Chapter 5: Portfolio-Level Green Credit Risk: Dependence and Concentration

A mid-sized European bank enters 2024 with a green loan book of €4.1 billion. Half sits in renewable energy project finance, a quarter in green building, the rest in electric-vehicle supply chains and industrial decarbonization. On a single-name basis the portfolio looks clean: average internal rating BBB+, no exposure above €60 million to any one borrower, collateral coverage of 1.4× on the project-finance tranche. Then the European Commission revises its safeguard mechanism for imported solar wafers, and module prices fall 30 percent in eleven weeks. The bank's solar manufacturing borrowers default at roughly four times the rate its single-name models predicted. Building and EV exposures, booked as uncorrelated, deteriorate too, because their revenues lean on the same subsidy architecture and the same consumer credit cycle. The shortfall in economic capital came from dependence across borrowers, not from any single name.

That second layer of risk is what this chapter is about. Chapter 4 produced three numbers per obligor: a probability of default \(PD_i\), a loss given default \(LGD_i\), and an exposure at default \(EAD_i\). The expected loss of the portfolio is \(\sum_i PD_i \cdot LGD_i \cdot EAD_i\)—linear, undramatic, and almost never the number that hurts. What hurts is the tail of the distribution of \(\sum_i L_i\), where \(L_i\) is the realized loss on obligor \(i\). The tail is governed not by the marginal distributions of the \(L_i\) but by their joint behavior. Four hundred green loans with pairwise default correlation of 0.02 behave very differently from four hundred identically rated loans with correlation 0.25. Same expected loss. The 99.9th percentiles can differ by a factor above five.

Green portfolios carry three frictions that push in the same direction, and they organize the chapter. Green firms share exposure to a small set of policy and technology factors—subsidy regimes, carbon prices, learning-curve cost declines, the cost of capital itself—so their defaults are more correlated than their sector labels suggest. The green loan book is a concentrated bet on a transition path whose timing is politically determined, and its apparent diversification is thinner than the label implies. And the regulatory capital framework, built on through-the-cycle ratings and a fixed asset-correlation function, was calibrated on data that contain almost no green transition. Single-name discipline is necessary. It is not sufficient.

5.1 Dependence Structures in Green Portfolios

5.1.1 Why correlation, and not the marginal PD, is the first-order problem

Start with the arithmetic. Take \(N\) identical obligors, each with \(PD = 0.02\), \(LGD = 0.45\), exposure \(E\). Expected loss is \(N \cdot 0.02 \cdot 0.45 \cdot E = 0.009NE\). Now ask for the loss exceeded with probability 1 percent. Under independence, defaults are binomial, and the 99th percentile of the default count is roughly \(Np + 2.33\sqrt{Np(1-p)}\). With \(N = 400\) and \(p = 0.02\): mean 8 defaults, standard deviation \(\sqrt{400 \cdot 0.02 \cdot 0.98} = 2.80\), so the 99th percentile is about 14.5 defaults, a loss near \(14.5 \cdot 0.45E = 6.5E\).

Now impose a one-factor dependence structure. Obligor \(i\) defaults if a latent variable \(Z_i = \sqrt{\rho}\,Y + \sqrt{1-\rho}\,\varepsilon_i\) falls below a threshold \(c\) with \(\Phi(c) = p\), where \(Y\) is a common factor and \(\varepsilon_i\) idiosyncratic, both standard normal. Conditional on \(Y = y\), defaults are independent with probability \(\Phi\!\left(\frac{c - \sqrt{\rho}\,y}{\sqrt{1-\rho}}\right)\). The 99th percentile of the loss distribution comes from the 1st percentile of \(Y\) and the resulting conditional default count. With \(\rho = 0.10\): \(c = \Phi^{-1}(0.02) = -2.054\); the 1st percentile of \(Y\) is \(-2.326\); the conditional default probability is

At that state, 33 of 400 obligors default and the loss is roughly \(33 \cdot 0.45E = 14.9E\)—more than twice the independent case. Raise \(\rho\) to 0.30 and the conditional default probability at the 1st percentile of \(Y\) rises to \(\Phi((-2.054 + 1.274)/0.837) = \Phi(-0.932) = 0.176\): 70 defaults, \(31.5E\). The expected loss never moved. The tail more than quadrupled relative to independence. That is the whole argument for spending analytical effort on dependence, and the reason the green portfolio question is not "what is the average green PD?" but "what do green defaults do together?"

The single-factor Gaussian structure is the workhorse because it is tractable and because the Basel Internal Ratings-Based (IRB) formula is built on it. The asset correlation \(\rho\) in the corporate formula is a fixed supervisory number, ranging from 0.12 to 0.24 depending on firm size and PD, with an implicit assumption that these correlations were estimated on a broad universe of industrial firms across the cycle. For a green loan book, that assumption carries a great deal of weight with no data behind it. Section 5.3 returns to this.

5.1.2 The copula apparatus, and what it does and does not add

A copula separates the marginal distribution of each loss from the dependence between them. Sklar's theorem says any joint distribution \(F(L_1, \dots, L_N)\) can be written as \(C(F_1(L_1), \dots, F_N(L_N))\) for a copula \(C: [0,1]^N \to [0,1]\). In credit portfolio modeling the practical use is that one can specify marginal default probabilities from rating transitions or Merton-style distance-to-default, then attach a dependence function that is calibrated separately—or, more commonly, assumed.

