# (Credit Risk/ Corporate Risk) Apply the Merton Model to Calculate Default Probability and the Distance to Default

Apply the Merton model to calculate default probability and the distance to default and describe the limitations of using the Merton model.

The Merton model, which is an example of a structural approach, is based on the premise that the technical event of default occurs only when the proprietary structure of the defaulting company is no longer considered worthwhile (V < D). Assuming that a default event is dependent on financial variables, default probability can be calculated using the Black- Scholes-Merton formula. The five relevant variables include the market risk interest rate, the maturity (when the debt expires), the debt face value (similar to an option strike price), the value of the borrowers assets, and the volatility of the assets value. The output provides the probability that the borrower will be insolvent.

In Mertons approach, the equity of a firm represents a call option on the market value of the assets. As such, the value of equity is a by-product of the market value and volatility of the assets, as well as the book value of liabilities; this implies that a firms asset volatility serves as the link between its business and financial risk. A firm’s risk structure is used to set its optimal financial structure, which in turn affects equity due to the probability of shareholders losing their investments due to default.
The default probability using the Merton approach and applying the Black-Scholes-Merton formula is as follows:

$$PD = N(\frac{ln(D)-ln(V_A) – r T + 0.5 \sigma_A^2 T}{\sigma_A \sqrt{T}})$$ $$= N(\frac{ln(\frac{D}{V_A}) – (r + 0.5 \sigma_A^2) T}{\sigma_A \sqrt{T}})$$

where:

In = the natural logarithm

D = debt face value

$$V_A$$ = firm asset value (market value of equity and net debt)

$$r$$  = expected return in the risky world

T = time to maturity remaining

$$\sigma_A$$ = volatility (standard deviation of asset values)

N = cumulated normal distribution

In the preceding equation, the components that lie within the brackets are seen as a standardized measure of the distance to the debt barrier. This distance represents a threshold beyond which a firm will enter into financial distress and subsequently default.

The distance to default (DtD) using the Merton approach (assuming T = 1) is as follows:

$$DtD = \frac{ln(\frac{D}{V_A})- r + 0.5 \sigma_A^2 -“other payouts”}{\sigma_A}$$

There are many challenges associated with using the Merton model. Neither the asset value itself nor its associated volatility are observed. The structure of the underlying debt is typically very complex, as it involves differing maturities, covenants, guarantees, and other specifications. Because variables change so frequently, the model must be recalibrated continuously. Also, its main limitation is that it only applies to liquid, publicly traded firms. Using this approach for unlisted companies can be problematic due to unobservable prices and challenges with finding comparable prices. Finally, due to high sensitivity to market movements and underlying variables, the model tends to fall short of fully reflecting the dependence of credit risk on business and credit cycles.