The Pipeline Under the Ceiling

Everyone prices the guarantee. Once a fiscal rule counts the contingent book, that is the wrong number — the rule that scores the book against the ceiling, not the ceiling itself, decides how much a country builds. Get it wrong, as almost every framework does, and you either strangle a sound pipeline or breach the ceiling in the first recession.

Abstract

Most of the fiscal-risk literature — and most of this series — warns that PPP guarantees are dangerous because they look free and stay off the recorded debt stock. A growing number of fiscal frameworks now do the opposite: they count the present value of firm and contingent PPP commitments against a hard ceiling. Peru's 2025 PPP law (Ley 32441) is the sharp case — the accumulated PV of firm and contingent commitments of the non-financial public sector may not exceed 12% of GDP. The moment a rule binds on that measure, a guarantee stops being a free good and becomes a claim on scarce ceiling headroom, and running a PPP programme stops being project-by-project approval and becomes a capital-budgeting problem under a constraint: which projects, with which guarantee packages, in what sequence, maximise programme value while the cumulative scored liability stays under the ceiling.

This paper builds that missing layer, and the finding is blunt: the binding constraint is not the ceiling — it is the arithmetic a ministry uses to add its guarantees up. Two decades of fiscal-risk practice taught governments to price a single guarantee; almost none score the book, and once a rule binds, the book is the only number that matters. A guarantee book can be scored three ways: at its expected value (cheap, and it under-counts the correlated tail); at its face / maximum exposure (expensive, and it charges the sum of every worst case as if they happened together); or at its portfolio expected shortfall — the correlated 1-in-20 tail of the whole book. The last is the only sub-additive rule: because a country's guarantees do not all pay out together, the tail of the book is less than the sum of the individual tails, so this rule is the only one that rewards a ministry for diversifying its guarantee portfolio.

On a worked nine-project medium-economy pipeline calibrated to a 12%-of-GDP ceiling, the difference is a whole project. The full pipeline scores at 6.1% of GDP under expected value, 9.3% under portfolio expected shortfall, and 12.0% under face value — the face-value book alone exhausts the ceiling. Solving the exact programme-value-maximising selection under each rule, portfolio-ES scoring admits US$25.5bn of programme value against US$23.7bn under face value — a gain of US$1.8bn (+8%), one entire project (an irrigation scheme), bought under the identical 12% ceiling purely by measuring the book at its correlated tail rather than the sum of its worst cases. And the cheap rule is worse than it looks: expected-value scoring admits the whole pipeline on paper, but its scored book of 6.1% of GDP understates that same pipeline's true correlated tail of 9.3% by US$8.9bn — it builds more today and strains the ceiling exactly when the macro draw arrives.

The result generalises across the ceiling architectures countries actually use. Running the same pipeline under a stock-PV cap, a funded contingency pool, a revenue-based flow cap, and a project-subsidy cap shows the scoring rule is decisive wherever the ceiling scores the portfolio — strongest for stock-PV caps and funded pools, present but distorted for flow caps (which reward stretching the concession tenor and bind procyclically), and absent for project-subsidy caps that never sum the book. And the value of the rule is hump-shaped in the ceiling's tightness: it is worth the most — here more than two projects — where a cap that disciplines without strangling should sit, an intermediate band of roughly 8–12% of GDP on the PV of firm-plus-contingent commitments.

The policy lead follows. A ministry that runs a programme under a liability-inclusive ceiling should (1) adopt a sub-additive scoring rule for guarantees — the portfolio tail, not the sum of faces and not the expected value; (2) prefer a stock-PV cap in the ~8–12%-of-GDP band, or a portfolio-ES-sized contingency fund, over flow or project-level caps; (3) publish the headroom ledger — the running total of scored liabilities against the ceiling path; (4) ration scarce headroom to the highest value per unit of headroom, not first-come-first-served; and (5) re-run the frontier each budget cycle, because the binding constraint moves with GDP, the guarantee book's correlation, and the pipeline.

Keywords: PPP programme · fiscal rule · contingent liabilities · ceiling headroom · guarantee scoring · portfolio value-at-risk · expected shortfall · sub-additivity · capital budgeting · Peru · Ley 32441 · Ministry of Finance.

1. The free good becomes a ration

This series has spent its length making one argument: a PPP guarantee is a priced liability that governments treat as free.¹⁰ The minimum-revenue guarantee that pays out when traffic disappoints, the exchange-rate backstop that pays out when the currency does, the availability commitment, the termination clause that prices like a written put — each is a real claim on the public purse, and each is dangerous precisely because it stays off the recorded debt stock and therefore outside the number a fiscal rule watches.¹ The remedy the series keeps reaching for is measurement: price the tail, disclose it, count it.

This paper breaks with that remedy in one respect, and it is the whole point. Pricing a guarantee — the task this series, PFRAM, and twenty years of fiscal-risk practice have laboured at — turns out to be the easy half. The hard half, the half that actually decides how much a country builds, is the rule that adds the priced guarantees into the single number a ceiling binds on. That rule is where fiscal space is won or lost, and it is the one almost no framework gets right. The field has been perfecting the price of a brick while the building was decided by the arithmetic no one was doing.

Some governments have started to. A fiscal framework can be written so that the present value of a PPP's firm and contingent commitments is counted against a debt or liability ceiling — so that a guarantee, the moment it is signed, consumes fiscal space the way a bond issue would. When it is, the entire problem inverts. The guarantee is no longer a free good that a line ministry can hand out to make a project bankable at no apparent cost. It is a draw on a scarce, shared budget — and every draw crowds out the next project's.¹²

Peru made this concrete in 2025. Under Ley 32441 and its regulation, the accumulated present value of the firm and contingent commitments of the non-financial public sector arising from Asociaciones Público-Privadas may not exceed 12% of GDP.² That is not a disclosure requirement; it is a hard cap on a measure that explicitly includes the contingent book. The country's PPP pipeline — the roads, ports, transmission lines, hospitals and irrigation schemes its investment agency has been assembling — must now fit, in cumulative scored liability, under a single 12%-of-GDP line.

