## _wp11188

## Source details

**Canonical URL:** [_wp11188](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11188.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11188.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11188.pdf.json)

---

### I. Introduction — context and motivation
- Regulatory reform after the recent financial crisis focused on increasing capital cushions; Basel III rules have doubled the minimal capital ratio and directed banks to hold excess capital as conservation and countercyclical buffers (BIS, 2010).
- Two classical arguments for higher capital:
  - Ex post: capital absorbs losses and reduces insolvency risk and systemic propagation.
  - Ex ante: capital increases shareholders’ “skin in the game”, reducing limited-liability-driven incentives to take excessive risk (Jensen and Meckling 1976, Holmstrom and Tirole 1997).
- Reasons for caution:
  - Banks are increasingly exposed to tail risk that materializes rarely but can exceed any plausible initial capital.
  - Under tail risk, banks do not internalize losses independently of initial capital, reducing buffer and incentive effects of capital.
  - Empirical anecdotes: some banks that appeared highly capitalized before the crisis still took excessive (often tail) risk; highly capitalized banks sought ways to put capital at risk to produce shareholder returns (Berger et al. 2008, Huang and Ratnovski 2009).

### Main contributions and key results (summary)
- Primary findings:
  - The traditional buffer and incentive ("skin in the game") effects of capital become less powerful when banks have access to tail-risk projects because tail realizations can wipe out almost any level of capital.
  - Excess risk-shifting incentives of bank shareholders may exist almost independently of the level of initial or required capital under tail risk.
  - Capital regulation also alters bank risk choices via capital adjustment costs when banks must raise equity to meet minimum capital ratios (Myers and Majluf, 1984; Jensen, 1986). These adjustment-cost incentives fall with higher bank capital because the probability of breaching the minimal capital ratio decreases.
  - The relationship between capital and risk can be non-monotonic under tail risk: higher capital can enable taking riskier projects by reducing the chance of breaching the minimal capital ratio in low (non-tail) returns, while tail realizations still cause insolvency regardless of initial capital.
  - The negative effect of extra capital on risk-taking becomes stronger when banks gain access to projects with even higher tail risk.
- Optimal capital choice and regulation:
  - A bank may choose to hold higher capital to create a cushion over the minimal capital requirement to be able to take tail risk without fear of corrective action for marginally negative project realizations.
  - Capital regulation must implement two bounds on bank capital:
    - A lower bound (a minimal capital ratio) to prevent ordinary risk-shifting.
    - An upper bound (practically, special attention devoted to banks with particularly high capital) to ensure they are not induced to take tail risk.
- Policy implications:
  - Traditional capital regulation has limitations in dealing with tail risk (similar to known limitations with correlation risk).
  - Banks with significant excess capital may be induced to take excess risk; relying solely on higher/excess capital for crisis prevention may produce a false sense of comfort.
  - Authorities should introduce complementary measures to target tail risk alongside pro-cyclical and conservation buffers; enhanced supervision focused on capturing tail risk may be essential.
  - Alternatives include prohibiting extreme bets or increasing their ex ante cost (e.g., prudential levies on strategies exposed to systemic risk, charging for extremely mismatched strategies or certain derivative positions).

### Model overview — agents, timing, preferences, and projects
- Time and preferences:
  - Three dates: 0, 1/2, 1; no discounting; everyone is risk-neutral.
- Bank balance at date 0:
  - Bank has capital C and deposits D with normalization C + D = 1.
  - Deposits are fully insured at no cost; they carry a 0 interest rate and need to be repaid at date 1.
- Investment opportunities (both cost 1 at date 0, return at date 1):
  - Safe project: return R_S > 1 (certain).
  - Risky project: three outcomes:
    - High: R_H > R_S, probability π.
    - Low: R_L, with 0 < R_L < 1, probability 1 − π − δ (notation δ used interchangeably with ϵ in parts of the text).
    - Extremely low (left tail): R_0 = 0, probability δ (ϵ).
- Key payoff assumptions:
  - R_S > π R_H + (1 − π − δ) R_L.  (equation (1))
  - R_S − 1 < π (R_H − 1).  (equation (2))
  - Bank chooses the safe project when indifferent.
- Information and regulatory timing:
  - Project choice is unobservable and unverifiable, but the return of the chosen project is observable and verifiable at date 1/2, allowing regulator corrective action before final returns at date 1.

### Capital regulation in the model — capital ratio, minimum, corrective action, and adjustment cost
- Capital ratio definition:
  - χ = (A − Δ) / A, with normalization C + D = 1 so date 0 capital ratio χ = C.
  - Date 1/2 and date 1 capital ratio: χ_i = (R_i − Δ) / R_i for i ∈ {S, H, L, 0}.
  - Date 0 minimal ratio satisfied: χ > χ_min with χ_min > 0.
  - Ordering: χ_H > χ_S > χ > χ_min for R_H > R_S > 1.
  - For R_0 = 0: χ_0 = −∞ < 0 < χ_min.
  - For R_L: χ_L may be > χ_min, 0 < χ_L < χ_min, or χ_L < 0 depending on initial capital.
- Corrective action at date 1/2:
  - If bank fails to satisfy χ_min at date 1/2, the banker has two options:
    - Surrender the bank to regulator → banker payoff = 0.
    - Attract additional capital to bring capital ratio to χ_min; attracting capital carries a cost T to existing shareholders (recapitalization cost).
  - In the main model, recapitalization cost is fixed at T.