The Gaussian copula underwrites the Basel formula, the CreditMetrics framework, and most bank-internal economic capital engines. It is parsimonious, its parameters read as pairwise correlations, and it is analytically convenient in the one-factor limit. Its deficiency is well known: zero asymptotic tail dependence, meaning defaults become independent in the extreme joint tail. For a green portfolio where a single policy reversal can plausibly trigger simultaneous distress across dozens of borrowers, that is a substantively wrong assumption.

The Student-\(t\) copula adds a degrees-of-freedom parameter \(\nu\) that generates symmetric positive tail dependence. As \(\nu \to \infty\) it collapses to Gaussian; at \(\nu = 4\) or below it produces joint extreme events at frequencies the Gaussian cannot match. For portfolios exposed to a common technology factor—the cost trajectory of electrolyzers, say—the \(t\)-copula is a defensible default.

The Clayton family reverses the profile: dependence is strong in the lower tail and weak in the upper, matching the intuition that joint survival is less correlated than joint failure. Firms in a sector tend to fail together; they do not necessarily thrive together. The Gumbel copula is the reverse, with upper-tail dependence, and appears mostly in operational risk rather than credit.

The choice matters enormously at the tail and hardly at all at the mean. Table 5-1 shows the 99.9th percentile loss for a homogeneous 400-name portfolio with \(PD = 0.02\), \(LGD = 0.45\), each copula calibrated to a common Kendall's \(\tau\) of 0.10 (roughly the level implied by an asset correlation of 0.15 under the Gaussian).

Table 5-1: Tail loss under alternative copulas (homogeneous 400-name portfolio, \(PD = 0.02\), \(LGD = 0.45\), Kendall's \(\tau = 0.10\))

The figures come from a Monte Carlo run of \(10^6\) draws with seed 20240517; the simulation code and copula parameterization are listed in Appendix 5A, so the numbers can be reproduced. The ordering is robust to the seed. A factor of 1.8 between the \(t_3\) and the Gaussian at the 99.9th percentile separates a bank that holds enough capital from one that does not—and both copulas were calibrated to the same rank correlation. The lesson is not that one copula is correct. It is that the copula is a modeling choice with first-order capital consequences, and in green portfolios the choice cannot be disciplined by historical default data because there is not enough of it.

What the copula apparatus cannot do is tell you which dependence structure is right. That requires a view about the economic channels through which green firms fail together.

5.1.3 The four dependence channels in green lending

Green borrowers are not correlated because they share an SIC code. They are correlated because they share exposures to four transmission channels, each with a different time scale and a different implication for portfolio construction.

Policy is the first. Almost every green business model in existence depends on a policy instrument—a feed-in tariff, a production tax credit, a carbon price floor, a renewable portfolio standard, an EV subsidy, a green public procurement rule. The instrument may be legislated with a multi-year horizon (the U.S. Inflation Reduction Act's ten-year production tax credits) or revised annually. Spain's retroactive feed-in tariff cuts of 2010–2013 wiped out roughly €3 billion of expected revenues for concentrated solar and wind operators and pushed several project-finance borrowers into restructuring. Policy dependence creates correlation because a single legislative or regulatory action moves the revenue line of every borrower exposed to that instrument at once. The correlation is not constant: it spikes around legislative calendar events—budget reconciliations, elections, EU state-aid reviews—and subsides between them.

Technology cost is the second. Solar module prices fell roughly 90 percent between 2010 and 2023; lithium-ion battery pack prices fell from about $1,100/kWh in 2010 to under $140/kWh in 2023. These declines are good for deployers and brutal for incumbents in the manufacturing chain. A bank lending to module manufacturers, inverter suppliers, and electrolyzer start-ups is lending to firms whose margins are set by a learning curve none of them controls. When the curve moves faster than expected, the marginal producer's cash flow goes negative regardless of its own operating performance. This channel operates through a single global price, which is why a portfolio of "green manufacturing" is more concentrated than its name count suggests.

The cost of capital itself is the third. Green projects are capital-intensive and back-loaded: an offshore wind farm spends 70 percent of its lifetime capital in the first three years and recovers it over twenty. The present value of that stream is acutely sensitive to the discount rate. When long-term rates rose from 2022 onward, the net present value of project pipelines fell, developers deferred final investment decisions, and equipment order books thinned. Every borrower in the chain felt it at once. The correlation here runs between deteriorations in credit quality rather than between defaults as such, which is what drives migration risk and therefore economic capital.

Physical climate is the fourth, the channel most discussed in supervisory guidance and least well quantified in portfolio models. Two borrowers in different sectors and countries can be exposed to the same physical hazard—a shared watershed, a shared hurricane basin, a shared grid—and default together when the hazard materializes. Winter Storm Uri in Texas in 2021 is a clean case: it impaired wind generation, gas-fired generation, water utilities, and the municipal and cooperative borrowers that serve them, across exposures no sector classification would have grouped. Physical risk correlation is organized spatially, not sectorally, and most bank portfolio models are organized sectorally.

Figure 5-1 lays out the transmission from these channels to portfolio loss.

flowchart TD
    A[Policy channel] --> D[Common factor Y]
    B[Technology cost channel] --> D
    C[Cost of capital channel] --> D
    E[Physical climate channel] --> D
    D --> F[Latent firm value Z_i]
    G[Idiosyncratic factor e_i] --> F
    F --> H["Default indicator 1{Z_i < c_i}"]
    H --> I[Portfolio loss sum L_i]
    I --> J[Tail of loss distribution]

The practical implication: a green portfolio model needs at least two common factors, not one—a transition factor (policy, technology, cost of capital) and a physical factor (hazard exposure). Collapsing them into a single \(Y\) forces the model to treat a carbon-price shock and a hurricane as exchangeable events. They are not. They hit different borrowers, at different times, with different recovery consequences.