The moment such a rule binds, the ministry of finance's job changes character. Project appraisal — is this one, alone, value for money? — is still necessary, but it is no longer sufficient, because approving a project now spends something that the next project needs. The right question becomes a programme question, and it is a capital-budgeting question under a constraint:

That is the question this paper answers. It is the missing layer between the single-project decision the rest of this series prices and the aggregate contingent tail that Stress-Testing PPP Portfolios measures.⁴ A small economy with three projects never has to ask it — three projects fit or they do not, and there is no interesting sequencing. A medium economy with a genuine pipeline and a binding ceiling has to ask it constantly, and the answer is not obvious, because the constraint is not the sum of the projects' costs. It is the scored value of their combined guarantee book — and, as the next sections show, that number depends entirely on a rule the ministry gets to choose.

Why a ceiling, if the appraisal is done well?

A natural objection precedes everything that follows. If every project is selected by a proper cost-benefit analysis, and its procurement mode chosen by a genuine value-for-money assessment, then each PPP in the pipeline is already, individually, worth doing and best done as a PPP. Why then impose a ceiling that rations projects the appraisal has already blessed? In a first-best world the objection is decisive — and, tellingly, that same world is one in which financing structure is irrelevant and there is no case for a PPP in the first place. The ceiling earns its place for reasons the project-level appraisal cannot reach, and they are worth stating before the mechanics, because they are the reason the mechanics matter.

The first reason is a horizon asymmetry in the budget process itself. A PPP's fiscal commitments run for twenty or thirty years; the medium-term expenditure framework that scrutinises every other claim on the budget runs for three to five. A traditional public work discharges its fiscal cost inside that window — it enters the budget, the MTEF and the deficit rule, and everyone sees it. A PPP commitment falls largely outside it. It receives less scrutiny than the traditional work it replaces not because it is smaller but because it is temporally invisible to the very instrument that disciplines everything else — fiscal illusion with a long maturity. This is also why the ceiling must be a stock cap on present value rather than a flow control: only a stock-PV measure pulls the whole thirty-year commitment back to a single number at the moment of signing, inside the window where it can still be weighed. A flow or MTEF-style control is horizon-limited by construction — it sees the near years and misses the tail, which is exactly where the commitment lives (§8).

The second reason is a measurement asymmetry within the PPP. A firm commitment — an availability payment — is a scheduled amount, easy to quantify. A contingent commitment — a guarantee — is hard to estimate without the right methodology, and therefore easy to understate or to leave out. Facing scrutiny on what is measured and none on what is not, a decision-maker substitutes toward the unmeasured: the hardest part of the PPP's cost to see is precisely the part that will be systematically under-reported. This gives the scoring rule of §3 a second, institutional edge over its efficiency one. Expected-value scoring is not merely the rule that under-provisions the tail; it is the rule a ministry that wished to hide the commitment would choose, because it produces the smallest, most comfortable number. Portfolio-ES scoring is the anti-concealment rule: it forces the correlated tail onto the books. Peru's law reaches for exactly this when it caps the quantifiable commitments — the ceiling compels the measurement it needs.¹¹

Disclosure alone does not answer the objection, which is why better contingent-liability reporting — necessary as it is — is not a substitute for a cap. Even a fully disclosed commitment beyond the electoral and MTEF horizon is under-weighted by the decision-maker who signs it, because the guarantee draws in a successor's term; seeing the number is not the same as bearing its weight. And disclosure measures the exposure without rationing it. The cap does both: it forces the commitment to be quantified, and it makes that quantity bind today.

Only once the commitment is visible and binding does the deeper, allocative reason apply — the one the appraisal genuinely cannot internalise. Each guarantee draws on a shared resource: the sovereign's capacity to absorb a correlated shock, its fiscal space and market access in a crisis. A project that passes cost-benefit and value-for-money in isolation still raises the probability and cost of the joint bad state for every other project in the book — a fiscal externality that is a property of the portfolio, not the project, and so is invisible to any project-level appraisal. The ceiling is the price of that externality. Put most sharply: the marginal value of public funds to a sovereign is state-contingent — it spikes in the very crisis when guarantees draw together — so the correct value-for-money test would price each guarantee at its crisis-state cost, which is its contribution to the portfolio tail. The ceiling, scored at that tail, is simply the decentralised way to force that state-contingent shadow price into the project decision. It does not override good appraisal; it supplies the price good appraisal was missing.

2. Headroom accounting — what a project costs the ceiling

Start with the accounting identity the ceiling imposes. Under a liability-inclusive rule, each candidate project consumes ceiling headroom equal to two components:

\[\text{headroom}_i \;=\; \underbrace{D_i}_{\text{direct}} \;+\; \underbrace{S(g_i)}_{\text{scored guarantees}}\]

The direct component $D_i$ is the on-balance-sheet financial liability the project puts on the public books in present value — the availability-payment stream of a government-pays PPP, the co-financing (VGF) the state commits, the sovereign's share of a hybrid structure. A pure user-pays toll concession with no state payment has $D_i = 0$; an availability hospital has $D_i$ equal to the PV of its unitary charge. This component is not controversial: it is a scheduled, unconditional obligation, and every framework counts it more or less the same way.

The scored-guarantee component $S(g_i)$ is where the paper lives. A minimum-revenue guarantee, an FX guarantee, a debt guarantee, a revenue-floor — each is a contingent obligation: it pays out only in a bad state, and its cost to the ceiling depends on how the rule chooses to value that contingency. Call the loss the guarantee pays if it is drawn its face $f_i$ (the maximum exposure, in PV), and the probability of a material draw over the contract life $p_i$. Then the same guarantee can be scored against the ceiling in three very different ways, developed in §3.

