We calculate a company's debt value using a similar approach to its equity value. Conditional on no default, it can be easily expressed as:

\text{DebtValue}_{\mathcal{P}}\left( t \right) = \mathbb{E}_{\mathcal{P}}\left\lbrack e^{- r\ \Delta t}\text{DebtValue}_{\mathcal{P}}\left( t + 1 \right)) \right\rbrack + DebtService\left( t \right) |

In the event of default:

\text{DebtValue}_{\mathcal{P}}\left( t \right) = \text{CFADS}_{\mathcal{P}}\left( t \right) + \text{Recovery}_{\mathcal{P}}\left( t \right) |

On the company's end date, we now have:

\text{DebtValue}_{\mathcal{P}}\left( t = T \right) = \min\left( \text{CFADS}_{\mathcal{P}}\left( t = T \right),\ DebtService\left( t = T \right) \right) |

As of the pricing date:

{DebtValue = DebtValue}_{\mathcal{P}}\left( t = 0 \right) |

Here \text{Recovery}_{\mathcal{P}}\left( t \right) is the recovery amount in the defaulting scenarios. Typical infrastructure lenders have strong control over the project. They can take many actions to protect their investment including restructuring the debt. The recovery depends on the future optimised cashflows as well as the current value of the whole project - that is the firm value (the sum of the equity and debt value).

Note that the cashflow in the equity value estimation is discounted by a factor equal to the risk-free rate curve and the equity premium, while the debt value is only discounted by the risk-free rate. This is because for the unlisted firms, the expected return of the equity is not reflected in the pricing process due to the extreme inefficiency of the market. However, most infrastructures borrow money from banks which, compared with raising money via the equity market, is much more competitive and comparable. The potential credit spread is already embedded in the possible defaulting scenarios under the measure \mathcal{P}.

### Debt Restructuring and Recovery

Now we turn to estimating recovery value. In the chart illustrated in the Equity Value section, each node represents one possible scenario of CFADS in the future, in which the equity and debt payout could be computed given the debt services. If the CFADS falls below a certain threshold, then the default mechanism will be activated. The lender can then consider the potential restructuring options to maximise their investment value.

That is, if the debt could be restructured successfully:

{Recovery}_{\mathcal{P}}\left( t \right) = \min\left( {DefaultedAmount}_{\mathcal{P}}\left( t \right),\ {RestructuredDebtValue}_{\mathcal{P}}\left( t \right) - {RestructCost}_{\mathcal{P}}\left( t \right) \right) |

Otherwise:

$${Recovery}_{\mathcal{P}}\left( t \right) = \min\left( {DefaultedAmount}_{\mathcal{P}}\left( t \right),\left( 1 - ExistCost \right)*{FirmValue}_{\mathcal{P}}\left( t \right) \right)\ \\

{DefaultedAmount}_{\mathcal{P}}\left( t \right) = OutstandingDebt\left( t \right) + DebtService\left( t \right) - {CFADS}_{\mathcal{P}}\left( t \right)$$

where:

{DebtService}\left( t \right) > {CFADS}_{\mathcal{P}}\left( t \right) |

{RestructCost}_{\mathcal{P}}\left( t \right) = {RstrCost*DefaultedAmount}_{\mathcal{P}}\left( t \right) |

ExistCost and RstrCost are the cost of liquidating company and restructuring debt respectively. They are assumed as two constants.

{RestructuredDebtValue}_{\mathcal{P}}\left( t \right) is the restructured debt value and is computed in the same way as the {DebtValue}_{\mathcal{P}}(t), but without restructuring mechanism i.e. No borrower expects their debt to be restructured again.

We consider various new debt arrangements which have same IRR as the existing one, but different maturities and coupon rates. The one with maximum value (including the existing debt schedule) is the candidate restructured debt schedule. If it could give better recovery than liquidating the company, then the restructuring would be successful. Otherwise, the lender would prefer to liquidate the company and exit the investment. However, if the shareholder would still receive a decent payout, and the full defaulted amount would be repaid after liquidation, then the lender would still prefer to restructure the debt.

The IPT also considers the 'soft default' where the restructuring process would be triggered when DSCR is lower than SOFT_DEFAULT_THRESHOLD. The corresponding equations are listed in the appendix.

### Soft defaults

The IPT also considers the soft default scenarios, where the lenders would take some actions to prevent the debt from getting worse. In the hard default scenarios, the recovery from restructured debt would have the cap of the defaulted amount. The restructured debt value would not be capped by the existing debt value in the soft default scenarios, as the lender is seeking potentially better debt performance than would be recouped by recovery of the outstanding debt. In the other words, in these scenarios, the debt value could be greater than the existing one after the restructuring process is triggered. Accordingly, we have the following modified equation for the debt value:

For \(DSCR > H_{s}\)

\[{DebtValue}_{\mathcal{P}}\left( t \right) = \mathbb{E}_{\mathcal{P}}\left\lbrack e^{- r\ \Delta t}{DebtValue}_{\mathcal{P}}\left( t + 1 \right)) \right\rbrack + DebtService\left( t \right)\]

For \(H_{s} > DSCR \geq H_{h}\)

\[{DebtValue}_{\mathcal{P}}\left( t \right) = {RestructuredValue}_{\mathcal{P}}(t) + DebtService\left( t \right)\]

For \(H_{h} > DSCR\)

\[{DebtValue}_{\mathcal{P}}\left( t \right) = {CFADS}_{\mathcal{P}}\left( t \right) + {Recovery}_{\mathcal{P}}\left( t \right)\]

The restructured debt value at the scenario of the time t \({RestructuredValue}_{\mathcal{P}}\left( t \right)\) is the bigger of the expected value of the existing debt at time t+1 \(\mathbb{E}_{\mathcal{P}}\left\lbrack e^{- r\ \Delta t}{DebtValue}_{\mathcal{P}}\left( t + 1 \right)) \right\rbrack\) and the restuctured value of the potential new debt \({RestructuredDebtValue}_{\mathcal{P}}\left( t \right) - {RestructCost}_{\mathcal{P}}\left( t \right)\).

The restructuring would happen if the restructured debt is more valuable in the soft defaulting scenarios. The cost of the debt restructuring is:

\[{RestructCost}_{\mathcal{P}}\left( t \right) = \left\{ \begin{matrix}

RstrCost*DebtOutstanding\left( t \right)\text{ if }H_{s} > DSCR \geq H_{h} \\

RstrCost*{DefaultedAmount}_{\mathcal{P}}\left( t \right)\text{ if }H_{h} > DSCR \\

\end{matrix} \right.\ \]