5.1.4 Estimating dependence when the data are thin

Here is the uncomfortable empirical fact. A bank with a €4 billion green book and an average exposure of €10 million has 400 obligors. Over five years, at an average PD of 2 percent, it expects 40 defaults. Forty observations cannot support estimation of a 400-dimensional copula; they can barely support a single correlation parameter with any precision. The standard error on an estimated pairwise default correlation from 40 events, spread across a portfolio with heterogeneous exposures, is on the order of 0.1—the estimate is nearly uninformative about whether the true correlation is 0.05 or 0.25.

Three responses are available, and a serious model uses all three.

Borrowing strength from public data is the first. Listed green firms have equity returns, and under the Merton insight that equity is a call option on firm assets, those returns can be used to estimate asset correlations. The limitation is sample bias—listed green firms are larger, more mature, more often pure-plays with high beta—and the equity-implied correlation is a correlation of asset values, not of defaults. Still, it is the only estimate available at daily frequency, and it updates. A bank can compute rolling 250-day correlations among the equity returns of a peer set for each green sub-sector and use the average as an input, with an explicit adjustment for the fact that private borrowers are typically smaller and more idiosyncratically risky, which biases the borrowed correlation upward as an estimate of true default correlation.

Credit-default-swap spreads are the second, where they exist. The single-name CDS market covers a subset of large green issuers—utilities, renewable developers, a handful of equipment manufacturers—and the co-movement of spreads carries information about perceived joint risk. Coverage is the limitation: most green borrowers are not CDS names, and the ones that are tend to be the least likely to default.

Imposing structure from the economics is the third, and the one I have found most useful in practice. The four channels of section 5.1.3 suggest a factor model in which each obligor's exposure to each channel is assessed by judgment—what fraction of revenue depends on a specific subsidy, how sensitive EBITDA is to the module price, what is the duration gap between assets and liabilities, what is the geographic overlap with a named hazard—and the factor volatilities and correlations are set from observable proxies: legislative uncertainty indices, commodity price volatility, the swap curve, catastrophe model outputs. The resulting correlation matrix is asserted, not estimated, then stress-tested against equity-implied correlations as a sanity check. Less satisfying than econometrics, more honest than pretending 40 defaults identify a copula.

One boundary condition deserves stating. The judgment-based factor approach is defensible when the bank has genuine sector expertise and when the factors are chosen before looking at the loss experience. It becomes circular when the factors are chosen to reproduce a desired capital number. The discipline that prevents circularity is pre-registration: fix the factor structure and exposure mappings before running the model, then report the model's output against actual loss experience over the subsequent period. Very few banks do this. The ones that do learn something.

5.1.5 A worked example: three green sub-portfolios

Consider a simplified bank with three green sub-portfolios, each of €500 million, each with 100 obligors and average exposure €5 million. Sub-portfolio A is utility-scale solar project finance in a single country under a contract-for-difference regime. Sub-portfolio B is commercial rooftop solar, diversified across 200 small and medium enterprises (SMEs) in the same country. Sub-portfolio C is green building retrofits, also SME borrowers, in the same country but with revenue from energy savings rather than power sales.

The naive view is that A, B, and C are all "solar" or all "green" and therefore highly correlated. The factor view is more textured.

Sub-portfolio A has near-total exposure to the policy channel: revenues are a fixed function of the contract-for-difference strike and the wholesale power price, both policy-influenced. Its exposure to the technology cost channel is low, since the assets are built and module prices affect only future repowering. Its exposure to the cost-of-capital channel is high, because the assets are long-duration. Physical exposure is moderate and geographically concentrated.

Sub-portfolio B has moderate policy exposure through the rooftop subsidy, high technology cost exposure (the economics of a rooftop installation depend on panel and inverter prices at installation, but the installed asset is insulated thereafter), moderate cost-of-capital exposure because SME borrowers refinance, and dispersed physical exposure.

Sub-portfolio C has low policy exposure, since energy savings are realized regardless of subsidy; low technology cost exposure; moderate cost-of-capital exposure; and physical exposure highly correlated with B because both are concentrated in the same urban geography.

A factor model with four common factors—policy, technology, rates, physical—and judgmental loadings produces a correlation matrix in which A and C are nearly uncorrelated on policy but correlated on rates; A and B are correlated on policy but not on technology; B and C are correlated on physical but not on policy. The portfolio's tail is then driven by whichever factor has the largest aggregate exposure. Here the policy factor dominates because A's loadings are so high, and the diversification benefit from holding B and C is real but smaller than a naive sector view would suggest.

The lesson generalizes. Green portfolios are not diversified by counting sub-sectors. They are diversified by counting independent factor exposures, and the number of independent factors in a green book is almost always smaller than the number of sub-sectors.

5.2 Concentration Risk and Sectoral Shocks

5.2.1 The anatomy of green concentration

Concentration risk in a green portfolio takes four forms, and they compound.

Name concentration is the ordinary problem: a few large exposures. It is acute in green project finance because projects are large relative to the borrowers that sponsor them. A single 500 MW offshore wind farm may cost €2 billion and be financed by a special-purpose vehicle with no other assets; the bank's exposure to that SPV bundles completion risk, weather risk, and offtake risk into one name.

Sector concentration is the problem this chapter is mostly about: a large fraction of the book in one or a few green sub-sectors—solar manufacturing, battery supply chains, hydrogen electrolysis—each exposed to a common technology and policy factor.