The programme is admissible if and only if the cumulative headroom stays under the ceiling in every period:

\[\sum_{i \in \text{admitted}} \Big(D_i + S(g_i)\Big) \;\le\; C_t \quad \text{for all } t,\]

where $C_t$ is the ceiling in period $t$ — a trajectory, not a point, because it moves with GDP (a 12%-of-GDP cap grows as the economy does) and because the scored book itself evolves as projects are commissioned, as guarantees amortise, and as draws are realised. The static version — does the whole book fit under the ceiling today — is the binding first cut, and Peru's cap is written as a stock cap on the accumulated PV, so the static problem is the one that bites first. The dynamic version — can the pipeline be sequenced so the cumulative path tracks under the growing ceiling — is the operating problem the ministry faces year to year, and §4 shows both.

Two consequences of this accounting are worth stating before the scoring rule, because they hold under any scoring rule and are half the paper's point:

Guarantees are now rationed, not free. Under the old regime, a line ministry could make a marginal project bankable by adding a minimum-revenue guarantee at no charge to any budget it recognised. Under a liability-inclusive ceiling, that guarantee consumes headroom that another project needs — so the marginal cost of over-insuring one deal is the next deal that no longer fits. Rationing changes the incentive: the leanest guarantee package that clears bankability is now strictly optimal, because every unit of unnecessary cover is a unit of foregone programme.

Value per unit of headroom, not value alone, is the ranking. Because headroom is the binding constraint, the programme should not be built in order of project value, nor in order of readiness, nor first-come-first-served. It should be built in order of value per unit of headroom consumed — the density

the public benefit $v_i$ each project delivers per point of ceiling it spends. A high-value project with a huge guarantee book can have a lower $\eta_i$ — and be a worse use of the ceiling — than two modest ones with lean guarantees. This is the standard result of capital budgeting under a budget constraint, applied to a resource, ceiling headroom, that most PPP units do not yet know they are spending. (Because $S(\cdot)$ is sub-additive under the ES rule, $\eta_i$ is not fixed — a project's headroom cost depends on what else is in the book — which is why §4 solves the selection exactly rather than greedily ranking by $\eta_i$.)

3. The scoring rule is the whole game

Everything now turns on $S(g_i)$: how a guarantee book is scored against the ceiling. There are three candidate rules, and the choice between them is not a technicality — it changes the measured liability of an identical pipeline by a factor of two, and it changes how much a country can build by a whole project.

Rule 1 — Expected value. Score each guarantee at its expected draw, $S^{\text{EV}}(g_i) = p_i f_i$, and the book at the sum. This is the cheapest rule: it charges the ceiling only the average cost of the contingency. Its virtue is that it does not over-provision for a tail that usually does not arrive; its fatal flaw is that the ceiling exists precisely to survive the tail, and expected value cannot see it. A book of guarantees each with a 15% draw probability scores at 15% of its face — and then, in the one-in-twenty macro crisis the ceiling is meant to withstand, pays out far more than that. Expected value is the rule that lets a programme look affordable on average and blow through its ceiling in a recession.

Rule 2 — Face / maximum exposure. Score each guarantee at its full face, $S^{\text{FACE}}(g_i) = f_i$, and the book at the sum of faces. This is the conservative rule, and it has the opposite flaw: it charges the ceiling the sum of every worst case as if they all happened at once. But a country's guarantees do not all pay out together — a demand-risk toll road, an FX-guaranteed transmission line, and an availability hospital fail in partly different states of the world. Charging the ceiling the sum of their faces treats a diversified book as if it were a single concentrated bet, and the effect is to make the ceiling bind far tighter than the real risk warrants — to leave the country building less than it safely could. These first two rules are not hypothetical constructions: the standard fiscal-risk tool a ministry already runs — the IMF–World Bank PPP Fiscal Risk Assessment Model (PFRAM) — reports for each guarantee precisely its maximum exposure and its expected cost, which are face value and expected value in this paper's terms.¹³ What PFRAM does not compute is the correlated tail of the whole book — and that, Rule 3, is the rule this paper argues a binding ceiling turns on.

Rule 3 — Portfolio expected shortfall. Score the book, not each guarantee, at its correlated tail: $S^{\text{ES}}(\text{book}) = D + \operatorname{ES}_{95}(\text{portfolio loss})$, where the portfolio loss is simulated from a two-factor model — a demand/recession factor that depresses traffic and throughput, and an FX/devaluation factor that hits the currency-guaranteed legs — the two correlated at the +0.16 measured from Peru's macro history (§7), with each guarantee loading on its factor through a systematic share; $\operatorname{ES}_{95}$ is the average loss in the worst 5% of outcomes — the one-in-twenty tail. Formally, for the portfolio loss $L$,

\[\operatorname{ES}_{\alpha}(L) \;=\; \mathbb{E}\!\left[\,L \mid L \ge \operatorname{VaR}_{\alpha}(L)\,\right], \qquad L \;=\; \sum_i f_i\,\mathbf{1}\!\left\{\ell_i > \Phi^{-1}(1-p_i)\right\}, \qquad \alpha = 0.95,\]

where $\ell_i$ is guarantee $i$'s latent draw variable (the two-factor model of Appendix A), $\mathbf{1}\{\cdot\}$ marks the states in which it draws, and $\Phi^{-1}$ is the inverse standard-normal CDF that maps the draw probability $p_i$ to a threshold. This is the rule Stress-Testing PPP Portfolios builds the machinery for, now used as the constraint measure rather than a disclosure.⁴ Its decisive property is sub-additivity: because the guarantees are not perfectly correlated, the expected shortfall of the book is less than the sum of the individual expected shortfalls —

\[\operatorname{ES}_{95}(\text{book}) \;\le\; \sum_i \operatorname{ES}_{95}(g_i) \;\le\; \sum_i f_i.\]

Sub-additivity is what makes portfolio-ES the only coherent risk measure of the three in the technical sense, and the only one that rewards a ministry for diversifying its guarantee book rather than punishing it.⁵ Under expected value, diversification is invisible (the sum of means is the sum of means regardless of correlation). Under face value, diversification is ignored (the sum of faces is the sum of faces). Only under portfolio-ES does spreading the guarantee book across uncorrelated sectors and currencies actually free headroom — which is exactly the behaviour a ministry should want to reward, because a diversified contingent book is a safer one.

