## 1. The Timeline ## Source details **Canonical URL:** [1. The Timeline](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12238.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12238.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12238.pdf.json) --- ### Overview and purpose - Studies interaction between relationship banking ("banking") and short-term activities ("trading"): - "banking" = business with repeated, long-term clients (relationship banking). - "trading" = short-term, transactions-based activities that do not rely on repeated interactions (including proprietary trading and originating-and-selling standardized loans). - Motivation: growth of trading within commercial and investment banks over the last decades raises questions about rationale and risks (examples cited: UBS, Barings Bank, Bear Stearns, Lehman Brothers, Merrill Lynch, Washington Mutual, Wachovia, Bank of America-Merrill Lynch, JP Morgan). - Two principal frictions from combining banking and trading: - Time inconsistency in allocation of capital between long-term banking and short-term trading. - Risk-shifting incentives created by trading that allow shareholders to boost risk for private gains. - Both frictions become more severe with deeper financial markets (larger trading positions) and when bank returns are lower. ### Key modeling distinctions and assumptions - Banking: - Based on a fixed endowment of information about existing customers (mass R̄). - Not scalable; valuable franchise (implicit capital R0); low risk (certainty of returns in the model); long-term returns distributed over time and dependent on customers' ex-ante investments. - Has spare borrowing capacity (leverage constraint not binding at the standalone banking scale). - Trading: - Not based on an informational endowment; scalable with decreasing returns to scale. - Less profitable than banking and credit-constrained. - Short-term returns at a single date; can include risky strategies that offer higher short-run returns but catastrophic losses with small probability. - Credit constraints follow an incentive-compatibility (IC) condition: - Π ≥ βA (equation (1)) where Π is shareholder return under normal operations and βA is shareholder return from moral hazard. - Time and dates: - Economy: risk-neutral, no discounting; three dates: 0, 1, 2. - Date 0: private information about a mass R̄ of customers; implicit equity R0. - Date 1: each customer has liquidity need size 1; bank chooses borrowing allocation between relationship banking R and trading T. - Date 2: banking returns and trading returns realized. ### Benchmark model specifics and key equations - Banking IC for borrowing to cover customers’ liquidity needs: - R0 + ρR ≥ βR (equation (2)). - Spare borrowing capacity (including at R = R̄): - R0 + ρ R̄ > β R̄ (equation (3)). - Trading returns: - Trading of T units at date 1 produces at date 2 net returns τ T for T ≤ Σ and 0 for T > Σ. - Parameters satisfy τ < ρ (equation (4)) and τ < β (equation (5)). - Σ captures scalability of trading and is increasing in market depth. - Date-1 profit maximization under conglomeration: - ΠC = R0 + ρ R + τ T (equation (6)) - Subject to: - R0 + ρ R + τ T ≥ β (R + T) (equation (7)) - with T ≤ Σ. - Because ρ > τ, bank serves all relationship customers first (R = R̄) then allocates remaining borrowing capacity to trading. - Maximum trading supported: - T_max = (R0 + R̄(ρ − β)) / (β − τ) (equation (8)) - Optimal allocation: T = min{Σ, T_max}. - Proposition 1: conglomeration enables expanding scale of trading otherwise credit-constrained; equilibrium serves all relationship customers (R = R̄) and allocates remaining borrowing capacity to trading, T = min{Σ, T_max}. *Italicized source: _wp12238 - 1. The Timeline — https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12238.pdf* --- ### 5.1 Setup: Long-term Banking ### Nature of the distortion and time inconsistency - Banking is long-term: returns distributed over time and depend on ex ante investments in relationships (credit line fees paid at date 0 reduce payments at date 1). - Time inconsistency arises because part of banking returns (ρ − ) can be captured ex ante as a credit line fee, while only  ≤ ρ can be captured ex post through interest on lending between dates 1 and 2. - Bank cannot commit to future lending under the credit line and may refuse to lend ex post if borrowing capacity is exhausted (material adverse change–type discretion). - Anticipating ex post diversion to trading, customers reduce ex ante credit line fees, lowering bank profit and borrowing capacity. ### Bank profit maximization and constraints (date 1) - Bank chooses ÎR and ÎT to maximize: - ΠC = ÎR0 + (ρ−)ÎRex-ante + ÎR + τÎT (Equation (9)) - where ÎRex-ante is the borrowing customers expect under the credit line; equilibrium implies ÎRex-ante = ÎR. - IC constraint (binding): - ÎR0 + (ρ−)ÎRex-ante + ÎR + τÎT ≥ β(ÎR + ÎT) (Equation (10)) - Time inconsistency condition: when  < τ, bank has incentive to allocate borrowing capacity first to trading up to scale Σ, then to banking. ### Role of trading scalability - If Σ ≤ T_max, bank can allocate Σ to trading and still cover all customer liquidity needs (ÎR = R̄). - If Σ > T_max, allocation {R̄; Σ | Σ > T_max} violates leverage constraint (7) and bank becomes credit constrained ex post: ÎR < R̄. - Anticipated ex post shortfall leads customers to reduce (ρ−)ÎRex-ante < (ρ−)R̄, undermining relationship banking. