## _wp16196

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---

### Introduction and policy background
- Shareholders have limited liability: limited downside but full upside from company value increases.
- Managers, acting on behalf of shareholders and incentivized via contracts, can engage in risk shifting: investing in risky projects with negative net present values.
- Debt markets partially correct risk shifting through credit spreads, but imperfectly because insured creditors are not incentivised to monitor credit risks.
- Post-crisis regulatory measures, particularly in the European Union (EU), have aimed to restrict risk shifting by bank executives, including a bonus cap.
- EU bonus cap specifics:
  - Capped bonuses at 100 percent of salary in general.
  - Provision to increase to 200 percent of salary with the approval of at least 65 percent of the firm’s shareholders (or 75 percent in the absence of a quorum).
- Legal event: The United Kingdom (UK) sued the EU in September 2013 over bonus cap-related issues.

### Research questions and approach
- Motivating questions:
  - To what extent can a bonus cap affect risk shifting and underinvestment?
  - How much of the impact of a bonus cap is offset by labor market adjustments?
- Approach:
  - Model the investment decisions of bank executives in a two-period setup (t=0,1).
  - Calibrate the model to observed data at region, country, and bank levels.
  - Simulate the impact of bonus caps while allowing for labor market adjustments.

### Model of banker investment decisions (key structure and comparative statics)
- Setup and compensation:
  - Two periods (t=0,1); banker receives fixed salary w0 at t=0 and may invest in a risky project; at t=1 banker receives bonus linked to realized portfolio return.
  - Performance threshold θ; incentive parameter β specifies sensitivity of compensation above θ.
  - Risky project return x at t=1 unknown at t=0; banker’s belief denoted ̃x with CDF G_x. Safe asset return x0 is known.
  - Banker chooses exposure s ∈ {0,1} to maximize E[compensation] − λ s where λ>0 is transaction cost.
  - Unregulated compensation: w(s, ̃x) ≡ max{w0 + β(s ̃x + (1−s)x0 − θ), w0}.
- Definition of ∆(x) (value of full risky investment relative to safe):
  - ∆(x) ≡ β ∫_θ^∞ (̃x−θ) dG_x(̃x) − λ if θ > x0
  - ∆(x) ≡ β ∫_θ^∞ (̃x−θ) dG_x(̃x) − β(x0 − θ) − λ if θ ≤ x0
- Comparative statics (select):
  - ∂∆(x)/∂θ = −β[1−G_x(θ)] if θ > x0; = βG_x(θ) if θ < x0.
  - ∂∆(x)/∂x0 = 0 if θ > x0; = −β if θ < x0.
  - ∂∆(x)/∂β = ∫_θ^∞ (̃x−θ) dG_x(̃x) if θ > x0; = ∫_θ^∞ (̃x−θ) dG_x(̃x) − (x0 − θ) if θ ≤ x0.

### Bonus cap modeling and analytic results
- Regulated compensation with cap multiple ρ (ρ>0) of fixed salary w0:
  - w(s, ̃x) ≡ max{w0 + min{β(s ̃x + (1−s)x0 − θ), ρ w0}, w0}.
  - Condition assumed unless specified: ρ w0 / β + θ ≥ x0.
  - If ρ w0 / β + θ < x0 and λ>0, banker prefers safe asset with zero exposure.
- ∆(x; ρ) expressions:
  - If θ > x0:
    ∆(x;ρ) = β ∫_{ρ w0/β + θ}^∞ (̃x − θ) dG_x(̃x) + ρ w0 (1 − G_x(ρ w0/β + θ)) − λ
  - If θ ≤ x0:
    ∆(x;ρ) = β ∫_{ρ w0/β + θ}^∞ (̃x − θ) dG_x(̃x) + ρ w0 (1 − G_x(ρ w0/β + θ)) − β(x0 − θ) − λ
- Comparative statics with cap:
  - ∂∆(x;ρ)/∂ρ = w0 (1 − G_x(ρ w0/β + θ)).
  - ∂∆(x;ρ)/∂w0 = ρ (1 − G_x(ρ w0/β + θ)).
  - Cross-partial: ∂^2 ∆(x;ρ) / ∂ρ ∂x = −w0 ∂G_x(ρ w0/β + θ)/∂x implies cap impact larger for higher-return projects under first-order stochastic dominance.
- Proposition 1 (summary):
  - A bonus cap reduces the banker’s incentive to invest in a risky asset.
  - If a higher-return project first-order stochastically dominates a lower-return project, the impact of a bonus cap is larger for the higher-return project.
- Implication: Bonus caps suppress risk taking for all projects and disproportionately for high-return projects, risking underinvestment while having limited effectiveness at preventing risk shifting.