### Mechanisms and intuition
- Tail risk undermines both buffer and incentive effects:
  - Tail realizations (R_0 = 0) wipe out equity regardless of initial capital; shareholders do not internalize a portion of losses.
- Capital-adjustment-cost deterrent:
  - If recapitalization cost T is significant, the risk of incurring T deters choosing projects that may breach χ_min in R_L realizations.
  - This deterrent weakens as initial/excess capital increases because probability of breaching χ_min falls.
- Non-monotonic relationship between capital and risk-taking:
  - For low capital, “skin in the game” and capital adjustment costs both can deter risk-taking.
  - For higher capital under tail risk, “skin in the game” is ineffective against catastrophic losses, while higher capital reduces the probability of incurring T in moderate losses, potentially enabling riskier project choices.
  - Thus, higher capital can increase risk-taking when tail-risk projects are available; effect strengthens with increased tail risk.

### Section III: "Skin in the game" and tail risk — analytical results (T = 0)
- Banker's payoff choosing the safe project:
  - Π_{T=0}^S = R_S − Δ = R_S − (1 − χ).  (Equation (3))
- Banker's payoff choosing the risky project:
  - Π_{T=0}^R = π·[R_H − (1 − χ)] + (1 − π − ϵ)·max{R_L − (1 − χ); 0}.  (Equation (4))
  - R_0 realization occurs with probability ϵ and carries zero payoff to the banker.
- Project choice condition (Π_{T=0}^S ≥ Π_{T=0}^R):
  - R_S − (1 − χ) ≥ π·[R_H − (1 − χ)] + (1 − π − ϵ)·max{[R_L − (1 − χ)]; 0}.  (Equation (5))
- Proposition 1 (dependence on initial capital χ):
  - Case (a): If R_S < πR_H + (1 − π)R_L, bank chooses safe project if
    - χ ≥ 1 − (R_S − πR_H − (1 − π − ϵ)R_L)/ϵ,
    - and chooses risky otherwise.
  - Case (b): If R_S ≥ πR_H + (1 − π)R_L, bank chooses safe project if
    - χ ≥ 1 − (R_S − πR_H)/(1 − π),
    - and chooses risky otherwise.
  - Focus is on Case (a) to study tail risk impact.
- Definition:
  - χ_{T=0} = 1 − (R_S − πR_H − (1 − π − ϵ)R_L)/ϵ.  (Equation (8))
  - χ_{T=0} is the threshold initial capital necessary to prevent risk-shifting under (6).
- Comparative statics (keeping E(R) constant):
  - Case 1 (shift mass from R_L to R_0 and R_H):
    - ∂χ_{T=0}/∂ϵ |_{E(R)=const} = (R_S − E(R))/ϵ^2 > 0.  (Equation (9))
  - Case 2 (higher ϵ compensated by higher R_L and R_H):
    - ∂χ_{T=0}/∂ϵ |_{E(R)=const} = (R_S − E(R))/ϵ^2 > 0.
  - Conclusion: required capital χ_{T=0} increases with tail risk ϵ; capital becomes progressively less effective as ϵ increases, especially for low ϵ.

### Quantitative calibration example (preserved numbers)
- Calibration parameters:
  - R_S = 1.03; R_H = 1.14; R_L = 0.92; π = 50%; ϵ = 1%.
- Computations:
  - Expected return on the safe project = 3%.
  - Expected return on the risky project = 2.1%.
  - Minimal capital necessary to prevent risk-shifting χ_{T=0} = 8%.
- Sensitivity:
  - An increase in ϵ from 1% to 1.1% raises χ_{T=0} from 8% to 16.4%.
  - Doubling ϵ to 2% raises required initial capital to 54%.
- Implication: preventing excess tail risk using minimal capital-based incentives alone may require implausibly high capital levels.

### Section IV: Tail risk and unintended effects of higher capital (introducing capital-adjustment costs T > 0)
- Key qualitative effects:
  - Marginally capitalized banks avoid risk because they fear breaching χ_min in R_L realizations and facing recapitalization cost T or abandonment.
  - Banks with higher capital may take more risk because their chance of breaching the ratio in R_L realizations is lower.
  - Unintended effect of higher capital is stronger with larger tail risk ϵ.
- Payoffs and recapitalization decision:
  - χ_{sufficient} = 1 − (1 − χ_{min}) R_L.  (Equation (10))
  - χ_{recapitalize} = 1 + T − R_L.  (Equation (12))
  - Focus on χ_{recapitalize} < χ_{sufficient}, which holds under:
    - T < χ_{min}/R_L.  (Equation (13))
  - Banker’s payoff in R_L:
    - Π_L = R_L − (1 − χ) if χ ≥ χ_{sufficient};
    - Π_L = R_L − (1 − χ) − T if χ_{recapitalize} < χ < χ_{sufficient};
    - Π_L = 0 if χ ≤ χ_{recapitalize}.  (Equation (14))
  - Overall payoff to risky project:
    - Π_R = π·[R_H − (1 − χ)] + (1 − π − ϵ)·Π_L.  (Equation (15))
- Project choice with recapitalization costs:
  - Bank chooses safe project when:
    - R_S − (1 − χ) ≥ π·[R_H − (1 − χ)] + (1 − π − ϵ)·Π_L.  (Equation (16))
- Two thresholds defined:
  - Ω = πR_H + (1 − π)R_L − ϵχ_{min}/R_L.  (Equation (17))
    - Ω is a threshold for risk-shifting presence in highly capitalized banks: for R_S < Ω, even banks with χ ≥ χ_{sufficient} may engage in risk-shifting.
  - Ζ = πR_H + (1 − π)(R_L − T) + ϵ(T − χ_{min}/R_L).  (Equation (18))
    - Ζ is a threshold for the capital-adjustment-cost effect: for R_S ≥ Ζ, there exist χ with χ_{recapitalize} < χ < χ_{sufficient} where the less-capitalized bank may choose the safe project to avoid recapitalization costs upon R_L realization.
- Result: capital-adjustment costs can create regions where higher capital unintentionally enables more risk-taking; these unintended effects are amplified with greater tail risk ϵ.