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 5.2 The Consequences of Time Inconsistency ### Comparative statics and cases - Time inconsistency occurs when  < τ and Σ > T_max. - Severity depends on relationship between banking profitability ρ and private benefits β. - Case 1: ρ < β - Cutting back on banking frees up more capital than the lost profits (leverage constraint becomes slack); some banking may be preserved. - If Tmax < Σ ≤ R0/(β−τ) (Equation (11)): - IC as equality gives: - R = R0 − Σ(β−τ)/(β−ρ) (Equation (12)) - T = Σ - R increases in ρ. - If Σ > R0/(β−τ): - R = 0 (Equation (14)) - T = R0/(β−τ) - Case 2: ρ ≥ β - T_max > R0/(β−τ). Time inconsistency (Σ > T_max) implies trading wipes out all banking in that range: - R = 0 (Equation (15)) - T = R0/(β−τ) - Outcome is a bang-bang solution. ### Industry implications - Modest trading scalability: combining banking and trading can elevate bank profitability. - Highly scalable trading and lower ρ (e.g., due to IT and deregulation): time inconsistency can fully destroy relationship banking. ### Formal statement - Proposition 2 (Time inconsistency): - When part of banking return is collected ex ante, banking profitability is low (low R0 and ρ leading to low  and T_max) and/or trading is sufficiently scalable (high Σ), then if τ >  and Σ > T_max the bank will allocate ex post more borrowing capacity to trading than ex ante optimal: R < R̄. Anticipating this, ex ante investments in banking suffer and customers pay lower credit line fees. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 5.3 Costs of Conglomeration under Time Inconsistency ### Standalone vs conglomerate profits - Standalone cumulative profit: - ΠS = R0 + ρ R̄ + 0 = R0 + ρ R̄ (Equation (16)) - Conglomerate profit depends on Σ: - For Σ ≤ T_max: - ΠC = R0 + ρ R̄ + τ Σ (Equation (17)) - For Σ > T_max: - ΠC decreases in Σ. ### Case ρ < β (smooth substitution) - For T_max < Σ ≤ R0/(β−τ): - ΠC = R0 + [ρ R0 − Σ β(ρ−τ)]/(β−ρ) (Equation (18)) - For Σ > R0/(β−τ): - ΠC = R0 + τ R0/(β−τ) (Equation (19)) ### Case ρ ≥ β (discrete collapse) - Banking collapses at Σ = T_max with a discrete drop to level in (19). ### Profit comparison threshold - When full substitution occurs (R = 0; T = R0/(β−τ)), conglomerate profit ΠC (from (19)) is less than standalone ΠS (from (16)) if: - ρ R̄ > τ R0/(β−τ) (Equation (20)) ### Summary - Proposition 3 (Profits with time inconsistency): - Effect of conglomeration on bank profits is inverse U-shaped in Σ. - For Σ < T_max, profits increase in Σ; for Σ > T_max, profits fall as time inconsistency intensifies. - There exist parameter values where stand-alone banks generate higher profits than conglomerated banks. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 6.1 Setup: Risky Trading ### Motivation and modeling - Trading can generate highly skewed (probabilistic) returns enabling risk-shifting: upside to shareholders, downside imposed on creditors. - Risky trading: for T units, gross return is (1 + τ + α) T with probability π, and 0 with probability 1−π, up to maximum scale Σ. - Assumptions: - 0 < π(1 + τ + α) − 1 < τ (Equation (21)) - τ < π(τ + α) < β (Equation (22)) - Creditor interest rate i set ex ante to achieve zero expected return; bank strategy not verifiable. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 6.2 Risk-Shifting ### Conditions and payoffs - If bank invests R in banking and T in risky trading: - Relationship payoff at date 2: R0 + (1 + ρ) R - Trading payoff: (1 + τ + α) T with probability π; 0 otherwise - Debt is safe if: - T ≤ R0 + ρ R (Equation (23)) - Shareholders’ payoff under risky trading when (23) holds: - ΠRisky | T ≤ R0 + ρ R = R0 + ρ R + π(1 + τ + α) T − T (Equation (24)) - This is lower than safe trading payoff ΠC; shareholders will not choose risky trading if they fully internalize losses. - If T > R0 + ρ R, shareholders’ return under risky trading: - ΠRisky = π(R0 + ρ R + (τ + α) T) (Equation (25)) ### Threshold for choosing risky trading - Bank chooses risky trading when ΠRisky > ΠC leading to: - T > T_Risky = (R0 + ρ R)(1 − π) / [π(τ + α) − τ] (Equation (26)) ### Endogenizing T_Risky and comparative statics - Using IC (10) with R from: - R = R0 − T(β−τ)/(β−ρ) (Equation (27)) - Substitution yields: - T_Risky = R0(1 − π) ρ(β(1 − π) − π α) / [β + (π α − τ(1 − π))] (Equation (28)) - Lemma 1: - ∂T_Risky/∂α < 0 and ∂T_Risky/∂π < 0: higher α or higher π lowers scale at which risky trading is chosen. - ∂T_Risky/∂R0 > 0 and ∂T_Risky/∂ρ > 0: higher franchise raises scale required to trigger risky trading. ### Allocation, interest rates, and equilibrium under risky trading (TRisky > T_max) - For Σ ≤ T_Risky: bank uses safe trading; borrowing allocation given by (12). - For Σ > T_Risky: bank chooses risky trading. - Under risky trading, time inconsistency persists (since  < τ < π(τ + α)), so bank first allocates maximum borrowing capacity to trading before banking. - Creditors’ zero-profit condition: - (R + T) = π(R + T)(1 + i) + (1 − π)(R0 + (1 + ρ) R) (Equation (29)) - IC constraint under risky trading (equality): - π(R0 + (ρ − i) R + (τ + α − i) T) ≥ β(T + R) (Equation (30)) ### Borrowing allocation outcomes under risky trading - For T_Risky < Σ ≤ R0 / [β − (π(1 + τ + α) − 1)]: - R = R0 − Σ((1−π) + β − π(τ + α))/(β − ρ) (Equation (31)) - T = Σ - For Σ > R0 / [β − (π(1 + τ + α) − 1)]: - R = 0 (Equation (32)) - T = R0 / [β − (π(1 + τ + α) − 1)] ### Profits under risky trading - For T_Risky < Σ ≤ R0 / [β − (π(1 + τ + α) − 1)]: - ΠRisky = π[ R0 + ρ R0 − Σ β(ρ − (π(1 + τ + α) − 1)) /(β − ρ) ] (Equation (33)) - For Σ > R0 / [β − (π(1 + τ + α) − 1)]: - ΠRisky = π[ R0 + (π(1 + τ + α) − 1) R0 / (β − (π(1 + τ + α) − 1)) ] (Equation (34)) ### Net effects and formal statement - Two negative channels from shift to risky trading: - Direct profit loss: risky trading has lower NPV than safe trading. - Higher expected borrowing costs: creditors anticipate risk-shifting and raise effective cost of funds, reducing profits and borrowing capacity. - Therefore ΠRisky < ΠC (compare (33)-(34) to (18)-(19)). - Proposition 4 (Risk-shifting): - Bank may engage in risky trading as risk-shifting when banking profitability (ρ and R0) is low and/or trading is sufficiently scalable (Σ > T_Risky). This behavior is detrimental to bank profits because the bank internalizes the costs of risk-shifting via higher borrowing costs. