10.3  Merton’s Model of a Distressed Firm

We now turn to Merton’s model to explain why GM can trade for a positive price even though shareholders equity is negative.    Although we are using the model for firms that are distressed, it can actually be applied to any firm; in fact, variations of the model are used to estimate default probabilities on debt even for companies with healthy balance sheets.

In its simplest form, suppose a firm has assets A and has issued a zero-coupon bond as debt that matures at time T.span style="mso-spacerun:yes">  Let F be the face value of this bond, so the company will have to pay back F at time T.  If it cannot, it will have to default.  LLet B be the value of the debt and E the shareholders equity, so today,

A = B + E, or Assets = Liabilities + Shareholders Equity

Even if E is negative, so A < B, the firm can continue to operate because it does not actually have to pay the liability until time T.span style="mso-spacerun:yes">  So if the assets grow sufficiently, it can avoid bankruptcy at time T.  AAs an aside, this is clearly a simplified story; if a firm has negative equity, it can be very difficult for that firm to obtain financing for everyday operations, in turns making it very difficult for it to operate without some protection from creditors.

So if at time T, Asub>T >= F, the firm can avoid bankruptcy, and in that case, the shareholders equity will be A_T -F.  If A_T <F, the shareholders equity will be negative and so the stock will be worth zero (because of the limited liability of stock holders).

So the value of a stock at time T, with everything measured on a per share basis, is

S_T = Max{A-F,0}

This is the same as the payoff from a call option; in Merton’s model, a stock is a call option on the assets of the firm.  The strike price (or exercise price) of the option is the face value of the debt.  IIt is a European option because the stock holder cannot exercise the option (which here would be akin to asking the firm to pay the difference between the assets and the liabilities to the stock holder).

The value of this option can be calculated from option pricing theory, specifically the Black-Scholes model.  This model requires two more inputs: the risk free interest rate and the volatility of the assets.  Let r be the risk free interest rate; we would typically use the yield on a Treasury instrument with maturity T.   The volatility of the assets presents more of a problem, but as you will see, it can be imputed from market prices.  Let s_A be the volatility of assets (or more correctly, the volatility of the asset return).

The stock value then follows from Black Scholes:

The relation between equity volatility and asset volatility is:

Here, N(.) is the cumulative normal distribution, and

If you have an estimate for /span> s_E, this equation can be solved for s_A (though it’s a bit more complicated than it looks because N(d₁) depends on s_A  as well).  In Valuation Tutor, we have two form of the Merton model; in the first, you have to specify s_A. In the second, you specify the equity volatility and Valuation Tutor solves for s_A.

This model can be further applied to estimating the “distance to default” which can be converted into a probability of default (Bharath and Schumway (2008)).