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An extended Merton problem with relaxed benchmark tracking

Lijun Bo, Yijie Huang, Xiang Yu

TL;DR

This paper extends Merton's consumption-portfolio problem by incorporating benchmark tracking against a stochastic process through a cost on the largest shortfall, enabled by a fictitious capital-injection mechanism. It develops a novel convex duality framework with a reflected auxiliary state and a dual process that reflects from above, yielding a rigorous connection between primal and dual value functions and enabling optimal controls in general utility settings. In the CRRA-GBM setting, the authors derive semi-analytical solutions for the primal controls and provide asymptotic insights, showing how capital injection induces more aggressive risk-taking and a subsistence consumption level. Numerical experiments and an empirical calibration illustrate the practical implications, including how the tracking incentive alters portfolio and consumption choices and the sensitivity to benchmark dynamics and injection costs.

Abstract

This paper studies Merton's problem in an extended formulation by incorporating the benchmark tracking on the wealth process. We consider a tracking formulation where the fund manager aims to maximize the trade-off between the expected utility of consumption and the expected largest shortfall of the wealth with reference to the benchmark level. Equivalently, the problem can be interpreted as a mixed stochastic control problem if a fictitious capital injection singular control is allowed, subjecting to the dynamic constraint that the wealth process compensated by the costly capital injection outperforms the benchmark at all times. By considering an auxiliary state process, we formulate an equivalent stochastic control problem with state reflections at zero. For general utility functions and Ito's diffusion benchmark process, we develop a convex duality theorem, new to the literature, to the auxiliary stochastic control problem with state reflections in which the dual process also exhibits reflections from above. For CRRA utility and geometric Brownian motion benchmark process, we further derive the optimal portfolio and consumption in feedback form using the new duality theorem, allowing us to discuss some interesting financial implications induced by the additional risk-taking from the capital injection and the goal of tracking.

An extended Merton problem with relaxed benchmark tracking

TL;DR

This paper extends Merton's consumption-portfolio problem by incorporating benchmark tracking against a stochastic process through a cost on the largest shortfall, enabled by a fictitious capital-injection mechanism. It develops a novel convex duality framework with a reflected auxiliary state and a dual process that reflects from above, yielding a rigorous connection between primal and dual value functions and enabling optimal controls in general utility settings. In the CRRA-GBM setting, the authors derive semi-analytical solutions for the primal controls and provide asymptotic insights, showing how capital injection induces more aggressive risk-taking and a subsistence consumption level. Numerical experiments and an empirical calibration illustrate the practical implications, including how the tracking incentive alters portfolio and consumption choices and the sensitivity to benchmark dynamics and injection costs.

Abstract

This paper studies Merton's problem in an extended formulation by incorporating the benchmark tracking on the wealth process. We consider a tracking formulation where the fund manager aims to maximize the trade-off between the expected utility of consumption and the expected largest shortfall of the wealth with reference to the benchmark level. Equivalently, the problem can be interpreted as a mixed stochastic control problem if a fictitious capital injection singular control is allowed, subjecting to the dynamic constraint that the wealth process compensated by the costly capital injection outperforms the benchmark at all times. By considering an auxiliary state process, we formulate an equivalent stochastic control problem with state reflections at zero. For general utility functions and Ito's diffusion benchmark process, we develop a convex duality theorem, new to the literature, to the auxiliary stochastic control problem with state reflections in which the dual process also exhibits reflections from above. For CRRA utility and geometric Brownian motion benchmark process, we further derive the optimal portfolio and consumption in feedback form using the new duality theorem, allowing us to discuss some interesting financial implications induced by the additional risk-taking from the capital injection and the goal of tracking.
Paper Structure (6 sections, 12 theorems, 116 equations, 10 figures, 3 tables)

This paper contains 6 sections, 12 theorems, 116 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Market Model and Problem Formulation
  3. Equivalent Control Problem and Duality Theorem
  4. The Case with CRRA Utility and GBM Benchmark
  5. Numerical Examples and Financial Implications
  6. Proofs

Key Result

Lemma 3.1

For value functions $\mathrm{w}$ defined in eq_orig_pb and $v$ defined in eq:u0, we have ${\rm w}(\mathrm{v},z)=v((\mathrm{v}-z)^+,z)$ for all $(\mathrm{v},z)\in\mathds{R}_+\times \mathds{R}$. Moreover, the value function $x\to v(x,z)$ defined in eq:u0 is non-decreasing and $\left|v(x_1,z)-v(x_2,z)\

Figures (10)

  • Figure 2: (a): The CRA level $\mu_Z\to \text{CRA}(\mu,\sigma,\mu_Z,\sigma_Z)$. (b): The CRA level $\sigma_Z\to \text{CRA}(\mu,\sigma,\mu_Z,\sigma_Z)$ . The model parameters are set as $\rho=5,~\mu=1,~\sigma=1,~p=-1,~\gamma=1,~\beta=1$.
  • Figure 5: The optimal portfolio $x\to \theta^*(x,1)$. The model parameters are set as $\rho=7,~\sigma=1,~\mu_Z=1,~\sigma_Z=1,~\gamma=1,~\beta=3$ and $\mu=0.1$ in panel (a), $\mu=1$ in panel (b).
  • Figure 6: The optimal consumption $x\to c^*(x,1)$. The model parameters are set as $\rho=7,~\mu=1,~\sigma=1,~\sigma_Z=1,~\gamma=1,~\beta=2$ and $\mu_Z=6.5$ in panel (a), $\mu_Z=2$ in panel (b).
  • Figure 7: (a): The variance value of optimal consumption. (b): The expected largest shortfall. The model parameters are set as $(x,z)=(1,0.5),\rho=5,~\mu=1,~\sigma=1,~\mu_Z=2,~\sigma_Z=1,~\gamma=1,~\beta=1$.
  • Figure 8: (a): The expectation of the total optimal discounted capital injection. (b): The optimal portfolio $x\to \theta^*(x,1)$. (c): The optimal consumption $x\to c^*(x,1)$. The model parameters are set as $z=1,~\rho=8,~\mu=1,~\sigma=1,~\sigma_Z=1,\gamma=1, ~p=0.5,~\beta=2$.
  • ...and 5 more figures

Theorems & Definitions (27)

  • Lemma 3.1
  • Lemma 3.2: Characterization of admissible portfolio and consumption
  • Remark 3.3
  • Theorem 3.4: Duality Theorem
  • Corollary 3.5
  • Lemma 3.6
  • Remark 3.7
  • Remark 3.8
  • Proposition 4.1
  • Corollary 4.2
  • ...and 17 more