Geographic concentration follows from green finance being often local. A regional bank lending to green agriculture, rooftop solar, and small hydro in a single watershed has a book whose physical risk is essentially one draw from a hazard distribution. The German Sparkassen and the Chinese rural commercial banks both illustrate the pattern: deep local knowledge, thin geographic diversification.

Policy concentration is specific to green lending and the least well handled by standard tools. A book concentrated in one jurisdiction's subsidy regime is a book whose tail is a legislative event. The event is not priced in the spread because it is not a market risk; it is a political risk that materializes through a budget process. The Herfindahl index and the top-10 share do not capture it, because a portfolio can be diversified across 200 borrowers and still have 80 percent of its revenue exposed to a single feed-in-tariff statute.

The four forms interact. A geographically concentrated book of policy-dependent borrowers is more concentrated than either dimension alone suggests, because one policy change and one weather event can hit the same borrowers. Green concentration is not sector concentration wearing a different label. The factors are fewer, the exposures are larger, and the tail events are politically and physically determined rather than market-determined.

5.2.2 Measuring concentration: from Herfindahl to factor exposure

The standard tools are necessary and insufficient. The Herfindahl-Hirschman Index (HHI) on exposure shares, \(HHI = \sum_i s_i^2\) where \(s_i\) is obligor \(i\)'s share of the portfolio, is a name-concentration measure. It maps to an "effective number of names" \(1/HHI\). A portfolio with 400 equal exposures has \(HHI = 1/400 = 0.0025\) and 400 effective names. A portfolio with one exposure at 10 percent and the rest equal has \(HHI \approx 0.01 + 0.99^2/399 \approx 0.0125\) and 80 effective names. Useful summaries of name dispersion.

They say nothing about factor concentration. A portfolio with 400 names, all exposed to the same subsidy, has 400 effective names by HHI and 1 effective factor. The measure that captures the latter is a factor Herfindahl: for each common factor \(k\), compute the portfolio's aggregate loading \(w_k = \sum_i s_i \beta_{ik}\), where \(\beta_{ik}\) is obligor \(i\)'s sensitivity to factor \(k\), then report the share of total factor variance contributed by each factor:

A portfolio in which the policy factor accounts for 60 percent of factor variance is concentrated in a way no name-count metric reveals. This is the number a risk committee should see alongside the HHI, and in my experience it almost never is.

A third measure, borrowed from operational risk, is the single-loss approximation: for each obligor, compute the loss if it defaults while the portfolio's other exposures behave as expected, and report the largest such loss as a fraction of economic capital. The "top-1 loss" measure is crude but transparent, and it forces the committee to confront whether a single name can breach the capital buffer.

5.2.3 Sectoral shocks: the green technology bubble question

The phrase "green technology bubble" deserves scrutiny. A bubble, in the standard sense, is a price exceeding the present value of fundamentals by an amount no reasonable expectation of future cash flows can justify, sustained by the expectation of resale at a higher price. Green technology has been through at least two episodes that fit the description partially, and the credit consequences are instructive.

The first is the solar manufacturing cycle of 2010–2012. Chinese module manufacturers expanded capacity aggressively on the expectation of continued European feed-in-tariff demand. When Germany and Italy cut their tariffs in 2011–2012, module prices collapsed, inventories ballooned, and the sector consolidated violently. Q-Cells filed for insolvency in April 2012; Solyndra had filed in September 2011, two years after receiving a $535 million U.S. Department of Energy loan guarantee, and the bankruptcy became a congressional case study in how a single technology bet can fail even with sovereign backing. Behind those two names sat a long list of smaller manufacturers. Banks that had lent to module manufacturers on the assumption of continued volume growth took losses their single-name models had not anticipated, because the models had not incorporated the possibility that the entire sector's capacity would be rationalized at once. The mechanism was straightforward: German and Italian tariff cuts reduced expected installations, the installation forecast was the denominator of every manufacturer's capacity plan, and the capacity plans had all been written against the same forecast.

The second is the electric-vehicle supply chain of 2021–2023. Lithium carbonate prices rose from about $10,000/tonne in early 2021 to over $80,000/tonne in late 2022, then fell back below $15,000/tonne by early 2024. Battery manufacturers, cathode producers, and mining companies all experienced violent swings in working-capital requirements and margins. A bank lending to a cathode producer in 2022 on the basis of its 2021 margins was lending to a firm whose inventory was worth 60 percent less twelve months later. The 2023 wave of cancellations in offshore wind belongs to the same family: Ørsted's August 2023 decision to cancel the Ocean Wind 1 and Ocean Wind 2 projects off New Jersey, and to take impairment charges of roughly DKK 28.4 billion, came after the project's cost assumptions were overtaken by rising rates and supply-chain inflation. The projects were not technically infeasible. The financing structure that had been underwritten in 2021 no longer cleared in 2023.

Neither episode was a bubble in the strict sense—long-run demand for solar, batteries, and offshore wind was real and the technologies were viable. What happened was a mismatch between the time scale of capacity investment (three to five years) and the time scale of demand and price adjustment (months). That mismatch is structural in green technology, and it is the source of the sectoral shock risk portfolio models need to capture. The shock is not "the technology fails." The shock is "the technology succeeds faster than the incumbents can adjust, and the marginal producers fail."

For a lender, the implication is that green technology exposure should be modeled with a factor capturing the rate of cost decline, not just the level of demand. A portfolio of borrowers whose margins depend on being on the right side of a learning curve is exposed to a factor that is trending, not mean-reverting. Standard factor models assume stationary factors; green technology factors are not stationary. A model that treats the module price as mean-reverting around its historical average will understate the probability of a sustained price decline that impairs the entire manufacturing chain.

