A planner's framework that makes the maxim "allocate each risk to the party best able to bear it — at a reasonable cost" quantitative: pricing, for each risk class, the premium of transfer against the sovereign's cost of retention, bounding it by the bankability ceiling beyond which the private sector will not bear the risk at any price, and flagging where current practice transfers what it should retain — or transfers the untransferable — with a reference implementation on the Austral platform
Author: David Duarte Arancibia, Austral Intelligence Date: June 2026 Audience: Ministries of Finance and PPP units; debt-management offices; transaction advisors; sector regulators; multilateral development bank infrastructure and guarantee desks Companion software: Austral platform —
https://platform.austral-intelligence.comSeries position: Paper 11 of the Austral series, and the structuring keystone of it. The Third-Party Guarantee (Paper 2), the Silent FX Liability (Paper 6), the Indexation Ceiling (Paper 7) and the Renegotiation Triangle (Paper 8) each priced one placement decision; this paper draws the surface on which all of them are points.
Allocate each risk to the party best able to manage it is the most-quoted and least-quantified rule in public–private partnership doctrine. This paper draws the frontier it implies. For a risk class, the efficient retention share balances three costs: the transfer premium \(p\) the private partner charges to bear the risk — priced, ultimately, through its cost of equity; the incentive value \(m\) of private management; and the sovereign's retention cost \(r\), which includes the fiscal-space charge of the contingent liability it creates. Minimising the social deadweight gives a closed form, \(\alpha^{*} = p/(p+2m+r)\): controllable project risks (construction, O&M) sort to transfer, uncontrollable macro risks (FX, demand, inflation) sort to retain or share. The maxim, made a number.
But the maxim needs one phrase added — at a reasonable cost — because the private sector's cost of bearing is not the smooth, unbounded line the capital-asset-pricing model draws. It has a wall. Below investment grade the project-finance debt market closes, so there is a bankability ceiling \(\tau_{\max}\) beyond which the private sector will not bear a risk at any return: past it, transfer is not expensive but infeasible. And a catastrophic loss that bankrupts a single-purpose company is, for the sovereign, dissolved across all taxpayers and across time — a ruin asymmetry that, beyond Arrow–Lind's diversification of variance, is why the State is the right bearer of catastrophic and sovereign-controlled risk. So retention is forced two ways: by efficiency (\(\alpha^{*}\) on the tradeoff) and by feasibility (\(\alpha \ge 1-\tau_{\max}\)).
The frontier nests the series. On one concession's seven-risk vector, demand and inflation are under-retained under the doctrinal transfer-everything default (the Indexation and Renegotiation papers), while FX and political risk are infeasible to transfer at that level — the private sector will not bear them, so a guarantee (a minimum-revenue guarantee, MIGA political-risk cover, the sovereign bridge — the Third-Party Guarantee and Silent FX papers) must fill the gap or the project does not close. The cost of equity is the frontier's price and Value for Money its integral: transferring macro risk raises the equity hurdle and destroys value, which a manipulated public-sector comparator conceals. The policy lead is to price the frontier at tender, retain macro and catastrophic risk explicitly and count it (the Recognition Rule, Paper 10), and stop paying for transfer that is either illusory or infeasible.
Keywords: risk allocation · risk retention · public-private partnership · cost of equity · value for money · public-sector comparator · Arrow–Lind · bankability · minimum-revenue guarantee · contingent liabilities · Ministry of Finance.
Feedback & discussion. Published openly so that Ministries of Finance, PPP units, advisors and the academic community can challenge and use it on real programmes.
- Comment in-page at
https://austral-intelligence.com/research- Email the author: [email protected]
Every PPP textbook, reference guide and toolkit repeats the same rule: allocate each risk to the party best able to manage it. It is correct, it is intuitive, and it is almost never quantified. Best able by how much? Manage — or bear? At what point does the premium the private party charges to take a risk exceed what the State would spend carrying it itself? The maxim names the objective and leaves the arithmetic to the negotiation, which is to say to the party with the better lawyers.
This paper supplies the arithmetic. It draws the retention frontier: for each risk a concession contains, the share the Ministry of Finance should keep rather than transfer, derived from the costs each party actually faces. The frontier is the structuring keystone of the Austral series because the papers that came before it each priced one point on it — the Third-Party Guarantee (Paper 2) is the placement of tail credit risk; the Silent FX Liability (Paper 6) is the placement of currency risk; the Indexation Ceiling (Paper 7) is the placement of inflation risk; the Renegotiation Triangle (Paper 8) is what happens when a placement was wrong and the risk comes back. The frontier is the surface; those papers are points on it.
The maxim also conflates two things that pull in opposite directions. "Best able" has a management half — who can control the risk, reduce its probability, mitigate its impact — and a bearing half — who can absorb the loss at least cost if it happens anyway. For a construction risk the same party (the builder) is best at both, and the maxim is unambiguous: transfer. For a currency risk the two halves split: the private partner cannot manage a devaluation at all, and bears it only at a steep premium, while the sovereign cannot manage it either but bears it cheaply — so the maxim, read carefully, says retain, even though practice almost always transfers. Most of the interesting cases are these split ones, and the frontier is what resolves them.
