Circularity in Debt Models
Executive Summary
Key Takeaways
- ✓ Circularity in debt models arises from the genuine financial simultaneity of interest expense and available cash within the same period. It is a structural consequence of accurate modelling, not a modelling error.
- ✓ The most common sources are average balance interest calculations and cash sweep mechanics that depend on the same period's interest charge.
- ✓ Opening balance interest is the preferred resolution for most institutional models: it eliminates the circularity with a marginal, immaterial accuracy trade-off.
- ✓ Debt sculpting algorithms provide a technically precise resolution for project finance models with DSCR-based repayment profiles.
- ✓ Where iterative calculation is retained, convergence parameters must be appropriate for the model's scale, a convergence check must be present, and the model must be tested for consistent convergence from different starting conditions.
- ✓ Models that produce different DSCR values depending on calculation state do not meet institutional standards for lender reliance.
Institutional Definition¶
Debt models used in project finance, infrastructure finance, and leveraged finance are structurally predisposed to circular references. The financial relationships within a debt model involve simultaneous dependencies: interest expense depends on the debt balance, the debt balance depends on repayment amounts, repayment amounts depend on available cash, and available cash depends on interest expense. When a model attempts to resolve all of these relationships in a single calculation period using the same period's values for each variable, it creates a dependency loop.
What Is Circularity in a Debt Model?¶
Circularity in a debt model is a circular reference that arises specifically from the interdependence of variables within the debt structure: interest, repayment, drawn balance, and cash available for debt service.
Unlike unintentional circular references (which arise from errors in formula construction), circularity in debt models typically arises from an attempt to model a financial relationship accurately. The modeller understands that interest in period T is calculated on the average of the opening and closing balance in period T, and that the closing balance in period T depends on the repayment in period T, and that the repayment in period T depends on the cash available after interest in period T. Each of these relationships is financially correct. Together, they create a dependency loop.
Sources of Circularity in Debt Models¶
1. Average Balance Interest Calculation¶
The most common source of circularity in debt models is the calculation of interest on the average of the opening and closing debt balance within a period.
The relationship is:
Average Balance = (Opening Balance + Closing Balance) / 2
Interest Charge = Average Balance × Interest Rate
Closing Balance = Opening Balance − Repayment
Repayment = f(Cash Available for Debt Service, Interest Charge)
Cash Available for Debt Service = Operating Cash Flow − Interest Charge
In this structure, Interest Charge depends on Average Balance, which depends on Closing Balance, which depends on Repayment, which depends on Cash Available, which depends on Interest Charge. This is a circular dependency.
2. Cash Sweep Mechanics¶
A cash sweep is a mechanism by which excess cash generated in a period is applied to repay outstanding debt, over and above any scheduled amortisation. In a cash sweep model:
Cash Available for Sweep = Cash After Scheduled Debt Service − Reserves − Distributions
Sweep Repayment = min(Cash Available for Sweep, Outstanding Debt Balance)
Outstanding Debt Balance (closing) = Outstanding Debt Balance (opening) − Scheduled Repayment − Sweep Repayment
If the interest charge for the period is calculated on the average balance (including the closing balance after sweep), and if the sweep amount is determined after interest has been deducted, the interest and the sweep amount are mutually dependent within the same period.
3. Revolver Drawdown to Fund Shortfalls¶
In models that include a revolving credit facility to fund cash shortfalls in any given period, the revolving balance depends on the shortfall, the shortfall depends on interest on the revolver, and interest on the revolver depends on the revolver balance. If the revolver interest is charged in the same period as the drawdown that funds the shortfall, the dependency is circular.
4. PIK Interest Accumulation¶
Payment-in-kind interest accumulates to the loan balance rather than being paid in cash. In a PIK structure, the closing balance includes the accrued PIK interest, and the PIK interest is calculated on the average or closing balance. If the PIK interest is calculated on a balance that includes the PIK interest itself, the calculation is circular.
Resolution Techniques¶
Technique 1 — Opening Balance Interest¶
The simplest and most widely used resolution is to calculate interest on the opening balance of the period rather than the average balance.