The tail mathematics here is not new. The sovereign contingent-liability literature already computes the expected and unexpected (tail) loss of a diversified guarantee portfolio — but it does so to size a provisioning buffer: how much a government should reserve against its guarantee book.¹⁴ The step this paper takes is to use that same portfolio tail as the scoring rule against a binding ceiling — a decision about how much a country can build, not how much it should set aside. Reserving and rationing are different questions, and the second one is where the scoring rule decides the programme.

Figure 2 puts all three rules on the guarantee book's own simulated loss distribution, and the geometry is the whole argument in one picture: expected value sits at the distribution's mean, far to the left of the tail the ceiling exists to survive; face value sits at the sum of every maximum, far to the right, as if the whole book drew at once; and portfolio-ES sits at the mean of the shaded worst-5% tail — the only one of the three that is both above the mean and below the sum of maxima, because it is the only one that counts the correlation of the book.

The three rules are not close. On the worked pipeline of §5 — nine projects, a 12%-of-GDP ceiling — the identical guarantee-and-liability book scores at:

Scoring rule	Full-pipeline scored liability	% of GDP
Expected value	US$16.4bn	6.1%
Portfolio ES₉₅ (correlated tail)	US$25.2bn	9.3%
Face / sum of maxima	US$32.5bn	12.0%

The same programme is a comfortable 6.1% of GDP, a snug-but-safe 9.3%, or an over-the-line 12.0% — depending only on the rule. Against the 12% ceiling, the face-value book breaches on its own; the portfolio-ES book fits with room; the expected-value book fits easily and lies about the tail. The sub-additivity is visible in the numbers: the guarantee portion of the ES score is US$11.5bn, against a sum of individual worst-case exposures of US$18.8bn — the diversification of the book is worth US$7.3bn of headroom, and only the ES rule counts it.

4. The optimiser — rationing the pipeline under the ceiling

With the scoring rule fixed, building the programme is an optimisation. The static problem — which subset of the pipeline maximises programme value subject to the scored book fitting under the ceiling — is a 0/1 knapsack:

\[\max_{x \in \{0,1\}^n} \sum_i v_i x_i \quad \text{s.t.} \quad S\!\Big(\textstyle\sum_i x_i g_i\Big) + \sum_i D_i x_i \;\le\; C,\]

where $v_i$ is the programme value the project delivers. Under the expected-value and face rules the constraint is linear (a sum of per-project scores), and this is the textbook knapsack; under portfolio-ES it is non-linear (the score of a subset is the correlated tail of that subset, which must be simulated), but for a real pipeline — nine, twelve, twenty projects — the problem is small enough to solve exactly by enumeration, and the reference optimiser does, so the results below are optima, not heuristics.

The dynamic problem adds time. Projects are commissioned over years; the ceiling $C_t$ grows with GDP; the scored book evolves as guarantees mature and as commissioned projects' liabilities amortise. Sequencing then becomes a multi-period knapsack: order and time the pipeline so the cumulative scored-liability path stays under the ceiling path, front-loading the highest value-per-headroom projects and timing the guarantee-heavy ones for years in which GDP growth has opened room. The static frontier is the binding cut and the one this draft quantifies; the sequencing layer is the operating manual, and it inherits the same lesson — the scoring rule decides how much of the path fits.

Three rationalisation levers operate inside the optimisation, and a ministry running the programme should pull all three:

5. A worked pipeline — the cost of the wrong rule

Consider a medium economy — GDP US$270bn, a 12%-of-GDP ceiling worth US$32.4bn of headroom — with a nine-project pipeline: two demand-risk toll roads, a regional airport, an FX-heavy metro line and transmission project, bulk water and a hospital network on availability terms, an irrigation scheme, and national broadband. Total programme value if all nine are built: US$25.5bn. The full inputs — each project's value, direct liability, guarantee face and draw probability, and the sector loadings on the two macro factors — are in the optimiser (pipeline_optimizer.py); the project economics are illustrative-but-plausible, but the demand↔FX correlation that drives the diversification result is estimated from Peru's macro history (+0.16 between the recession and devaluation shocks, 2001–2024; see §7 and estimate_correlations.py), not chosen.

Solving the exact value-maximising programme under each scoring rule, against the identical 12% ceiling:

Scoring rule	Programme value admitted	Projects	Headroom used
Face / sum of maxima	US$23.7bn	8 / 9	US$29.2bn
Portfolio ES₉₅	US$25.5bn	9 / 9	US$25.2bn
Expected value	US$25.5bn (on paper)	9 / 9	US$16.4bn

Read the first two rows against each other, because they are the paper's result. Under face-value scoring, the 12% ceiling cannot fit the whole pipeline — the sum of faces (US$32.5bn) exceeds it — so the optimiser drops the lowest value-per-headroom project, the irrigation scheme, and the country builds US$23.7bn of value. Under portfolio-ES scoring, the same ceiling fits all nine projects, because the book's correlated tail (US$25.2bn) leaves room the sum of faces did not — and the country builds the full US$25.5bn.

That +8% is the reading at Peru's 12%-of-GDP cap, which sits at the loose edge of the binding range; it is one project on a nine-project book. The gain is larger where the cap binds harder — §7 sweeps the ceiling and shows the value of the scoring rule is hump-shaped, worth as much as US$4.2bn (two projects) at a cap near 8% of GDP before falling away as the cap loosens enough to fit everything. It also grows as the book grows and as its guarantees become less correlated (more diversified books have more tail to reclaim). A country whose ceiling binds in that intermediate zone, or whose pipeline is longer, leaves proportionally more value on the table by scoring at face.