### Proposition 2 and non-monotonic choice thresholds (preserved symbols and structure)
- Proposition 2: Bank project choice characterized by thresholds χ^* and χ^{**}.
- Case (a) (Ζ ≤ R_S < Ω):
  - There exist χ^* < χ_{sufficient} and χ^{**} > χ_{sufficient} such that:
    - For χ < χ^*: bank chooses risky project and may abandon or recapitalize upon R_L.
    - For χ^* ≤ χ < χ_{sufficient}: bank chooses safe project to avoid abandonment/recapitalization (capital adjustment cost effect).
    - For χ_{sufficient} ≤ χ < χ^{**}: bank chooses risky project because capital is high enough to avoid breaching χ_min in R_L (risk-shifting enabled by higher capital).
    - For χ ≥ χ^{**}: bank chooses safe project because skin in the game prevents risk-shifting.
- Case (b) (R_S < Ζ):
  - There exists χ^{**} > χ_{sufficient} such that for χ < χ^{**} the bank chooses risky project and for χ ≥ χ^{**} the safe project; only risk-shifting effect.
- Case (c) (R_S ≥ Ω):
  - There exists χ^* < χ_{sufficient} such that for χ < χ^* bank chooses risky project, and for χ ≥ χ^* the safe project; only capital adjustment cost effect.
- Exact threshold expressions:
  - χ^* = 1 − [R_S − πR_H − (1 − π − ϵ)(R_L − T)]/ϵ.  (Equation (19))
  - χ^{**} = 1 − [R_S − πR_H − (1 − π − ϵ) R_L]/ϵ.  (Equation (20))
- Main qualitative result (Case (a)):
  - Non-monotonic relationship between bank capital and risk-taking in the presence of tail risk and capital adjustment cost:
    - χ < χ^*: risky project chosen.
    - χ^* ≤ χ < χ_{sufficient}: safe project chosen (avoid recapitalization).
    - χ_{sufficient} ≤ χ < χ^{**}: risky project chosen (risk-shifting enabled by higher capital).
    - χ ≥ χ^{**}: safe project chosen (sufficient skin in the game).
  - Note: under tail risk χ^{**} may be prohibitively very high.

### Comparative statics with tail risk (Case (a) focus)
- Case 1 (higher ϵ compensated by higher π, keeping E(R) constant):
  - χ_{sufficient} unaffected by distributional change (determined by χ_min and R_L per (10)).
  - Critical threshold χ^{**} = χ_{T=0}.
  - ∂χ^{**}/∂ϵ (with E(R) constant) = (R_S − E(R))/ϵ^2 > 0.
  - Interpretation: when tail risk increases, interval [χ_{sufficient}, χ^{**}) expands because higher-capital banks begin taking more risk.
- Case 2 (higher ϵ compensated by higher R_L and R_H):
  - χ_{sufficient} decreases in R_L (hence in ϵ).
  - χ^{**} increases in ϵ.
  - Interval [χ_{sufficient}, χ^{**}) widens more than in Case 1 → both more and less capitalized banks start choosing risky projects.

### Bank private capital choice and optimal regulation (Section V)
- Bank private cost of holding capital: θ, θ > 1 (Assumption A).
- Assumption B: cost of bank capital becomes prohibitive for high values, making χ_min = χ^{**} impossible to implement (χ^{**} can be very high under tail risk).
- Lemma 1:
  - Setting χ_min < χ^* is never optimal (bank will always choose χ_min and a risky project); therefore χ_min ≥ χ^*.
- Proposition 3 (bank’s private capital choice):
  - For χ_min ≥ χ^*, the bank’s private capital choice is either minimal capital χ_min or the level sufficient to avoid recapitalization costs in R_L realization, χ_{sufficient}.
  - There exists θ^* > 1:
    - θ^* = 1 + ϵ R_L (1 − R_L)^{−1} − [R_S − π R_H − (1 − π − ϵ) R_L][(1 − χ_min)(1 − R_L)]^{−1}.  (Preserved expression structure as in (21))
    - Bank chooses χ_{sufficient} for θ < θ^* and χ_min otherwise.
  - Intuition:
    - Because capital is costly, bank picks the lowest capital possible on each interval.
    - Banks may choose to accumulate capital to be able to take tail risk (i.e., voluntarily hold higher capital to reach χ_{sufficient}).
- Implication for regulation:
  - Since χ^{**} may be implausibly high, feasible regulatory objective is to ensure bank capital is in [χ^*, χ_{sufficient}) where banks take the safe project due to aversion to capital adjustment costs.
  - Both below and above this interval, banks undertake the risky project.
  - Two instruments:
    - Set minimal capital requirement at χ_min = χ^*.
    - Effective constraint (or special supervisory attention) on banks’ excess capital above χ_{sufficient} to prevent tail risk-taking.
  - Conjecture 1:
    - In the presence of tail risk, optimal capital regulation combines a minimal capital requirement χ_min = χ^* and effective constraints on banks with excess capital (above χ_{sufficient}) to prevent them from taking tail risk.

