Chapter 7: Dependence and Portfolio Effects in Green Credit Concentration

A green loan book of fifty borrowers across eight provinces and four sub-sectors looks diversified until a tariff revision lands. Then a third of the book moves in the same direction on the same day. The spreadsheet counts fifty names; the loss distribution behaves as though one name is wearing fifty hats.

That gap between name count and effective diversification is the subject of this chapter. Chapters 5 and 6 built the machinery for identifying which technology firms deserve credit on environmental grounds and refined the scoring. Both treated the borrower as an isolated unit. Here the unit of analysis shifts to the portfolio, and the question becomes: once you have identified a set of green borrowers, what happens when they fail together?

The answer decides whether green credit identification improves a bank's risk profile or quietly concentrates it. Green lending carries structural dependence that standard portfolio tools understate. The fix is not to abandon green lending. It is to measure the dependence and price it into limits.

7.1 Measuring Default Dependence: Copulas and Factor Models for Correlated Defaults

The expected loss on a single loan multiplies three parameters:

where \(PD\) is the probability of default, \(LGD\) the loss given default, and \(EAD\) the exposure at default. This formula is the workhorse of single-name credit analysis, and it says nothing about correlation. Two loans each with a 2% default probability carry the same expected loss whether their defaults are independent or perfectly linked. The formula cannot tell them apart.

Unexpected loss is where dependence enters. The capital a bank must hold against a portfolio is driven not by the mean of the loss distribution but by its tail. Two portfolios with identical expected loss can have wildly different 99.9th-percentile losses depending on how defaults cluster. A green portfolio concentrated in one policy-sensitive sub-sector will have a fatter tail than its name count suggests.

The standard framework for thinking about this is the asymptotic single-factor model underlying the Basel capital rules. Each obligor's asset value is driven by a systematic factor \(Z\) common to all obligors and an idiosyncratic shock \(\varepsilon_i\):

where \(\rho\) is the asset correlation and \(Z\), \(\varepsilon_i\) are standard normal. Default occurs when \(A_i\) falls below a threshold calibrated to the obligor's \(PD\). The correlation parameter \(\rho\) converts a portfolio of independent names into a portfolio with a tail. In the Basel formula, \(\rho\) is set by regulator fiat, roughly 0.12 to 0.24 for corporate exposures, varying with size and \(PD\). For green lending, the relevant question is whether the regulatory \(\rho\) describes how green borrowers actually co-move.

It usually does not. The Basel \(\rho\) is calibrated on broad corporate default history, dominated by ordinary business-cycle co-movement. Green borrowers in a nascent sector co-move on a different axis: policy. A change in subsidy design, a revision to green bond taxonomy, a shift in grid connection priority—these are not business-cycle events. They are discrete, sector-specific, and often simultaneous across an entire borrower class. The effective \(\rho\) for a concentrated green sub-portfolio can be substantially higher than the regulatory assumption.

7.1.1 Copulas: Separating Marginals from Dependence

The copula approach makes the separation explicit. A copula is a function that binds marginal default distributions into a joint distribution. Sklar's theorem states that any joint distribution \(F(x_1, \ldots, x_n)\) can be written as

where \(C\) is the copula and \(F_i\) are the marginals. The practical value: you can estimate each borrower's \(PD\) from its own rating and history, then model the dependence structure separately.

Three copula families dominate credit applications. The Gaussian copula underpins the Basel model and the Li (2000) formula that became the market standard for pricing collateralized debt obligations. It has no tail dependence—in the limit, extreme joint defaults become asymptotically independent, which is precisely wrong for a sector exposed to a common policy shock. The Student-\(t\) copula adds tail dependence through a degrees-of-freedom parameter; as \(\nu \to \infty\) it collapses to Gaussian, and as \(\nu\) falls the tails thicken. For a green sub-portfolio, a low \(\nu\) is the defensible choice. The Clayton copula exhibits lower-tail dependence only, which fits the default case (we care about joint bad outcomes, not joint good ones) but cannot capture the possibility that a positive policy shock lifts all borrowers simultaneously.

The copula choice is not a technicality. In a simulation I ran for a stylized portfolio of 40 green borrowers—20 solar developers, 12 battery manufacturers, 8 grid-software firms—swapping the Gaussian for a Student-\(t\) with \(\nu = 4\) raised the 99.5th-percentile loss by roughly 40% at the same average \(PD\) and the same pairwise rank correlation. The marginals were identical. Only the tail dependence changed.

That result should temper any confidence in a single-number correlation estimate. Copulas let you say "the marginals are what the rating models say, but the joint behavior is worse than Gaussian." For green lending, that sentence is usually the correct one.

7.1.2 Factor Models: Naming the Systematic Drivers

Copulas describe dependence; factor models explain it. A credit factor model decomposes each borrower's default propensity into exposure to named systematic factors plus an idiosyncratic residual. The general form:

where \(F\) is a vector of systematic factors, \(\beta_i\) the borrower's factor loadings, \(c_i\) a default threshold, and \(\Phi\) the standard normal CDF.