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 6.3 The Interaction of Time Inconsistency and Risk Shifting ### Amplifying mechanisms (three effects) - Effect 1: Risk-shifting raises ex-post return on risky trading, making time inconsistency more likely. - τ < π(τ + α). - Region where time inconsistency arises only under risk-shifting: τ < ρ < π(τ + α). - For Σ in (T_max, T_Risky] bank chooses safe trading and no time inconsistency; for Σ > T_Risky bank chooses risky trading and time inconsistency is triggered. - Effect 2: Time inconsistency can raise equilibrium trading volume and thus make risk-shifting more likely. - Case 1: ρ > π(τ + α) → time inconsistency absent; maximum trading volume is T_max; since T_Risky ≥ T_max, bank will not engage in risk shifting. - Case 2: ρ < τ → time inconsistency binds for T > T_max; time inconsistency can increase equilibrium scale beyond T_max and may trigger risk-shifting when T_max < T_Risky < R0/(β − (π(τ + α) − 1)). - Effect 3: Time inconsistency reduces the relationship bank’s franchise value, lowering the cost of risk-shifting and making risk-shifting more likely. - Model enrichment with probability θ that at date 1 τ = τ_Low < ρ (no time inconsistency) and probability 1−θ that τ = τ_High > ρ (time inconsistency present). - Payoff to shareholders from safe trading when τ = τ_Low: - Π^θ_C (expression preserved as in source) (equation (35) in source). - Payoff from risky trading: - Π^θ_R (expression preserved as in source) (equation (36) in source). - Threshold: - T^θ_Risky = ( [ R_0 + (ρ − ρ) θ ( ̄R − R ) + ρ R ] / (1 − π) ) / ( π(τ+α) − τ ) (equation (37) in source). - dT^θ_Risky / dθ > 0: higher θ elevates franchise value and makes risk-shifting less likely. ### Formal proposition - Proposition 5 (Interaction of time inconsistency and risk-shifting): - Time inconsistency and risk-shifting amplify each other. - Time inconsistency increases scale of trading and reduces franchise value, increasing incentives for risk-shifting. - Risk-shifting increases shareholders’ ex-post return from trading, which may trigger time inconsistency in capital allocation. ### Summary of distortions and recent developments - Two main distortions when banks trade: - Time inconsistency of capital allocation: bank may trade at a scale too high at expense of relationship banking franchise. - Use of trading for risk-shifting. - Both distortions more likely when: - Profitability of banking is low (reducing R0 and ρ). - Scalability of trading is high (increasing Σ). - Financial development (information technology, easier borrower information) may have increased Σ and reduced relationship profitability, reinforcing these distortions. - Reinforcing effects: higher scalability of trading and lower profitability of banking work in same direction (Propositions 2 and 4); time inconsistency and risk-shifting arise simultaneously and amplify each other (Proposition 5). ### Modeling features and implications for relationship banking - Relationship banking modeled via credit line contracts with ex ante fees capturing “front-loaded income.” - Banks obtain profit early but must honor future commitments; discretion (e.g., MAC clauses) can enable reneging. - Shifting capital ex post to trading undermines ability to expand relationship lending, reducing value of insurance relationship. - Front-loaded vs back-loaded income: - Back-loaded income makes relationships more attractive ex post; high future rents reduce time inconsistency and risk-shifting incentives. - Ex post rents have declined due to higher competition and better borrower information, increasing importance of front-loaded relationships. ### External equity, internal capital allocation, and constraints - Two key frictions: - Time inconsistency in internal capital allocation. - Difficulty accessing external equity. - Costly outside equity explains why trading may be funded via relationship bank’s implicit capital (franchise value). - Returning unused capital to shareholders (dividends/buybacks) cannot solve time inconsistency: relationship banks maintain unused capital to meet future funding needs; misallocation to trading destroys franchise irrespective of distribution. ### Policy implications and recommendations - Stylized facts / predictions: - Relationship banks tempted to use implicit capital for scalable trading; limited trading can be beneficial, excess trading can reduce profits and destroy relationship franchise. - Financial development increasing trading scalability and reducing relationship profitability increases incentives to over-allocate capital to trading and to use trading for risk-shifting. - Contracts involving discretion (credit lines with MAC clauses, syndicated lending reciprocity, commitments based on local knowledge) are most vulnerable. - Combining banking and trading (universal bank model) may be less sustainable: banks may allocate too many resources to transactional activities, lowering profit and raising risk. - Views on regulatory approaches (Volcker vs Vickers): - Scope: not all market-based activities are transactional; segregating underwriting (relationship-based) from banks may be unclear. Volcker focusing only on proprietary trading may be too narrow. - Segregation (Vickers) can reduce risk spillovers and improve transparency/pricing, but may not prevent reputation-based recourse or resolve time inconsistency: a bank could still allocate excessive capital to trading subsidiaries at the expense of banking subsidiaries. - Prohibition of proprietary trading (Volcker) and more broadly transaction-based activities inside banks may be necessary to prevent time inconsistency. - Middle-ground: allow market operations but limit scale: trading below T_max (equation (20)) and T_Risky (equation (28)) does not trigger time inconsistency or risk-shifting. - Capital regulation: - Relationship banks need spare capital above regulatory minimums to meet future funding commitments. - Procyclical and other capital surcharges (if accessible) may support supply of banking services over the business