### Regional model, measures, and labor market equilibrium
- Representative regional bank j:
  - Average exposure: s*_j ≡ ∫_{−∞}^∞ s*_j(x) dF_j(x). (Equation (1))
  - Average ex-post compensation: w*_j ≡ ∫_{−∞}^∞ w_j(s*_j(x), x) dF_j(x). (Equation (2))
  - Risk shifting measure: rs*_j ≡ ∫_{−∞}^{x0} s*_j(x) dF_j(x). (Equation (3))
  - Underinvestment measure: ui*_j ≡ ∫_{x0}^{∞} (1 − s*_j(x)) dF_j(x). (Equation (4))
- Labor market indifference condition across J regions:
  - u* ≡ w*_j − λ s*_j − c_j for all 1 ≤ j ≤ J. (Equation (5)); c_j is entry cost to region j.
- Regulation and labor mobility:
  - With mobility restrictions, ex-ante expectations unchanged immediately after cap; to prevent migration, banker who leaves must be paid v*_j − u^C_j(ˆρ_j).
  - Without mobility restrictions, regulators must increase compensation to restore banker participation:
    - Adjustment (i): Increase fixed salary only; ˆw0j solves Equation (7).
    - Adjustment (ii): Increase fixed salary and performance threshold with ˆθ_j(ˆw0j) = θ_j + (ˆw0j − w0j)/β and ˆw0j solves Equation (8).

### Calibration inputs and estimated parameters
- Data and proxies:
  - Aggregate ex-post compensation and average fixed salary of bank CEOs in five regions (2005–2013) from S&P Capital IQ: {w0j, w*j}.
  - Aggregate exposure s*_j recovered from market beta of banking sector by region, adjusting for leverage: s*_j = beta_j L_Mj / L_Bj.
  - Safe asset return x0 proxied by average three-month U.S. Treasury bill rate 2005–2013 = 1.5 percent; robustness check used five-year U.S. Treasury bill rate average 2005–2013 = 2.5 percent.
  - Distribution F_j constructed from BIS consolidated banking statistics, accounting data, and ROA distributions from Datastream.
  - Bankers’ belief G_x assumed normal around true return with SD based on historical coefficient of variation of constituent ROA.
- Estimation steps:
  - Step 1: Use (w*_j − w*_1) = λ(s*_j − s*_1) + (c_j − c_1) to estimate λ and relative entry costs (c_1 = 0).
  - Step 2: Estimate {θ_j, β_j} by minimizing objective matching simulated aggregate exposures and compensations to observed values via Monte Carlo (K = 1000 region-investment samples and R = 100 forecast errors per draw); constraint β_j ≥ 0.
- First-step estimates:
  - λ = 3,593,384
  - c_AS ex U.S. = 1,962,863
  - c_U.S. = 3,746,119
  - c_CE = 623,313
  - c_EMEA = 821,428
- Second-step estimates:
  - β_ASIA = 5.25×10^7
  - β_AS ex U.S. = 3.04×10^7
  - β_U.S. = 3.91×10^7
  - β_CE = 3.93×10^7
  - β_EMEA = 4.76×10^7
  - θ_ASIA = 1.80
  - θ_AS ex U.S. = -3.18
  - θ_U.S. = -3.09
  - θ_CE = 2.20
  - θ_EMEA = 0.60

### Model fit and status-quo outcomes (Monte Carlo)
- Estimated versus actual (selected):
  - Asia exposure: Actual 26.52 percent, Estimated 28.0; Compensation Actual $1,930,123, Estimated $1,920,500.
  - Anglo-Saxon ex U.S. exposure: Actual 24.3 percent, Estimated 24.2; Compensation Actual $3,812,937, Estimated $3,847,000.
  - U.S. exposure: Actual 30.8 percent, Estimated 30.8; Compensation Actual $5,829,346, Estimated $5,867,600.
  - Continental Europe exposure: Actual 23.1 percent, Estimated 23.1; Compensation Actual $2,429,083, Estimated $2,397,800.
  - EMEA exposure: Actual 38.9 percent, Estimated 41.0; Compensation Actual $3,196,900, Estimated $3,185,400.
- Status-quo insights:
  - Risk shifting is not proportional to exposure to risky assets (e.g., U.S. has higher exposure but lower risk shifting than continental Europe).
  - Risk shifting peaks around 2008 across regions.
  - There is substantial underinvestment across regions.
- Robustness: Qualitative results similar when x0 = 2.5 percent (five-year average).

