10.6  The Extended Merton Model

In the extended Merton model, the asset volatility is replaced as an input by the equity volatility.  The relationship between the two volatilities is:

 

Given the equity volatility, we can try and solve this equation for the asset volatility.  Note,  however, that a solution does not always exist; if it does not, then Valuation Tutor responds with “N/A” in its calculated fields.

The standard Merton Model above already tells you the equity volatility needed to match the asset volatility; is 3.48221 in the example we just completed.  But that is not the point; we want the equity volatility to be the input and see what the resulting asset volatility and stock price are.

We calculated the historical volatility of GM in 2008 using daily data for the year; the equity volatility was 118%.  Entering this into the advanced Merton model yields:

 

This has an implied asset volatility of about 51%, which is much lower than the 65% needed to match the stock price.  But this is to be expected; we have assumed that the entire debt of GM is due to be paid within 6 months.  A more detailed analysis would look at the structure of the liabilities and determine what the appropriate time period is.  For example, if we assume the debt is to be paid off in four years, the values match the prices of GM quite closely:

 

 Hopefully you can see the power of the model: any model that simply projects current earnings or dividends into the future cannot be expected to value such firms.  The Merton model provides an elegant solution to this problem.