### Robustness, proofs, and extensions (high-level preserved technical points)
- Proofs and detailed derivations are in Appendix I and II (cases labeled (T1a’), (T1a), (T2a), (T2b), (T3a); scenarios S1–S4 and variants S1’–S4’, S1”–S3”).
- Key identities preserved:
  - Ζ = πR_H + (1 − π)(R_L − T) + ϵ(T − χ_min/R_L); Ω = πR_H + (1 − π) R_L − ϵ χ_min/R_L. (from (18) and (17))
  - Under assumption (13): Ζ < B < Ω, with B = πR_H + (1 − π)(R_L − T).
- Extensions (Appendix II):
  - Charter value V > 0:
    - χ_{Recapitalize}^V = 1 + T − R_L − V. (24); condition V < 1 + T − R_L. (25)
    - Adjusted thresholds χ^*_V and χ^{**}_V:
      - χ^*_V = 1 − V − [R_S − πR_H − (1 − π − ϵ)(R_L − T)]/ϵ. (28)
      - χ^{**}_V = 1 − V − [R_S − πR_H − (1 − π − ϵ) R_L]/ϵ. (29)
    - Conclusion: positive charter value reduces relevant thresholds (except χ_{sufficient}), partially counterbalancing risk-taking incentives of well-capitalized banks.
  - Concave capital adjustment cost (fixed + variable component):
    - Cost(c, R_L) = T + β( (1 − χ)/(1 − χ_min) − R_L ). (31)
    - Bank closure threshold:
      - χ_{Recapitalize}^{CC} = 1 + (T − R_L (1 + β)) / (1 + β/(1 − χ_min)). (33)
    - New upper-bound parameter:
      - B_{CC} = πR_H + (1 − π) R_L (1 + β) − T / (1 + β/(1 − χ_min)). (34)
    - Thresholds with concave cost:
      - χ^*_{CC} = 1 − [R_S − πR_H − (1 − π − ϵ)[R_L (1 + β) − T]]/ϵ − β (1 − π − ϵ)/(1 − χ_min). (35)
      - χ^{**}_{CC} = χ^{**} (unchanged).
    - Conclusion: concave cost shrinks interval where recapitalization occurs but leaves (χ_{sufficient}, χ^{**}) boundaries relevant for risk incentives of well-capitalized banks unchanged.
- Figures and timeline (not reproduced here) illustrate the mechanisms: non-monotonic choice, two opposite effects of capital, tail risk impact on required capital, recapitalization decision and payoffs, and heavier left tail widening interval where well-capitalized banks choose risky project.

*Italic: Content extracted from the PDF _wp11188 - References .............................................................................................................*

### References .............................................................................................................

### _wp11188 - References .............................................................................................................

### I. INTRODUCTION — context and motivation
- Regulatory reform after the recent financial crisis focused on increasing capital cushions of financial intermediaries; Basel III rules have doubled the minimal capital ratio and directed banks to hold excess capital as conservation and countercyclical buffers (BIS, 2010).
- Two classical arguments for higher capital:
  - Ex post: capital absorbs losses and reduces insolvency risk and systemic propagation.
  - Ex ante: capital increases shareholders’ “skin in the game”, reducing limited-liability-driven incentives to take excessive risk (Jensen and Meckling 1976, Holmstrom and Tirole 1997).
- Reasons for caution:
  - Banks are increasingly exposed to tail risk that materializes rarely but can exceed any plausible initial capital (examples: carry trades reliant on short-term wholesale funding; reckless underwriting of contingent liabilities on systemic risk; undiversified exposures to inflated housing markets).
  - Under tail risk, banks do not internalize losses independently of initial capital, reducing buffer and incentive effects of capital.
  - Empirical anecdotes: some banks that appeared highly capitalized before the crisis still took excessive (often tail) risk; highly capitalized banks sought ways to put capital at risk to produce shareholder returns (Berger et al. 2008, Huang and Ratnovski 2009).

### Main contributions and key results (summary)
- Primary findings:
  - The traditional buffer and incentive ("skin in the game") effects of capital become less powerful when banks have access to tail-risk projects because tail realizations can wipe out almost any level of capital.
  - Excess risk-shifting incentives of bank shareholders may exist almost independently of the level of initial or required capital under tail risk.
  - Capital regulation also alters bank risk choices via capital adjustment costs when banks must raise equity to meet minimum capital ratios (Myers and Majluf, 1984; Jensen, 1986). These adjustment-cost incentives fall with higher bank capital because the probability of breaching the minimal capital ratio decreases.
  - The relationship between capital and risk can be non-monotonic under tail risk: higher capital can enable taking riskier projects by reducing the chance of breaching the minimal capital ratio in low (non-tail) returns, while tail realizations still cause insolvency regardless of initial capital.
  - The negative effect of extra capital on risk-taking becomes stronger when banks gain access to projects with even higher tail risk.
- Optimal capital choice and regulation:
  - A bank may choose to hold higher capital to create a cushion over the minimal capital requirement to be able to take tail risk without fear of corrective action for marginally negative project realizations.
  - Capital regulation must implement two bounds on bank capital:
    - A lower bound (a minimal capital ratio) to prevent ordinary risk-shifting.
    - An upper bound (practically, special attention devoted to banks with particularly high capital) to ensure they are not induced to take tail risk.
- Policy implications:
  - Traditional capital regulation has limitations in dealing with tail risk (similar to known limitations with correlation risk).
  - Banks with significant excess capital may be induced to take excess risk; relying solely on higher/excess capital for crisis prevention may produce a false sense of comfort.
  - Authorities should introduce complementary measures to target tail risk alongside pro-cyclical and conservation buffers; enhanced supervision focused on capturing tail risk may be essential.
  - Alternatives include prohibiting extreme bets or increasing their ex ante cost (e.g., prudential levies on strategies exposed to systemic risk, charging for extremely mismatched strategies or certain derivative positions).