### Counterfactual simulations of bonus caps (2005–2013) and labor mobility regimes
- Simulation design:
  - Bonus cap expressed as multiple of fixed salary; lower cap → more severe.
  - Two labor mobility regimes:
    - No labor mobility (restricted): bankers cannot leave and ex-ante expectations unchanged immediately after cap.
    - Free labor mobility (not restricted): compensation adjustments (i) increase fixed salary only, or (ii) increase fixed salary and performance threshold, to restore ex-ante utility.
- Main simulation results:
  - With labor mobility restricted:
    - Risk shifting substantially reduced in all regions with a moderate cap.
    - Trade-off: reduction in risk shifting associated with increase in underinvestment across regions.
    - Largest regulatory impact in emerging regions because fixed salary is low and cap effects are stronger.
  - With labor mobility not restricted:
    - Labor market adjustments substantially offset regulatory impact on risk shifting and underinvestment for most regions.
    - For Asia, Anglo-Saxon ex U.S., U.S., and Continental Europe, only very tight caps (5 × fixed wage or less) reduce risk shifting.
    - Exception under adjustment type (ii): moderate caps eliminate risk shifting for EMEA due to shift in performance threshold.
- 2008 crisis period:
  - Simulated impact of bonus cap on risk shifting in 2008 largely offset by labor market adjustments for most regions under both adjustment (i) and (ii).
  - Caveat: actual labor market rigidity during crises might limit adjustments relative to simulated equilibrium.

### Policy implications, trade-offs, and extensions
- Key policy implications:
  - Effectiveness of bonus caps is constrained by labor market responses that can offset regulatory intent via increases in base pay.
  - Bonus caps reduce incentives for risky investment and are more effective for high-return projects, but labor market adjustments can largely offset intended regulatory effects.
  - Bonus caps substantially foster underinvestment; eliminating labor market adjustments does not resolve underinvestment concern.
  - To substantially reduce risk shifting, regulators would need to restrict labor mobility or apply caps across sectors bankers can move to.
- Regulatory design considerations:
  - Assess regional heterogeneity in risk shifting before setting targets; simulated risk shifting is lower in the U.S. than in other regions.
  - Cross-jurisdictional coordination or broader sectoral caps may enhance impact on risk shifting but do not eliminate underinvestment trade-off.
- Model caveats and extensions:
  - A dynamic model could alter predictions if future earnings threat affects behavior (may reduce risk shifting).
  - Including banker monitoring effort could change results:
    - If monitoring negatively correlated with profitability, the model may underestimate underinvestment and overestimate risk shifting.
    - If positively correlated, opposite biases may occur.
  - Allowing heterogeneous costs by project type could change predictions on underinvestment and risk shifting.

*Source: _wp16196 — model of banker investment decisions, regional bank characterization, calibration, simulations of bonus cap impacts, and policy discussion.*

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

### _wp16196 - References

### Introduction
- Shareholders have limited liability: limited downside but full upside from company value increases.
- Managers, acting on behalf of shareholders and incentivized via contracts, can engage in risk shifting: investing in risky projects with negative net present values.
- Debt markets partially correct risk shifting through credit spreads, but imperfectly because insured creditors are not incentivised to monitor credit risks (Demirgüç-Kunt and Kane, 2002).
- Bank executives’ risk shifting has been argued to play an important role in the global financial crisis (Boyd and Hakenes, 2014).

### Policy background and regulatory details
- Post-crisis regulatory measures, particularly in the European Union (EU), have aimed to restrict risk shifting by bank executives, including a bonus cap.
- EU bonus cap specifics as described:
  - Capped bonuses at 100 percent of salary in general.
  - Provision to increase to 200 percent of salary with the approval of at least 65 percent of the firm’s shareholders (or 75 percent in the absence of a quorum).
- Legal and political events:
  - The United Kingdom (UK) sued the EU in September 2013 over bonus cap-related issues.