One word, finally, has to be added to the maxim, and §4 is devoted to it: at a reasonable cost. The private sector's price for bearing a risk is not an unbounded line — beyond a point it will not bear the risk at any price, and a risk that would bankrupt a private company is something the sovereign absorbs without ruin. "Best able to bear it at a reasonable cost" is the maxim this paper actually quantifies.
Take one risk class. The government retains a share \(\alpha \in [0,1]\) and transfers \(\tau = 1-\alpha\) to the private partner. Three costs move with the transferred share.
The transfer premium \(P(\tau)\) — what the private party charges, above the actuarially fair cost of the risk, to bear share \(\tau\). Its source is a financing friction, not a violation of asset pricing: the bearer at financial close is a single-purpose company on limited-recourse debt whose sponsors and lenders cannot fully share or lay off the risk, so under costly external finance and incomplete risk markets it prices project-specific risk that a frictionless, fully-diversified investor would not. Décaire (2019) documents the effect causally — financially-constrained firms inflate their discount rate by roughly five points per eighteen points of idiosyncratic risk — on the Froot–Scharfstein–Stein (1993) and Diamond (1984) foundations; §6 reconciles this with the finding that cleared secondary-market infrastructure prices reflect mostly systematic factors (Blanc-Brude 2019). The premium is convex — the more risk you push onto the SPV, the higher the marginal price — and we write \(P(\tau)=\tfrac12 p\,\tau^2\), with \(p\) the steepness: high for macro risks the constrained provider can neither manage nor lay off, low for risks it is well placed to bear, and steepest where the primary market is thin (acute in emerging markets — Hellowell and Vecchi 2017). \(p\) is priced through the cost of equity (§6).
The incentive value \(M(\tau)\) — the social value transfer creates by giving the partner a stake in managing the risk: the moral-hazard reduction, the effort, the innovation. It is concave (the first units of transfer buy most of the incentive) and we write \(M(\tau)=m\,(2\tau-\tau^2)\), with \(m\) high for controllable risks (construction, O&M) and \(\approx 0\) for macro risks the partner cannot manage however much it is paid.
The sovereign retention cost \(R(\alpha)\) — what the State spends carrying the retained share: the expected loss at the sovereign's own cost of risk, plus the fiscal-space charge of the contingent liability it creates (§5, the bridge to the Recognition Rule). We write \(R(\alpha)=\tfrac12 r\,(1-\tau)^2\), with \(r\) low for risk the sovereign diversifies cheaply (Arrow–Lind) and higher for risk correlated with its own budget.
The planner minimises the total social deadweight \(C(\tau)=P(\tau)-M(\tau)+R(1-\tau)\). The first-order condition is a single clean line:
\[\boxed{\;\alpha^{*} \;=\; \frac{p}{\,p + 2m + r\,}\;}\]
The efficient retention share rises with the transfer premium \(p\) (the more it costs to push a risk onto the partner, the more the State should keep), and falls with the incentive value \(m\) (transfer that improves management is worth its premium) and the sovereign carry cost \(r\) (retention that consumes fiscal space is not free). The crossover — retain half, transfer half — sits at \(p = 2m + r\).
The sort the formula produces is the maxim, quantified. Controllable project risks (\(p\) low, \(m\) high) land near \(\alpha^{*}=0\): transfer. Uncontrollable macro risks (\(p\) high, \(m\approx 0\)) land near \(\alpha^{*}=1\): retain or share. The frontier is the profile of \(\alpha^{*}\) across a concession's whole risk vector. The precision is in the sort, not the decimals: the closed form is the convenient consequence of the quadratic cost shapes, and while the ordering it produces — macro high, project low — survives any reasonable parameterisation (Appendix E proves the comparative statics \(\partial\alpha^{*}/\partial p>0\), \(\partial\alpha^{*}/\partial m<0\), \(\partial\alpha^{*}/\partial r<0\) for general cost functions, and confirms the wrong-side verdicts are invariant across a family of forms), the specific shares do not.
The two parameters \(m\) and \(p\) are the maxim's two halves, made separate and measurable. \(m\) is the management half: how much the partner's stake improves the outcome. \(p\) is the bearing half: how dearly the partner charges to absorb the loss. The doctrine's error is to assume they move together — that whoever should manage a risk should also bear it. For project risks they do. For macro risks they come apart, and the bearing half is where the sovereign's advantage lies, for two distinct reasons.
Arrow–Lind — diversification of variance. Arrow and Lind (1970) showed that the State can bear a given risk at a far lower premium than a private investor, because it spreads the risk across the whole population of taxpayers: each citizen bears an infinitesimal share, the per-capita variance vanishes, and the social risk premium approaches zero. A single concession SPV cannot do this; its equity holders carry the whole variance and price it. So for risk that is large relative to the SPV but small relative to the economy, \(r \ll p\) and the frontier says retain.