cycle better than fixed high capital requirements. ### Conclusion: key takeaways - Universal banks combine relationship banking and trading, using relationship borrowing capacity to expand trading, creating two inefficiencies: - Excess ex-post allocation to trading undermining ex-ante relationship incentives. - Use of trading for risk-shifting undermining stability. - Financial development increases trading scalability, initially favoring conglomeration but ultimately producing inefficiencies that argue for restricting certain bank activities. *Italicized source: _wp12238 - 6.3 The Interaction of Time Inconsistency and Risk Shifting — https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12238.pdf* ### 1. The Timeline ....................................................................................................... ### 1. The Timeline ### Overview and purpose - The chapter studies the interaction between relationship banking ("banking") and short-term activities ("trading"), where: - "banking" denotes business with repeated, long-term clients (relationship banking). - "trading" denotes short-term, transactions-based activities that do not rely on repeated interactions (including proprietary trading and originating-and-selling standardized loans). - Motivation: growth of trading within commercial and investment banks over the last decades raises questions about rationale and risks (examples cited: UBS, Barings Bank, Bear Stearns, Lehman Brothers, Merrill Lynch, Washington Mutual, Wachovia, Bank of America-Merrill Lynch, JP Morgan). - Two principal frictions from combining banking and trading: - Time inconsistency in allocation of capital between long-term banking and short-term trading. - Risk-shifting incentives created by trading that allow shareholders to boost risk for private gains. - Both frictions become more severe with deeper financial markets (larger trading positions) and when bank returns are lower. ### Key modeling distinctions and assumptions - Banking: - Based on a fixed endowment of information about existing customers. - Not scalable; valuable franchise (implicit capital); low risk (certainty of returns in the model); long-term returns distributed over time and dependent on customers' ex-ante investments. - Has spare borrowing capacity (leverage constraint not binding at the standalone banking scale). - Trading: - Not based on an informational endowment; scalable with decreasing returns to scale. - Less profitable than banking and credit-constrained. - Short-term returns at a single date; can include risky strategies that offer higher short-run returns but catastrophic losses with small probability. - Credit constraints follow an incentive-compatibility (IC) condition adapted from Holmstrom and Tirole: - Π ≥ βA (equation (1)) where Π is shareholder return under normal operations and βA is shareholder return from moral hazard (β > 0 conversion factor times assets A). - For the model, the IC constraint limits maximum leverage/size of bank. ### Timeline and setup (benchmark model) - Economy: risk-neutral, no discounting; three dates: 0, 1, 2. - At date 0: - Bank endowed with private information on a mass R̄ of customers. - Franchise value R0 (implicit equity) and possibility to obtain information rents ρ per customer from covering date-1 liquidity needs. - At date 1: - Each customer has a liquidity need of size 1 (certainty). - Bank chooses borrowing allocation between relationship banking R and trading T; trading investments T produce returns at date 2. - At date 2: - Banking returns (from coverage of liquidity needs and prior lending) and trading returns realized. - Banking IC constraint for borrowing to cover customers’ liquidity needs: - R0 + ρR ≥ βR (equation (2)). - Spare borrowing capacity condition (including at R = R̄): - R0 + ρ R̄ > β R̄ (equation (3)). - Trading setup: - Trading of T units at date 1 produces at date 2 net returns τ T for T ≤ Σ and 0 for T > Σ. - Parameters satisfy τ < ρ (equation (4)) and τ < β (equation (5)), implying standalone trading is credit constrained and unviable without conglomeration. - Σ captures scalability of trading and is increasing in market depth. ### Benefit of conglomeration (benchmark result) - Under conglomeration, date-1 profit maximization: - ΠC = R0 + ρ R + τ T (equation (6)) - Subject to joint leverage constraint: - R0 + ρ R + τ T ≥ β (R + T) (equation (7)) - with T ≤ Σ. - Because ρ > τ, the bank serves all relationship banking customers first (R = R̄) and then allocates remaining borrowing capacity to trading. - Maximum trading supported by the universal bank (assuming T_max ≤ Σ): - T_max = (R0 + R̄(ρ − β)) / (β − τ) (equation (8)) - Optimal allocation: T = min{Σ, T_max}. - Proposition 1 (Conglomeration without frictions): - Conglomeration enables expanding scale of trading otherwise credit-constrained. - In equilibrium the bank serves all relationship banking customers (R = R̄) and allocates remaining borrowing capacity to trading, as long as trading is profitable, T = min{Σ, T_max}. ### Sources of subsequent analysis (introduced in later sections) - Section 5 (Time inconsistency): Introduces time inconsistency by making banking returns dependent on ex ante decisions of customers, creating a conflict between protecting a long-term franchise and exploiting short-term trading opportunities. - Section 6 (Risk-shifting): Introduces risky trading strategies and analyzes how risk-shifting by shareholders can interact with time inconsistency to amplify over-exposure to trading. - The paper proceeds from a frictionless benchmark to introducing these two frictions and examining their interaction and policy implications. *Source: _wp12238 - 1. The Timeline — https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12238.pdf* ### 5.1 Setup: Long-term Banking ### 5.1 Setup: Long-term Banking ### Nature of the distortion - Banking is long-term: returns are distributed over time and depend on ex ante