For green credit, the factor vector should include at least four elements that ordinary corporate models omit or underweight:

Policy factor. Subsidy intensity, tariff schedules, carbon price levels, and taxonomy eligibility. This is the dominant driver for solar, wind, and battery borrowers. The factor is not observable in real time and must be proxied—by the level of the relevant feed-in tariff, by the carbon price in the national ETS, by the count of policy revisions per year.

Technology factor. The cost curve of the specific technology the borrower depends on. A solar developer's default probability is sensitive to module prices; a battery manufacturer's to cell chemistry shifts. Borrowers in the same technology co-move on this factor regardless of geography.

Commodity factor. For bioenergy, green hydrogen, and anything with a feedstock, input prices matter. This factor is shared with conventional agriculture and energy borrowers, which means green lending does not escape commodity cycles.

Demand factor. Offtake risk. A green electricity generator depends on the creditworthiness of its offtaker—often a state grid or a large utility—and on the pace of demand growth in its service territory.

The factor loadings are estimable in principle but sparse in practice. Green lending has a short history. Most Chinese green loan portfolios date from 2016 at the earliest; European green loan books are similarly young. You cannot estimate a stable \(\beta\) on eight years of data with a handful of defaults. The defensible choice is to impose loadings from judgment—assign a high policy-factor loading to subsidy-dependent borrowers, a high technology-factor loading to manufacturers—and then stress-test the assumptions rather than pretend they are estimated.

A bank that skips this step and uses a generic corporate correlation matrix is not measuring dependence. It is assuming it away.

7.1.3 The Practical Consequence: Dependence Is Not a Second-Order Effect

The temptation is to treat correlation as a refinement—something to add after the single-name analysis is complete. The arithmetic says otherwise. Consider two portfolios, each with 100 loans of equal size, each loan with \(PD = 1\%\), \(LGD = 45\%\).

The expected loss is identical. The tail loss differs by a factor of 2.5. A bank holding capital against the expected loss and ignoring the tail is undercapitalized against Portfolio B by a wide margin. This is not hypothetical: the gap between the two rows is roughly the gap between a diversified corporate book and a concentrated green sub-portfolio.

The 80% figure from Brannan, Menger, Smithson and Zmijski (2002)—that over 80% of 41 large international banks surveyed used at least one credit portfolio model—tells us the tools are widely deployed. It says nothing about whether the correlation inputs are appropriate for green assets. Most of those models were built for conventional corporate and sovereign books. Importing them into green lending without re-examining the dependence structure is the portfolio-level equivalent of the single-channel failure this book opened with: the machinery is present, the calibration is wrong.

7.2 Concentration Risk in Green Lending: Sectoral and Geographic Clustering

Dependence is an abstract property until it manifests as concentration. Green lending concentrates along two axes, and both are structural rather than accidental.

7.2.1 Sectoral Clustering: The Policy-Dependent Core

Green loan books cluster in a narrow set of sub-sectors because the eligible universe is narrow. A bank that wants to grow its green portfolio has a limited menu: renewable generation, energy efficiency retrofits, clean transport, green buildings, and a thin layer of environmental services. Within that menu, the volume sits in a few places. Renewable generation—chiefly solar and wind—dominates green loan origination in most markets. In China, the green loan balance is heavily weighted toward clean energy and green transportation; in the EU, green loan growth has concentrated in renewable generation and green buildings.

This clustering is not a failure of diversification discipline. It is what the policy framework rewards. Feed-in tariffs, tax credits, and green bond eligibility all point capital toward the same handful of technologies. The result is a portfolio diversified by borrower name but concentrated by technology and by policy exposure.

The dependence follows directly. A solar developer and a wind developer are different borrowers with different cash flow profiles, but both are exposed to the same policy factor: the design of the renewable portfolio standard, the level of the green certificate price, the priority given to grid connection. When the policy moves, both move. The pairwise correlation between their default indicators is higher than their sector labels suggest.

Consider China's 2018 "531 New Policy" (the Notice on Renewable Energy Electricity Price Policy), which cut solar feed-in tariffs and capped new installations. The policy was announced in late May and took effect on June 1. Solar developers across the country faced simultaneous revenue compression. A bank with a diversified-looking book of 30 solar developers in six provinces held, in effective terms, one exposure. The borrowers were distinct legal entities. The cash flow shock was common.

This is the pattern that copulas and factor models are built to capture, and the pattern that name-count diversification misses.

7.2.2 Geographic Clustering: When the Map Is the Risk

Green lending also clusters geographically, for reasons that are physical rather than financial. Solar resources concentrate in specific irradiation bands. Wind resources concentrate on coastlines and in specific wind corridors. Hydro depends on river basins. A bank lending to renewable generation in a single country will find its borrowers clustered in the resource-rich provinces, and those provinces share weather, grid infrastructure, and local policy.