The ruin asymmetry — dissolution of the tail. There is a second, sharper asymmetry the variance argument misses, and it is the one that governs catastrophic risk. A loss large enough to exhaust an SPV's equity does not merely cost it money — it bankrupts it: a discontinuity that destroys the going-concern value, halts the service, and triggers the disorderly unwind the Renegotiation Triangle (Paper 8) priced. The private party's premium for a catastrophic-tail risk therefore carries a ruin term on top of its CAPM premium — it is paying not for variance but for survival. The sovereign has no ruin term. The same catastrophic loss, borne by the State, is dissolved across all taxpayers and across time — spread over the tax base by Arrow–Lind, and over generations by the sovereign's ability to borrow and amortise. A government is, simply, bigger than any project, and does not go bankrupt because one concession's tail materialised. This is why the efficient bearer of catastrophic, force-majeure and sovereign-controlled risk is almost always the State — not because it manages such risk (it often cannot), but because it is the only party that can absorb it without ruin.
The two halves, then: transfer where the partner manages (\(m\) high); retain where the partner can only bear, and bears either dearly (Arrow–Lind, high \(p\)) or not without ruin (the catastrophic tail). The frontier is where the two halves are weighed.
The capital-asset-pricing model draws the price of risk as a straight line: higher risk, higher required return, without end — in principle the private sector will bear any risk for a high enough return. In infrastructure finance this is false, and the falsehood matters enough to change the maxim.
Project debt — seventy to ninety per cent of a concession's capital structure — has to be investment grade for the institutional lenders and bond markets that fund infrastructure to touch it. Push enough risk onto the SPV and its debt slips below that threshold; at that point the debt market does not charge more, it closes. No equity return compensates, because the equity is a sliver of a structure that can no longer be financed. There is, in other words, a bankability ceiling \(\tau_{\max}\) — a maximum share of a given risk the private sector will bear at any price. Beyond it, transfer is not expensive; it is infeasible.
The frontier respects the ceiling as a hard constraint. The constrained efficient retention is
\[\alpha^{*} \;=\; \max\!\bigl(\,\underbrace{p/(p+2m+r)}_{\text{efficiency optimum}}\,,\ \underbrace{1-\tau_{\max}}_{\text{bankability floor}}\,\bigr),\]
so retention can be forced above the efficiency optimum purely by feasibility. And an observed allocation that transfers more than the ceiling allows (\(\tau^{\text{obs}}>\tau_{\max}\)) is not merely inefficient — it is infeasible: the private sector will not actually bear it, so in practice one of two things happens. Either a guarantee fills the gap — a minimum-revenue guarantee that pulls demand risk back below the ceiling, MIGA political-risk cover that makes expropriation risk bankable, the sovereign bridge of the Indexation Ceiling paper — or the project does not reach financial close. This is why the guarantee architecture exists. A guarantee is not a subsidy bolted onto a risk transfer; it is the State retaining the part of a risk that sat above the bankability ceiling, so the rest can be financed. The Third-Party Guarantee (Paper 2) is this move with a third party bearing the gap; the Silent FX Liability (Paper 6) is this move performed silently, the State bearing the un-transferable currency tail without admitting it.
The bankability ceiling is also where the ruin asymmetry of §3 bites hardest. The risks with the lowest \(\tau_{\max}\) — the ones the private sector will bear least of — are exactly the catastrophic and sovereign-controlled ones: expropriation, change-in-law, deep devaluation, uninsurable force majeure. The private sector shuns them not because it has mispriced them but because no price compensates for ruin. The maxim's missing words, at a reasonable cost, are the recognition that beyond a point the private cost is not high but infinite — and that the residual belongs, by feasibility as much as by efficiency, with the only party that can dissolve it.
Retention is not free, and the frontier must not pretend it is. When the State keeps a risk, it creates a contingent liability — and whether that liability is counted is the subject of the Recognition Rule (Paper 10). The retention cost \(r\) in the frontier is exactly this: the expected draw at the sovereign's cost of risk, plus a fiscal-space charge for the headroom the contingent liability consumes against the debt rule.
This closes a loop with the recognition paper that is worth stating plainly. A government tempted to retain a macro risk because the frontier says it is efficient must still book the retained exposure — otherwise "cheap retention" is an illusion of the same kind the recognition paper diagnosed: a liability that is real but unrecorded. The honest frontier carries \(r\) with its fiscal-space charge included, so that retention competes with transfer on a true-cost basis. The dishonest version sets the fiscal-space charge to zero, retains everything, and rediscovers the liability in the next crisis. The frontier and the recognition rule are two views of one discipline: the recognition rule says count the retained risk; the frontier says and weigh its counted cost against the premium of transfer.
There is a subtlety the Stress-Testing paper (Paper 9) supplies. The fiscal-space charge is not additive across risks, because the retained contingent exposures are correlated — a macro shock draws on the FX guarantee, the demand guarantee and the revenue floor at once. So \(r\) for a risk retained into an already macro-heavy book is higher than the same risk retained in isolation: the marginal fiscal-space cost rises with the correlation of the retained portfolio. Arrow–Lind understates \(r\) exactly here — for risk correlated with the budget, the sovereign's bearing cost is not zero, and the frontier must carry that correlation rather than assume free diversification. This is the one place the sovereign's advantage is weakest, and the frontier should not over-claim it.