### Model overview — agents, timing, preferences, and projects
- Time and preferences:
  - Three dates: 0, 1/2, 1; no discounting; everyone is risk-neutral.
- Bank balance at date 0:
  - Bank has capital C and deposits D with normalization C + D = 1.
  - Deposits are fully insured at no cost; they carry a 0 interest rate and need to be repaid at date 1.
- Investment opportunities (both cost 1 at date 0, return at date 1):
  - Safe project: return R_S > 1 (certain).
  - Risky project: three outcomes to capture variance and skewness (left tail):
    - High: R_H > R_S, probability π.
    - Low: R_L, with 0 < R_L < 1, probability 1 − π − δ.
    - Extremely low (left tail): R_0 = 0, probability δ.
- Assumptions on payoffs and incentives:
  - NPV of safe project is higher than risky project:
    - R_S > π R_H + (1 − π − δ) R_L. (equation (1))
  - Return on safe asset not too high so banker may choose risky project for low initial capital:
    - R_S − 1 < π (R_H − 1). (equation (2))
  - Bank chooses the safe project when indifferent.
  - Bank has no continuation value beyond date 1 (impact of positive continuation value discussed in Appendix; it reduces bank risk-taking but does not affect qualitative results).
- Information and regulatory timing:
  - Project choice is unobservable and unverifiable, but the return of the chosen project is observable and verifiable at date 1/2, allowing regulator corrective action before final returns at date 1.

### Capital regulation in the model — capital ratio, minimum, corrective action, and adjustment cost
- Capital ratio definition:
  - χ = (A − Δ) / A, where A is asset value and Δ is face value of deposits; economic capital = A − Δ.
  - Date 0 capital ratio: χ = C / (C + D) = C (given C + D = 1).
  - Date 1/2 and date 1 capital ratio: χ_i = (R_i − Δ) / R_i, for i ∈ {S, H, L, 0} depending on project and realization.
  - Date 0 minimal ratio satisfied: χ > χ_min with χ_min > 0.
  - Ordering for realizations with R_H and R_S: χ_H > χ_S > χ > χ_min since R_H > R_S > 1.
  - For R_0 = 0 (extreme low outcome), minimal ratio never satisfied: χ_0 = −∞ < 0 < χ_min.
  - For R_L, sufficiency ambiguous: depending on initial capital, χ_L may be > χ_min, or 0 < χ_L < χ_min, or χ_L < 0.
- Corrective action at date 1/2:
  - If bank fails to satisfy χ_min at date 1/2, the banker has two options:
    - Surrender the bank to regulator → equity value wiped out, banker payoff = 0.
    - Attract additional capital to bring capital ratio to χ_min; attracting capital carries a cost T to existing shareholders (recapitalization cost), reflecting dilution or adverse selection under urgency.
  - In the main model, recapitalization cost is fixed at T.

### Mechanisms and intuition
- Tail risk undermines both buffer and incentive effects of capital:
  - Tail realizations (R_0 = 0) wipe out equity regardless of initial capital, so shareholders do not internalize a portion of losses; “skin in the game” is restricted.
- Capital adjustment costs provide another deterrent to risk-taking:
  - If banks face a significant cost T to raise equity when undercapitalized, the risk of incurring T discourages choosing projects that may breach χ_min in non-tail adverse realizations.
  - However, this deterrent weakens as initial/excess capital increases because probability of breaching χ_min falls, enabling riskier project choices without fear of raising equity.
- Non-monotonic relationship between capital and risk-taking:
  - For low capital, “skin in the game” and capital adjustment costs both can deter risk-taking.
  - For higher capital under tail risk, “skin in the game” is ineffective against rare catastrophic losses, while higher capital reduces the probability of incurring T in moderate losses, potentially enabling riskier projects.
  - Thus, higher capital can increase risk-taking when tail-risk projects are available; effect strengthens with increased tail risk.

### Policy recommendations and implications (enumerated)
- Recognize limitations of traditional capital regulation in the presence of tail risk.
- Do not rely solely on higher or "excess" capital as crisis prevention; excessive reliance may induce undesirable risk-taking by well-capitalized banks.
- Introduce complementary measures specifically targeting tail risk, such as:
  - Prohibiting extreme bets that exploit limited liability via skewed payoffs.
  - Increasing ex ante costs of tail-risk strategies (e.g., prudential levies on systemic-exposure strategies).
  - Enhanced supervision focused on capturing tail risk exposures.
- Implement regulatory attention not just via minimum capital ratios but also on banks with particularly high capital (practical upper-bound measures—surveillance, special attention).

### Relation to literature and empirical evidence
- Builds on literature on unintended effects of bank capital regulation (Kahane 1977, Kim and Santomero 1988, Koehn and Santomero 1980; Boot and Greenbaum 1993; Blum 1999; Caminal and Matutes 2002; Flannery 1989; Hellman et al. 2000).
- Empirical studies show positive correlation between capital levels and bank risk-taking: Angora et al. (2009), Bichsel and Blum (2004).
- Relates to recent literature arguing for new regulatory tools to handle sophisticated intermediaries and new risks (Acharya and Yorulmazer 2007; Acharya et al. 2010; Brunnermeier and Pedersen 2008; Huang and Ratnovski 2011; Perotti and Suarez 2009, 2010).

### Paper structure and appendices (content map)
- Section 2: theoretical model.
- Section 3: traditional "skin in the game" effect of capital on risk-taking.
- Section 4: how higher capital can enable risk-taking with tail-risk projects.
- Section 5: endogenizes bank’s choice of initial capital; implications for optimal capital regulation.
- Section 6: conclusions.
- Appendix contains:
  - 1. Proofs (starts page 22).
  - 2. Extensions (starts page 28).
  - Figures (pages 32–38) including:
    - The Timeline.
    - The Two Opposite Effects of Capital on Bank Risk-taking.
    - Tail Risk and the Initial Capital Required to Prevent Risk-shifting.
    - Bank’s Recapitalization Decision and Payoffs.
    - Bank’s Project Choice and variants when the risky project has a heavier left tail (Case 1 and Case 2).