### Mechanisms: how bonus caps interact with incentives and labor markets
- Labor market adjustments:
  - If restrictions on variable pay induce bank managers to move abroad to avoid regulation, banks may respond to a shortage of qualified managers by increasing base pay (International Monetary Fund (IMF), 2014; Murphy, 2013).
  - An increase in base pay raises the manager’s expected payoff from a risky project by raising the compensation floor (the compensation received when a risky project yields a low return).
  - This can undo the intended effect of the cap by increasing the attractiveness of undertaking risky projects.
- Impact on good vs. bad risk taking:
  - Bonus caps can reduce good risk taking by excessively discouraging investment into risky projects with positive net present values.
  - Because the cap affects payoffs associated with high returns, it reduces expected payoff from high-return risky investments more than from low-return ones, potentially yielding underinvestment.

### Research questions and methodology
- Two motivating questions:
  1. To what extent can a bonus cap affect risk shifting and underinvestment?
  2. How much of the impact of a bonus cap is offset by labor market adjustments?
- Approach described:
  - Model the investment decisions of bank executives.
  - Calibrate the model by matching theoretical predictions with observed data.
  - Use the estimated model to simulate the impact of bonus caps while allowing for labor market adjustments.

### Key findings and model predictions
- Model predictions:
  - Bonus capping reduces risk taking for all levels of profitability, although it is less effective for unprofitable projects.
  - The policy can aggravate underinvestment while having only a limited impact on the prevention of risk shifting.
- Numerical simulation results (summary):
  - Bonus caps substantially foster underinvestment.
  - Bonus caps are only useful in the absence of labor market adjustments.
  - Regulatory efforts to eliminate labor market adjustments (for example, setting caps in all sectors bank executives may move to, possibly at the international level) strengthen the effect of bonus capping on risk shifting to some extent, but they never fully prevent underinvestment.

### Implications for regulation and policy design
- Effectiveness of bonus caps is constrained by labor market responses that can offset regulatory intent via increases in base pay.
- Risk of underinvestment is a significant adverse effect of bonus caps, even when they reduce some forms of risk taking.
- Cross-jurisdictional coordination or broader sectoral caps may enhance the impact on risk shifting but do not eliminate the underinvestment trade-off.

*Source: _wp16196 - References . . . . . . . . . . . . . . . . . . . .*

### Section 3 describes the model of investment decisions by bankers. Section 4 characterizes the

### _wp16196 - Section 3 describes the model of investment decisions by bankers. Section 4 characterizes the

### Literature review
- No empirical studies have yet assessed the impact of bonus caps because regulatory reforms following the Global Financial Crisis are recent.
- Existing empirical compensation literature (International Monetary Fund (IMF) (2014)) finds:
  - Performance-linked compensation in the form of options tends to be associated with more risk.
  - Implications for other forms of bonus instruments are mixed.
- Limitations of existing empirical studies:
  - Data do not distinguish the effect on risk shifting from the effect on risk taking.
  - Data do not reveal the effect on underinvestment.
  - Reduction in observed risk measures may correspond to decreased risk shifting or increased underinvestment; welfare impact remains unclear.
- Structural studies (Dittmann, Maug, and Zhang (2011); Llense (2010)) calibrate firm reactions in US and France and find bonus caps have only moderate impact on shareholder value due to labor market adjustments.
- This paper’s approach:
  - Parsimonious, uses easily observable data at region, country, and bank levels to simulate policies for various objectives and enables international analysis.
  - Focuses on sensitivity of variable pay to performance (in the spirit of Gollier, Koehl, and Rochet (1997)).
  - Does not evaluate differences across bonus instruments (cash, equity, options).