Implementation. In each period, interest is calculated as:
Interest = Opening Balance × (Interest Rate / Periods per Year)
Impact on accuracy. Calculating interest on the opening balance rather than the average balance introduces a small understatement of interest in periods where the balance is declining. The understatement equals half the period's repayment multiplied by the interest rate. For quarterly periods, this is typically less than 0.05% of the total interest charge per period, which is immaterial for financial modelling purposes.
Trade-off. This technique is an approximation. For transactions where the loan agreement specifies interest on the average balance, the model's output will not exactly match the contractual interest calculation. In such cases, the model's developer should note the approximation in the model's documentation.
Preferred in. General project finance and leveraged finance models where the marginal accuracy difference is not material to the DSCR or covenant calculation.
Technique 2 — Debt Sculpting Algorithm¶
In project finance models that use debt sculpting to size the repayment profile, the circularity in the cash sweep calculation is resolved by deriving the debt repayment schedule from the projected cash flows before the circularity arises, rather than from the residual cash after the circular calculation.
A debt sculpting algorithm calculates the repayment amount in each period as the amount required to reduce the debt balance such that the DSCR in each period equals a defined target, or as the amount that consumes a defined proportion of the available cash flow. This repayment amount is calculated from the opening balance and the projected operating cash flows, without reference to a closing balance that has not yet been determined.
Implementation. The sculpting algorithm typically uses a GOALSEEK function or a direct formula to derive the repayment. In models that avoid GOALSEEK (for auditability reasons), the sculpting is achieved through a direct calculation of the proportional share of available cash flow allocated to debt service.
Preferred in. Project finance models where debt sizing is based on DSCR targets and where the repayment profile is derived from projected cash flows rather than fixed amortisation.
Technique 3 — Lag by One Sub-Period¶
Where the model requires the average balance calculation and the sculpting technique is not appropriate, the circularity can be resolved by calculating the closing balance using the cash available after the opening balance interest charge, and then adjusting the next period's calculation for any residual difference.
This technique is less common and more complex to implement correctly. It is mentioned for completeness but is not recommended for standard institutional use.
Technique 4 — Controlled Iterative Calculation¶
Where none of the elimination techniques is appropriate for the specific financial relationship being modelled, iterative calculation may be used with the following controls:
- Maximum Iterations set to at least 500 (default of 100 is frequently insufficient for convergence in complex models)
- Maximum Change set to no more than 0.01 (in absolute value terms; for models with large absolute values, a lower threshold is required)
- A convergence check cell that confirms the residual between the circular cells on the last two iterations
- A test protocol confirming that the model converges to the same output when opened fresh versus when calculated from a previously converged state
- Documentation in the model's assumption log confirming the convergence settings and the test results
Audit Checks for Circularity in Debt Models¶
When a project finance or leveraged finance model is submitted for independent audit, the following checks are applied to the debt calculation structure.
Presence check. The audit engine identifies all circular references in the model, including those spanning multiple worksheets. Any circularity in the debt schedule tabs, interest calculation schedules, or cash waterfall is flagged.
Convergence check. For models using iterative calculation, the audit verifies that the convergence settings are appropriate for the model's scale and that a convergence check mechanism is present. Where the model does not include a convergence check, the audit applies its own convergence test.
Opening vs average balance check. The audit verifies which approach is used for the interest calculation base and notes whether the approach is consistent with the loan agreement's interest definition.
Repayment algorithm check. In models using debt sculpting, the audit verifies that the sculpting algorithm correctly derives the repayment amount from the available cash flows and that the DSCR calculation in each period is consistent with the repayment derived by the algorithm.
Cash waterfall integrity check. The audit traces the cash waterfall to confirm that each application of cash (interest, scheduled repayment, sweep, reserves, and distributions) is correctly ordered and that the cash available at each step of the waterfall is correctly calculated from the step above it.
Consistency under scenario switching. The audit verifies that the circularity handling is consistent across all scenarios. A model that correctly handles circularity in the base case but introduces convergence problems in a stress scenario is a material audit finding.