*Italic: Content extracted from the PDF _wp11188 - References .............................................................................................................*

### Appendix we discuss a specification with concave cost (i.e., thecost of recapitalization falls with

### _wp11188 - Appendix we discuss a specification with concave cost (i.e., thecost of recapitalization falls with

### Timeline and model setup
- Timeline: model outcomes and sequence of events are depicted in Figure 1 (figure not reproduced here).
- The banker chooses to abandon rather than recapitalize the bank when indifferent.
- The return function with three outcomes (R_H, R_L, R_0) is used to capture:
  - A marginal bad (low) realization R_L;
  - An extreme bad (tail) realization R_0 (zero payoff to the banker);
  - A good realization R_H.
- The model is solved backwards: first derive payoffs given project choice, then derive project choice; comparative statics and a calibration exercise follow.

### Intuition: "skin in the game" versus capital-adjustment cost and tail risk
- Classic Myers and Majluf (1984) limited-liability channel: limited liability subsidizes risk-taking and tilts banks toward the risky project.
- Two opposing capital effects:
  - Skin-in-the-game effect: higher capital makes shareholders internalize more losses, discouraging risk-taking (left panel intuition).
  - Capital-adjustment-cost effect: higher capital reduces the probability of breaching minimal capital ratio and thus reduces expected costly corrective action, which can encourage risk-taking (center panel intuition).
- When combined with tail risk (right panel intuition):
  - A risky project can both marginally breach the minimal capital ratio (triggering adjustment cost) and result in an extreme negative outcome R_0 (tail risk under limited liability).
  - The relationship between bank capital and risk-taking may become non-linear: banks with higher capital may choose inefficiently high risk when tail risk is significant.

### Section III: "SKIN IN THE GAME" AND TAIL RISK — main analytical results
- Setup abstracting from capital adjustment costs (T = 0).
- Banker's payoff choosing the safe project:
  - Π_{T=0}^S = R_S − Δ = R_S − (1 − χ).  (Equation (3))
- Banker's payoff choosing the risky project:
  - Π_{T=0}^R = π·[R_H − (1 − χ)] + (1 − π − ϵ)·max{R_L − (1 − χ); 0}.  (Equation (4))
  - The R_0 realization occurs with probability ϵ and carries zero payoff to the banker.
- Project choice condition:
  - Bank chooses safe project when Π_{T=0}^S ≥ Π_{T=0}^R, equivalent to:
    - R_S − (1 − χ) ≥ π·[R_H − (1 − χ)] + (1 − π − ϵ)·max{[R_L − (1 − χ)]; 0}.  (Equation (5))

Proposition 1 (dependence of project choice on initial capital χ)
- Case (a): If R_S < πR_H + (1 − π)R_L, the bank chooses the safe project if
  - χ ≥ 1 − (R_S − πR_H − (1 − π − ϵ)R_L)/ϵ,
  - and chooses the risky project otherwise.
- Case (b): If R_S ≥ πR_H + (1 − π)R_L, the bank chooses the safe project if
  - χ ≥ 1 − (R_S − πR_H)/(1 − π),
  - and chooses the risky project otherwise.
- Interpretation:
  - Case (b) represents negligible tail risk; project choice not affected by ϵ.
  - Focus is on Case (a) to study impact of tail risk.

Definition
- χ_{T=0} = 1 − (R_S − πR_H − (1 − π − ϵ)R_L)/ϵ.  (Equation (8))
  - χ_{T=0} is the threshold initial capital necessary to prevent risk-shifting under (6).

### Comparative statics: how χ_{T=0} responds to tail risk ϵ
- Maintain constant expected value E(R) of the risky project; consider two parameter interpretations that increase ϵ while keeping E(R) fixed.

Case 1 (shift mass from R_L to R_0 and R_H)
- Increase ϵ and π at the expense of (1 − π − ϵ); to keep E(R) constant, π increases by R_L/(R_H − R_L)·Δϵ.
- Using (8):
  - ∂χ_{T=0}/∂ϵ |_{E(R)=const} = (R_S − E(R))/ϵ^2 > 0.  (Equation (9))
- Conclusion: required capital χ_{T=0} increases with tail risk ϵ.

Case 2 (higher ϵ compensated by higher R_L and R_H)
- Increase ϵ and raise R_L and R_H so E(R) remains constant; required changes imply R_L and R_H increase by R_L·Δϵ/(1 − ϵ − Δϵ).
- Result:
  - ∂χ_{T=0}/∂ϵ |_{E(R)=const} = (R_S − E(R))/ϵ^2 > 0.
- In both cases:
  - χ_{T=0} grows logarithmically in ϵ; capital becomes progressively less effective at controlling risk-taking as ϵ increases, with the effect most pronounced for low values of ϵ.
  - Policy result: tail risk limits the effectiveness of capital regulation in dealing with bank risk-taking incentives.

### Quantitative calibration example
- Calibration parameters: R_S = 1.03; R_H = 1.14; R_L = 0.92; π = 50%; ϵ = 1%.
- Computations:
  - Expected return on the safe project = 3%.
  - Expected return on the risky project = 2.1%.
  - Minimal capital necessary to prevent risk-shifting χ_{T=0} = 8%.
- Sensitivity:
  - An increase in ϵ from 1% to 1.1% raises χ_{T=0} from 8% to 16.4%.
  - Doubling ϵ to 2% raises required initial capital to 54%.
- Implication: preventing banks from taking excess tail risk using minimal capital-based incentives alone may require implausibly high capital levels.

### Section IV: Tail risk and unintended effects of higher capital (introducing capital-adjustment costs)
- Introducing capital-adjustment costs T produces stronger and potentially perverse effects:
  - Marginally capitalized banks do not take risk because they are averse to breaching minimal capital ratio in R_L realizations (they face costly recapitalization or abandonment).
  - Banks with higher capital may take more risk because their chance of breaching the ratio in R_L realizations is lower.
  - Comparative statics: unintended effect of higher capital is stronger when tail risk ϵ is higher.

