How Banks Optimise the Balance Sheet: Capital, Liquidity, RWA, Funding, Profitability, Linear Programming and Shadow Prices

Quick answer: bank balance-sheet optimisation asks how a bank should allocate scarce capital, liquidity, funding and balance-sheet capacity across loans, securities, deposits, hedges and other positions while obeying risk, regulatory, customer and operational constraints. The mathematical version is a constrained optimisation problem: choose asset and liability quantities to maximise a defined objective—such as risk-adjusted profit or economic value—subject to minimum capital ratios, leverage limits, liquidity requirements, stable-funding requirements, concentration limits, funding capacity and risk appetite. The most useful output is not only the “optimal portfolio.” It is also the shadow price of each binding constraint: how much extra value the bank could create if it had one more unit of CET1, HQLA, stable funding or another scarce resource.

A bank does not optimise one ratio. It optimises a system in which improving one constraint can make another one tighter.

Page role: allocation under simultaneous constraints

Bukit Timah Tutor already explains individual components such as bank capital models, liquidity stress testing, funds transfer pricing and collateral optimisation.

This article owns the portfolio-level question that begins after those measurements exist: given many profitable opportunities and many binding constraints, how should the whole bank allocate its scarce balance-sheet resources?

The OCC describes balance-sheet management as covering investment securities, liquidity and interest-rate risk and emphasises that banks must manage liquidity so obligations can be met at reasonable cost. See OCC Balance Sheet Management and OCC Liquidity.

1. Define decision variables before defining the objective

Suppose the bank is deciding how much of several activities to hold or originate:

• x₁ = mortgages;
• x₂ = commercial loans;
• x₃ = government securities;
• x₄ = other liquid securities;
• x₅ = wholesale funding;
• x₆ = term debt;
• x₇ = deposits or deposit-pricing buckets;
• x₈ = hedge positions.

In a real optimiser the vector can contain thousands of product, tenor, currency, legal-entity and customer segments. The first modelling decision is therefore resolution: too coarse and the optimiser hides important constraints; too granular and the model becomes unstable, slow or data-hungry.

2. Profit is not one number either

A naive objective might maximise net interest income. That can produce a fragile answer by preferring high-yielding assets without charging for credit risk, funding, liquidity or capital.

A more useful contribution margin for activity i can be written schematically as:

π_i = revenue − FTP/funding cost − expected loss − operating cost − liquidity cost − capital charge − other attributable costs.

The optimiser can then maximise:

Maximise Σ π_ix_i.

This turns the loan-pricing logic in How Banks Price Loans for Risk-Adjusted Return into a portfolio allocation problem.

3. The balance-sheet identity is the first hard constraint

The optimiser cannot create assets without funding them. At the accounting level:

Assets = Liabilities + Equity.

If the bank originates another S$1 billion of loans, the model must decide what provides the corresponding funding/equity capacity. A portfolio optimiser that adds assets without solving the liability side is not optimising a bank balance sheet; it is optimising an asset list.

4. Risk-based capital turns RWA into a scarce resource

If CET1 must remain above a required ratio k relative to risk-weighted assets:

CET1 / RWA ≥ k.

Equivalently:

RWA ≤ CET1 / k.

That inequality creates an RWA budget. If one product consumes more RWA per dollar of exposure, it must generate enough incremental return to justify using more of that scarce budget.

This is why a balance-sheet optimiser needs risk-weighted profitability rather than nominal yield. A 7% loan with high expected loss and heavy capital usage can be economically inferior to a 5% loan with low loss and low capital consumption.

5. The leverage ratio creates a second capital constraint

Risk-based capital is not the only capital constraint. Basel’s leverage ratio is deliberately non-risk-based and acts as a backstop against excessive on- and off-balance-sheet leverage. See the BIS Basel III leverage ratio executive summary.

A simplified constraint is:

Tier 1 capital / leverage exposure ≥ minimum.

This matters because a low-risk asset can consume little RWA but still consume leverage exposure. Once the leverage ratio becomes binding, the optimiser may value balance-sheet size itself as scarce even for low-risk positions.