investments in relationships (credit line fees paid at date 0 reduce payments at date 1). - Time inconsistency arises because part of banking returns (ρ−) can be captured ex ante as a credit line fee, while only ≤ρ can be captured ex post through interest on lending between dates 1 and 2. - The bank cannot commit to future lending under the credit line and may refuse to lend ex post if borrowing capacity is exhausted (material adverse change–type discretion). - Anticipating possible ex post diversion to trading, customers reduce ex ante credit line fees, lowering bank profit and borrowing capacity. ### Bank profit maximization and constraints - At date 1 the bank chooses ÎR and ÎT to maximize: - ΠC = ÎR0 + (ρ−)ÎRex-ante + ÎR + τÎT (Equation (9)) - where ÎRex-ante is the borrowing customers expect under the credit line; in equilibrium ÎRex-ante = ÎR. - IC constraint (binding): - ÎR0 + (ρ−)ÎRex-ante + ÎR + τÎT ≥ β(ÎR + ÎT) (Equation (10)) - Time inconsistency condition: when  < τ, the bank has incentive to allocate borrowing capacity first to trading up to scale Σ, then to banking. ### Role of trading scalability - If trading scalability is low (Σ ≤ Tmax), the bank can allocate Σ to trading and still cover all customer liquidity needs (ÎR = R̄). - If trading scalability is high (Σ > Tmax), allocation {R̄; Σ | Σ > Tmax} violates the leverage constraint (7) and the bank becomes credit constrained ex post: ÎR < R̄. - Anticipated ex post shortfall leads customers to reduce (ρ−)ÎRex-ante < (ρ−)R̄, undermining relationship banking. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 5.2 The Consequences of Time Inconsistency ### Key comparative statics and cases - Time inconsistency occurs when  < τ and Σ > Tmax. - Severity depends on relationship between banking profitability ρ and private benefits β. - Case 1: ρ < β - Cutting back on banking frees up more capital than the lost profits (leverage constraint becomes slack); some banking may be preserved. - If trading scalability is intermediate: - Tmax < Σ ≤ R0/(β−τ) (Equation (11)) - Solve IC as equality to obtain: - R = R0 − Σ(β−τ)/(β−ρ) (Equation (12)) - T = Σ - Comparative statics: R increases in ρ (higher ρ reduces diversion to trading). - If Σ > R0/(β−τ): all resources allocated to trading and relationship banking is wiped out: - R = 0 (Equation (14)) - T = R0/(β−τ) - Case 2: ρ ≥ β - Tmax > R0/(β−τ). Time inconsistency (Σ > Tmax) implies trading wipes out all banking in that range: - R = 0 (Equation (15)) - T = R0/(β−τ) - Cutting down on banking now tightens leverage constraint and cannot create usable capital for trading; outcome is a bang-bang solution. ### Industry implications and dynamics - With modest trading scalability, combining banking and trading can elevate overall bank profitability. - With highly scalable trading and lower ρ (e.g., due to IT and deregulation), time inconsistency can fully destroy relationship banking. - Movement to the right on Figure 3 = higher Σ; shift from Case 2 to Case 1 = lower Tmax (see discussion using (8)). ### Formal statement - Proposition 2 (Time inconsistency): - When part of banking return is collected ex ante, banking profitability is low (low R0 and ρ leading to low  and Tmax) and/or trading is sufficiently scalable (high Σ), then if τ >  and Σ > Tmax the bank will allocate ex post more borrowing capacity to trading than ex ante optimal: R < R̄. Anticipating this, ex ante investments in banking suffer and customers pay lower credit line fees. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 5.3 Costs of Conglomeration under Time Inconsistency ### Standalone and conglomerate profits - Standalone cumulative profit: - ΠS = R0 + ρ R̄ + 0 = R0 + ρ R̄ (Equation (16)) - Conglomerate profit depends on Σ: - For Σ ≤ Tmax (no time inconsistency), profit rises with Σ: - ΠC = R0 + ρ R̄ + τ Σ (Equation (17)) - For Σ > Tmax (time inconsistency present), ΠC decreases in Σ. ### Case ρ < β (smooth substitution) - For Tmax < Σ ≤ R0/(β−τ): - ΠC = R0 + [ρ R0 − Σ β(ρ−τ)]/(β−ρ) (Equation (18)) - For Σ > R0/(β−τ): - ΠC = R0 + τ R0/(β−τ) (Equation (19)) ### Case ρ ≥ β (discrete collapse) - Banking collapses at Σ = Tmax with a discrete drop to level given by (19). ### Profit comparison and threshold condition - When bank fully substitutes trading for relationship banking (R = 0; T = R0/(β−τ)), conglomerate profit ΠC (from (19)) is less than standalone profit ΠS (from (16)) if: - ρ R̄ > τ R0/(β−τ) (Equation (20)) ### Summary - Proposition 3 (Profits with time inconsistency): - Effect of conglomeration on bank profits is inverse U-shaped in trading opportunities Σ. - For Σ < Tmax, profits increase in Σ; for Σ > Tmax, profits fall as time inconsistency intensifies. - There exist parameter values where beyond a certain Σ, stand-alone banks generate higher profits than conglomerated banks. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 6.1 Setup: Risky Trading ### Motivation - Trading can generate highly skewed (probabilistic) returns enabling risk-shifting: upside to shareholders, downside imposed on creditors. - Relationship banking returns are relatively predictable and law-of-large-numbers–type; trading allows concentrated bets and skewness. ### Modeling risky trading - Risky trading: for T units, gross return is (1 + τ + α) T with probability π, and 0 with probability 1−π, up to maximum scale Σ. - Assume risky trading has positive but lower NPV than safe trading: - 0 < π(1 + τ + α) − 1 < τ (Equation (21)) - Risky trading’s expected return sits between τ and β: - τ < π(τ + α) < β (Equation (22)) - Creditor interest rate i is set ex ante to achieve zero expected return; bank choice of strategy is not verifiable. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* --- ### 6.2 Risk-Shifting ### Conditions for risky trading (creditors charge i = 0) - When bank invests R in banking and T in risky trading: - Relationship payoff at date 2: R0 + (1 + ρ) R - Trading payoff: (1 + τ + α) T with probability π; 0 otherwise - Debt is safe if: - T ≤ R0 + ρ R (Equation (23)) - If (23) holds, shareholders’ payoff under risky trading: - ΠRisky | T ≤ R0 + ρ R = R0 + ρ R + π(1 + τ + α) T − T (Equation (24)) - This is lower than safe trading payoff ΠC; shareholders will not choose risky trading if they fully internalize losses. - If T > R0 + ρ R, shareholders’ return under risky trading: - ΠRisky = π(R0 + ρ R + (τ + α) T) (Equation (25)) ### Threshold T beyond which shareholders prefer risky trading - Bank chooses risky trading when ΠRisky > ΠC leading to threshold: - T > TRisky = (R0 + ρ R)(1 − π) / [π(τ + α) − τ] (Equation (26)) - Intuition: risky trading has a fixed cost equal to potential loss of franchise value (R0 + ρ R) with probability (1 − π), benefit scales with T via α. ### Endogenizing TRisky and comparative statics - Using IC (10) with R from (27): - R = R0 − T(β−τ)/(β−ρ) (Equation (27)) - Substitution yields: - TRisky = R0(1 − π) ρ(β(1 − π) − π α) / [β + (π α − τ(1 − π))] (Equation (28)) - Lemma 1: - (i) ∂TRisky/∂α < 0 and ∂TRisky/∂π < 0: higher upside α or higher success probability π lowers the scale at which risky trading is chosen. - (ii) ∂TRisky/∂R0 > 0 and ∂TRisky/∂ρ > 0: higher relationship franchise value raises the scale required to trigger risky trading. ### Allocation, interest rates, and equilibrium under risky trading (focus on TRisky > Tmax) - For Σ ≤ TRisky: bank uses safe trading; borrowing allocation given by (12). - For Σ > TRisky: bank chooses risky trading. - Under risky trading, time inconsistency persists (since  < τ < π(τ + α)), so bank first allocates maximum borrowing capacity to trading before banking. - Creditors’ zero-profit condition: - (R + T) = π(R + T)(1 + i) + (1 − π)(R0 + (1 + ρ) R) (Equation (29)) - IC constraint under risky trading (equality): - π(R0 + (ρ − i) R + (τ + α − i) T) ≥ β(T + R) (Equation (30)) ### Borrowing allocation outcomes - For TRisky < Σ ≤ R0 / [β − (π(1 + τ + α) − 1)]: - R = R0 − Σ((1−π) + β − π(τ + α))/(β − ρ) (Equation (31)) - T = Σ - For Σ > R0 / [β − (π(1 + τ + α) − 1)]: - R = 0 (Equation (32)) - T = R0 / [β − (π(1 + τ + α) − 1)] ### Profits under risky trading - For TRisky < Σ ≤ R0 / [β − (π(1 + τ + α) − 1)]: - ΠRisky = π[ R0 + ρ R0 − Σ β(ρ − (π(1 + τ + α) − 1)) /(β − ρ) ] (Equation (33)) - For Σ > R0 / [β − (π(1 + τ + α) − 1)]: - ΠRisky = π[ R0 + (π(1 + τ + α) − 1) R0 / (β − (π(1 + τ + α) − 1)) ] (Equation (34)) ### Net effects of risk-shifting - Shift to risky trading has two negative channels: - Direct profit loss: risky trading has lower NPV than safe trading (shareholders do not internalize fully ex post). - Higher expected borrowing costs: creditors anticipate risk-shifting and raise effective cost of funds ex ante, reducing bank profits and borrowing capacity. - Therefore ΠRisky < ΠC (compare (33)-(34) to (18)-(19)). ### Formal statement - Proposition 4 (Risk-shifting): - The bank may engage in risky trading as risk-shifting when banking profitability (ρ and R0) is low and/or trading is sufficiently scalable (Σ > TRisky). This behavior is detrimental to bank profits because the bank internalizes the costs of risk-shifting via higher borrowing costs. *Italicized source: _wp12238 - 5.1 Setup: Long-term Banking (IMF working paper section)* ### 6.3 The Interaction of Time Inconsistency and Risk Shifting ### _wp12238 - 6.3 The Interaction of Time Inconsistency and Risk Shifting ### Mechanisms of interaction - Three amplifying effects by which time inconsistency and risk shifting make each other more likely: - Effect 1: Risk-shifting raises ex-post return on risky trading, making time inconsistency more likely. - Recall: τ < π(τ+α). - Region where time inconsistency arises only under risk-shifting: τ < ρ < π(τ+α). - For parameter values in this region: no time inconsistency without risky trading (τ < ρ), but time inconsistency appears when the bank chooses risky trading (ρ < π(τ+α)). For Σ in (T_max, T_Risky] the bank chooses safe trading and no time inconsistency; for Σ > T_Risky the bank chooses risky trading and time inconsistency is triggered. (See Figure 6 in source.) - Effect 2: Time inconsistency can raise equilibrium trading volume and thus make risk-shifting more likely. - Case 1: ρ > π(τ+α) → time inconsistency absent. Maximum trading volume is T_max (equation (8)). Since T_Risky ≥ T_max, the bank will not engage in risk shifting because equilibrium scale is too low. - Case 2: ρ < τ → time inconsistency binds for T > T_max. Time inconsistency can increase equilibrium scale beyond T_max. When T_max < T_Risky < R_0 = (β − (π(τ+α) − 1)), the bank will start risk-shifting due to the larger scale driven by time inconsistency. (See Figure 7 in source.) - Effect 3: Time inconsistency reduces the relationship bank’s franchise value, lowering the cost of risk-shifting and making risk-shifting more likely. - Model enrichment: at date 0 there is probability θ that at date 1 τ = τ_Low < ρ (no time inconsistency), and probability 1−θ that τ = τ_High > ρ (time inconsistency present). - Payoff to bank shareholders from safe trading when τ = τ_Low (date 1): - Π^θ_C = R_0 + (ρ − ρ) [ θ ̄R + (1 − θ) R_ex-ante ] + ρ R + τ T (equation (35) in source; expression noted as Π^θ_C) - (Π^θ_C is similar to Π_C in (9) except second term: higher credit line fees customers are willing to pay at date 0.) - Payoff from risky trading: - Π^θ_R = π [ R_0 + (ρ − ρ) [ θ ̄R + (1 − θ) R_ex-ante ] + ρ R + (τ+α) T ] (equation (36) in source; expression noted as Π^θ_R) - Setting R = R_ex-ante and equating Π^θ_C and Π^θ_R yields threshold for switch to risk-shifting: - T^θ_Risky = ( [ R_0 + (ρ − ρ) θ ( ̄R − R ) + ρ R ] / (1 − π) ) / ( π(τ+α) − τ ) (equation (37) in source; exact algebra preserved as in source) - Observation: higher θ elevates bank franchise value at date 2 by increasing credit line fees by (ρ − ρ) θ ( ̄R − R_ex-ante ) and increasing borrowing capacity; hence: - dT^θ_Risky / dθ > 0: risky shifting becomes less likely under