And the third row is the trap. Expected-value scoring admits the whole pipeline too — but for the wrong reason, and dangerously. Its scored book is only US$16.4bn (6.1% of GDP), because it charges the ceiling the average draw rather than the tail. That same admitted pipeline's true correlated tail is US$25.2bn (9.3% of GDP) — the expected-value rule understates the real one-in-twenty exposure of its own programme by US$8.9bn. A ministry scoring at expected value would look at a comfortable 6.1%-of-GDP contingent book, approve the whole pipeline with headroom to spare, and discover in the next recession that the book it actually holds is 9.3% — a US$8.9bn hole opening against a ceiling it thought it was well inside. Expected value builds the most today and strains the ceiling exactly when the macro draw arrives. It is the rule most likely to be in use, and the rule most likely to fail the country when the rule is supposed to protect it.

6. Peru, in practice — a live 12%-of-GDP ceiling

This is not a hypothetical constraint. Peru's Ley 32441 (2025), Article 31.1, sets the ceiling at 12% of GDP on the accumulated stock of quantifiable firm and contingent commitments, net of income, of the non-financial public sector arising from APPs, at present value — reviewable every three years.² Two features of that wording matter for this paper. It counts the commitments as quantifiable — the law presumes each contingent guarantee is assigned a number, which is exactly the scoring choice §3 is about. And it counts them net of income — the measure is the net fiscal exposure, which makes the correlated-tail question sharper still, because netting a guarantee's expected inflows against its draw only makes sense at a coherent measure of the draw. The law was passed to unblock a large investment pipeline, and it was accompanied by exactly the concern this paper formalises: commentators warned that the framework could expose the state to contingent obligations of up to 12% of GDP — the ceiling read as a risk, not just a limit.

Peru therefore has all three ingredients that make the programme problem bind: a real, long pipeline assembled by its investment agency; a hard ceiling that counts the contingent book; and a legal requirement to quantify that book contract by contract. What the law does not yet fix — and what this paper argues is the decisive choice — is how the firm-and-contingent PV is scored against the 12%. If the quantification is done at expected value, the country will over-build against an understated tail (§5's third row); if at face or the sum of maxima, it will under-build against an overstated one (§5's first row); if at the correlated portfolio tail, it will build the most the ceiling can safely carry (§5's second row). The ceiling is set in law; the scoring rule that determines what the ceiling means is a regulatory and methodological choice still in play — and it is worth, on the numbers here, a whole project. This is not a rhetorical "in play": Article 31.3 of the same law delegates the indicators, parameters and calculation methodologies for the firm-and-contingent commitments to a later MEF ministerial resolution, and the Reglamento (Decreto Supremo N° 316-2025-EF, December 2025) approved the framework without fixing that methodology — so the choice among expected value, face and portfolio tail is, as a matter of Peruvian law, genuinely open and pending the MEF's own hand.⁹

Colombia offers a second live case with a different architecture: APP vigencias futuras are capped, and contingent liabilities are provisioned through the Fondo de Contingencias Contractuales de las Entidades Estatales against a fiscal rule.⁶ The mechanics differ — a funded contingency pool rather than a single PV ceiling — but the analytical question is identical: the amount a project must set aside, and the amount the programme can carry, depend on whether the contingency is scored at its expected value or its correlated tail. Any framework that counts contingent PPP liabilities against a limit inherits the result of §3.

7. The exercise across ceiling types

Peru's is one architecture — a stock cap on the present value of firm and contingent commitments, as a share of GDP. Countries write the ceiling in several forms, and the natural question is whether the scoring-rule result survives the translation. It does, with instructive variations. Running the same nine-project pipeline under each architecture (reproducible in ceiling_types.py) shows where the result bites hardest, and exposes a defect or two that some architectures carry and others do not.⁷

Stock-PV cap (Peru 12%, Honduras 5%) — and the scoring rule is worth most where a sensible cap sits. Sweep the stock cap from 4% to 15% of GDP and measure the programme value that sub-additive (ES) scoring admits over face-value scoring. The gain is not monotone in tightness — it is hump-shaped (Figure 4, left). When the cap is loose (≥13% of GDP here), the whole pipeline fits under either rule and the scoring choice is worth nothing. When it is punishingly tight (4–5%), so little fits under any rule that the reclaimed value is bounded by what can be built at all. The scoring rule is worth the most — up to US$4.2bn, more than two projects, at a cap of about 8% of GDP — in the intermediate binding zone, precisely where a cap that disciplines without strangling ought to sit. Peru's 12% sits at the loose upper edge of that zone (the +US$1.8bn of §5); Honduras's 5% sits deep inside it, which is exactly why Honduras, whose actual exposure has already reached 6.6% of GDP against a 5% cap, is the live cautionary case — a tight cap makes the scoring rule decisive, and the wrong rule there is the difference between a strangled pipeline and a prudent one.⁸

Funded contingency pool (Colombia, Indonesia) — the same rule sizes the fund. Where the framework is a contingency fund the state must pre-capitalise rather than a headroom ceiling, the scored contingent book is the fund. The three rules size it very differently (Figure 4, right): at expected value the fund is US$2.7bn (1.0% of GDP) — and is empty when the correlated draw arrives, under-capitalised against the true tail by the same US$8.9bn; at face value it is US$18.8bn (7.0% of GDP) — idle cash the diversification of the book never required; at portfolio-ES it is US$11.5bn (4.3% of GDP), which both covers the one-in-twenty tail and minimises the fund. The sub-additive rule is the cheapest adequate provision — a rare case where the prudent choice is also the less expensive one.