6. LCR converts short-term liquidity into another budget

The Basel Liquidity Coverage Ratio (LCR) is designed so a bank holds enough high-quality liquid assets (HQLA) to cover net cash outflows in a 30-calendar-day stress scenario. In simplified form:

LCR = HQLA / 30-day stressed net cash outflows ≥ 100% under the standard requirement outside permitted stress usage.

See BIS — Liquidity Coverage Ratio executive summary.

Now a new loan can affect the optimiser through more than RWA. It may create contingent drawdowns or funding outflows, changing the LCR denominator. A government bond may earn less income but contribute HQLA to the numerator.

This makes liquidity value state-dependent. The low-yielding security can be more valuable when the HQLA constraint is binding than when the bank has abundant liquidity headroom.

7. NSFR adds a longer-horizon funding constraint

The Net Stable Funding Ratio (NSFR) promotes a stable funding profile over a longer horizon. In simplified form:

NSFR = Available Stable Funding / Required Stable Funding ≥ 100%.

See BIS — Net Stable Funding Ratio executive summary.

A long-dated illiquid asset generally requires more stable funding than a short liquid asset. The optimiser therefore has to consider not only “Can I fund this asset today?” but “Does the funding structure remain sufficiently stable for the asset profile?”

8. A miniature linear programme

Suppose a teaching bank has three possible asset buckets:

Asset	Risk-adjusted contribution	RWA per S$1	HQLA contribution	Stable funding required
Mortgage	1.8%	0.35	0	0.85
SME loan	2.6%	0.75	0	0.90
Government security	0.8%	0.05	1.00	0.05

The bank wants the highest risk-adjusted contribution, but it also has:

• a limited RWA budget;
• a minimum HQLA need;
• a finite amount of stable funding;
• product-demand caps.

The optimiser may choose fewer SME loans than their headline 2.6% contribution suggests because they consume more RWA and stable funding. It may hold government securities even at lower income because they satisfy the liquidity constraint.

There is no contradiction. The portfolio optimum values each activity by its marginal contribution after the scarce resources it consumes.

9. Shadow prices explain what is actually scarce

Linear programming has a particularly useful output: the dual variable or shadow price associated with a binding constraint.

If the RWA constraint has shadow price λ_RWA = 2 cents per unit, then—locally and under the model assumptions—one extra unit of permitted RWA capacity would improve the objective by about 2 cents.

Similarly:

• shadow price of CET1 → marginal value of extra capital;
• shadow price of HQLA → marginal value of extra liquid assets or lower stressed outflows;
• shadow price of stable funding → marginal value of additional long-term/stable funding;
• shadow price of a concentration limit → cost of being unable to add exposure in that segment.

This is powerful because it turns ratios into economic signals. The optimiser can tell management not merely that “LCR is binding,” but approximately how much value would be unlocked if the liquidity constraint were relaxed by one small unit.

10. Internal pricing can pass shadow prices back to businesses

If stable funding is scarce, the bank can reflect that scarcity in funds transfer pricing. If CET1 is scarce, businesses can be charged a capital cost. If HQLA is scarce, contingent liquidity usage can be priced.

In idealised form:

marginal internal price = market funding cost + liquidity shadow cost + capital shadow cost + other scarce-resource costs.

This connects the central optimiser to decentralised decisions. A relationship manager does not need to solve the whole bank optimisation problem for every loan if internal prices faithfully carry the current scarcity signals.

The weak link is obvious: stale transfer prices can tell businesses that a scarce resource is cheap long after the bank’s constraint has tightened.

11. Concentration constraints stop the optimiser from choosing one apparently dominant trade

If SME lending appears to have the highest risk-adjusted margin, an unconstrained optimiser may allocate too much to one sector or economic factor. Concentration limits introduce inequalities such as:

Exposure to sector j ≤ limit_j.

The limit can reflect single-name exposure, sector, geography, collateral type, counterparty or another common factor.

See How Banks Measure Credit-Portfolio Concentration. The optimiser should consume the concentration model’s output rather than invent its own unrelated definition of diversification.

12. Interest-rate risk adds vector constraints, not one total

Balance-sheet growth changes duration, net-interest-income sensitivity and key-rate exposure. A portfolio can satisfy capital and liquidity requirements while becoming excessively exposed to one part of the yield curve.