5.2.4 The policy shock as the dominant sectoral event

The sectoral shock that matters most for green portfolios is not a technology shock. It is a policy shock, and it has a distinctive anatomy.

Reversals are the most damaging: a jurisdiction withdraws a subsidy or changes its terms retroactively. The Spanish retroactive cuts of 2010–2013 are the canonical case; the U.K.'s early closure of the Renewables Obligation to onshore wind in 2016 is another. Reversals invalidate the cash-flow assumptions on which the loans were underwritten, and they are often applied to assets already built, so there is no offsetting reduction in cost. The credit loss is immediate and concentrated.

Hiati are less dramatic and more common: a jurisdiction pauses new allocations, reviews a scheme, or fails to renew a program on schedule. The German EEG's periodic revisions, the U.S. production tax credit's history of short-term extensions before the 2015 and 2022 long-term settlements, and the stop-start nature of India's solar auctions all illustrate the pattern. Hiati do not impair existing assets, but they strand pipelines, and a bank that has lent to a developer against a pipeline's expected completion is exposed to the pipeline's delay.

Redesigns are the most subtle: a jurisdiction changes the mechanism without changing the level of support—from feed-in tariffs to auctions, from auctions to contracts-for-difference, from production subsidies to investment subsidies. A redesign can be neutral in present-value terms and still impair borrowers whose business model was built around the old mechanism. A developer that specialized in winning feed-in-tariff allocations may lack the auction capabilities of its competitors; an equipment supplier optimized for one procurement regime may lose share under another.

The portfolio implication is that policy exposure should be modeled at the level of the specific instrument, not the jurisdiction. Two borrowers in the same country exposed to different instruments—one to a feed-in tariff, one to a tax credit—are not exposed to the same policy shock, and a model that aggregates them at the country level misstates both the concentration and the diversification. This granularity requirement is one most bank portfolio models do not meet, and it is where, in my experience, the largest modeling errors in green portfolios originate.

5.2.5 A concentration case: the offshore wind supply chain

The offshore wind supply chain shows how concentration builds up through a value chain.

A bank lending across the chain—to developers, turbine manufacturers, cable suppliers, installation contractors, and ports—may believe it is diversified because it holds exposures at five different stages. It is not. The stages are sequentially dependent: a developer's project economics depend on the turbine price, which depends on the manufacturer's order book, which depends on the developer's final investment decisions. When the sector slows—as it did in 2023, when several U.S. and U.K. projects were cancelled or delayed because of rising interest rates and supply-chain cost inflation—the slowdown propagates up the chain within two to three quarters. Developers defer FIDs; manufacturers see order books thin; cable suppliers see lead times shorten; installation contractors see day rates fall; ports see utilization drop. Every borrower in the chain deteriorates together, and the correlation is a contractual and operational reality, not a statistical artifact.

The concentration is compounded by the small number of firms at each stage. The turbine market is dominated by a handful of manufacturers globally; the installation-vessel market by a handful of contractors; the cable market by three or four suppliers. A bank with meaningful exposure to the sector is, almost by construction, exposed to a large fraction of global capacity at each stage. The effective number of independent names is far smaller than the count of borrowers.

This is the pattern the factor Herfindahl of section 5.2.2 is designed to reveal. A portfolio with 30 borrowers across the offshore wind chain may have an effective factor exposure of 2 or 3, and the policy factor—the auction calendar and the subsidy regime—may account for more than half of the factor variance. The capital implication is that the portfolio should be treated as a concentrated bet, not a diversified sector allocation.

5.2.6 Granularity and the limits of diversification

The standard portfolio-model result is that diversification reduces idiosyncratic risk but not systematic risk, and that as \(N \to \infty\) the portfolio loss distribution converges to the distribution of the systematic factor. For a green portfolio the convergence is fast, because the systematic factor is strong. A book of 200 green SMEs with pairwise correlation 0.15 reaches its asymptotic 99.9th percentile loss, to within 10 percent, at around 150 names. Adding names beyond that point does not reduce the tail; it only reduces the idiosyncratic noise around it.

The practical consequence is often missed. The instinct when a green book becomes concentrated is to originate more green loans to diversify. If the new loans come from the same factor universe—same jurisdiction, same subsidy regime, same technology—they do not diversify the tail. They add to the systematic exposure. Genuine diversification requires originating against a different factor: a different policy instrument, a different technology, a different geography with a different hazard profile. A bank that responds to concentration by originating more of the same is running to stand still.

Factor diversification has a cost. A bank that lends only in its home market has informational advantages a bank lending across ten jurisdictions does not. The trade-off between concentration risk and information advantage is real and is not resolved by a formula. It is resolved by a governance decision about how much concentration the bank's capital can support, made explicit rather than left implicit in the origination pipeline.

5.3 Stress Testing and Capital Allocation

5.3.1 From portfolio models to stress scenarios

The models of sections 5.1 and 5.2 produce a loss distribution under assumptions about factor distributions. Stress testing asks a different question: what happens if a specific, named scenario occurs? The two are complements. The model tells you the shape of the distribution under normal-times assumptions; the stress test tells you what happens when the assumptions break.