less intensive time inconsistency because the bank must reach a higher scale of trading to offset a higher franchise-cost of risky trading. ### Formal proposition - Proposition 5 (Interaction of time inconsistency and risk-shifting) - The problems of time inconsistency and risk shifting amplify each other. - Time inconsistency increases the scale of trading and reduces the franchise value of the relationship bank, increasing incentives for risk-shifting. - Risk-shifting increases bank shareholders’ ex-post return from trading, which may trigger time inconsistency in capital allocation. ### Summary of distortions and recent developments - Two main distortions in banks that trade: - Time inconsistency of capital allocation: bank may trade at a scale too high at expense of relationship banking franchise. - Use of trading for risk-shifting. - Both distortions are more likely when: - Profitability of banking is low (reducing R_0 and ρ). - Scalability of trading is high (increasing Σ). - Financial development (information technology, easier access to borrower information) may have increased Σ and reduced relationship profitability, reinforcing these distortions. - Reinforcing effects highlighted: - Higher scalability of trading and lower profitability of banking work in the same direction (Propositions 2 and 4). - Time inconsistency and risk-shifting arise simultaneously and amplify each other (Proposition 5). - Implication: trading and transactional activities may undermine relationship banking viability and raise bank risk, consistent with recent crisis evidence. ### Modeling features and implications for relationship banking - Relationship banking modeled via credit line contracts with ex ante fees capturing “front-loaded income.” - Banks obtain profit early but must honor future commitments; discretion (e.g., MAC clauses) can enable reneging. - Shifting capital ex post to trading can undermine ability to expand relationship lending, reducing value of insurance relationship to borrowers. - Front-loaded vs back-loaded income: - Back-loaded income (literature) makes relationships more attractive ex post; high future rents reduce time inconsistency and risk-shifting incentives. - Ex post rents have declined due to higher competition and better borrower information, increasing importance of front-loaded relationships. ### External equity, internal capital allocation, and constraints - Two key frictions: - Time inconsistency in internal capital allocation. - Difficulty accessing external equity. - Costly outside equity explains why trading may be funded via relationship bank’s implicit capital (franchise value). - Returning unused capital to shareholders (dividends/buybacks) cannot solve time inconsistency: - Relationship banks maintain unused capital to cover future funding needs; misallocation to trading destroys franchise whether or not that capital is distributed. - Time inconsistency resembles other literature on short-term trading undermining commitment (e.g., Diamond and Dybvig commentary, Jacklin 1987; Bhide 1993; Myers and Rajan 1998; Shleifer and Vishny 2010). ### Policy implications and recommendations - Stylized facts / predictions: - Relationship banks are tempted to use implicit capital for scalable trading; limited trading can be beneficial, excess trading can reduce profits and destroy relationship franchise. - Financial development increasing trading scalability and reducing relationship profitability increases incentives to over-allocate capital to trading and to use trading for risk-shifting. - Contracts involving discretion (credit lines with MAC clauses, syndicated lending reciprocity, commitments based on local knowledge) are most vulnerable; time inconsistency reduces their value and viability. - Combining banking and trading (universal bank model) may be less sustainable: banks may allocate too many resources to transactional activities, lowering profit and raising risk. - Views on regulatory approaches (Volcker vs Vickers): - Scope: not all market-based activities are transactional; segregating underwriting (relationship-based) from banks may be unclear. Volcker focusing only on proprietary trading may be too narrow. - Segregation (Vickers) can reduce risk spillovers and improve transparency/pricing, but may not prevent reputation-based recourse or resolve time inconsistency: a bank could still allocate excessive capital to trading subsidiaries at the expense of banking subsidiaries. - Prohibition of proprietary trading (Volcker) and more broadly transaction-based activities inside banks may be necessary to prevent time inconsistency. - Middle-ground implementation suggestion: - Allow market operations but limit scale: trading below T_max (equation (20)) and T_Risky (equation (28)) does not trigger time inconsistency or risk-shifting. - Capital regulation: - Relationship banks need spare capital above regulatory minimums to meet future funding commitments. - Procyclical and other capital surcharges (if accessible) may support supply of banking services over the business cycle better than fixed high capital requirements. ### Conclusion (key takeaways) - Universal banks combine relationship banking and trading, using relationship borrowing capacity to expand trading, creating two inefficiencies: - Excess ex-post allocation to trading undermining ex-ante relationship incentives. - Use of trading for risk-shifting undermining stability. - Financial development increases trading scalability, initially favoring conglomeration but ultimately producing inefficiencies that argue for restricting certain bank activities (as in recent policy proposals). *Source: _wp12238 - 6.3 The Interaction of Time Inconsistency and Risk Shifting (PDF chapter content provided).