### Model of banker investment decisions (Section 3)
- Setup:
  - Two periods (t=0,1). At t=0 banker receives fixed salary w0 and may invest in risky project; at t=1 banker receives bonus linked to realized portfolio return.
  - Performance threshold denoted by θ.
  - Incentive parameter β specifies how compensation increases with performance above θ.
  - Risky project return x at t=1 unknown at t=0; banker’s belief denoted ̃x with CDF Gx. Safe asset return x0 is known.
  - Banker chooses exposure s ∈ {0,1} to maximize expected compensation net of transaction cost λs (λ>0).
- Compensation (unregulated):
  - w(s, ̃x) ≡ max{w0 + β(s ̃x + (1−s)x0 − θ), w0}.
  - Expected chosen exposure s*(x) = arg max_{s∈{0,1}} E_x[w(s, ̃x)] − λs.
  - Expected compensation w*(x) ≡ E_x[w(s*(x), ̃x)].
- Comparative statics:
  - Define ∆(x) as the value of fully investing in risky asset relative to safe asset:
    - ∆(x) ≡ β ∫_θ^∞ (̃x−θ) dG_x(̃x) − λ if θ > x0
    - ∆(x) ≡ β ∫_θ^∞ (̃x−θ) dG_x(̃x) − β(x0 − θ) − λ if θ ≤ x0
  - Effects:
    - ∂∆(x)/∂θ = −β[1−G_x(θ)] if θ > x0; = βG_x(θ) if θ < x0.
      - Intuition: incentive for risk taking rises when θ is closer to the risk free rate x0.
    - ∂∆(x)/∂x0 = 0 if θ > x0; = −β if θ < x0.
      - Higher risk free rate discourages risk taking when θ < x0.
    - ∂∆(x)/∂β = ∫_θ^∞ (̃x−θ) dG_x(̃x) if θ > x0; = ∫_θ^∞ (̃x−θ) dG_x(̃x) − (x0 − θ) if θ ≤ x0.
      - Increase in β can both encourage and discourage risk taking depending on relation between θ and x0.

### Bonus capping (end of Section 3)
- Regulation modeled by cap multiple ρ (ρ>0) of fixed salary w0:
  - Regulated compensation: w(s, ̃x) ≡ max{w0 + min{β(s ̃x + (1−s)x0 − θ), ρ w0}, w0}.
  - Condition for banker not earning max bonus at risk free rate: ρ w0 / β + θ ≥ x0 (assumed unless specified).
  - If ρ w0 / β + θ < x0 and λ>0, banker prefers safe asset and has zero exposure.
- Key analytic results:
  - Marginal value of increasing exposure is lower under a bonus cap; bonus cap reduces incentive for risk taking.
  - Define ∆(x; ρ) as the value of risk taking with cap ρ:
    - If θ > x0:
      ∆(x;ρ) = β ∫_{ρ w0/β + θ}^∞ (̃x − θ) dG_x(̃x) + ρ w0 (1 − G_x(ρ w0/β + θ)) − λ
    - If θ ≤ x0:
      ∆(x;ρ) = β ∫_{ρ w0/β + θ}^∞ (̃x − θ) dG_x(̃x) + ρ w0 (1 − G_x(ρ w0/β + θ)) − β(x0 − θ) − λ
  - Comparative statics with cap:
    - ∂∆(x;ρ)/∂ρ = w0 (1 − G_x(ρ w0/β + θ)) (so ∆ increases in ρ).
    - ∂∆(x;ρ)/∂w0 = ρ (1 − G_x(ρ w0/β + θ)).
    - ∂∆(x;ρ)/∂θ given as in text with split cases; θ affects ∆ similarly as without cap.
    - Cross-partial: ∂^2 ∆(x;ρ) / ∂ρ ∂x = −w0 ∂G_x(ρ w0/β + θ)/∂x; implies cap impact larger for higher-return projects under first-order stochastic dominance.
- Proposition 1 (summary):
  - A bonus cap reduces the banker’s incentive to invest in a risky asset.
  - If a higher-return project first-order stochastically dominates a lower-return project, the impact of a bonus cap is larger for the higher-return project.
- Implications:
  - Bonus caps suppress risk taking for any project, more so for high-return projects.
  - Bonus caps may be ineffective at preventing risk shifting and may cause underinvestment by suppressing high-return risky projects.

