Merton Value of Equity Formula

[bpdulog]

Hi,

I have seen the Merton value of equity formula given as:



However in the text, it is provided as follows:



What is the significance of the Pt(T) variable? It is defined as the price of a $1 0 coupon bond that matures at time T. Does it act as some kind of discount factor?

 

[David Harper CFA FRM]

Hi @bpdulog Above the formula in the text is "Pt(T) the price at t of a zero-coupon bond that pays $1 at T." So we followed Stulz here with P(T) = F*exp(-rT); i.e., the face value of the debt discounted to the present value as a risk-free price. More broadly, this formula is the essence of the first step in Merton (see my long note here for explication on both steps https://forum.bionicturtle.com/threads/merton-model-a-summary-of-the-issues.5646/ ): equity is treated as a call option on the firm's assets. Because if you own all of the equity in the firm, you can own all of the assets by payoff off all of the debt. So you have an option with a strike price equal to the face value of the debt, and this formula is the Black-Scholes but where V = firm value = S(0) in a more familiar symbol; and F is the more familiar strike price, K. I hope that helps!