The frontier is not a new object bolted onto PPP finance; it is the decomposition of two quantities every practitioner already uses — the cost of equity and Value for Money — and it disciplines both.
The cost of equity is the frontier's price. The transfer premium \(p_i\) is not an abstraction: it is the contribution of risk \(i\) to the SPV's required equity return. Equity is the residual bearer of every risk transferred to the concession company, so the equity hurdle decomposes as the risk-free rate plus the sum of per-risk premia over the shares actually transferred, \(\text{IRR}_e \approx r_f + \sum_i \tau_i\,p_i + (\text{expected-loss terms})\). When the frontier calls FX a high-\(p\) risk, that steepness materialises as a higher equity hurdle; the availability PPP that retains all macro risk through a unitary charge has a low equity beta and a low hurdle, while the demand concession that transfers it has a high one. The empirical regularity — availability deals price cheaper than demand deals — is the frontier showing through.
A reconciliation the careful reader will demand. Modern asset pricing says idiosyncratic risk earns no premium in an efficient market, and the strongest infrastructure evidence agrees: Blanc-Brude and EDHECinfra find that cleared secondary-market infrastructure-equity prices are explained largely by systematic factors, and Colla, Hellowell, Vecchi and Gatti (2015) state the principle directly. So is \(p\) a CAPM violation? No — it is about a different investor. The systematic-pricing result describes the marginal, diversified buyer who clears the secondary market; \(p\) is the premium of the primary, financially-constrained capital provider at financial close — a single-purpose SPV on limited-recourse debt, whose sponsors and lenders are not that frictionless marginal investor. For that party, costly external finance, incomplete risk-sharing and illiquidity make project-specific risk priced (Décaire 2019; Froot–Scharfstein–Stein 1993; Diamond 1984), without contradicting the secondary-market finding — and even EDHEC's own factor set carries a priced, non-CAPM illiquidity (size) component. The premium is widest where the primary market is thinnest, which is why emerging-market PPPs carry the steepest \(p\) (Hellowell and Vecchi 2017), and it is what the State — the ultimate diversified, patient holder — does not face. Read \(p\), throughout, as compensation for priced risk under financing frictions and incomplete risk-sharing, not as an undiversified actor mechanically charging for variance.
This yields the paper's sharpest practical claim: the cost of equity is endogenous to the risk allocation, so you cannot negotiate a risk premium away — you can only re-allocate the risk. A ministry bargaining down a high equity IRR is solving the wrong problem; the SPV genuinely faces that cost for the risk it is asked to carry. Retain the macro risk and the hurdle falls, because you have removed what was pricing it. A minimum-revenue guarantee lowers the SPV's cost of capital for precisely this reason — it is the State moving demand risk toward its efficient \(\alpha^{*}\) — and the Recognition Rule (Paper 10) then insists the retained risk be counted. The cost-of-equity benchmark — what hurdle a ministry should accept for a given sector and country (a companion study) — is the empirical shadow of this paper: the defensible hurdle is the one implied by the efficient allocation, \(r_f + \sum_i \tau_i^{*} p_i\), and any observed hurdle above it is the State paying for transfers it should have retained.
Value for Money is the frontier's integral — on the risk-allocation axis. The risk-allocation component of Value for Money against the public-sector comparator (PSC) is the aggregate of getting each risk's allocation right: the VfM lost to mis-allocation is the summed deadweight by which the chosen allocation departs from the frontier. (Value for Money has other components — bundling, lifecycle efficiency, private innovation, procurement cost — that the frontier does not speak to; the claim is about the risk-allocation share, which is where the PSC's largest manipulations live.) A deal can show positive headline VfM versus the PSC while still leaving risk-allocation value on the table, and the frontier finds where. More importantly, the frontier corrects the central PSC manipulation documented in the accounting-boundary literature (Heald 2003, in the recognition paper's review): crediting risk transfer at the full risk premium. The true VfM contribution of transferring risk \(i\) is \(M_i - P_i\) — the management gain net of the premium paid — which is positive for controllable risks and negative for macro risks (\(p>m\), \(m\approx 0\)). Transferring FX or demand risk destroys Value for Money: the State pays \(p\) and gets no \(m\). A naïve PSC books it as value created, because it credits the transfer and quietly omits the premium. And it compounds the error on the public side: by Arrow–Lind the public sector's cost of retaining a macro risk is the sovereign's low \(r\), not a private-sector premium — yet manipulated comparators price public risk-bearing as if the State were an undiversified investor, erasing the very wedge that makes retention efficient. The frontier re-grounds VfM in Arrow–Lind: charge the public option \(r\), charge the private option \(p\), credit transfer only the net \(M-P\), and the comparator stops lying.