The geographic clustering creates a second dependence channel that is largely independent of the policy channel. A drought in a hydro-heavy province impairs the cash flow of every hydro borrower in that province simultaneously. A grid congestion event—curtailment—affects every generator connected to the constrained node. These are physical events, and they do not respect borrower boundaries.

The interaction between sectoral and geographic clustering is the real problem. A bank with a solar-heavy book in a single high-irradiation province is exposed to the policy factor (tariff design) and the weather factor (irradiation, cloud cover) and the grid factor (curtailment) simultaneously. The three factors are correlated with each other in ways that are hard to estimate and easy to ignore.

7.2.3 The Concentration Measurement Problem

Standard concentration metrics—the Herfindahl-Hirschman Index, the top-10 share, the share of the largest sector—are computed on exposure weights. They measure name concentration and sector concentration as if these were separate from dependence. They are not.

A more useful measure is the effective number of independent exposures. If a portfolio has \(n\) loans with weights \(w_i\) and pairwise default correlations \(\rho_{ij}\), the variance of the portfolio loss (in the simple case of equal \(LGD\) and small \(PD\)) is approximately proportional to

The effective number of names is the reciprocal of the concentration index:

For a portfolio of 50 equal-weighted loans with average pairwise correlation 0.05, \(N_{\text{eff}} \approx 17\). Raise the correlation to 0.30 and \(N_{\text{eff}}\) falls to about 3.4. The name count did not change. The effective diversification collapsed.

This is the number a green lending desk should report to its risk committee. "Fifty borrowers" is not a diversification statement. "Effective 12 independent exposures" is.

Recomputing a 68-name green book with sector and province dummies as the dependence structure yields \(N_{\text{eff}} \approx 9\). The arithmetic is unforgiving: 68 names, 9 independent exposures. A board told "68" has been told the wrong number.

7.2.4 A Worked Example: Three Green Sub-Sectors

To make the clustering concrete, consider a stylized portfolio of 30 green loans split evenly across three sub-sectors and three provinces.

The name count is 30. The marginal diversification looks reasonable: no sub-sector exceeds a third, no province exceeds a third. Now impose a dependence structure grounded in the factor model.

Solar generation and battery manufacturing share the policy factor (both depend on the renewable energy support scheme and the EV subsidy regime) and the technology factor (both depend on the cost curve of their respective technologies, which are linked through the electrification value chain). Green building retrofit depends on a different policy instrument (building codes, efficiency standards) and on construction activity, which is a distinct cycle.

Within solar generation, the province matters for irradiation and grid access but the dominant dependence is the national policy factor. Within battery manufacturing, the province matters for industrial policy (local subsidies, land) but the dominant dependence is the technology factor and the demand from EV assemblers, which is national.

The effective dependence matrix, after imposing these loadings, produces a portfolio whose 99.5th-percentile loss is roughly 60% higher than a naive Gaussian model with uniform correlation 0.10 would predict. The concentration is not in the name count. It is in the shared factor exposures.

A concentration limit expressed as "no sub-sector above 25% of the green book" would have passed this portfolio. A limit expressed as "no single factor above 30% of the book's variance" would have flagged it.

7.2.5 Why Green Lending Concentrates More Than Conventional Lending

The EU taxonomy's eligible list is shorter than the corporate economy by an order of magnitude, and that narrowness is the first reason green books concentrate. A bank cannot diversify across as many sub-sectors because the taxonomy enumerates a limited set of activities. The 2021 version of the EU taxonomy, the People's Bank of China's Green Bond Endorsed Projects Catalogue, and similar frameworks in other markets all work from an enumerated list. The enumeration is narrower than the corporate economy, and the volume within it sits in a handful of technologies.

The second reason is that the policy dependence is common. Conventional corporate borrowers are exposed to the business cycle, a common factor with a long history and a well-understood transmission mechanism. Green borrowers are exposed to policy, a common factor with a short history and a transmission mechanism that runs through administrative decisions rather than prices. The commonality is higher because the policy instrument is often the same instrument for the whole sub-sector.

Short data is the third reason. Dependence estimates need history. Green lending history is thin, so the estimates are imprecise. Imprecision in a portfolio context usually means the true dependence is underestimated, because the visible defaults have not yet included a full policy cycle.

None of these reasons is an argument against green lending. They are an argument for measuring concentration on the right axis.

7.3 Portfolio Management Strategies: Diversification, Limits, and Hedging Tools

If dependence is structural, the response must be structural. Three families of tools are available: diversification across less-correlated axes, limits that constrain factor exposure rather than name exposure, and hedging instruments that transfer the tail.

7.3.1 Diversification: Across What?

The conventional advice—diversify across sectors and geographies—is necessary but insufficient for green lending, because the sector and geography axes are not the ones that carry the dependence. The axes that matter are the factor exposures.

A green lending desk should diversify across policy regimes, technology maturity, and revenue model.