### Regional model (Section 4)
- Representative regional bank j:
  - Distribution of risky project returns characterized by F_j.
  - Average exposure to risky projects: s*_j ≡ ∫_{−∞}^∞ s*_j(x) dF_j(x). (Equation (1))
  - Average ex-post compensation: w*_j ≡ ∫_{−∞}^∞ w_j(s*_j(x), x) dF_j(x). (Equation (2))
  - Risk shifting measure: rs*_j ≡ ∫_{−∞}^{x0} s*_j(x) dF_j(x). (Equation (3))
  - Underinvestment measure: ui*_j ≡ ∫_{x0}^{∞} (1 − s*_j(x)) dF_j(x). (Equation (4))
- Labor market equilibrium (J regions):
  - Banker indifference condition: u* ≡ w*_j − λ s*_j − c_j for all 1 ≤ j ≤ J. (Equation (5)); c_j is entry cost to region j.
- Regulation with restriction on labor mobility:
  - Ex-post reservation value: v*_j ≡ ∫_{−∞}^∞ w*_j(x) dF_j(x) − λ s*_j − c_j. (Equation (6))
  - Ex-ante expected utility after cap ˆρ_j: u^C_j(ˆρ_j) ≡ w_j(0; ˆρ_j) + ∫_{−∞}^∞ 1[∆_j(x_j; ˆρ_j) > 0] ∆_j(x_j; ˆρ_j) dF_j(x_j) − c_j.
  - To prevent migration, banker who leaves must be paid v*_j − u^C_j(ˆρ_j).
- Regulation without restriction on labor mobility:
  - Typically u^C_j(ˆρ_j) < v*_j; compensation must be adjusted upward to maintain participation.
  - Two adjustment types:
    - (i) Increase fixed salary only. New fixed salary ˆw0j solves Equation (7): v*_j = w_j(0; ˆρ_j; ˆw0j, θ_j) + ∫ 1[∆_j(x_j; ˆρ_j; ˆw0j, θ_j) > 0] ∆_j(...) dF_j(x_j) − c_j.
      - Unique solution exists due to continuity and monotonicity in ˆw0j.
    - (ii) Increase both fixed salary and performance threshold. New threshold: ˆθ_j(ˆw0j) = θ_j + (ˆw0j − w0j)/β. ˆw0j solves Equation (8). Uniqueness not analytically guaranteed; numerical search found a single solution in simulations.
  - Anticipated outcome: labor market adjustments offset regulatory impact; unclear net effect on risk shifting and underinvestment without calibration.

### Calibration (Section 5)
- Data sources and construction:
  - Aggregate ex-post compensation and average fixed salary of bank CEOs in five regions (2005–2013) from S&P Capital IQ ({w0j, w*j}).
  - Aggregate exposure to risky assets ({s*_j}) recovered from market beta of banking sector by region, adjusting for leverage differential: s*_j = beta_j L_Mj / L_Bj.
  - Return on safe assets x0 proxied by average three-month U.S. Treasury bill rate 2005–2013 = 1.5 percent.
  - Distribution of risky returns F_j constructed from BIS consolidated banking statistics and accounting data, combined with geographical breakdown of bank assets and ROA distributions of local equity index constituents (Datastream).
  - Bankers’ belief about risky return G_x assumed normal around true return with standard deviation based on historical coefficient of variation of constituent ROA.
- Estimation strategy:
  - Step 1: Use linear relation (Equation (9)) (w*_j − w*_1) = λ(s*_j − s*_1) + (c_j − c_1) to estimate λ and relative entry costs c_j (set c_1 = 0).
  - Step 2: Estimate {θ_j, β_j} by minimizing objective (Equation (10)) matching simulated aggregate exposures and compensations to observed values via Monte Carlo simulation:
    - Draw K = 1000 region-investment samples and R = 100 forecast errors per draw.
    - Constraints β_j ≥ 0.
- Estimated parameters (reported in Tables 6 and 7):
  - First-step estimates:
    - λ = 3,593,384
    - c_AS ex U.S. = 1,962,863
    - c_U.S. = 3,746,119
    - c_CE = 623,313
    - c_EMEA = 821,428
  - Second-step estimates:
    - β_ASIA = 5.25×10^7
    - β_AS ex U.S. = 3.04×10^7
    - β_U.S. = 3.91×10^7
    - β_CE = 3.93×10^7
    - β_EMEA = 4.76×10^7
    - θ_ASIA = 1.80
    - θ_AS ex U.S. = -3.18
    - θ_U.S. = -3.09
    - θ_CE = 2.20
    - θ_EMEA = 0.60
- Model fit (Table 8) shows estimated versus actual:
  - Asia exposure: Actual 26.52 percent, Estimated 28.0; Compensation Actual $1,930,123, Estimated $1,920,500.
  - Anglo-Saxon ex U.S. exposure: Actual 24.3 percent, Estimated 24.2; Compensation Actual $3,812,937, Estimated $3,847,000.
  - U.S. exposure: Actual 30.8 percent, Estimated 30.8; Compensation Actual $5,829,346, Estimated $5,867,600.
  - Continental Europe exposure: Actual 23.1 percent, Estimated 23.1; Compensation Actual $2,429,083, Estimated $2,397,800.
  - EMEA exposure: Actual 38.9 percent, Estimated 41.0; Compensation Actual $3,196,900, Estimated $3,185,400.
- Status-quo estimates (Monte Carlo):
  - Risk shifting is not proportional to exposure to risky assets (e.g., U.S. has higher exposure but lower risk shifting than continental Europe).
  - Risk shifting peaks around 2008 across regions.
  - There is substantial underinvestment across regions.
- Robustness: Similar qualitative results when using five-year U.S. Treasury bill rate average 2005–2013 = 2.5 percent as x0 (Figures 6 and 7).