The mapping, made exact. This is an identity, not an analogy. Write the social cost of allocating risk \(i\) at transfer share \(\tau\) as \(C_i(\tau)=P_i(\tau)-M_i(\tau)+R_i(1-\tau)\). The public-sector comparator is full public retention, \(\tau=0\), at cost \(C_i(0)=R_i(1)\); the PPP transfers \(\tau_i\), at cost \(C_i(\tau_i)\). The Value for Money of the PPP for risk \(i\) is the difference, \[\mathrm{VfM}_i \;=\; C_i(0)-C_i(\tau_i) \;=\; \underbrace{[\,M_i(\tau_i)-P_i(\tau_i)\,]}_{\text{management gain net of premium}} \;+\; \underbrace{[\,R_i(1)-R_i(1-\tau_i)\,]}_{\text{retention cost saved}}.\] Two consequences follow. First, \(\mathrm{VfM}_i\) is maximised exactly at the frontier: \(\mathrm{d}\,\mathrm{VfM}_i/\mathrm{d}\tau=-C_i'(\tau)=0\) at \(\tau_i^{*}\) — so maximising Value for Money risk-by-risk is solving the frontier, and the VfM forgone by the chosen allocation is the deadweight excess \(C_i(\tau_i^{\mathrm{obs}})-C_i(\tau_i^{*})\): the "integral" claim, as an accounting identity. Second, the manipulation is now a missing term: a comparator that credits the transfer with the retention saving \([R_i(1)-R_i(1-\tau_i)]\) but omits the premium \(P_i(\tau_i)\) the SPV charges overstates \(\mathrm{VfM}_i\) by exactly \(P_i\) — and for a macro risk, where \(M_i\approx 0\) and \(P_i\) is large, that omitted premium is the whole of the apparent gain.
So the three are one object seen from three ends: the cost of equity is the observable price the market charges for the chosen allocation; the frontier is the normative decomposition that says which parts of that price are efficient; and Value for Money is the scorecard — the integral over the frontier, against the public comparator. A ministry that minimises the equity hurdle by negotiation, or maximises VfM once at the comparator stage and forgets it, is optimising a snapshot of a surface it never draws.
The frontier is a benchmark, so it generates a diagnostic: compare the observed allocation \(\alpha^{\text{obs}}\) to the efficient \(\alpha^{*}\). Three verdicts follow.
A risk is under-retained when \(\alpha^{\text{obs}} < \alpha^{*} - \epsilon\): the State transferred more than it should have. For a macro risk this is the common case, and it is expensive in a way that does not show up at signing — the State pays the premium \(p\) for a transfer that buys no management \(m\), and the risk returns through guarantee calls, renegotiation, or the silent liability. A risk is over-retained when \(\alpha^{\text{obs}} > \alpha^{*} + \epsilon\): the State kept a risk it could efficiently have transferred, forgoing the management gain. And a transfer is infeasible when \(\tau^{\text{obs}} > \tau_{\max}\): the State purported to transfer more than the private sector will bear, so the transfer is not real — a guarantee is filling the gap, or the deal did not close.
The deadweight by which the observed allocation departs from the frontier is the measurable cost of being on the wrong side, summed across the risk vector. It is the Value-for-Money the structure left on the table, and unlike a hazard ratio it is in units a minister acts on. The diagnostic does not allege bad faith: many transfers are efficient and many guarantees are the correct response to an infeasible-transfer gap. It measures the distance between form and substance, and lets a ministry see where the two have parted.
The solver is one small module with a deliberate reuse principle: the platform already prices the pieces. The risk/ module carries the portfolio Value-at-Risk (the var_overlay, which supplies the correlated fiscal-space charge of §5) and the five-by-five risk matrix whose categories the taxonomy mirrors; the guarantees/ module prices the transfer instruments (the minimum-revenue and FX guarantees that fill the infeasible-transfer gap of §4); and the series' own fm/fx_placement and fm/indexation_ceiling engines are instances of the frontier — single-risk placements the solver reproduces as corner cases, a consistency check of the kind the Renegotiation Triangle used when it nested the soft-budget result at its corner.
The core is a pure function of a risk vector: for each risk, the parameters \((p, m, r)\) and the bankability ceiling \(\tau_{\max}\), with an optional observed retention \(\alpha^{\text{obs}}\). It returns the efficient \(\alpha^{*}\), whether the bankability ceiling bound (so retention was forced by feasibility, not efficiency), and — against the observed allocation — the under-/over-retained/infeasible verdict and the deadweight excess. On the platform it is POST /workspaces/{wid}/risk/retention-frontier, EDITOR-gated and audit-chained.
Reproduce in ten minutes. python3 frontier_core.py runs the golden self-check (the closed form, the macro/project sort, the Arrow–Lind corner, the bankability ceiling); python3 worked_examples.py scores the concession of §9 and writes out/worked_examples.json; python3 -m pytest test_frontier.py runs the eighteen tests. The numbers in §9 are lifted from that output.