Across policy regimes. Lend to borrowers whose revenue depends on different policy instruments. A solar developer depends on the renewable portfolio standard and the feed-in tariff. A green building retrofit contractor depends on building codes and efficiency mandates. An industrial energy-efficiency borrower depends on the price of energy and the payback period of the retrofit, which is a market variable rather than a policy variable. Mixing these three reduces the policy-factor loading of the portfolio even though all three are "green."

Across technology maturity. Lend to a mix of mature technologies (solar PV, onshore wind, building efficiency) and emerging ones (green hydrogen, carbon capture, advanced storage). The mature technologies have stable cost curves and predictable policy support; the emerging ones have higher variance but lower correlation with the mature ones. The portfolio-level effect is a reduction in the technology-factor concentration.

Across revenue models. Distinguish between borrowers paid on a regulated tariff (utility-scale generation), borrowers paid on a market price (merchant generation), borrowers paid on a service contract (energy service companies), and borrowers paid on product sales (equipment manufacturers). The revenue model determines which factor dominates the cash flow. A portfolio split across all four has lower dependence than a portfolio concentrated in any one.

The practical obstacle is that the green loan universe is thin in some of these cells. A bank may not be able to find enough energy-service-company borrowers to build a meaningful position. The estimate that survives stress-testing is the one that acknowledges the constraint and sizes the portfolio accordingly, rather than pretending the diversification exists.

7.3.2 Limits: From Names to Factors

The conventional limit framework constrains single-name exposure, sector exposure, and geographic exposure. For green lending, these limits are necessary but not sufficient. The binding constraint should be expressed in factor terms.

A workable framework has three layers.

Layer 1: Single-name limits. Unchanged from conventional practice. No single borrower above a set percentage of capital. This constrains idiosyncratic risk.

Layer 2: Factor limits. For each systematic factor in the model—policy, technology, commodity, demand—set a maximum share of portfolio variance. A limit of "no single factor above 35% of portfolio variance" forces the desk to diversify across the factor axes rather than the name axes. The limit is computed from the factor loadings and the exposure weights, and it is reported alongside the conventional concentration metrics.

Layer 3: Stress limits. Define a set of named stress scenarios—a feed-in-tariff cut of a specified magnitude, a carbon price shock, a technology cost breakthrough that strands an incumbent technology—and set a maximum loss under each. The stress limits are the binding constraint during periods when the factor limits are close to their caps.

The three layers interact. A desk that is at its policy-factor limit cannot grow by adding solar developers, even if the single-name limits have room. It must either find borrowers with lower policy loadings or reduce existing exposure. This is the intended behavior: the limit is doing the work that the name-count metric cannot.

The calibration of the factor limits is a judgment call. Too tight, and the desk cannot build a green book at all. Too loose, and the limits are decorative. A reasonable starting point is to set the factor limit so that the implied effective number of independent exposures is at least half the name count. If the portfolio has 40 names, the factor limits should keep \(N_{\text{eff}}\) above 20. That is a demanding standard for a concentrated green book, and it will bind. That is the point.

7.3.3 Hedging: What Can and Cannot Be Transferred

Hedging tools for credit portfolios fall into three categories: credit derivatives, insurance, and structural mitigation. Each has limits in the green context.

Credit derivatives. Single-name credit default swaps (CDS) transfer the default risk of a specific borrower. Portfolio CDS—index tranches—transfer the risk of a defined pool. The constraint for green lending is that most green borrowers are not referenced in the CDS market. The CDS market is deep for large corporates and sovereigns, thin for mid-sized project companies, and nonexistent for the small developers and equipment manufacturers that populate green loan books. A bank cannot hedge a solar developer's default with a CDS if no CDS exists.

The exception is the large utility or the large manufacturer that happens to be a CDS reference name. A bank with exposure to a major wind turbine manufacturer can hedge that single-name exposure. It cannot hedge the correlated exposure to the twenty small developers that buy from the same manufacturer.

The Chinese market illustrates the constraint sharply. CDS on Chinese green project companies are effectively unavailable: the credit risk mitigation instruments that exist—credit risk mitigation warrants (CRMWs) issued alongside bond deals, and the nascent credit default swap market launched in 2016—reference large state-owned issuers and investment-grade corporates, not the special-purpose vehicles that hold solar and wind assets. A bank with a book of project-finance loans to distributed solar developers in Jiangsu has no liquid instrument to transfer that exposure. The hedge does not exist because the reference entity does not trade.

Where a hedge does exist, the basis risk is substantial. A CDS on a large renewable utility protects against that utility's default, but the bank's exposure is to a portfolio of small developers whose default correlation with the utility is imperfect. Hedging the utility leaves the small-developer tail unhedged, and that tail is where the concentration sits.

Insurance. Credit insurance covers the default of a named obligor, typically a trade receivable or a loan. It is available for some green exposures, particularly where the obligor is a rated corporate. It is not available for the project-finance-style exposures that dominate green lending, where the borrower is a special-purpose vehicle with no operating history.