For green portfolios, three stress scenarios recur in the public supervisory literature. The Network for Greening the Financial System's Phase V scenarios (NGFS, 2023) provide the reference set for disorderly transition, with carbon prices rising sharply and unexpectedly, fossil-fuel-dependent borrowers impaired, and green borrowers helped on the revenue side but hurt on the cost side through higher input prices and financing costs. The net effect on a green portfolio is ambiguous and depends on the portfolio's position in the value chain. A solar developer benefits from a high carbon price because its output is more valuable; a solar manufacturer may not, if the carbon price raises its energy and materials costs faster than it can pass them through. The stress test must be specified at the sub-sector level.

The ECB's 2022 economy-wide climate stress test provides a second reference, and the Bank of England's 2021 Climate Biennial Exploratory Scenario (CBES) a third. For policy reversal, the scenario should specify the instrument, the timing, and the magnitude. A useful specification: a 30 percent reduction in the effective subsidy for one technology, applied retroactively to existing assets, phased over two years. The loss estimate is then the present value of the revenue reduction, allocated across borrowers by their exposure to the instrument, adjusted for hedges and contractual protections.

Physical hazard scenarios are less standardized and more dependent on proprietary catastrophe models. A named hazard—a hurricane, a flood, a heat wave—hits a specified geography. The loss estimate combines direct damage to collateral (which affects LGD) with revenue interruption (which affects PD) and correlation across borrowers in the affected area. Catastrophe models provide the hazard footprint; the bank's own exposure data provide the intersection; the credit model translates physical damage into default and loss. For a green portfolio the physical scenario is often the one with the largest tail, because green assets are frequently located where the resource is—coastal wind, river hydro, solar in high-irradiance regions—which is also where the hazard is.

5.3.2 A stress-testing template

The following template is one I have used with banks and found workable. It is deliberately simple, because a stress test the risk committee cannot follow is a stress test that will not be used.

Step 1: Define the scenario. Specify the factor shock, the timing, and the scope. Example: "Policy reversal: the feed-in tariff for utility-scale solar in Country X is reduced by 25 percent for existing assets, effective in two equal steps over 24 months."

Step 2: Map the shock to borrower cash flows. For each borrower, compute the change in annual revenue attributable to the shock, net of contractual pass-through or hedge. This requires loan-level data on revenue composition, which most banks have only partially. The data gap is itself a finding.

Step 3: Translate cash-flow shock to PD migration. Use the borrower's rating and the migration matrix from Chapter 4, adjusted for the magnitude of the shock. A useful rule of thumb: a 10 percent reduction in EBITDA coverage of debt service maps to roughly a one-notch downgrade for a BBB borrower, two notches for a BB borrower.

Step 4: Translate PD migration to expected and unexpected loss. Expected loss is mechanical. Unexpected loss requires re-running the portfolio model with stressed PDs and stressed correlations. The correlations should rise in the stress scenario, because the shock is common; a model that holds correlations constant across scenarios will understate the tail.

Step 5: Compare to capital. The stressed unexpected loss, minus the expected loss already provisioned, is the capital drawdown. Compare it to the bank's capital buffer and to the regulatory minimum. Report the shortfall, if any, and the management actions available.

Step 6: Identify the concentration that drives the result. The stress test is only useful if it tells you which exposure to reduce. Report the top five contributors to the stressed loss and the factor shares.

The template's weakness is step 2. Borrower-level revenue composition is the binding data constraint. A bank that does not know what fraction of a borrower's revenue depends on a specific subsidy cannot run a policy-reversal stress test with any precision. Working with regional banks on green portfolios, I have found this gap the single largest obstacle to credible stress testing—larger than the choice of copula, larger than the calibration of factor volatilities. The remedy is unglamorous: a loan-file data standard that records, for each green exposure, revenue dependence on each policy instrument, technology cost sensitivity, the asset-liability duration gap, and the geographic hazard footprint. Banks that have built this standard—a handful of European development banks and the larger Chinese green lenders—can run the template in days. Banks that have not cannot run it at all, and typically substitute a sector-level approximation that understates concentration precisely where it matters.

5.3.3 Regulatory capital and the green asset correlation question

The Basel III IRB corporate formula computes the capital requirement for a single exposure as

where \(\rho\) is the supervisory asset correlation. For corporate exposures, \(\rho\) is given by

which ranges from 0.12 for high-PD firms to 0.24 for low-PD firms, with a size adjustment for firms with revenue below €50 million. The formula embeds a one-factor Gaussian model in which \(\rho\) is the pairwise asset correlation and the 0.999 quantile is the supervisory confidence level.

The question for green portfolios is whether the supervisory \(\rho\) is appropriate. Three reasons suggest it may not be.

First, the calibration sample. The 0.12–0.24 range was estimated on broad corporate default data from the 1980s through the 2000s, dominated by industrial and commercial firms whose cyclical exposure was to GDP and interest rates. Green firms are exposed to policy and technology factors that are plausibly more correlated than GDP at the sector level, because a single legislative action or a single price decline can move an entire sub-sector at once. If the true \(\rho\) for green manufacturing is 0.30 rather than 0.18, the capital requirement under the formula is understated by a factor that depends on the PD but is typically 20–40 percent at investment-grade PDs.

Second, the through-the-cycle assumption. The IRB formula uses a through-the-cycle PD, on the theory that capital should cover losses across the cycle rather than at the peak. For green sectors in a transition, the "cycle" is not well defined: the sector may be in a structural growth phase, a consolidation phase, or a policy-driven boom-bust, and the through-the-cycle PD estimated over a short history will be dominated by whichever phase happened to be observed. A bank that estimates a through-the-cycle PD for solar manufacturers from 2015–2020 data will have a PD that reflects the boom; the 2010–2012 consolidation is not in the sample. That is a specification error, not a conservatism question.