Flow cap on annual commitments (Brazil, 5% of net current revenue) — the scoring logic transfers, but two distortions attach. A cap on annual commitments rather than a PV stock inherits the EV-vs-face-vs-ES logic unchanged. But it carries two defects a stock cap does not. First, a tenor distortion: because the commitment is spread over the concession's life, stretching the tenor thins the annual number and buys headroom the underlying risk has not fallen — a gaming margin the stock-PV measure closes. Second, and worse, a flow cap denominated in revenue is procyclical: net current revenue falls in the recession exactly as contingent guarantees draw, so the ceiling can bind hardest at the moment the country most needs to honour its book. A flow cap is workable, but it should be paired with a stock check and scored, like the rest, at the correlated tail.

Project-level subsidy cap (India, VGF ≤ 20% of project cost) — the wrong quantity. A cap on each project's upfront subsidy disciplines single-deal generosity, but it never sums the guarantee book: the correlated contingent tail — US$11.5bn, 4.3% of GDP, on this pipeline — is invisible to it. The scoring-rule result does not bite here, not because the tail is small, but because this architecture never scores the portfolio at all. It is a useful complement to a ceiling and a dangerous substitute for one.

8. Which ceiling, and how tight — a ranking and reasonable ranges

The exercise supports a normative ordering. The right question for a ministry designing or reforming its framework is not only how tight the ceiling should be, but what architecture it should take and which rule should score the book against it. Ranked from most to least fit for controlling PPP fiscal risk:

The through-line is the paper's: the ceiling is not the binding constraint — the rule that scores the book against it is. Choose a stock-PV cap in the 8–12%-of-GDP band (or a portfolio-ES-sized fund), score the guarantee book at its correlated tail, publish the headroom ledger, and re-run the frontier each cycle. The architecture and the number matter; the scoring rule matters more.

9. Limitations

(1) The contribution is a frontier, not a prescription. Real pipelines are sequenced by politics, readiness, and regional equity, not by an optimiser. This paper does not claim ministries do — or should mechanically — build the value-maximising subset. It establishes the benchmark — the most programme value a ceiling can safely carry under a given scoring rule — and the cost of deviating from it, so that a deviation (building a lower value-per-headroom project first for regional reasons) can be priced rather than made blindly. The frontier is the discipline, not the plan.

(2) The result is conditional on the scoring rule, which is a policy choice. The paper's entire point is that the scoring rule drives everything — which means the numbers are conditional on it, not observed facts. There is no "true" scored liability independent of the rule; there is only expected value, face, or portfolio-ES, and the paper argues normatively for the third on the grounds of coherence and incentive. A ministry that rejects sub-additive scoring rejects the paper's headline, and it should do so explicitly.

(3) The pipeline economics are illustrative; the correlation that drives the result is not. The nine-project pipeline's values, direct liabilities, guarantee faces and draw probabilities are plausible but synthetic — a real programme substitutes its own book. The one parameter that actually moves the portfolio-ES advantage, though — the correlation between the demand and FX guarantee shocks — is estimated, not assumed. It is anchored to Peru's own macro history: over 2001–2024 (World Bank), the correlation between Peru's recession shock (a shortfall of real GDP growth) and its devaluation shock (PEN/USD depreciation above trend) is a modest +0.16 — Peru's busts and its currency weakness co-move, but far from lockstep — which maps to an implied demand↔FX guarantee-draw correlation of about +0.10 once each guarantee's idiosyncratic share is netted out.³ This is lower than the near-lockstep a single-common-factor model would impose, so the measured diversification benefit (US$7.3bn of headroom, the +8%) is if anything larger than a naïve single-factor calibration would show. The estimation is reproducible in estimate_correlations.py, published alongside the optimiser. What remains genuinely calibration-dependent is the scale of the pipeline (bigger books, more tail to reclaim) and the systematic-share loadings within each shock; the sign and rough size of the result rest on a number measured from Peru, not chosen. The method follows Stress-Testing PPP Portfolios, which estimates the same correlation structure from macro history.⁴

(4) The static ceiling is the first cut, not the whole problem. Peru's cap is a stock cap, so the static knapsack binds first; but the operating problem is dynamic (sequencing under a growing ceiling), and the draw-timing of guarantees — which realise over decades — interacts with the amortisation of direct liabilities and GDP growth in ways this draft sketches (§4) but does not fully solve. The multi-period optimiser is the natural extension.

(5) The measure must match the rule the country actually writes. The paper assumes the ceiling counts both firm and contingent PV; frameworks that count only firm liabilities, or only a sub-set of guarantees, change the constraint and must be modelled as written. Mis-stating which liabilities a given ceiling counts would sink the premise, so the Peru rule is stated precisely and no claim of generality beyond liability-inclusive ceilings is made.

10. The policy lead

For a ministry of finance running a PPP programme under a fiscal rule that counts contingent liabilities, four moves follow directly from the arithmetic.

Adopt a sub-additive scoring rule. Score the guarantee book against the ceiling at its correlated expected shortfall, not the sum of faces and not the expected value. The sum of faces leaves the country building less than it safely could (§5: a whole project foregone); the expected value builds more than it safely can and hides the tail until it arrives (§5: a US$8.9bn understatement). Only the portfolio tail counts diversification, rewards a safer book, and states the real exposure — and it is the measure the country needs to hold anyway to know what it owes.

Publish the headroom ledger. Maintain and disclose the running total of scored liabilities against the ceiling trajectory — the fiscal-space equivalent of a budget execution report. A programme that cannot see its cumulative headroom cannot ration it, and a rule that binds on a number no one publishes is a rule waiting to be breached by accretion.

Ration to value per unit of headroom. When the ceiling binds, admit projects in order of the public value they deliver per point of ceiling they consume — not first-come-first-served, not by project size, not by sponsor readiness. Grant the leanest guarantee that clears bankability, because under a binding ceiling every unit of unnecessary cover is a foregone project. And assemble the book for diversification, because under a sub-additive rule an uncorrelated guarantee is cheaper headroom than a correlated one.