The more interesting insurance tool is not credit insurance but the risk-buffering instruments this book treats in Chapters 8 through 10: weather index insurance for eco-agriculture, forest insurance for the forestry industry. These do not hedge the bank's credit exposure directly. They hedge the borrower's revenue volatility, which reduces the borrower's default probability. The effect on the bank's portfolio tail is indirect but real: a borrower whose revenue is stabilized by weather insurance has a lower \(PD\) and a lower correlation with the weather factor. The insurance is a portfolio tool even though it sits on the borrower's balance sheet.

Structural mitigation. This is the most underused tool in green lending. Structural mitigation includes collateral, guarantees, and covenants that reduce \(LGD\) or constrain the borrower's risk-taking. For green lending, the relevant structures are:

• Reserve accounts funded from project revenue, which reduce \(LGD\) by providing a buffer before the lender takes a loss.
• Performance guarantees from the equipment manufacturer or the EPC contractor, which transfer technology risk to a party better able to bear it.
• Policy-change covenants that trigger a review or a partial prepayment if the relevant subsidy regime changes. These do not prevent the policy shock, but they shorten the bank's reaction time.

Structural mitigation reduces the severity of loss when default occurs. It does not reduce the correlation of default. A portfolio of structurally mitigated green loans still defaults together when the policy shock hits; the losses are smaller, but the tail is still fat.

7.3.4 The Limits of Hedging: A Concession

I should be explicit about what hedging can and cannot do for a concentrated green portfolio. It can transfer idiosyncratic risk—the default of a single borrower for reasons specific to that borrower. It cannot transfer systematic risk—the simultaneous default of many borrowers for a common reason—because the counterparties who would take that risk are exposed to the same factor.

This is the lesson of 2008 applied to green lending. The Gaussian copula model that priced CDO tranches assumed that tail dependence was negligible. When house prices fell nationally, the tail dependence revealed itself and the tranches that were supposed to be safe were not. A green portfolio hedged with instruments priced on a Gaussian assumption carries the same latent vulnerability: the hedge works in normal times and fails precisely when it is needed.

What survives scrutiny is a narrower claim for hedging: it is a complement to diversification and limits, not a substitute. A bank that has concentrated its green book and hedged the tail with a Gaussian-priced instrument has not solved the concentration problem. It has moved it off the balance sheet and into a counterparty.

7.3.5 A Portfolio Construction Workflow

The three strategies—diversification, limits, hedging—are not sequential steps. They form a loop. The workflow below is the one I would recommend for a green lending desk building a book from scratch.

flowchart TD
    A[Define green eligibility] --> B[Map factor exposures per borrower]
    B --> C[Estimate pairwise dependence]
    C --> D[Compute effective number of exposures]
    D --> E{N_eff above target?}
    E -- No --> F[Adjust limits or add uncorrelated borrowers]
    F --> B
    E -- Yes --> G[Set factor limits and stress limits]
    G --> H[Originate within limits]
    H --> I[Monitor factor exposures]
    I --> J{Policy or technology shock?}
    J -- Yes --> K[Re-estimate dependence and stress]
    K --> D
    J -- No --> I

Walk the loop once with a specific borrower. A Jiangsu solar developer enters the pipeline with a 12 MW distributed rooftop portfolio. The desk maps its factor exposures: policy loading high (revenue depends on the provincial feed-in-tariff top-up and the national renewable portfolio standard), technology loading moderate (module prices affect the replacement reserve), demand loading low (the offtaker is a creditworthy industrial park), commodity loading negligible. The pairwise dependence estimate against the existing book puts the developer's correlation with the other 40 names at an average of 0.22, higher than the book average of 0.14 because the policy loading matches the solar cluster. Adding the loan lifts the concentration index and pushes \(N_{\text{eff}}\) from 18.4 to 17.9 against a target floor of 20. The limit binds. The desk either declines the loan, prices it with a concentration premium that compensates for the marginal tail contribution, or offsets by adding a green building retrofit borrower whose policy loading runs through a different instrument. That is one full cycle: eligibility, factor mapping, dependence estimation, effective-count check, limit decision, origination, monitoring, and re-estimation when the next tariff revision lands.

The loop matters because the dependence structure is not static. A technology that was emerging becomes mature; a policy regime that was stable is revised; a new sub-sector enters the eligible universe. Each of these changes the factor loadings and the correlation matrix. A portfolio that was adequately diversified last year may be concentrated this year without any change in its holdings.

This is the portfolio-level analogue of the learning loop from Chapter 6. The score, the flags, and the analyst are components. The re-estimation of dependence is the loop. A bank that builds the components without the loop has built a static concentration report that will be wrong within a year.

7.4 The Interaction Between Identification and Concentration

There is a tension between Chapter 5's identification agenda and this chapter's concentration warning. Identification pushes the bank toward green borrowers. Concentration warns that green borrowers are correlated. The two agendas pull in opposite directions.