Common Mistakes¶
| Common Mistake | Why It Matters |
|---|---|
| Enabling iterative calculation without setting appropriate convergence parameters | The default Excel settings of 100 iterations and a maximum change of 0.001 are frequently insufficient for project finance models. A model that appears to converge under default settings may produce materially different outputs if recalculated from scratch versus recalculated from a previously converged state. |
| Failing to include a convergence check cell | A model that uses iterative calculation without a convergence check provides no in-model indication of whether it has actually converged. This makes it impossible for the model's users or an auditor to confirm convergence without external testing. |
| Assuming that the absence of a circular reference warning means the model is correct | If iterative calculation has been set globally in the Excel application settings (not embedded in the model file), the circular reference warning will not appear even on a fresh installation. A user who receives the model without the correct application settings will see zeros in the circular cells without any warning. |
| Using average balance interest in conjunction with a cash sweep without recognising the compound circularity | Models that combine average balance interest with a cash sweep mechanism have a more complex circular dependency than models with either feature alone. The additional interaction between the interest charge and the sweep amount requires explicit testing. |
| Not testing circularity handling under stress scenarios | The circular dependency in a debt model may converge reliably under base case assumptions but fail to converge under stress scenarios where debt service coverage is close to the minimum. This is the scenario where accurate convergence matters most for lender purposes. |
Best Practices¶
| Best Practice | Why It Matters |
|---|---|
| Default to opening balance interest for institutional models | The accuracy difference is immaterial for most financial modelling purposes, the structural simplicity is significant, and the model's portability and auditability are substantially improved. |
| Use a debt sculpting algorithm in project finance models that derive their repayment profile from DSCR targets | This is the technically correct approach for sculpted debt repayment and eliminates the primary source of circularity in project finance models. |
| Document the circularity handling approach in the model's assumption log | Any model submitted for lender review should state clearly whether it uses opening balance interest, sculpting, or iterative calculation, and if iterative calculation is used, what the convergence parameters are and what testing was performed. |
| Embed iterative calculation settings in the model file if they are required | This ensures that the model's behaviour is consistent across different users and machines, regardless of their individual Excel application settings. |
| Include a convergence check cell and format it conditionally | A red fill on the convergence check cell when the residual exceeds the defined threshold provides a persistent in-model signal if convergence has failed. |
Regulatory and Industry Context
Lenders and their technical advisers in project finance transactions routinely examine the circularity handling in submitted financial models as part of pre-financial-close model audit processes. A model that uses uncontrolled iterative calculation, or that produces different DSCR values depending on calculation state, will not satisfy the lender's requirement for a stable, verifiable model.
The ICAEW Financial Modelling Code and the FAST Standard both recommend against circular references where avoidable. Where they are unavoidable, documented convergence controls are required.
In syndicated project finance transactions, where the financial model is shared among multiple lenders with different technical capabilities, a model that relies on iterative calculation creates a specific risk: different lenders may open the model in different states and observe different DSCR outputs. This is a material transaction risk that has caused deal delays in practice.
Worked Example
Scenario. A 500 MW solar project finance model calculates quarterly debt service over a 20-year loan term. The senior debt facility pays interest quarterly at a floating rate on the average balance. A cash sweep applies 50% of excess cash flow after scheduled debt service to accelerate repayment.
Audit finding. The automated audit identifies a four-cell circular reference: the interest charge cell references the average balance cell, the average balance cell references the closing balance cell, the closing balance cell references the sweep repayment cell, and the sweep repayment cell references the net cash after interest cell, which references the interest charge cell.
Iterative calculation is enabled in the file settings with default parameters (100 iterations, maximum change 0.001). No convergence check cell is present. Testing reveals that the DSCR in Year 3 Quarter 2 differs by 0.03x between a fresh calculation and a calculation from a previously converged state.
Remediation option 1 — Opening balance interest. The interest calculation is revised to apply the quarterly rate to the opening balance. The circularity is eliminated. The DSCR in all periods is stable and reproducible. The difference in quarterly interest calculated under this approach versus the average balance approach is less than 0.02% of the total interest charge in any period.
Remediation option 2 — Enhanced iterative controls. Where the lender's technical adviser specifically requires average balance interest: Maximum Iterations is increased to 1,000, Maximum Change is reduced to 0.000001, a convergence check cell is added, and a documented test confirms that the model converges to the same DSCR values when opened fresh and when calculated from a previously converged state. The convergence settings are embedded in the model via a Workbook_Open macro.