A. Payoffs and recapitalization decision (with T > 0)
- Banker's payoff from safe project:
  - Π^S = R_S − Δ = R_S − (1 − χ).
- Under R_H: banker obtains R_H − (1 − χ).
- Under R_0: banker obtains 0.
- Under R_L, three capital states depending on χ relative to thresholds:
  - Positive and sufficient capital if χ ≥ χ_{sufficient} where
    - χ_{sufficient} = 1 − (1 − χ_{min})R_L.  (Equation (10))
    - In this case banker continues, repays depositors, gets R_L − (1 − χ).
  - Positive but insufficient capital (χ_{recapitalize} < χ < χ_{sufficient}): banker must choose between recapitalizing at cost T or abandoning.
    - Banker chooses to recapitalize if R_L − (1 − χ) − T > 0, equivalent to
      - χ > χ_{recapitalize} = 1 + T − R_L.  (Equation (12))
  - Negative or too small capital χ ≤ χ_{recapitalize}: banker abandons and payoff is 0.
- Focus on case χ_{recapitalize} < χ_{sufficient}, which holds under:
  - T < χ_{min}/R_L.  (Equation (13))
- Banker’s payoff in R_L:
  - Π_L = R_L − (1 − χ) if χ ≥ χ_{sufficient};
  - Π_L = R_L − (1 − χ) − T if χ_{recapitalize} < χ < χ_{sufficient};
  - Π_L = 0 if χ ≤ χ_{recapitalize}.  (Equation (14))
- Overall payoff to risky project:
  - Π_R = π·[R_H − (1 − χ)] + (1 − π − ϵ)·Π_L.  (Equation (15))

B. Project choice with recapitalization costs
- Bank chooses safe project when:
  - R_S − (1 − χ) ≥ π·[R_H − (1 − χ)] + (1 − π − ϵ)·Π_L.  (Equation (16))
- Define two thresholds:
  - Ω = πR_H + (1 − π)R_L − ϵχ_{min}/R_L.  (Equation (17))
    - Ω is a threshold for risk-shifting presence in highly capitalized banks: for R_S < Ω, even banks with χ ≥ χ_{sufficient} may engage in risk-shifting.
  - Ζ = πR_H + (1 − π)(R_L − T) + ϵ(T − χ_{min}/R_L).  (Equation (18))
    - Ζ is a threshold for the capital-adjustment-cost effect: for R_S ≥ Ζ, there exist values of χ with χ_{recapitalize} < χ < χ_{sufficient} where the less-capitalized bank may choose the safe project to avoid recapitalization costs upon R_L realization.
- The analysis shows that capital-adjustment costs can create regions where higher capital unintentionally enables more risk-taking, and these unintended effects are amplified with greater tail risk ϵ.

*Source: _wp11188 - Appendix we discuss a specification with concave cost (i.e., thecost of recapitalization falls with — IMF PDF content provided in the input.*

### Appendix also verifies that.

### _wp11188 - Appendix also verifies that.

### Proposition 2: Bank project choice characterized by thresholds 
- Proposition 2 states the bank’s project choice is characterized by thresholds 
  ∗ and 
  ∗∗.
- Case structure (exact wording and symbols preserved):
  - (a) For ≤
     , there exist 
    ∗  
     , and 
    ∗∗  
     , such that
    - For 
      ∗ the bank chooses the risky project and may abandon or recapitalize it upon the 
       realization;
    - For 
      ∗ ≤
       the bank chooses the safe project to avoid abandonment or recapitalization upon the 
       realization; the choice of the safe project here represents the capital adjustment cost effect;
    - For 
       ≤
      ∗∗ the bank chooses the risky project because its capital is high enough to avoid breaching the minimal capital ratio in the 
       realization; the choice of the risky project here represents risk-shifting enabled by higher capital;
    - For ≥
      ∗∗ the bank chooses the safe project because its capital is high enough to prevent risk-shifting;
  - (b) For 
     , there exists 
    ∗∗  
     such that for 
    ∗∗ the bank chooses the risky project and for ≥
    ∗∗ the safe project; there is only a risk-shifting effect: a bank with 
     never chooses a safe project to avoid recapitalization cost;
  - (c) for 
     ≥, there exists 
    ∗  
     such that for initial capital 
    ∗ the bank chooses a risky project, and for ≥
    ∗ the safe project; there is only a capital adjustment cost effect: a bank with 
     never engages in risk-shifting.
- Proof reference: See Appendix.

### Exact threshold expressions (as derived in Appendix)
- 
  ∗ = 1 −
  
  
  − 
  
  − (1 −  − )(
  
  − )
    ; (19)
- 
  ∗∗ = 1 −
  
  
  − 
  
  − (1 −  − )
  
    . (20)

### Main qualitative result (Case (a) summary)
- Non-monotonic relationship between bank capital and risk-taking in presence of tail risk and capital adjustment cost:
  - If 
    ∗ : strong risk-shifting incentives and low abandonment cost → chooses risky project.
  - If 
    ∗ ≤
     : equity value higher → chooses safe project to avoid abandoning/recapitalizing (capital adjustment cost effect).
  - If 
     ≤
    ∗∗ : capital adjustment cost not binding → switches back to risky project (risk-shifting enabled by higher capital).
  - If ≥
    ∗∗ : sufficient skin in the game → risk-shifting incentives not binding; safe project chosen.
- Note: under tail risk 
  ∗∗ may be prohibitively very high.