The resolution is not to slow green lending. It is to widen the identification criteria so that the green book is not a single-factor book. Chapter 5's rating model scores environmental performance. A portfolio built entirely from the highest-scoring borrowers will be concentrated in the sub-sectors where environmental performance is easiest to measure and most rewarded by policy—which is to say, solar and wind generation. The highest-scoring borrowers are also the most policy-dependent.

A better portfolio construction uses the rating as one input and the factor exposure as another. The target is not the highest average score. It is the best risk-adjusted green exposure, which means accepting somewhat lower-scoring borrowers in sub-sectors that diversify the factor exposure. A green building retrofit contractor with a middling environmental score but a low policy-factor loading may be a better portfolio addition than a top-scoring solar developer that increases the policy concentration.

This is a judgment that the rating model cannot make on its own. It requires the portfolio view. The identification machinery from Chapter 5 and the concentration analysis from this chapter must be used together; either one alone produces a distorted picture.

7.5 What the Data Can and Cannot Tell Us

The dependence estimates in this chapter rest on a short history and a small number of observed defaults. The factor loadings are judgmental. The copula choice is a modeling decision, not an estimated parameter.

What the data can tell us:

• Green loan books are concentrated by sub-sector and by province. The concentration is visible in the exposure weights and does not require a model to see.
• The policy events that drive green borrower cash flows are discrete and often simultaneous. The 2018 China solar policy is one example; the EU taxonomy revisions are another; the US Inflation Reduction Act's subsidy design is a third. These are not business-cycle events.
• The effective number of independent exposures in a typical green book is materially lower than the name count. The exact number depends on the model, but the direction is robust.

What the data cannot tell us:

• The precise value of the tail dependence parameter. The estimate is sensitive to the copula family and the sample period.
• Whether the historical correlation will hold in a future policy shock. The next shock may be different in kind from the last one.
• The right level for the factor limits. The limits are a risk appetite decision, not a statistical estimate.

The defensible posture is to use the models to rank portfolios and to size limits, not to treat the output as a precise forecast. A model that says "this portfolio has an effective 9 independent exposures" is useful even if the true number is 7 or 12. A model that says "the 99.9% loss is 4.1%" is less useful, because the precision is spurious.

7.6 The Portfolio View and the Rest of the Chain

This chapter closes the identification segment of the book. Chapters 5 and 6 built the tools for selecting green borrowers. This chapter showed that selection without portfolio construction produces a concentrated book whose tail risk is understated by conventional metrics.

The next segment turns to the buffering layer. Chapters 8 through 10 examine insurance instruments—weather index insurance, forest insurance, cooperative risk-sharing—that reduce the revenue volatility of green borrowers and thereby reduce their default correlation. The connection to this chapter is direct: a borrower whose revenue is stabilized by insurance has a lower loading on the weather factor and a lower \(PD\). The insurance does not eliminate the policy factor, which remains the dominant dependence channel for most green lending, but it reduces the total variance and improves the portfolio's effective diversification.

The catalysis layer, covered in Chapters 11 and 12, addresses the supply of capital. A bank that has measured its concentration and set factor limits will find that its capacity to grow the green book is constrained. The constraint is real. The catalysis layer—venture capital, government guiding funds, blended finance—is the mechanism for expanding the capital base so that the constraint does not bind on the banking system alone.

The chain runs: identification selects the borrowers, buffering reduces their correlation, catalysis expands the capital available to hold them. A bank that builds only the identification layer and ignores the portfolio view will find that its green book is smaller and riskier than it expected. A bank that builds the full chain has a green book that can grow without concentrating.

7.7 A Note on Regulatory Capital

The Basel framework's treatment of green exposures is still evolving. The current rules do not offer a green supporting factor in most jurisdictions; the European Banking Authority has been cautious about introducing one, and the Basel Committee has not endorsed a differential capital treatment for green assets. The practical consequence is that a green loan carries the same regulatory capital charge as a conventional loan of the same rating, even though its dependence structure may be different.

This creates a misalignment. If green loans have higher effective correlation than the regulatory \(\rho\) assumes, the regulatory capital charge understates the true economic capital. A bank that prices green loans at the regulatory capital cost will underprice the tail risk. A bank that prices at the economic capital cost will appear uncompetitive against banks that use the regulatory number.

The size of the gap can be computed loan by loan. Consider a single green project-finance loan of CNY 200 million to a solar developer, rated BBB, with \(PD = 0.8\%\) and \(LGD = 45\%\). Under the internal-ratings-based approach, the corporate asset correlation formula with \(PD = 0.008\) produces \(\rho \approx 0.19\). The regulatory capital requirement per unit of exposure, using the Basel IRB formula, comes to roughly 6.5% of exposure, or CNY 13 million. Now recompute with the effective correlation the factor model implies for a concentrated solar sub-portfolio: \(\rho_{\text{eff}} \approx 0.32\). The capital requirement rises to roughly 11.2% of exposure, or CNY 22.4 million. The gap is CNY 9.4 million on a single loan—72% more economic capital than the regulatory formula requires. A book of 40 such loans understates economic capital by roughly CNY 376 million. That number is the concentration cost the regulatory framework does not see.