The optimiser can therefore constrain a vector of sensitivities:

|DV01_2Y| ≤ limit, |DV01_5Y| ≤ limit, …

or include scenario-loss penalties in the objective. Hedging variables can then be solved jointly with asset/liability quantities.

This connects to How Banks Construct Interest-Rate Hedges.

13. Why the problem is often not truly linear

Linear programming is a useful teaching skeleton because many balance-sheet rules can be approximated as linear constraints. Real banks contain nonlinearities:

• deposit rates change customer behaviour;
• credit loss rises nonlinearly with concentration;
• market liquidity deteriorates as position size grows;
• capital rules can contain thresholds, floors and netting effects;
• hedges contain convexity;
• funding cost can increase when issuance volume rises;
• business volumes may be indivisible or subject to fixed costs.

The appropriate engine may therefore be nonlinear programming, mixed-integer programming, stochastic programming or robust optimisation rather than a simple LP.

14. Uncertainty changes “optimal” into “robust enough”

An optimiser built on single-point forecasts can produce a brittle portfolio. Suppose it assumes deposit runoff of 5%, credit losses of 1%, stable wholesale funding and normal market liquidity. If all four assumptions worsen together, the “optimal” solution may violate LCR, capital or risk appetite very quickly.

Three common alternatives are:

• scenario optimisation — require acceptable performance across several specified states;
• stochastic optimisation — optimise expected objective across a probability distribution;
• robust optimisation — optimise against a bounded set of adverse parameter values without requiring precise probabilities.

The best formulation depends on whether probabilities are credible and which failure modes matter most.

15. Stress-test constraints can be embedded directly

A bank may require that a candidate balance sheet remains above capital and liquidity thresholds under several stress scenarios, not merely in the base case.

For scenario s:

CET1 ratio(x, s) ≥ internal stress floor

LCR(x, s) ≥ stress tolerance

Now the optimiser can reject a portfolio that maximises normal-year profit but becomes infeasible under a severe recession or deposit outflow.

See How Banks Stress-Test Capital Under Macroeconomic Scenarios.

16. Management actions must be operationally feasible

An optimiser can generate impossible solutions if it assumes the bank can instantaneously:

• raise unlimited deposits at the same rate;
• issue term debt without market impact;
• sell illiquid loans at book value;
• originate a new loan portfolio overnight;
• hedge any amount without collateral consequences;
• move capital or liquidity freely between legal entities.

Operational constraints therefore belong in the mathematics. Add issuance capacity, origination speed, asset-sale discounts, legal-entity transfer limits and hedge-market depth where material.

17. Multiple objectives require explicit trade-offs

A bank can care about more than short-run profit:

• risk-adjusted return;
• capital resilience;
• liquidity resilience;
• customer lending capacity;
• earnings stability;
• strategic franchise value;
• concentration reduction.

A multiobjective optimiser can combine these with weights or construct a Pareto frontier: solutions where improving one objective necessarily worsens another.

The frontier is often more honest than one “optimal” answer because it exposes the trade-off management must choose rather than burying the choice inside an arbitrary weight.

18. Evidence polarity: constraints can become less or more valuable

A good optimiser should allow the economic value of resources to change with evidence. Examples:

• Deposit runoff rises → HQLA and stable funding shadow prices can increase.
• Credit spreads widen → term funding becomes more expensive.
• Capital rises through retained earnings → RWA scarcity can fall.
• Loan demand weakens → origination constraints become less binding.
• Interest-rate exposure grows → hedge capacity becomes more valuable.

If the optimiser’s shadow prices remain unchanged while the bank’s economic state changes sharply, either the constraints are not truly binding or the model is stale.

19. Counterexamples that break single-ratio optimisation

• “Maximise RAROC.” Counterexample: the portfolio has excellent RAROC but insufficient HQLA.
• “Maximise LCR.” Counterexample: the bank holds excessive low-return liquidity and destroys sustainable earnings.
• “Minimise RWA.” Counterexample: low-RWA assets still consume leverage exposure and funding capacity.
• “Use the cheapest funding.” Counterexample: short wholesale funding improves current spread but breaches stable-funding resilience.
• “Optimise the base case.” Counterexample: the solution becomes infeasible under a modest deposit or credit stress.