Re-run the frontier every budget cycle. The binding constraint is not static: it moves with GDP (the ceiling grows), with the guarantee book's correlation (the tail moves), and with the pipeline (the candidate set changes). The optimal programme this year is not the optimal programme next year. The frontier is a standing calculation, not a one-time study — which is precisely why it should run on published code a ministry can re-execute on its own book, quarter after quarter.

The deeper point is the one that closes the series' argument about guarantees. For most of this series, the danger of a guarantee was that it stayed off the measure a rule watches. Under a liability-inclusive ceiling, the guarantee is on the measure — and the new danger is subtler: that the country scores it with the wrong rule, and either under-builds against a phantom risk or over-builds against a hidden one. Counting the liability was step one. Counting it coherently — at the correlated tail, so the number rewards diversification and states the real exposure — is what lets a ministry build the most infrastructure a fiscal rule can safely carry. That, and not the ceiling, is the parameter that decides how much a country gets to build.

Governments will go on arguing about where to set the ceiling. The one that wants to build the most its fiscal rule can safely carry should stop arguing about the number, and start arguing about how it adds its guarantees up. The ceiling is not the constraint. The arithmetic is.

Appendices

Appendix A — the optimiser, in full

The reference implementation is pipeline_optimizer.py, published alongside this paper and runnable with python3 pipeline_optimizer.py (pure numpy, no external services). It:

The two-factor draw model, formally. Two standard-normal macro factors drive the book: a demand/recession factor $F_d$ and an FX/devaluation factor $F_{fx}$, correlated at the estimated twin-shock coefficient $\rho = 0.16$,

\[F_d \sim \mathcal N(0,1), \qquad F_{fx} \;=\; \rho\,F_d + \sqrt{1-\rho^{2}}\;Z, \quad Z \sim \mathcal N(0,1)\ \text{independent}.\]

Each guarantee $i$ loads on its sector's factor $F_{k(i)}$ (with $k(i)\in\{d,fx\}$) through a systematic share $w_i$, the remainder idiosyncratic, giving a latent draw variable and a draw indicator

\[\ell_i \;=\; \sqrt{w_i}\;F_{k(i)} \;+\; \sqrt{1-w_i}\;\varepsilon_i, \quad \varepsilon_i \sim \mathcal N(0,1), \qquad \text{guarantee } i \text{ draws} \iff \ell_i > \Phi^{-1}(1-p_i),\]

so that $\Pr(\text{draw}_i)=p_i$ by construction and the loss is $L=\sum_i f_i\,\mathbf 1\{\ell_i>\Phi^{-1}(1-p_i)\}$. The implied demand↔FX draw correlation is $\sqrt{w_d\,w_{fx}}\,\rho \approx 0.10$ — far below the $\rho=1$ a single-common-factor model imposes, which is why that earlier draft understated the diversification benefit.

Every number in §3 and §5 is printed by that script; re-running it reproduces the paper's headline (portfolio-ES admits +US$1.8bn / +8% of programme value, one project, over face value under the identical 12% ceiling; expected value understates its own admitted book's tail by US$8.9bn). The demand↔FX factor correlation is estimated from Peru's macro history in the companion estimate_correlations.py (World Bank real GDP growth and PEN/USD, 2001–2024) rather than assumed; a real application substitutes its own pipeline economics and, if desired, re-estimates the correlation on its own currency and cycle.

Appendix B — the exercise across ceiling types

ceiling_types.py reuses the same pipeline and scoring machinery to run the four ceiling architectures of §7 (python3 ceiling_types.py). It (a) sweeps the stock-PV cap from 4% to 15% of GDP and measures the value sub-additive scoring reclaims over face value — the hump-shaped curve of Figure 4, peaking near +US$4.2bn at ~8% of GDP; (b) sizes the funded contingency pool under each rule (EV US$2.7bn, ES US$11.5bn, face US$18.8bn); (c) annualises commitments to show a flow cap's tenor distortion and procyclicality; and (d) shows a project-subsidy cap never scores the portfolio tail (US$11.5bn, 4.3% of GDP, invisible to it). The country anchors for each architecture are collected, with sources, in CEILING_LANDSCAPE.md. Figure 4 is produced by make_ceiling_figure.py, which imports its numbers directly from these scripts so the figure and text cannot drift apart.

