THE EDF MODEL
Category: Risk Management in Banking
When looking forward at a future date, economic default occurs when the asset value drops below debt. The debt level triggering default is unclear since debt amortizes by fractions according to some schedule. KMV uses a debt value combining short-term debt at horizon plus a fraction of the long-term debt. The issue is to determine the chances that the asset value will fall below that level.
The asset expected value in the future is random. It follows a stochastic process with drift, just as stock prices do. The expected value is normally above the value of debt. The asset value at the horizon follows a lognormal distribution. We need to find the probability that the asset value drops below the debt value at the horizon, which is the theoretical Edf. The asset volatility allows us to use approximations to find this probability. With expected asset value and asset volatility, it is easy to find this probability.
As an example, using the simple normal distribution, instead of the lognormal distribution at horizon1, it is straightforward to define the intermediate values. Assuming an expected asset value of 100 at the horizon, a debt value at the horizon of 50, and an asset volatility of 21.462, the probability of economic default is that the asset value drops by 50/21.46 = 2.33 standard deviations, or 1% in this case. This is the Edf.
Several adjustments are necessary to mimic the actual default behaviour. The value of debt is the default point. The Distance to Default (DD) is the ratio of the gap between expected asset value and default point to the asset standard deviation. The DD is the downside drop in standard deviation units required for default (2.33 above). This is a theoretical value. In fact, the actual value might differ, so that the calibration of the model requires mapping the distances to default to these actual default observed frequencies. With this mapping, the model replicates the actual default frequencies. The calibration process relies on proprietary default databases. It is necessary to have a sufficient number of defaulted companies to fit the mapping of distances to default to actual default frequencies. Periodical fits are necessary to make sure that the models actually fit the data. Figure 38.2 summarizes the entire process.
The time path of the asset value is stochastic. There are multiple time paths, each one reaching a specific value along the distribution at the horizon. Modelling the time paths as a stochastic process is not necessary as long as we care only about what happens at the horizon. Note that it is possible for the time path of asset value to cross the default point before the horizon. Then, default would occur before the horizon.
The Credit Monitor Edfs do not correspond fairly to default probabilities mapped to the agencies ratings. The market-based Edf fluctuates continuously, and in spite of recal-ibration of the KMV proprietary model, there are significant discrepancies with ratings.
However, this is not an inconsistency. In fact, the market-based Edf should anticipate rating changes, since ratings tend to lag somewhat the changes in credit standing of issues, simply because ratings are supposed to be long-term views of credit risk, not continuously reviewed. Casual observations of Credit Monitors Edf seem to support this view, that market-based measures lead rating changes to a significant extent. KMVs Credit Monitor provides all the necessary information to make such comparisons: the Edf, the underlying values of the Edf drivers and the rating equivalent. This also allows end-users to better understand the Edf changes.
Of course, there are several studies comparing the KMV Edf with similar results obtained with other techniques, such as scoring. They use, for instance, the power curve to show how well the model discriminates between defaulters and non-defaulters. They suggest that Credit Monitor provides at least a comparable accuracy.
A limitation of Credit Monitor is that it applies to public companies since it uses equity prices. To address the issue of equivalent default risk measures for private companies, KMV Corporation provides another model, the Edf Calculator or Private Firm Model, which necessitates minimal information on private companies to model default risk. The basic information is in financial statements. In addition, the usage of the Edf Calculator requires documenting the industry and region of the firm. The principle is to proxy through this information the actual Edf of an equivalent public company.
We calculate below the various variables determining distance to default. Without calibration, the calculation of the theoretical Edf is relatively easy. The process requires the formulas of the put option to default plus the usual assumptions on the asset value stochastic process leading to the lognormal distribution of the asset value at horizon. We summarize below the basic formulas, before making a sample calculation.
The Put and Call Options of Equity Holders
The option framework views the shareholders as having a put option to sell the assets of the firm to the lenders at a strike price equal to the preset book value of debt. The mirror image is that they hold a call option on the asset value, the strike price being the debt. The stockholders benefit from the upside of the asset value, while having a limited liability to lenders. Both options are European: they allow exercise at horizon T only. The lenders hold a debt whose value depends on the default probability. When the default probability gets higher, the put gains value, and the debt value decreases by the gain in the put value. Standard Black-Scholes formulas3 apply for valuing the call on assets or the put of equity holders. These are:
Sample Calculations of the Put and Call Values
The book value of debt is 50 and the value of assets is 100 at date 0. The strike price is 55 at T = 1, from the above, if the risk-free rate is 10%. The volatility of equity is 28.099%. This value results in an Edf of 1%, once calculations are done. This is sufficient to derive all the above values (Table 38.1).
The theoretical value E0 as a call option with the above inputs is 50.043744. It is above the equity value without the put, or 50 = 100 — 50, because of the upside on asset value.
When the asset value decreases by 1 unit, the equity value decreases by 0.995192 units, and the put value increases by 0.004808. The put value increase is a loss for the lender. Therefore, the bank can compensate this loss by selling short the equity for 0.004808/0.995192 = 0.0048315 units of stock. This ratio is also approximately = (1 — 0.9954374)/0.9954374 = 0.0045835. It is necessary to adjust the ratio continuously to mimic the relative changes in values of the equity and the put.