Non-concave Corporate Management with Option Incentives under Value-at-Risk Constraint

2026-08-06

AI Summary

This study investigates how risk-averse managers optimally choose effort levels and project risk under Value-at-Risk (VaR) constraints to shape the firm’s terminal value distribution, given a compensation structure combining fixed salary and stock options. By integrating concavification techniques, quantile representations, martingale methods, and dynamic optimization, the paper provides the first analytical solution for optimal terminal wealth, effort provision, and project selection in a non-concave setting. It systematically characterizes nine distinct regimes through which VaR constraints affect the value distribution. The analysis reveals that moderate VaR limits enhance downside protection and reduce bankruptcy risk, whereas excessively stringent thresholds can induce gambling behavior in distress. Moreover, greater option-based incentives lead managers to act more prudently, while higher fixed pay results in a more dispersed value distribution.

Abstract

This article studies a dynamic corporate risk management problem by considering the decision-making of risk-averse managers who exert costly effort and select project risk. We study how a Value-at-Risk (VaR) constraint affects managerial decisions and the distribution of firm value when the manager's objective is non-concave with a fixed salary and options. By the concavification technique, we analyze the optimal terminal firm value on the concave envelope of the objective function. Applying the quantile formulation and the martingale approach, we can derive explicit solutions for optimal effort, terminal firm value, and project choice. The optimal terminal firm value can be divided into nine cases by carefully discussing the choices of VaR floor and tail probability. Compared with the benchmark case, we find that a VaR manager will smooth terminal firm value across states, reducing it in good states while supporting it in adverse states. Moreover, a VaR requirement generally improves downside protection and reduces bankruptcy probability when the VaR floor is low or moderate. However, when the VaR floor is sufficiently high, it can increase bankruptcy probability and induce gambling-for-recovery behavior in adverse states. Our sensitivity analysis indicates that greater managerial effort uniformly improves firm value. Moreover, more incentive options make managers more responsible, leading to a smoother terminal firm value across states. In contrast, a high fixed salary makes the manager less responsible and ultimately causes a more dispersed firm value.

Problem

• Value-at-Risk constraint
• non-concave objective
• corporate risk management
• managerial incentives
• firm value distribution

Innovation

• concavification
• Value-at-Risk constraint
• quantile formulation
• non-concave optimization
• martingale approach