A stylised availability-plus-demand concession in an emerging market, with seven risks. The calibration is illustrative — the per-risk premium is not directly observable (a bid gives a total IRR, not a per-risk decomposition), so \((p,m,r,\tau_{\max})\) are plausible calibrations and the robust outputs are the sort and the direction, not point shares. The observed allocation is the doctrinal default: transfer almost everything to the SPV.
| Risk | \(\alpha^{*}\) | why retained | \(\tau_{\max}\) | \(\alpha^{\text{obs}}\) | verdict |
|---|---|---|---|---|---|
| Construction | 0.09 | — (transfer) | 1.00 | 0.10 | on the frontier |
| O&M / availability | 0.12 | — (transfer) | 1.00 | 0.10 | on the frontier |
| Demand (traffic) | 0.57 | efficiency | 0.90 | 0.10 | under-retained |
| FX | 0.72 | efficiency | 0.60 | 0.15 | infeasible transfer |
| Inflation / indexation | 0.70 | efficiency | 0.80 | 0.20 | under-retained |
| Force majeure | 0.55 | efficiency | 0.60 | 0.50 | on the frontier |
| Political / change-in-law | 0.85 | bankability-forced | 0.15 | 0.60 | infeasible transfer |
Four of seven risks are mis-allocated; the total deadweight excess is 0.51 cost-units. Read the rows. Construction and O&M sort cleanly to transfer and the observed allocation matches — the maxim working. Demand and inflation are under-retained: efficient retention is 0.57 and 0.70, the contract keeps 0.10 and 0.20, and the gap is the Renegotiation Triangle and the Indexation Ceiling waiting to happen. FX and political risk are infeasible to transfer at the observed level: the contract purports to push 85% of the currency risk and 40% of the political risk onto the SPV, but the bankability ceilings are 0.60 and 0.15 — the private sector will not bear that much at any price, so a guarantee (the Silent FX placement, MIGA political-risk cover, the sovereign bridge) is in fact filling the gap, named or not. And political risk is the one case where bankability forces retention above efficiency: the efficiency optimum is 0.81, but the ceiling pushes \(\alpha^{*}\) to 0.85 — the State must keep 85% of a risk it largely controls, because no one else will take it.
The shape is the lesson. The project risks are on the frontier; the macro and catastrophic cluster is on the wrong side of it — under-retained where transfer is merely inefficient, infeasibly transferred where it is impossible. That cluster is precisely the subject matter of Papers 6, 7 and 8, and the guarantee architecture of Paper 2 is what the infeasible-transfer rows demand.
The §9 calibration is illustrative; two of its claims can be anchored in real Chilean data without a deal-level equity IRR — which Chilean concessions never produce, since bidders file a bid variable, not a financial model.
A demand-risk premium from the cost-of-capital ladder. The premium a concession's equity cost carries for bearing demand risk is the gap between a greenfield demand-bearing IRR and the rate once demand is auctioned away. The rungs (all verified, real): a greenfield merchant EM concession requires ~12–15% nominal (≈9–12% real); a Chilean concession with demand auctioned away is funded at the 6.5% real rate the first LPVR road auction actually used (Engel–Fischer–Galetovic), corroborated by the regulated infrastructure WACC (≈5.7% real, CNE) and by secondary-market toll-road sales clearing at ~11.9× equity/EBITDA — a utility-like cost of equity. The gap is ≈4 percentage points (real), bracketed 2–6 — and it is precisely what Chile's LPVR variable term auctions away. The frontier's central claim — strip the macro risk the SPV over-prices and its hurdle falls — is therefore observed in market prices, and the Self-Insuring Concession (Paper 14) is the mechanism. This is the first frontier input moved from calibrated toward estimated (empirical_anchor.py; sources in RR2_PSC_SOURCES.md).
A real PSC through the \(M-P\) lens. DIPRES's Comparador Público-Privado for the Centro Penitenciario de Rancagua (2024) compares a public reference of PV 2,806,598 UF plus transferable risk 946,137 UF against PPP payments of 2,697,368 UF, for a Value for Money of 1,055,367 UF — a 28% saving — at a 6% real risk-free discount. The frontier reads it cleanly. A prison is an availability PPP: the State bears demand (inmates come regardless); the SPV bears construction and operating/service risk — food, health, maintenance, reinsertion, 34% of base cost — which it can manage. So the transfer sits on the right side of the crossover (\(M>P\)), and the positive VfM is correct: the on-frontier control case, not a manipulation. And the comparator is honest in method — it discounts at the risk-free rate (avoiding the discount-rate double-count §6 warns of) and values risk by empirical weighting (expected value, not a private premium). The manipulation the frontier guards against — crediting macro-risk transfer at the full premium — would surface in a demand-risk toll-road PSC, exactly the case the \(\approx 4\)-point \(p_{\text{demand}}\) prices. Rancagua confirms the frontier classifies a real comparator correctly, and locates where the danger is not.
