Equity Value

Our structural approach to credit risk views the equity of the firm as an option, the payout of which is conditional on the firm's ability to repay its debt, given the evolution of its revenue and CFADS.

We assume that you can calibrate and estimate credit risk metrics given a company's capital structure and future cashflows.

Following the option pricing model, the company's equity price is calculated as:

$\begin{array}{l}\displaystyle EquityValue = \mathbb{E}_{\mathcal{P}}\left\lbrack \sum_{t = 1}^{T}{\text{EquityCF}_{\mathcal{P}}\left( t \right)e^{- \left( r + \mu \right)t}} \right\rbrack\end{array}$

Then, conditional on no default:

$\begin{array}{l}\displaystyle {EquityCF}_{\mathcal{P}}\left( t \right) = \left( 1 - RetentionRate(t) \right)*\left( \text{CFADS}_{\mathcal{P}}\left( t \right) - DebtService\left( t \right) \right)\end{array}$

And in the event of default:

$\begin{array}{l}\displaystyle {EquityCF}_{\mathcal{P}}\left( t \right) = 0\end{array}$

where the $\begin{array}{l}DebtService\end{array}$ and $\begin{array}{l}RetentionRate\end{array}$ are definite for the time $\begin{array}{l}t\end{array}$ , $\begin{array}{l}r\end{array}$ is the risk free rate and $\begin{array}{l}\mu\end{array}$ is the expected equity premium, $\begin{array}{l}\mathcal{P}\end{array}$ is the measure of CFADS evolution and could be described by the $\begin{array}{l}CFADS_t\end{array}$ estimation and the forward-looking volatility of CFADS, $\begin{array}{l}\sigma_{CFADS}\end{array}$ . $\begin{array}{l}\text{EquityCF}_{\mathcal{P}}\left( t \right)\end{array}$ is the equity payout in the different CFADS scenarios under the measure $\begin{array}{l}\mathcal{P}\end{array}$ .

The company defaults in the scenarios where:

$\begin{array}{l}\displaystyle \text{CFADS}_{\mathcal{P}}\left( t \right) < DebtService\left( t \right)\end{array}$

Which we can rewrite $\begin{array}{l}EquityValue\end{array}$ in the recurrent format for convenience as:

$\begin{array}{l}\displaystyle \text{EquityValue}_{\mathcal{P}}\left( t \right) = \mathbb{E}_{\mathcal{P}}\left\lbrack e^{- (r + \mu)\ \Delta t}\text{EquityValue}_{\mathcal{P}}\left( t + 1 \right) \right\rbrack + \text{EquityCF}_{\mathcal{P}}\left( t \right)\end{array}$

On the company's end date (maturity), we have:

$\begin{array}{l}\displaystyle \text{EquityValue}_{\mathcal{P}}\left( t = T \right) = \text{EquityCF}_{\mathcal{P}}\left( t = T \right)\end{array}$

As of the pricing date:

$\begin{array}{l}\displaystyle {EquityValue = EquityValue}_{\mathcal{P}}\left( t = 0 \right)\end{array}$

Default and non-default scenarios

The debt services are discounted only in the non-defaulting scenarios. The debt value in the defaulting scenarios is reset to its recovery value plus CFADS.

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Default and non-default scenarios