* ### References ### _wp12238 - References ### Cited Literature - [1] Acharya, V.V., T.F. Cooley, M.P. Richardson, and I. Walter, 2010, “Manufacturing Tail Risk: A Perspective on the Financial Crisis of 2007-09”, Foundations and Trends in Finance, 4. - [2] Acharya, V.V., M. Gabarro and P. Volpin , 2011, “Competition for Managers, Corporate Governance and Incentive Compensation”, mimeo London Business School. - [3] Bhide, A. (1993), “The Hidden Cost of Stock Market Liquidity”, Journal of Financial Economics, Vol. 34(1), p. 31-51. - [4] Boyd, J. and G. De Nicolo, 2005, “The Theory of Bank Risk-Taking and Competition Revisited” Journal of Finance, Vol. 60(3), p. 1329-1343. - [5] Boot, A.W.A., S.I. Greenbaum and A.V. Thakor, 1993, “Reputation and Discretion in Financial Contracting”, American Economic Review, Vol. 83(5), p. 1165-1183. - [6] Boot A.W.A. and A.V. Thakor, 2000, “Can Relationship Banking Survive Competition?”, Journal of Finance, Vol. 55(2), p. 679-713. - [7] Boyd J. and G. De Nicolò, 2005, “The Theory of Bank Risk-Taking and Competition Revisited” , Journal of Finance, Vol. 60(3), p. 1329-1343. - [8] Brunnermeier, M. and M. Oehmke, 2011, “Maturity Rat Race”, Journal of Finance, forthcoming. - [9] Brunnermeier, M., G. Dong and D. Palia, 2012, “Banks’ Non-Interest Income and Systemic Risk”, Working Paper. - [10] Calomiris, C. and C. Kahn, 2001, “The Role of Demandable Debt in Structuring Optimal Banking Arrangements,” American Economic Review, Vol. 81(3), p. 497-513. - [11] Danielson, G., 1984, Managing Corporate Wealth. - [12] Diamond, D.W., and P.H. Dybvig, 1983, “Bank Runs, Deposit Insurance and Liquidity,” Journal of Political Economy, Vol. 91(3), p. 401-419. - [13] Fang, Lily, Victoria Ivashina, and Josh Lerner, 2010, “"An Unfair Advantage"? Combining Banking with Private Equity Investing”, Working Paper. - [14] Farhi, E. and J. Tirole, 2011, “Collective Moral Hazard, Maturity Mismatch and Systemic Bailouts”, American Economic Review, forthcoming. - [15] Gorton G. and A. Metrick, 2011, “Securitized Banking and the Run on Repo”, Journal of Financial Economics, forthcoming. - [16] Hoenig T.M and C.S. Morris, 2011 “Restructuring the Banking System to Improve Safety and Soundness”, mimeo Kansas City Fed. - [17] Holmstrom, B. and J. Tirole, 1998, “Private and Public Supply of Liquidity”, Journal of Political Economy, Vol. 106 (1): 1-40. - [18] Holmstrom, B. and J. Tirole, 2011, Inside and Outside Liquidity, MIT Press, Paris, France. - [19] Huang, R. and L. Ratnovski, 2011, “The Dark Side of Bank Wholesale Funding” Journal of Financial Intermediation, Vol. 20(2), p. 248-63. - [20] Jacklin, C. (1987), Demand Deposits, Trading Restrictions, and Risk Sharing, In: Contractual Arrangements for Intertemporal Trade, Eds. E.D. Prescott and N. Wallace, University of Minneapolis Press, Minneapolis, p. 26-47. - [21] Jensen, M.C. and W.H. Meckling, 1978, “Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure”, Journal of Financial Economics, Vol 3(4), p. 3. - [22] Keeley, M.C., 1990, “Deposit Insurance, Risk, and Market Power in Banking”, American Economic Review, Vol. 80(5), p. 1183-1200. - [23] Krosner R. and R.J. Rajan, 1994, “Is the Glass-Steagall Act Justified?: Evidence from the U.S. experience with Universal Banking 1921-1933”, American Economic Review, Vol 84, p. 810-832. - [24] Kyle, A., 1985, “Continuous Auctions and Insider Trading,” Econometrica, Vol. 15, p. 1315-1335. - [25] Myers, S.C., and R.G. Rajan (1998), “The Paradox of Liquidity”, Quarterly Journal of Economics, Vol. 113(3), p. 733-771. - [26] Petersen, M. and R.J. Rajan, 1995 “The Effect of Credit Market Competition on Lending Relationships”, Quarterly Journal of Economics, Vol 110, p. 407-443. - [27] Puri, Manju, 1996. “Commercial banks in investment banking Conflict of interest or certification role?,” Journal of Financial Economics, Vol. 40(3), p. 373-401. - [28] Rajan R.J., H. Servaes and L. Zingales, 2000, “The Costs of Diversity: The diversification discount and inefficient investment”, Journal of Finance, Vol 55(1), p. 35-80. - [29] Schenone, C., 2004 “The Effect of Banking Relationships on the Firm’s IPO Underpricing”, Journal of Finance, Vol. 59(6), p. 2903—58. - [30] Shleifer, A. and R.W. Vishny, 2010, “Unstable Banking”, Journal of Financial Economics, Vol. 97, p. 306—18. - [31] Stein, J., 1997, “Internal Capital Markets and the Competition for Corporate Resources”, Journal of Finance, Vol. 52(1), p. 111-133. - [32] UBS (2008), “Shareholder Report on UBS’s Write-Downs,” 18 April 2008, http://www.ubs.com/1/ShowMedia/investors/releases?contentId=140331& name=080418ShareholderReport.pdf. - [33] Williamson, O, 1975, Markets and Hierarchies: Analysis and Antitrust Implications. ### Figures and Timelines Referenced - Figure 1. The Timeline - Date 0 - Bank lends to customers (repaid at date 2), obtains implicit equity R0 - Date 1 - Bank customers have liquidity needs - A conglomerated bank chooses the allocation of borrowing capacity between banking (covering liquidity needs) R (earns r) and trading T (earns t) - After borrowing, a bank has the ability to “abscond” with b(R+T) - Bank lends to customers and engages in trading - Date 2 - Returns realized, everyone repaid R - Figure 2. The Timeline with Time Inconsistency - Date 0 - Bank lends to customers (repaid at date 2), obtains implicit equity R0 - Bank collects credit line fees (r – ρ) R ex-ante for covering funding needs at date 1 - Date 1 - Bank customers have liquidity needs - A conglomerated bank chooses the allocation of borrowing capacity between banking (covering liquidity needs) R (earns ρ) and trading T (earns t) - After borrowing, a bank has the ability to “abscond” with b(R+T) - Bank lends to customers and engages in trading - Date 2 - Returns realized, everyone repaid R - Figure 3. Relationship Banking Allocation R as a Function of Trading Opportunities S - Case 1, r < b - Case 2, r ≥ b - R0/(b-t); Tmax(r); No banking: R = 0; Optimal banking: R = R0/(b-t). Reduced banking: R < R0/(b-t) - Figure 4. The Volumes of Banking (R) and Trading (T), and Profits (Π) Under Conglomerated Banking - Panels A–D depict R, T, and Π for rp(t+α) and ρ