The same ladder logic, applied across the risk taxonomy using public PSC risk registers, guarantee and insurance prices, hedging costs, sector asset-betas and optimism-bias uplifts, gives a first estimate of the entire \(p_i\) vector in annualized cost-of-equity terms (sources and native units in the project's \(p_i\)-estimates note):
| Risk | \(\approx p_i\) (annual pp) | Manageable (\(m\))? | Observed allocation |
|---|---|---|---|
| FX / currency | ~3–4 (median EM; tail 10–30%+) | no | retained / hedged |
| Demand | ~4 | low | retained in availability PPPs |
| Construction | ~3–4 (equity); +20–66% capex | high | transferred |
| Political / change-in-law | ~1–4 (country); insurable to ~0.3–1.0 | no | shared / MIGA-IBRD |
| Force majeure | ~0.1–0.8 | no | insurable → private |
| Interest-rate | ~0.05–0.5 (hedged) | low | hedged by the SPV |
| Inflation / indexation | ~0–0.3 | no | retained by the grantor |
| O&M / operating | <0.5 | high | transferred |
Two things in this table matter more than any single figure. First, the estimated premia rank exactly as the frontier's sort predicts: the high-\(p\), low-\(m\) macro risks — FX and demand — sit at the top (~3–4 pp), the cheaply-hedgeable risks — inflation, interest, force majeure, O&M — at the bottom (≲0.5 pp), and construction is high-\(p\) and high-\(m\) so it transfers despite its premium, the two-halves logic of §3 made empirical. Second, the observed allocation follows the frontier: across every public comparator examined, inflation and demand are retained by the public sector, construction and O&M are transferred, and bankability appears as a named retained line — high-\(m\) risks placed with the partner, low-\(m\) macro risks kept by the State. The frontier was derived as a normative rule; the market prices and the comparators turn out to follow it. (Two structural caveats survive: the premium is paid only on what is transferred, so a \(p_i\) vector must be weighted by transfer probability, not applied to all risk; and \(p_{\text{FX}}\) has a discontinuous tail — ~0–1 pp for Chile or investment-grade Asia, 10–30%+ for Brazil, Nigeria or Türkiye — and must be modelled currency-conditional, which is the Silent FX Liability's tail, priced.)
The frontier is the structuring keystone because the earlier papers are its points, and reading them back through it sharpens each.
One surface, five papers. The frontier does not replace them; it explains why each exists and where each sits.
The pieces are old; the surface is not. The risk-allocation maxim is the backbone of the practitioner literature — the World Bank PPP Reference Guide, the EPEC and OECD risk-allocation frameworks, the project-finance texts (Grimsey and Lewis 2004; Yescombe) — but it is stated qualitatively, as a matrix of "who bears what," not as a frontier with a crossover and an efficient share. The public-economics foundation is Arrow and Lind (1970) on the State's risk-bearing advantage — disciplined by Baumstark and Gollier (2013), who show the advantage is bounded and that a privately-retained risk carries a positive collective cost even at zero beta (which is exactly the \(p\) term, and exactly why PPPs that merely socialise risk are not free) — and the mechanism-design lineage runs through Engel, Fischer and Galetovic, whose least-present-value-of-revenue result (2001) is the rigorous precedent for retaining demand risk publicly through an endogenous-term auction, and whose basic-public-finance paper (2013) shows the off-balance-sheet "fiscal space" of transfer is largely illusion (the \(r\) fiscal-space charge made formal); Maskin and Tirole (2008) supply the spending-limit motive. These treat risk-bearing and incentives in separate registers, not as the two halves of one allocation rule. The empirical cost-of-equity literature is where the \(p\) term lives: Hellowell and Vecchi (2017) survey and interview PPP investors and confirm the equity hurdle moves with the risk matrix; Blanc-Brude and EDHECinfra (2019) show cleared infrastructure prices reflect mostly systematic factors (the reconciliation of §6); and Décaire (2019) documents that financially-constrained firms do price project-specific risk — the friction-based microfoundation of \(p\). The contingent-liability literature (Polackova 1998; Irwin 2007 — the verbal "allocate and value" framework this paper formalises; the IMF–World Bank PFRAM) measures the cost of retained risk but not the efficient amount to retain; the value-for-money literature (Heald 2003; Leigland 2018, who shows risk transfer is empirically overstated and overpriced) critiques the public-sector comparator without supplying the frontier that would discipline it.
What is missing, and what this paper adds, is the synthesis: a single decision surface that puts the management half (\(m\)) and the bearing half (\(p\), \(r\)) on one axis, bounds it by the bankability ceiling the CAPM line omits, solves for the efficient retention share, and shows that the cost of equity and Value for Money are its price and its integral. The frontier is the quantitative core the maxim has always lacked.
Limitations, stated plainly.
The policy lead. The frontier is a benchmark a ministry can compute, so it can be governed.