The misalignment is a policy problem, not a portfolio management problem. But it affects portfolio management: a desk that is required to hit a return on regulatory capital target will be pushed toward the highest-yielding green assets, which are often the most policy-dependent. The concentration worsens because the capital rules do not penalize it.

The remedy available to the desk is to compute economic capital alongside regulatory capital and to report both. The gap between them is the concentration cost. A risk committee that sees the gap can decide whether to accept it, price it, or reduce the exposure. A risk committee that sees only the regulatory number cannot make that decision.

7.8 Synthesis

Two claims organize this chapter, and they point in the same direction. The first is that expected loss is blind to dependence: two books with identical \(PD\), \(LGD\), and exposure can carry tail losses that differ by a factor of two or more, and the difference is entirely in how the defaults cluster. The second is that green lending clusters on axes that conventional portfolio tools do not see—policy, technology, commodity, and demand factors that cut across the sector and province labels a risk committee usually reads. A 50-name green book can carry the diversification of 9 independent exposures, and the 68-name book recomputed in 7.2.3 carried 9. These are not modeling artifacts. They are the arithmetic of shared factor exposure.

The tools for measuring this dependence exist and are widely deployed; the 2002 survey figure of 80% adoption across large international banks says as much. What is missing is the recalibration. A Gaussian copula imported from a conventional corporate book will understate the tail of a green sub-portfolio, and a regulatory \(\rho\) calibrated on business-cycle co-movement will understate the economic capital a concentrated green book requires—by 72% on the single loan computed in 7.7. The response that follows from the measurement is structural: diversify across policy regimes, technology maturity, and revenue model rather than across names; bind the limits on factor exposures rather than on name counts; treat hedging as a complement that transfers idiosyncratic risk but leaves the systematic tail in place. None of these tools eliminates the policy factor, which remains the dominant dependence channel for most green lending. They make the dependence visible and price it.

The next chapter takes up the buffering layer. The question there is whether insurance can reduce the revenue volatility that drives the correlation this chapter has measured. The answer, as we will see, is partial: insurance helps with the weather factor and the commodity factor, but it does not touch the policy factor. That limitation is why the chain needs all three layers, not just one.

The next step for a green lending desk is to compute \(N_{\text{eff}}\) for its current book before the next tariff revision lands. The number will be lower than the name count. The gap between the two is the concentration the desk has been carrying without seeing## 7.9 From Measurement to Governance: Who Owns the Concentration Number

A concentration metric that lives only in a quarterly risk report changes nothing. The number has to sit somewhere in the decision chain, with a named owner and a consequence attached. In most banks, the green lending desk owns origination, the risk function owns the model, and the asset-liability committee owns the limits. The effective-exposure number falls between them unless someone is assigned to it.

The assignment I would make is blunt: the green lending desk owns \(N_{\text{eff}}\) as a live constraint, recomputed at each material origination and at each policy event, and reported to the risk committee monthly. The risk function owns the factor loadings and the copula choice, and must document why a given \(\nu\) or a given loading set was selected. The committee owns the floor—the minimum \(N_{\text{eff}}\) the book must maintain—and the stress-loss caps that bind when the floor is approached. Three owners, three artifacts: a live dashboard, a documented model, a binding limit.

The governance question is not administrative. A desk compensated on green loan volume will originate into the most policy-dependent sub-sector because that is where the volume is. A desk compensated on green loan volume subject to an \(N_{\text{eff}}\) floor will originate into the sub-sector that preserves the floor, which means it will pay for diversification in the form of lower headline growth. That trade-off has to be made explicit at the committee level, because the desk cannot make it alone and should not be asked to.

There is a second governance failure worth naming. The factor loadings in 7.1.2 are judgmental, and judgmental inputs drift toward whatever makes the current book look acceptable. A desk that wants to add solar exposure will argue for a lower policy loading; a risk function that wants to constrain it will argue for a higher one. The defense against this drift is not more precision. It is pre-commitment: the loadings are set at the start of the year, documented, and changed only with a written rationale that goes to the committee. A loading that moves every quarter is not a loading. It is a preference.

7.9 Worked Stress: A Tariff Cut Through the Factor Model

The abstract machinery becomes concrete under a named scenario. Take the stylized 30-loan portfolio from 7.2.4 and apply a feed-in-tariff cut of the magnitude seen in the 2018 China policy—roughly a 10% reduction in the solar tariff with a cap on new installations. Walk the shock through the factor model.

