🏦 Private infrastructure loans
Private infrastructure loans do not trade on a public market, so observable prices and spreads are scarce. The credit spread of each instrument is therefore modelled rather than read off a screen.
⚙️ Spread modelling and Kalman filter
The model expresses a borrower's credit spread as a combination of its observable risk characteristics; the more risk a lender bears, the wider the spread it demands. Rather than fixing this relationship once, the model continuously updates it using a Kalman filter, so the weight placed on each risk factor adapts as market conditions change. Crucially, each date's estimate uses only information available up to that date; the filter is one-sided, with no look-ahead, making the historical series reproducible and free of hindsight bias.
📈 Valuation application
The modelled spread then flows into valuation: each instrument's contractual cashflows are discounted on a risk free curve plus its credit spread. The same model is applied consistently across the eligible universe, so that valuations differ only where the data justifies, reflecting each instrument's own risk profile.
⚙️ The variables and their impact
Each risk factor enters the spread through an economically interpretable channel. Together they capture the credit, structural and market drivers of what a lender charges.
Probability of default
Credit risk
Sourced from the companion infrastructure credit risk model, it anchors the spread to the borrower's fundamental likelihood of default. It is the dominant driver in the model and the primary measure of credit quality in the framework.
⌛ Maturity
Term structure
The remaining life of the instrument, shaping how the spread varies across the term structure of a borrower's debt.
💸 Benchmark rate
Rate environment
The prevailing reference interest rate. It lets the credit spread respond to the wider rate environment rather than being modelled in isolation from it.
🏗️ Facility size
Liquidity & scale
Face value of the facility, a proxy for the borrower's scale and the relative liquidity of the exposure.
🏭 Sector
Common factor
The industrial sector includes power, roads, wind and solar, energy and water resources, and environmental services. A sector acts as a common systematic factor: borrowers in the same sector tend to move together, and different sectors sit at different spread levels. This shared movement remains when spreads are aggregated across many borrowers. We selected these sectors because each has enough project‑finance constituents to estimate a stable, statistically significant loading. Sparse sectors or those dominated by a single outlier are folded into a common baseline rather than given their own factor, ensuring every sector effect is supported by data.
💰 Business model
Revenue certainty
Whether revenue is contracted or merchant (market exposed). Merchant exposure changes the cashflow risk profile of a borrower, which the model prices through this factor.
Because the coefficients vary over time, the strength of each channel can change through the cycle. For example, sector effects may widen during stress and compress in calmer markets, while each factor’s economic direction remains interpretable.
🛡️ Robustness of the model
Because prices for these instruments are not continuously observable, the framework is validated on the properties that matter for a fundamentally valued asset class: the economic soundness of its factors, the statistical behaviour of the estimates, and the discipline of the data and process behind them.
💸 Economic soundness
|
Feature |
Details |
|---|---|
|
Grounded factors |
Every driver carries the sign and role that credit intuition expects, with credit risk the dominant driver and clear sector effects. |
|
Independent credit anchor |
Default probability is supplied by a separately governed credit risk model, so credit quality is an external input rather than a circular one. |
|
Interpretable throughout |
Each coefficient maps to a clear economic story, so the output can always be explained, not just produced. |
📊 Statistical behaviour
|
Feature |
Details |
|---|---|
|
No look ahead |
The one sided Kalman filter uses only past information at each point, so the history is reproducible and free of hindsight. |
|
Stable coefficients |
The time varying estimates drift smoothly; they do not jump erratically or flip sign, indicating a well identified model. |
|
Full cycle exposure
|
The estimation history spans several credit and interest rate regimes, stress and calm alike, so relationships are not fitted to a single benign period. |
🗄 Data & process
|
Feature |
Details |
|---|---|
|
Breadth of data. |
A long history and a wide cross section of borrowers across many sectors, geographies and currencies. |
|
Output discipline. |
Spreads are monitored each cycle for range, dispersion and outliers; they remain bounded and economically sensible, with no negative or extreme values. |
|
Transparent & governed. |
The process is rules based with no discretionary overrides, and its inputs, credit risk model, rate curves and spread model, are maintained as independent, reviewed components. |
🔧 Diagnostics from the model
📊 Model residual distribution
📋 Pricing error summary
|
Pricing error, % |
Mean |
Median |
5th |
95th |
|---|---|---|---|---|
|
Modelled less observed |
-1.2 |
-3.7 |
-36.4 |
40.9 |
The calibration report's histogram of model residuals, reproduced here: each residual is modelled less observed, divided by the model's own standard deviation, shown across three standard deviations against a standard normal. The table expresses the same residuals as a pricing error, in percent of the observed spread (modelled less observed), across the full estimation set. The residuals sit close to the standard normal, indicating well calibrated uncertainty.
📋 Scope of the framework.
This is a fundamental valuation model for instruments without continuous observable prices. Its purpose is to produce a stable, economically grounded spread and a consistent cross sectional ordering of risk. Valuations are model estimates and should be interpreted as such.
📈 Backtesting
Because the Kalman filter is one sided, each date's modelled spread uses only information available up to that date. The historical series is therefore an out of sample record, reproducible and free of hindsight.
Each cycle the specification is chosen against competing specifications on out of sample error, and this specification is the selected model.
The chart compares the modelled spread with the spread subsequently observed from 2020 onward for the project finance universe, as the annual average across the universe.
📊 Modelled spread vs. observed, mean
Annual average of the actual and modelled spread, averaged in the model's log space as in the model selection report, using all observations. Spreads in basis points.
🔬 Model performance summary
The modelled spread tracks the observed spread closely across the cycle. The model is a fundamental estimate, so it captures the level and movement of the spread across the universe rather than the moment to moment spread of any single loan.