20. The balance-sheet optimisation pipeline

1. Define decision variables at an appropriate product/tenor resolution.
2. Reconcile starting assets, liabilities and capital.
3. Estimate risk-adjusted contribution by activity.
4. Map RWA, leverage exposure, HQLA, ASF/RSF and funding usage.
5. Add concentration, interest-rate, legal-entity and operational limits.
6. Choose the objective function and risk appetite.
7. Formulate the linear/nonlinear optimisation problem.
8. Solve the base-case allocation.
9. Inspect which constraints are binding.
10. Read shadow prices as scarcity signals.
11. Run scenario, stochastic or robust alternatives.
12. Stress capital, liquidity and funding simultaneously.
13. Check management-action feasibility and execution costs.
14. Feed updated scarcity prices into FTP, product pricing and planning.

21. Failure modes and weak links

• Objective-function myopia. One-period profit excludes risk or resilience.
• Constraint omission. A legal, leverage, liquidity or concentration limit is missing, so the mathematical optimum is operationally impossible.
• Double charging. Capital/liquidity costs are embedded in both contribution margin and constraints without consistent definitions.
• Stale coefficients. RWA, runoff, funding or profitability inputs lag current conditions.
• Perfect-market assumption. The model assumes unlimited deposits, debt issuance or asset-sale liquidity at one price.
• Base-case overfit. Tiny input changes produce large portfolio reallocations.
• Black-box shadow price. Internal businesses receive a resource charge nobody can trace to the binding constraint.
• Legal-entity blindness. Group-level liquidity appears available but cannot move to the entity that needs it.

22. Diagnostics and falsifiers

• Which three constraints have the highest shadow prices?
• How much profit changes if CET1 increases by 1%?
• Does the optimiser choose a radically different portfolio after a modest runoff stress?
• Which product looks profitable before but unattractive after capital and liquidity scarcity are charged?
• Does the solution remain feasible under RWA inflation or funding-spread widening?
• Which management action is mathematically selected but operationally too slow?
• How much objective value is lost by adding concentration protection?
• What observation would prove the current “optimal” portfolio is not robust?

Suppose someone claims, “The best balance sheet is the one with the highest net interest margin.” A falsifier is a higher-margin portfolio that breaches capital, LCR or NSFR constraints, or suffers unacceptable losses under stress. A single accounting spread cannot define a feasible bank.

23. Verification and update triggers

• reconcile optimiser inputs to regulatory and financial reporting;
• independently reproduce key ratios from the proposed solution;
• compare optimiser recommendations with simple heuristic portfolios;
• stress all binding constraints jointly;
• test sensitivity to contribution-margin and runoff assumptions;
• revalidate after major regulatory or funding-market changes;
• compare predicted shadow prices with observed marginal business decisions;
• keep an execution ledger showing whether planned balance-sheet actions actually occurred at assumed prices and speeds.

Connections across the finance-and-banking algorithms lane

• Bank capital models — supply capital and RWA constraints.
• Liquidity stress testing — supplies runoff and survival constraints.
• Funds transfer pricing — distributes funding/liquidity scarcity into product economics.
• Collateral optimisation — solves a narrower resource-allocation problem inside the broader balance sheet.

Research anchors

• Bank for International Settlements — current consolidated Basel Framework.
• BIS — Liquidity Coverage Ratio executive summary.
• BIS — Net Stable Funding Ratio executive summary.
• BIS — Basel III leverage ratio framework executive summary.
• OCC — Balance Sheet Management.

The deeper lesson

Bank balance-sheet optimisation is the mathematics of scarcity made explicit. Capital is scarce. HQLA is scarce. Stable funding is scarce. Leverage capacity is scarce. Customer demand, concentration limits and hedge capacity impose still more boundaries. The optimiser’s job is not to make every ratio beautiful. It is to find a feasible allocation whose return survives the constraints—and to reveal which constraint is actually costing the bank the next dollar of opportunity. That is why shadow prices are as educationally important as the final portfolio.

Educational note: This article explains public banking and operations-research concepts. It is not investment advice, bank-management advice, capital-planning advice or institution-specific regulatory guidance.