Explicit Solutions for Debt Sizing with Sculpted Repayment in Project Finance Modeling

This research note presents explicit formulas for debt and repayment sizing. It provides a direct method to determine the target Debt Service Coverage Ratio (DSCR) and the corresponding sculpted repayment schedule based on a specific debt size. Conversely, it also offers a formula to calculate the maximum debt size for a given target DSCR. The availability of these explicit formulas streamlines the construction of a robust and light-weight project financial model, eliminating the need for macros, circular references, or goal seek functions.

Keywords: Project Finance, Debt Sizing, Sculpted Repayment, Target DSCR

Notation:

Denote by

• $C$: Cash Flows Available for Debt Service (CFADS)
• $O$: Outstanding Bond (Debt)
• $g$: Guarantee fee rate (if quarterly payment, it's a quarterly rate)
• $E$: Other bond issuance cost
• $D$: Debt Service
• $P$: Principal repayment
• $N$: Number of (re)payments
• $r$: Bond interest rate at each payment date (if quarterly payment, it's quarterly rate)
• $x$: Inverse of target DSCR
• $n$: Moratorium period of non-repayment

Assumption:

1. (1) Other bond issuance cost $E_t$ is assumed independent of debt outstanding amount $D_t$. In practice, the bond issuance cost may depend on the bond outstanding amount; however, we assume the variation due to any change in outstanding amount is negligible.
2. (2) Guarantee fee $g_t = \bar{g}$ is constant.

Formulas:

Given a debt amount of $O_0$, the target DSCR is given by:
$$DSRC = \frac{\sum_{i=n+1}^{N} C_i^* \prod_{j=i+1}^{N} R_j^*}{O_0 \prod_{i=n+1}^{N} R_i^* +\sum_{i=n+1}^{N} E_i^* \prod_{j=i+1}^{N} R_j^* } ,$$

where $R_t^* := \frac{1+r_t}{1-\bar{g}}; E_t^* := \frac{E_t}{1-\bar{g}}; C_t^* := \frac{C_t}{1-\bar{g}}$.

Given a target DSCR, the maximum debt amount can be determined as follow:
$$O_0 = \frac{x \sum_{i=1}^{N} C_i^* \prod_{j=i+1}^{N} R_j^* - \sum_{i=1}^{N} E_i^* \prod_{j=i+1}^{N} R_j^* }{\prod_{i=1}^{N} R_i^*}$$

where $R_t^* := \frac{1+r_t}{1-\bar{g}}; E_t^* := \frac{E_t}{1-\bar{g}}; C_t^* := \frac{C_t}{1-\bar{g}}$.

Proof:

The debt service at payment date $t$ is given by
$$\begin{equation} D_t = O_{t-1} r_t + O_t g_t + E_t + P_t \end{equation}$$
Also, we have
$$\begin{equation} O_t = O_{t-1} - P_t \end{equation}$$

Noting that $D_t = x C_t$, we add (1) and (2) to get
$$x C_t = O_{t-1} (r_t +1) - O_t (1-\bar{g}) + E_t$$
Re-arranging, we obtain

$$O_t = O_{t-1} \frac{1+r_t}{1-\bar{g}}+ \frac{E_t}{1-\bar{g}} - \frac{C_t}{1-\bar{g}}x$$

Therefore,
$$\begin{equation} O_t = O_{t-1} R_t^* + E_t^* - C_t^* x. \end{equation}$$
For $t=1$,
$$O_1 = O_{0} R_1^* + E_1^* - C_1^* x.$$
Substituting it into $O_2$, we get
$$O_2 = O_{0} R_1^* R_2^* + (E_1^* R_2^* + E_2^*) - (C_1^*R_2^* + C_2^*) x .$$
Subsituting it into $O_3$, we get
$$O_3 = O_{0} R_1^* R_2^* R_3^* + (E_1^* R_2^* R_3^* + E_2^* R_3^* + E_3^*) - (C_1^*R_2^*R_3^* + C_2^* R_3^* + C_3^*) x .$$
By continuing to substitute progressively, we obtain
$$O_N = O_{0} \prod_{i=1}^{N} R_i^* + \sum_{i=1}^{N} E_i^* \prod_{j=i+1}^{N} R_j^* - x \sum_{i=1}^{N} C_i^* \prod_{j=i+1}^{N} R_j^*$$

Knowing that $O_N = 0$, we can derive $x$ as below:
$$x = \frac{O_0 \prod_{i=1}^{N} R_i^* +\sum_{i=1}^{N} E_i^* \prod_{j=i+1}^{N} R_j^* }{\sum_{i=1}^{N} C_i^* \prod_{j=i+1}^{N} R_j^*}.$$
In case of a moratorium period of $n$ periods, $O_n = O_{n-1} = ...=O_0$, and therefore,
$$x = \frac{O_0 \prod_{i=n+1}^{N} R_i^* +\sum_{i=n+1}^{N} E_i^* \prod_{j=i+1}^{N} R_j^* }{\sum_{i=n+1}^{N} C_i^* \prod_{j=i+1}^{N} R_j^*}.$$