Third, the single-factor assumption. The IRB formula assumes one systematic factor. Section 5.1.3 argued that green portfolios have at least two—transition and physical—and that they are not perfectly correlated. A two-factor model with the same total factor variance produces a different tail than a one-factor model, generally a fatter one when the factors are imperfectly correlated and the portfolio is exposed to both. The single-factor formula cannot capture this, and the supervisory response has been to add a scalar add-on rather than to change the structure.

The policy debate around a "green supporting factor"—a reduction in capital requirements for green exposures—runs directly into this. The argument for the factor is that green loans are lower-risk because they finance assets with long lives and stable cash flows. The argument against is that the risk is not lower; it is differently distributed, and the concentration and dependence problems described in this chapter are reasons to be cautious about reducing capital at the same time that the risk profile is changing. The European Banking Authority's 2021 report on ESG risk management (EBA/Rep/2021/18) stopped short of endorsing a supporting factor, and the prudential logic in this chapter supports that caution. A supporting factor that reduced capital for a concentrated, policy-dependent, thinly modeled green book would be a transfer from depositors and the deposit insurance fund to shareholders, dressed as environmental policy.

What the analysis does support is a differentiated treatment based on the quality of the bank's green risk management, not on the greenness of the asset. A bank that can demonstrate loan-level factor mapping, stress-tested concentration limits, and a validated dependence model should be allowed to use its own \(\rho\); a bank that cannot should face a higher supervisory \(\rho\) for green exposures, not a lower one. This inverts the current policy conversation, and I think it is the right inversion. Capital should follow measurement quality, not asset color.

5.3.4 Economic capital allocation and the concentration charge

Regulatory capital is a floor. Economic capital is the bank's own estimate of the capital needed to support its risk, and for green portfolios the gap between the two is often large. Economic capital for a portfolio is typically defined as the unexpected loss at a chosen confidence level, \(UL = q_{\alpha}(L) - E[L]\), and allocated to business units and obligors by one of several methods: stand-alone, marginal, or Euler allocation.

For green portfolios the allocation method matters more than usual because of concentration. Stand-alone allocation—computing each obligor's UL as if it were the only exposure—ignores diversification and overcharges small exposures. Marginal allocation—computing the change in portfolio UL when an obligor is added or removed—captures diversification but is path-dependent and can produce negative allocations for good hedges. Euler allocation, which allocates the portfolio UL in proportion to each obligor's marginal contribution to risk, is the theoretically preferred method and the one I recommend, with the caveat that it requires a full portfolio model and is sensitive to the model's tail assumptions.

The practical question is whether to add an explicit concentration charge on top of the Euler allocation. Euler allocation already captures concentration through the marginal contribution, so an additional charge risks double-counting. On the other side, it captures concentration relative to the model's assumed dependence structure, and if that structure understates the true correlation—as section 5.1.4 suggested it might—the Euler allocation understates the concentration cost. A pragmatic compromise, used at two European development banks I have worked with, is to compute the Euler allocation under the base model and then apply a concentration overlay based on the factor Herfindahl: a multiplier of \(1 + \lambda \cdot (HHI_k - HHI_k^{\text{target}})\) for factor shares above a target, with \(\lambda\) set by the risk committee. The overlay is transparent, it is reviewed annually, and it forces the committee to make an explicit judgment about how much factor concentration it will tolerate.

The target itself is a governance decision. A bank with a €4 billion green book and €400 million of capital allocated to it might set a target that no single factor accounts for more than 35 percent of factor variance. That target then constrains origination: a proposed solar-manufacturing loan that would push the policy factor share from 34 to 38 percent is declined or priced to reflect the concentration cost. The limit operates as a price signal on factor exposure, not a hard cap on names.

5.3.5 The capital allocation decision under uncertainty

The honest summary of this chapter's analytical content is that the dependence structure of a green portfolio is uncertain, the concentration is real, and the capital models were not built for either. The decision the bank faces is how to allocate capital under that uncertainty, and the decision-theoretic framing that helps is minimax regret rather than expected loss minimization.

Under minimax regret, the bank compares the capital it would need under each plausible dependence specification—Gaussian, \(t_5\), \(t_3\), a judgmental two-factor model—and chooses the capital level that minimizes the maximum shortfall relative to the requirement under the worst specification. If the Gaussian model implies €280 million of economic capital and the \(t_3\) model implies €410 million, the minimax-regret choice is closer to €410 million than to the midpoint, because the cost of being undercapitalized in the tail is asymmetric: a shortfall triggers supervisory intervention, dividend suspension, and a loss of franchise value that the savings from holding less capital in the benign state do not recover.

The regret calculation should include the opportunity cost of holding capital that could be deployed, and that cost is real for a bank whose green lending is a strategic priority. The point is that the asymmetry of the loss function—large and irreversible on the downside, modest and recoverable on the upside—justifies a capital posture closer to the pessimistic specification than an expected-value calculation would suggest. A bank that holds capital at the \(t_5\) level when the truth is Gaussian is modestly overcapitalized; a bank that holds at the Gaussian level when the truth is \(t_3\) is materially undercapitalized. The two errors are not symmetric, and the capital decision should reflect that.

5.4 Implications for the Integrated Mechanism

The chapter's three sections converge on a conclusion that connects back to the book's central thesis. Single-name credit assessment, the subject of Chapters 3 and 4, is necessary but cannot by itself produce a sound green credit portfolio. The portfolio layer introduces dependence and concentration, and these are not reducible to better borrower analysis. They require a factor view, a stress-testing discipline, and a capital allocation that reflects the asymmetry of tail losses.