### Counterfactual analysis (Section 6)
- Simulation design:
  - Simulate average impact of bonus cap (cap expressed as multiple of fixed salary; lower cap → more severe) on risk shifting and underinvestment over 2005–2013 for each region.
  - Two labor mobility regimes:
    - No labor mobility (labor mobility restricted): bankers cannot leave; ex-ante expectations unchanged immediately after cap.
    - Free labor mobility (labor mobility not restricted): fixed salary adjustments (i) or fixed salary and performance threshold adjustments (ii) are used to restore bankers’ ex-ante utility.
- Main results:
  - With labor mobility restricted:
    - Risk shifting substantially reduced in all regions with a moderate cap.
    - Trade-off: reduction in risk shifting is associated with increase in underinvestment across regions.
    - Regulatory impact largest in emerging regions because fixed salary is low and cap effects are stronger.
  - With labor mobility not restricted (labor market adjustments):
    - Labor market adjustments substantially offset regulatory impact on risk shifting and underinvestment for most regions.
    - For Asia, Anglo-Saxon ex U.S., U.S., and Continental Europe, only very tight caps (5 × fixed wage or less) reduce risk shifting.
    - Exception: under adjustment type (ii) (simultaneous increases in w0 and θ), moderate caps eliminate risk shifting for EMEA due to shift in performance threshold.
- Crisis period (2008) results:
  - Simulated impact of bonus cap on risk shifting in 2008 largely offset by labor market adjustments for most regions under both adjustment (i) and (ii).
  - Note: actual labor market rigidity during crises might limit adjustments relative to simulated equilibrium.

### Discussion and policy implications (Section 7)
- Heterogeneity:
  - Substantial variation in risk shifting levels across regions under status-quo; regulators should assess regional level of risk shifting before setting targets.
  - Risk shifting lower in the U.S. than in other regions in simulations.
- Effectiveness of bonus caps:
  - Bonus caps can be effective at eliminating risk shifting only if labor market adjustments are prevented.
  - In practice, to substantially reduce risk shifting, regulators would need to restrict labor mobility or apply caps across sectors bankers can move to.
  - Example: European Banking Authority (EBA) decision to limit exemptions on bonus cap rules to ensure broader coverage.
- Trade-offs:
  - Bonus caps aggravate underinvestment; eliminating labor market adjustments does not resolve underinvestment concern.
- Extensions and caveats:
  - A dynamic model could alter predictions if future earnings threat affects behavior (may reduce risk shifting).
  - Including banker monitoring effort could change results:
    - If monitoring negatively correlated with profitability, model may underestimate underinvestment and overestimate risk shifting.
    - If positively correlated, opposite biases may occur.
  - Allowing heterogeneous costs by project type could change predictions on underinvestment and risk shifting.
- Key takeaway for policymakers:
  - Bonus caps reduce incentives for risky investment and are more effective for high-return projects, but labor market adjustments can largely offset intended regulatory effects and caps can increase underinvestment; careful calibration and consideration of labor mobility and cross-sector coverage are essential.

### Appendix (selected analytical results)
- Any finite bonus cap ρ′ reduces option value of risk-taking:
  - ∆(x; ρ′) − ∆(x; ∞) = −∫_{ρ′}^∞ ∂∆(x; y)/∂y dy = −∫_{ρ′}^∞ w0 (1 − G_x(y w0/β + θ)) dy < 0.
- Under first-order stochastic dominance (G_x′(.) < G_x′′(.) if x′ > x′′), reduction in option value is larger for higher-return investments:
  - (∆(x′; ρ′) − ∆(x′; ∞)) − (∆(x′′; ρ′) − ∆(x′′; ∞)) = ∫_{ρ′}^∞ ∫_{x′′}^{x′} w0 ∂G_z(y w0/β + θ)/∂x dz dy < 0.

*Italic: Source — _wp16196 (PDF): model of banker investment decisions, regional bank characterization, calibration, simulations of bonus cap impacts, and policy discussion.*

### REFERENCES

### _wp16196 - REFERENCES

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*Source: _wp16196 - REFERENCES*

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