### Comparative statics (Case (a) focus)
- Objective: show how increasing tail risk changes incentives, especially for well-capitalized banks.
- Case 1 (higher  compensated by higher , keeping E(R) constant):
  - Both 
    ∗ and 
    ∗∗ affected.
  - Consider interval [
     , 
    ∗∗ ), where bank undertakes risky project.
  - 
     unaffected by distributional change (determined by 
    min and 
     per (10)).
  - Critical threshold: 
    ∗∗ = 
    =0 .
  - Impact:
    - ∂
      ∗∗ / ∂  (with E(R) constant) = (
       − E(R)) / ^2  > 0.
  - Interpretation: when tail risk increases, interval [
     , 
    ∗∗ ) expands because higher-capital banks begin taking more risk.
- Case 2 (higher  compensated by higher 
   and 
  ):
  - Affects 
    ∗, 
    ∗∗, and 
     .
  - 
     is decreasing in 
     and hence in .
  - 
    ∗∗ is increasing in .
  - Interval [
     , 
    ∗∗ ) widens more than in Case 1 → both more and less capitalized banks start choosing risky projects.

### Bank capital choices and optimal capital regulation (Section V)
- Bank private cost of holding capital: , >1 (Assumption A).
  - Interpreted as alternative cost of banker’s own funds elsewhere.
- Assumption B: cost of bank capital becomes prohibitive for high values, making 
  min = 
  ∗∗ impossible to implement (recall 
  ∗∗ can be very high under tail risk).
- Lemma 1:
  - Setting 
    min < 
    ∗ is never optimal (bank will always choose 
    min and a risky project); therefore 
    min ≥ 
    ∗.
- Proposition 3 (bank’s private capital choice):
  - For 
    min ≥ 
    ∗ , the bank’s private capital choice is either minimal capital 
    min or the level sufficient to avoid recapitalization costs in 
     realization, 
     .
  - There exists 
    ∗ > 1:
    - 
      ∗ = 1 + 
      
      
      1 − 
      
      −
      
      
      − 
      
      − (1 −  − )
      
      (1 − 
      min)(1 − 
       )  (21)
    - Bank chooses 
       for  < 
      ∗ and 
      min otherwise.
  - Intuition:
    - Because capital is costly, bank picks lowest capital possible on each interval.
    - Compare profits at 
      min and 
       ; banker prefers 
       if cost  is not too high.
    - Banks may choose to accumulate capital to be able to take tail risk (i.e., voluntarily hold higher capital to enable risky project when 
       is reached).
- Implication for regulation:
  - Since 
    ∗∗ (preventing risk-taking via "skin-in-the-game") may be implausibly high, feasible regulatory objective is to ensure bank capital is in [
    ∗ , 
     ) where banks take safe project due to aversion to capital adjustment costs.
  - Both below and above this interval, banks undertake the risky project.
  - Two instruments to implement outcome:
    - Set minimal capital requirement at 
      min = 
      ∗ (no reason to set 
      min > 
      ∗ since capital is costly).
    - Effective constraint (or special supervisory attention) on banks’ excess capital above 
       to prevent them from taking tail risk.
  - Conjecture 1 (preserved wording):
    - In the presence of tail risk, optimal capital regulation combines a minimal capital requirement 
      min = 
      ∗ and effective constraints on banks with excess capital (above 
       ) to prevent them from taking tail risk.

### Robustness, proofs, and extensions (high-level bullets of preserved technical points)
- Proofs and further derivations are in Appendix I and II (exact case analyses, conditions labelled (T1a’), (T1a), (T2a), (T2b), (T3a), and scenarios S1–S4 and variants S1’–S4’, S1”–S3”).
- Key definitions and identities preserved:
  -  = 
    
    + (1 − )(
    
    − ) + ( − 
    min
    
    
    ) ;  = 
    
    + (1 − )
    
    − 
    min
    
     (from (18) and (17)).
  - Under assumption (13): , with  = 
    
    + (1 − )(
    
    − ).
- Extensions (Appendix II) with preserved statements:
  - Charter value (>0):
    - New thresholds and adjustments: 
      
       = 1 +  − 
       −  (24); condition  < 1 +  − 
       (25).
    - Modified thresholds 
      ∗
       and 
      ∗∗
       with exact expressions:
      - 
        ∗
         = 1 −  −
        
        
        − 
        
        − (1 −  − )(
        
        − )
          (28)
      - 
        ∗∗
         = 1 −  −
        
        
        − 
        
        − (1 −  − )
        
          (29)
    - Conclusion: positive charter value reduces relevant thresholds (except 
      ), partially counterbalancing risk-taking incentives of well-capitalized banks.
  - Concave capital adjustment cost (fixed + variable component):
    - Recapitalization cost function:
      - Cost(c, 
         ) =  + ( 1− 1−
        min
        − 
         )  . (31)
    - Bank closure threshold:
      - 
        
         = 1 + ( − 
        
        (1 + )) / (1 +  1−
        min )  . (33)
    - New upper bound parameter:
      - 
         = 
        
        + (1 − ) 
        
        (1 + ) − 
        1 +  1 − 
        min  . (34)
    - Thresholds with concave cost:
      - 
        ∗
         = 1 −
        
        
        − 
        
        − (1 −  − )[
        
        (1 + ) − ]
         − 
        1 − 
        min
        (1 −  − )  . (35)
      - 
        ∗∗
         = 
        ∗∗  (unchanged from (20)).
    - Conclusion: concave cost specification shrinks interval where recapitalization occurs but leaves (
       , 
      ∗∗ ) boundaries relevant for risk incentives of well-capitalized banks unchanged.
- Figures and timeline descriptions (Figure 1–6) illustrate the mechanisms: non-monotonic choice, two opposite effects of capital (limited liability vs capital adjustment cost), tail risk impact on required capital, recapitalization decision and payoffs, and how heavier left tail widens interval where well-capitalized banks choose risky project.

*Italic: Source — _wp11188 - Appendix also verifies that.*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11188.pdf_