards.
DD = (ln(V/F) + (μ - 0.5 * σV ^2 ) T) / ( σV √T)
- V = Total value of the firm (unobservable, must be inferred)
- μ = Expected continuously compounded return on V
- F = Face value of firm’s debt
- T = Time to maturity
- σV = Volatility of underlying firm, unobservable, calculated using the following relationship between the firm and its equity:
- σE = (V/E) N(d1) σV
- Where E is market cap and N(d1) is the cumulative standard normal distribution function on d1 (for those really keen, the formula for d1 is available within the second linked source at the end. As another complex formula, I had to draw the line somewhere!).
- While the volatility of the equity can be calculated, the volatility of the underlying firm can only estimated.
The formula essentially expresses a measure of the difference between a firms asset value and the face value of its debt, all scaled by the standard deviation of the firms asset value. The DD calculation provides us with the number of standard deviations the firm’s value is away from default, therefore the smaller the value of DD, the larger the probability of default.
Similar to Ohlson, Merton doesn’t explicitly apply his solution to an out of sample set of data in his original piece, so again we will review its accuracy through subsequent papers. A review of Merton’s DD in the journal ‘The Review of Financial Studies’ found that the model is able to classify 64.9% of defaulting firms in the highest probability decile at the beginning of the quarter in which they default. The researchers are able to boost this percentage by combining the technique with traditional accounting measures to reach 75.8% of firms failing in the top decile of probability.
The model also performed well in a study applying the method to the prediction of cost of debt, another good market proxy for default risk. This application is discussed in a study by Virginia Tech, if the models are properly capturing risk then when applied to a set of companies and sorted into deciles of probability, the credit spreads should increase at the highest risk deciles. With this technique, Merton’s DD performs well across a wide range of industries and in different periods of stress and volatility in the markets, coming second only to the CHS model.
In conclusion we can say that Merton’s DD is more useful when applied to highly leveraged firms using credit spread data and for short-term prediction. For the basic analysis of industrial firms, the accountancy based measures of Altman and Ohlson are more than adequate for solid forecasting of bankruptcy. The CHS model meanwhile is the best all rounder with proven results in forecasting across several applications, best applied to non-financial firms in the growth and value sectors. For this reason, we will be taking a look at potentially complementing the existing Z-Score with the CHS model on the Stockopedia company reports at some time in the future.
From the Source
- Original paper here
- University of Michigan paper here (with login)
- Virginia Tech comparison study here
Filed Under: Short Selling,