Further Reading¶
- ICAEW, Financial Modelling Code, Institute of Chartered Accountants in England and Wales
- FAST Standard Organisation, FAST Standard for Financial Modelling
- World Bank, PPP Fiscal Risk Assessment Model, World Bank Group
Continue Reading¶
Prerequisites¶
- Project Finance Model Audit — the parent pillar
Related Technical Guides¶
- Circular References — the general guide to circular references in financial models
- Debt Sculpting Mechanics — the technique for deriving repayment profiles that resolves circularity in project finance models
Related Glossary¶
- DSCR — the key coverage metric affected by circularity in debt models
- Debt Service — the component of the cash waterfall that circular references most commonly affect
- Cash Waterfall — the payment priority structure within which debt service circularity occurs
Related Products¶
- Financial Model Audit Engine (FMAE) — deterministic structural auditing referenced throughout this guide
How OXXON tests thisRun a free structural check with FMAE
Frequently Asked Questions
Is circularity in a debt model always an error?
No. Circularity arising from genuine financial simultaneity (interest expense and available cash in the same period) is not an error in the model's logic, but it does create a structural risk in the model's calculation that must be managed. The question is how to handle the circularity, not whether it represents a modelling mistake.
What is the most common approach used in institutional project finance models?
Opening balance interest is the most common approach in models that have been built to institutional standards. It avoids circularity, is simple to implement, and is accepted by lenders and their technical advisers as a standard approximation.
Will a model using opening balance interest pass a lender's model audit?
Yes, provided the choice of opening balance interest is documented and consistent throughout the model. Lenders and their technical advisers are familiar with this convention and accept it as a standard approach.
Can the cash sweep circularity be resolved without changing the interest calculation?
Yes. The sweep amount can be calculated based on the cash flow available before the sweep, using the interest charge on the opening balance rather than the average balance. This resolves the circularity in the sweep calculation without changing the interest convention.
What happens if my model uses iterative calculation but a recipient's Excel settings have iteration disabled?
The circular cells will return zero, which will cascade through the debt schedule and produce zero or incorrect values for all debt-dependent outputs including DSCR. This is a material model transmission risk.
How does debt sculpting eliminate circularity?
Debt sculpting calculates the repayment amount as a function of the projected cash available for debt service and the target DSCR. Because the repayment is derived from a cash flow that is calculated from the opening balance interest (not the average balance), the dependency loop is broken: repayment no longer depends on a closing balance that depends on the repayment.
Related Articles
Circular References in Financial Models
A circular reference in a financial model occurs when a formula in one cell depends, directly or through a chain of intermediate cells, on its own value. In Excel, circular references are flagged by default and cause the affected cells to display zero rather than a calculated result. When iterative calculation is enabled, Excel resolves circular references by repeatedly recalculating the sheet until a convergence threshold is met, which can produce different results depending on the starting conditions and may mask non-convergence. In financial models, circular references arise most frequently in interest-on-drawn-debt calculations, cash sweep mechanics, and tax shield computations.
Debt Sculpting Mechanics in Project Finance Models
Debt sculpting is a technique used in project finance financial models to derive the periodic debt repayment schedule from the projected cash flows available for debt service, rather than from a fixed amortisation schedule. The repayment in each period is sized such that the debt service coverage ratio (DSCR) in that period equals a defined target, or such that a defined proportion of available cash flow is applied to debt service. Sculpting shapes the repayment profile to match the project's cash flow profile, front-loading repayment in high-cash-flow periods and reducing repayment in lower-cash-flow periods, which increases the project's ability to service debt throughout the loan life.
Debt Service
Debt service is the total periodic payment obligation on a loan facility, comprising interest payable in the period and scheduled principal repayment due in the period. In project finance, debt service is the denominator of the debt service coverage ratio (DSCR). The DSCR measures the ratio of cash available for debt service (CADS) to total debt service, and must exceed the minimum threshold specified in the loan agreement throughout the loan life. Debt service is applied at a defined step in the cash waterfall, after operating costs and before reserve contributions and equity distributions.
Cash Waterfall
A cash waterfall is the contractually defined priority sequence in which cash generated by a project is allocated to successive payment obligations. In a project finance structure, the cash waterfall determines the order in which operating costs, debt service (interest and principal), reserve contributions, and equity distributions are paid from the project's revenue. Senior obligations are paid first; junior obligations and distributions are paid only after senior obligations are fully satisfied. The DSCR and other coverage covenants are calculated at specific points within the waterfall to determine whether cash can flow to the next level.