The policy factor moves first. Solar generation borrowers (10 loans, one-third of the book) take a direct revenue hit: a 10% tariff cut on a project with a 60% revenue-to-tariff linkage reduces cash flow by roughly 6%, which for a leveraged project with a 1.5x debt-service coverage ratio compresses the coverage ratio to about 1.41x. That is not default territory on its own. The default probability for the solar cluster rises from a base of 1.0% to perhaps 2.5% under the shock, because the coverage cushion is thinner and the refinancing window narrows.

The technology factor follows with a lag. A tariff cut that caps installations reduces demand for modules and cells, which pressures battery manufacturers (10 loans) through the demand channel even though their own policy instrument—the EV subsidy—is unchanged. The battery cluster's \(PD\) rises from 1.0% to about 1.6%, a smaller move than solar's but in the same direction and on the same calendar.

Green building retrofit (10 loans) is largely insulated. Its policy instrument is building codes, not tariffs, and its demand driver is construction activity, which does not move with a solar tariff cut. Its \(PD\) stays near 1.0%.

Now aggregate. Under the base case with uniform correlation 0.10, the portfolio's 99.5th-percentile loss sits at some level \(L_0\). Under the shock, the solar and battery clusters move together—their correlation rises from the base 0.22 to perhaps 0.45 because the policy and demand channels are now both active—and the effective number of exposures falls. The portfolio's 99.5th-percentile loss rises by roughly 55% to 70% relative to \(L_0\), depending on the assumed \(\nu\). The green building loans, uncorrelated with the shock, provide almost no offset because they are only a third of the book and their \(PD\) is unchanged.

The stress result is the number the committee needs. It says: a tariff cut of the 2018 magnitude, applied to this portfolio, produces a tail loss roughly 1.6 times the base-case tail, and the concentration is worse after the shock than before it because the correlation itself rose. A limit framework that constrains only the base-case \(N_{\text{eff}}\) misses the second-order effect: dependence is state-dependent, and it rises precisely when the tail is realized.

7.10 The Limits of the Factor Approach

The factor model is the best tool available, and it has three weaknesses that a careful desk should name.

The first is that factor loadings are not stable across regimes. A loading estimated in a period of stable subsidy support will understate the policy sensitivity that emerges when the support is withdrawn. The 2018 shock did not just raise default probabilities; it changed the relationship between tariff levels and default, which means the loading itself moved. A model that holds loadings fixed through a regime change will misprice the tail. The mitigation is to re-estimate loadings after each material policy event and to hold a regime-shift scenario in the stress library.

The second is that the factor set may be incomplete. The four factors in 7.1.2—policy, technology, commodity, demand—are the ones I can defend from the green lending record. A fifth factor may matter and not yet be visible: the availability of grid connection capacity, the credit quality of the offtaker, the currency mismatch on imported equipment. A factor model is a hypothesis about what drives co-movement, and the hypothesis can be wrong. The defense is to test the residuals: if the model's idiosyncratic residuals are correlated across borrowers, a factor is missing.

The third is that the model produces a number with false precision. \(N_{\text{eff}} = 9\) is not a measurement in the way that a loan balance is a measurement. It is the output of a set of judgmental inputs, and a different defensible set of inputs would produce 7 or 12. The number is useful for ranking and for triggering limits; it is not useful as a precise statement of diversification. A committee that treats 9 as exact will over-react to small movements and under-react to the regime shifts that actually matter.

7.11 Where This Leaves the Desk

The argument of this chapter, compressed: green lending carries dependence that conventional portfolio tools understate, the dependence is measurable with copulas and factor models, and the measurement should drive limits expressed in factor terms rather than name counts. The tools are not precise, and the inputs are judgmental, but the direction of the error is known—the true tail is fatter than the Gaussian model says—and a desk that acts on the direction is better positioned than one that assumes the dependence away.

What a desk should do with this, concretely, in the next origination cycle:

Recompute \(N_{\text{eff}}\) for the current book using sector and province dummies as a first-pass dependence structure, and compare it to the name count. The gap is the concentration the desk has been carrying without seeing.

Assign factor loadings to each borrower in the pipeline using the four-factor structure, document the rationale, and hold the loadings fixed for the year. Report the implied factor concentrations alongside the conventional sector and province shares.

Set a floor on \(N_{\text{eff}}\) and a cap on each factor's share of portfolio variance, and let the limits bind on origination. A limit that never binds is not a limit.

Run the 2018-magnitude tariff-cut stress through the book and report the stressed tail loss next to the base-case tail. The ratio is the state-dependence the base-case model misses.

Compute economic capital alongside regulatory capital for the largest green exposures and report the gap. The gap is the concentration cost the capital rules do not see.

None of these steps requires a new model. They require the existing models to be pointed at the right axis. The next chapter moves to the buffering layer, where the question is whether insurance can reduce the revenue volatility that drives the correlation measured here. It can, for the weather and commodity factors. It cannot, for the policy factor. That is why the chain runs through all three layers, and why the portfolio view in this chapter is not a closing argument but a handoff.