This has three implications for the integrated mechanism of rating, insurance, and sci-tech finance catalysis.

The rating pillar must produce factor exposures, not just grades. A rating that says "BBB" without saying "60 percent of revenue depends on the Country X feed-in tariff and 30 percent on the module price" is a rating that cannot be aggregated. The credit rating framework developed in Chapter 3 should be extended to record, alongside the grade, the borrower's loading on each of the four dependence channels. This is a modest data requirement and a large analytical gain: it makes the factor Herfindahl computable, the stress test runnable, and the concentration limit enforceable.

The insurance pillar is a portfolio tool as well as a single-name tool. Weather index insurance and forest insurance, the subjects of Chapters 6 and 7, do not only reduce the PD of the insured borrower; they reduce the correlation between borrowers exposed to the same physical hazard. A portfolio of agricultural loans in a single watershed is concentrated in the physical factor; if every borrower is insured against the dominant hazard, the portfolio's physical factor loading falls and the tail thins. That diversification benefit is invisible in single-name analysis and should be priced into the insurance premium as a portfolio-level value. The linkage between insurance and credit developed in Chapter 8 is, at the portfolio level, a correlation-reduction mechanism.

The sci-tech finance pillar—the catalysis of green innovation through venture capital, guarantee funds, and intermediary systems—is a concentration risk in disguise if it is not managed. A policy-guided fund that concentrates in a single technology bet is a fund whose losses are correlated with the failure of that technology. The performance evaluation framework of Chapter 11 should therefore include a factor-concentration metric alongside the return metric, so that a fund achieving high returns by concentrating in one winning technology is not rewarded as if it had achieved those returns through diversified skill.

The boundary condition on all three is data. None of the portfolio-level analysis in this chapter can be executed without loan-level factor exposures, and most banks do not have them. The digital infrastructure agenda of Chapter 12 is, in this light, a precondition for the portfolio discipline this chapter describes. A bank that cannot map its green book to factors cannot measure its concentration, cannot stress-test it credibly, and cannot allocate capital to it with any confidence. The first investment is a loan-file data standard, not a better copula.

5.5 What This Chapter Does Not Claim

Three caveats are owed.

The chapter does not claim that green portfolios are riskier than brown ones. The comparison is not meaningful without specifying the horizon and the scenario. A green portfolio is exposed to transition and physical factors; a brown portfolio is exposed to the same factors in different proportions. The claim is that green portfolios have a dependence structure different from the one embedded in standard models, and that the difference is large enough to matter for capital.

It does not claim that the factor model is calibrated. The loadings in section 5.1.5 are judgmental, the factor volatilities are proxies, and the copula choice is a modeling decision. The value of the framework is that it makes the assumptions explicit and testable, not that it produces a number that is right. A bank that adopts the framework and then validates it against realized losses over three to five years will learn more than a bank that adopts a vendor model and never questions it.

It does not claim that concentration is always bad. A bank with deep expertise in a single green sub-sector—a specialist lender to offshore wind, say—may have an information advantage that more than offsets the concentration cost, because it can price and monitor the risk better than a diversified lender. The framework in this chapter does not resolve that trade-off; it makes it visible. The resolution is a strategic decision for the bank's board, and it should be made with the factor Herfindahl and the stress-test results on the table.

5.6 The Next Layer

The portfolio analysis in this chapter treats the loss given default of each obligor as a parameter, stressed in the scenarios but not modeled in its dependence on the portfolio's state. Insurance is the next layer because it changes LGD and PD in ways that are correlated across borrowers. A weather index insurance contract that pays out when rainfall falls below a threshold does not just reduce the insured farmer's default probability; it reduces it precisely in the states of the world when the portfolio's other farmers are also in distress. That is a dependence-modifying instrument, and it is the closest thing the financial system has to a tool that directly attacks the correlation problem this chapter has described.

Chapter 6 takes up the design of that contract. The specific structure to be examined is a rainfall-deficit index written on a gridded weather product, with a strike at the 20th percentile of the local historical distribution, a linear payout slope up to the 5th percentile, and a cap at the sum insured. The questions are how the basis risk between the grid cell and the individual farm's yield behaves across climate states, and whether the contract's payout triggers coincide with the states in which the lender's agricultural portfolio is most concentrated. Both can be tested against the same loan-file data standard this chapter has argued for, which is why the two chapters belong together.

5.7 Appendix 5A: Reproducibility Note for Table 5-1

The figures in Table 5-1 were produced by Monte Carlo simulation with \(10^6\) draws, random seed 20240517, using NumPy's default PCG64 generator. The simulation code is available in the book's companion repository under chapter5/copula_tail_table.py. Each copula was calibrated to a common Kendall's \(\tau\) of 0.10 by solving for the parameter that reproduces the target rank correlation on a separate calibration run of \(10^7\) draws (seed 20240518). Marginal default probabilities were set to \(PD = 0.02\) for all 400 obligors, with \(LGD = 0.45\) and equal exposures. The Gaussian and Student-\(t\) copulas were simulated via the latent-variable representation; the Clayton and Gumbel copulas via the Marshall–Olkin algorithm. Standard errors on the 99.9th percentile estimates are approximately 0.3 loss units at \(10^6\) draws, so the ratios in the final column are stable to two decimal places. The parameter file chapter5/copula_params.json records the calibrated parameter for each family and the realized Kendall's \(\tau\) on the calibration run.