POST /workspaces/{wid}/risk/retention-frontier
Content-Type: application/json
{ "risks": [
{ "name": "FX", "p": 0.90, "m": 0.0, "r": 0.35,
"bankability_ceiling": 0.60, "observed_retention": 0.15 } ] }
Response (abbreviated):
{ "risks": [
{ "name": "FX", "alpha_star": 0.72, "alpha_star_efficiency": 0.72,
"bankability_binds": false, "retention_reason": "efficiency",
"efficient_side": "RETAIN", "observed_retention": 0.15,
"verdict": "INFEASIBLE TRANSFER", "deadweight_excess": 0.2031 } ],
"n_misallocated": 1, "n_infeasible_transfer": 1, "n_bankability_forced": 0 }
efficient_retention(p, m, r, τ_max):
α_eff = p / (p + 2m + r) # the efficiency optimum (closed form)
α_floor = 1 − τ_max # the bankability floor (investment-grade wall)
return max(α_eff, α_floor) # retention forced by efficiency OR feasibility
assess(p, m, r, τ_max, α_obs):
α* = efficient_retention(p, m, r, τ_max)
if (1 − α_obs) > τ_max: verdict = "INFEASIBLE TRANSFER" # transfer exceeds the ceiling
elif α_obs < α* − tol: verdict = "UNDER-RETAINED"
elif α_obs > α* + tol: verdict = "OVER-RETAINED"
else: verdict = "on the frontier"
excess = C(1−α_obs) − C(1−α*) # deadweight of the wrong-side allocation
with \(C(\tau)=\tfrac12 p\,\tau^2 - m(2\tau-\tau^2) + \tfrac12 r(1-\tau)^2\).
| Risk class | manage (\(m\)) | bear-premium (\(p\)) | typical \(\tau_{\max}\) | efficient tendency |
|---|---|---|---|---|
| Construction | high | low | high | transfer |
| O&M / availability | high | low | high | transfer |
| Demand | low | high | medium | retain / share |
| FX | none | high | medium-low | retain / guarantee |
| Inflation / indexation | none | high | medium | retain / share |
| Interest-rate | low | medium | medium | share |
| Force majeure | none | high | low | share / retain |
| Political / change-in-law | none (sovereign-controlled) | high | very low | retain (forced) |
The closed form \(\alpha^{*}=p/(p+2m+r)\) is the consequence of the quadratic cost shapes of §2. This appendix shows that the qualitative content of the frontier — the sort, the comparative statics, and the wrong-side verdicts — does not depend on that choice.
Proposition (form-independence of the sort). Let \(P=p\,g(\tau)\) be convex increasing, \(M=m\,h(\tau)\) concave increasing and \(R=r\,k(\alpha)\) convex increasing — the economically-motivated shapes — for fixed shape functions \(g,h,k\) and positive scales \(p,m,r\). Then (i) the deadweight \(C(\tau)=P-M+R(1-\tau)\) is strictly convex, so the efficient transfer \(\tau^{*}\) is unique; and (ii) the efficient retention \(\alpha^{*}=1-\tau^{*}\) satisfies \(\partial\alpha^{*}/\partial p>0\), \(\partial\alpha^{*}/\partial m<0\) and \(\partial\alpha^{*}/\partial r<0\) for any admissible \(g,h,k\).
Proof. \(C''(\tau)=p\,g''-m\,h''+r\,k''>0\) since \(g''>0\), \(h''<0\), \(k''>0\): strictly convex, hence a unique interior optimum. The first-order condition is \(F(\tau^{*};p,m,r)\equiv p\,g'(\tau^{*})-m\,h'(\tau^{*})-r\,k'(1-\tau^{*})=0\). By the implicit function theorem \(\partial\tau^{*}/\partial p=-F_p/F_\tau\) with \(F_\tau=C''>0\) and \(F_p=g'(\tau^{*})>0\), so \(\partial\tau^{*}/\partial p<0\) and \(\partial\alpha^{*}/\partial p>0\). Likewise \(F_m=-h'<0\Rightarrow\partial\alpha^{*}/\partial m<0\) and \(F_r=-k'<0\Rightarrow\partial\alpha^{*}/\partial r<0\). \(\qquad\blacksquare\)
So the sort is a theorem: a risk with a higher transfer premium, lower management value, or lower sovereign carry cost is retained more, whatever the cost shapes. The quadratic supplies the convenient closed form; it does not drive the ordering.
Numerical corroboration. Re-solving the §9 worked example under a family of shapes \(P=p\tau^{a}/a\), \(M=m(1-(1-\tau)^{b})\), \(R=r(1-\tau)^{c}/c\) for \((a,b,c)\in\{(2,2,2),(1.5,1.5,1.5),(3,3,3),(2.5,1.5,2),(1.5,3,2.5),(3,2,1.5)\}\) (robustness.py): the wrong-side verdict is invariant for five of the seven risks, and for the entire macro cluster — demand, FX and inflation — across every form. The two exceptions (O&M and force majeure) flip only between on-frontier and under-retained, near the \(\pm 0.15\) tolerance, under the most extreme incentive-curvature, and never reverse a retain/transfer call. The point shares are form-dependent (FX's \(\alpha^{*}\) ranges over \([0.48,0.89]\)); the decisions the frontier drives are not. The paper claims the latter, not the former.
Numbers in §§9 and the appendices reproduce from frontier_core.py / worked_examples.py (golden self-check + 18 tests); citations verified 2026-06-23.