The Austral PPP series: the third-party-guarantee decision (#2); the silent FX liability (#6); the indexation ceiling (#7); stress-testing PPP portfolios (#9); the recognition rule (#10); the optimal risk-retention frontier (#11). Each prices a single contingent instrument; this paper rations a whole book of them under a binding ceiling. ↩
Peru, Ley N° 32441 (2025), Ley que regula la promoción de la inversión privada mediante Asociaciones Público-Privadas y Proyectos en Activos (deroga el D.Leg. 1362), Artículo 31.1: "el stock acumulado de compromisos firmes y contingentes cuantificables, neto de ingresos, asumidos por el Sector Público No Financiero en contratos de APP, calculado a valor presente, no puede exceder el 12% del producto bruto interno," reviewable every three years; the Informe Multianual de Inversiones en APP quantifies each signed contract's firm and contingent commitments. Verified against the official text hosted by OSITRAN (ositran.gob.pe, 2026) and the passage-time coverage (Infobae, La República, 16 Sep 2025), which reported the 12% both as a limit and as a contingent-exposure risk ("obligaciones contingentes de hasta 12% del PBI"). The reglamento (Decreto Supremo, MEF) sets the quantification methodology — the scoring choice this paper argues is decisive. ↩↩
Estimated from World Bank data for Peru, 2001–2024: real GDP growth (NY.GDP.MKTP.KD.ZG) and the period-average PEN/USD exchange rate (PA.NUS.FCRF). Defining the recession shock as the shortfall of growth below its own mean and the devaluation shock as depreciation above its mean, corr(recession, devaluation) = +0.16 — the "twin shock" of an EMDE, present but weak. In the paper's two-factor guarantee model this factor correlation, combined with each guarantee's systematic share, implies a demand↔FX draw correlation of roughly +0.10, well below the +1.0 a single-common-factor model implicitly imposes. Full derivation in estimate_correlations.py, published with the optimiser. ↩
D. Duarte Arancibia (2026), Stress-Testing PPP Portfolios — Macro Shocks at Programme Level, Austral (series #9): the correlated value-at-risk / expected shortfall of a guarantee book, decomposed to project and risk factor, with a common-factor simulation calibrated to macro history. This paper uses that measure as the ceiling's constraint rather than as a disclosure. ↩↩↩
Sub-additivity — that the risk of a combined book is no greater than the sum of the parts' risks — is one of the axioms of a coherent risk measure (Artzner, Delbaen, Eber & Heath, 1999). Expected shortfall is coherent; value-at-risk in general is not, and face/sum-of-maxima is super-additive in the diversification it ignores. Coherence is exactly the property that makes the measure reward, rather than penalise, a diversified guarantee book — the behaviour a ministry should want. ↩
Colombia caps APP vigencias futuras and provisions contingent liabilities through the Fondo de Contingencias Contractuales de las Entidades Estatales, within the national fiscal rule. The architecture (a funded contingency pool) differs from Peru's single PV ceiling, but the scoring question — expected value versus correlated tail — is identical. ↩
The country architectures and their anchors are collected in the companion note CEILING_LANDSCAPE.md, web-verified July 2026: stock-PV caps (Peru 12% of GDP, Honduras 5%); flow caps on revenue (Brazil 5% of net current revenue, Lei 11.079 Art. 28; Paraguay; Colombia 0.4% of GDP annual authorisations; Pakistan ~4% of portfolio PV); funded pools (Colombia's Fondo de Contingencias, Indonesia's IIGF); disclosure without a cap (Chile's expected-value-and-maximum-exposure reporting); Treasury guarantee limits (Turkey, Philippines); project-subsidy caps (India VGF ≤20% of cost); and discretionary approval gates (Kenya, South Africa). Sources: OSITRAN; IADB (Fiscal Impact of PPPs in LAC; PPPs in Honduras); PPIAF (Managing Contingent Liabilities — Australia, Chile, South Africa); ADB (Philippines); India Department of Economic Affairs; APMG PPP Certification Guide §1.8.6. ↩
Honduras caps the cumulative present value of all quantifiable firm and contingent PPP commitments at 5% of GDP (raisable only by presidential request in the Council of Ministers); the IADB reports actual exposure already at 6.6% of GDP — a live breach of a stock-PV cap, and the closest analogue to this paper's concern outside Peru. It is the reason the intermediate-tightness result matters: a 5% cap sits deep in the binding zone, where the scoring rule is decisive. ↩
Peru, Ley N° 32441, Article 31.3: the indicators, parameters and calculation methodologies for the quantifiable firm and contingent commitments, available fiscal space and liabilities of entities holding APP projects are established by resolución ministerial of the Ministry of Economy and Finance (MEF). The Reglamento, approved by Decreto Supremo N° 316-2025-EF (published 24 December 2025, El Peruano), sets the framework but not the valuation methodology, which remains delegated to the MEF resolution. As of this writing that resolution had not fixed the scoring rule — the choice this paper analyses is therefore legally live, not yet settled. ↩
T. C. Irwin (2007), Government Guarantees: Allocating and Valuing Risk in Privately Financed Infrastructure Projects, World Bank (Directions in Development). The foundational statement that guarantees are priced risk whose costs "do not show up in the government's accounts," so governments are tempted to grant too many — the over-issuance the liability-inclusive ceiling is designed to discipline. ↩
The point that PPPs can be used to move spending off-budget and evade the intertemporal budget constraint, rather than because they are genuinely more efficient, is the central caution of E. Engel, R. Fischer & A. Galetovic (2014), The Economics of Public–Private Partnerships: A Basic Guide, Cambridge University Press: a PPP relaxes the budget today only by committing the state tomorrow, so a framework that does not count the commitment invites over-use. The horizon and measurement asymmetries in the text are why a value-for-money gate, however well run, does not by itself discipline the contingent book — and why the ceiling and the case for PPPs stand or fall together. ↩
A. Cebotari (2008), Contingent Liabilities: Issues and Practice, IMF Working Paper WP/08/245, sets out the firm/contingent taxonomy and the practice of capping PPP exposure through separate stock and flow ceilings. That literature uses the ceiling as a limit — a device to bound exposure. This paper takes the ceiling as given and solves the dual: the most programme value it can carry, and how the scoring rule moves that frontier. ↩
IMF & World Bank (2019), PPP Fiscal Risk Assessment Model (PFRAM) 2.0 — User Manual: PFRAM reports each guarantee's maximum exposure (worst case) and expected cost, and assesses fiscal impact "under the worst-case scenario." It is a project-level fiscal-impact tool, not a ceiling-constrained portfolio optimiser, and it does not net the correlated tail across the book. This paper extends that standard — the author was a co-developer of PFRAM — by adding the sub-additive portfolio rule and making it the constraint measure. ↩
See e.g. World Bank (2016), Contingent Liabilities Risk Management: A Credit Risk Analysis Framework for Sovereign Guarantees (Policy Research Working Paper 7538, F. F. Bachmair), and Inter-American Development Bank (2015), Toolkit for the Identification, Measurement, Monitoring and Risk Management of Contingent Sovereign Liabilities — both decompose a guarantee portfolio into expected loss and unexpected (tail) loss on a diversified book, for the purpose of provisioning and reserving. The mathematics of the diversified tail is theirs; the use of it as a ceiling-scoring rule that decides admissible programme value is this paper's. ↩