Source-linked AI summary
Time consistent portfolio management
Ivar Ekeland, Oumar Mbodji, Traian A. Pirvu
TL;DR
The paper studies how non-constant discounting makes investment, consumption, and life-insurance strategies time-inconsistent and develops policies that successive selves can implement with only infinitesimal commitment. It characterizes these policies through an integral-equation framework and finds that hyperbolic discounting can generate hump-shaped consumption, while aggregation weights affect insurance outcomes.
Problem
Non-constant discounting makes conventional optimal strategies time-inconsistent and non-implementable, including when consumption and life insurance have different discounting or aggregation weights vary over time.
Method
The paper models successive selves as a leader-follower game, defines policies through infinitesimal-commitment deviations, and characterizes them with an integral equation under CRRA preferences.
Results
Hyperbolic discounting can produce hump-shaped consumption unlike exponential discounting, and numerical experiments examine how time-varying aggregation weights affect insurance premiums.
Takeaways & Limitations
Time-consistent policies extend portfolio management beyond the classical optimization paradigm to settings with hyperbolic discounting and multiple decision-makers.
Takeaways & Limitations
The paper has no general existence or uniqueness theory for the integral equations and PDEs replacing the classical HJB equation, using an Ansatz to sidestep this difficulty.
Abstract
from arXiv · showhide
This paper considers the portfolio management problem of optimal investment, consumption and life insurance. We are concerned with time inconsistency of optimal strategies. Natural assumptions, like different discount rates for consumption and life insurance, or a time varying aggregation rate lead to time inconsistency. As a consequence, the optimal strategies are not implementable. We focus on hyperbolic discounting, which has received much attention lately, especially in the area of behavioural finance. Following [10], we consider the resulting problem as a leader-follower game between successive selves, each of whom can commit for an infinitesimally small amount of time. We then define policies as subgame perfect equilibrium strategies. Policies are characterized by an integral equation which is shown to have a solution. Although we work on CRRA preference paradigm, our results can be extended for more general preferences as long as the equations admit solutions. Numerical simulations reveal that for the Merton problem with hyperbolic discounting, the consumption increases up to a certain time, after which it decreases; this pattern does not occur in the case of exponential discounting, and is therefore known in the litterature as the "consumption puzzle". Other numerical experiments explore the effect of time varying aggregation rate on the insurance premium.
1 Introduction
The paper extends portfolio management with investment, consumption, and life insurance to non-constant discounting, addressing time inconsistency through infinitesimal-commitment policies. It focuses on hyperbolic discounting and shows that it can generate hump-shaped consumption patterns and affects insurance decisions.
- Motivation: Hyperbolic discounting is motivated by evidence that people are more sensitive to delays occurring earlier, producing the common difference effect.Its discount factor is h(t) = (1 + at)^−b, with a, b > 0.
- Motivation: Non-constant discount rates make optimal strategies time-inconsistent and therefore potentially non-implementable without commitment.Different selves may prefer different strategies over the same future interval.
- Approach: The paper characterizes time-consistent strategies as policies using an integral-equation approach that extends earlier BSDE-based results to more general discount rates and problems.Under CRRA utilities, the problem is reduced to a one-dimensional integral equation solved by a fixed-point argument.
- Results: Hyperbolic discounting can produce hump-shaped consumption, unlike the smoothly growing or declining consumption predicted by the Merton model.The paper identifies this as the consumption puzzle and reports it as a time-consistent strategy in certain cases.
- Results: Numerical simulations examine how the insurer’s weight assigned to beneficiaries affects the life insurance process.The study compares the effects of discounting and aggregation on portfolio-management outcomes.
2 The Model
The model combines risky investment, consumption, life insurance, random lifetime, and terminal wealth under a non-exponential expected-utility criterion. Because discounting is non-constant, the paper replaces ordinary optimality with policies requiring only infinitesimal deviations to be unprofitable.
- 2.1 The decisions: The investor trades a risk-free asset and one stock, consumes, buys life insurance, and receives continuous deterministic income.The stock is modeled through Brownian motion, with excess return μ = α − r > 0.
- 2.1 The decisions: Life insurance is modeled as infinitesimal term contracts paying l(t) to beneficiaries upon immediate death, while p(t) denotes the insurance amount.The legacy combines η(t)X(t) and l(t)p(t), and negative p(t) permits selling insurance.
- 2.3 The intertemporal utility: The expected-utility criterion values intertemporal consumption, terminal wealth, and legacy, with discount and weighting functions that need not be exponential.The coefficient n weights terminal wealth relative to continuous consumption, while h(t) discounts utility.
- 2.4 Time-consistent strategies: Non-exponential discounting makes a strategy optimal at one time potentially unacceptable later, so it will not be implemented without commitment.This motivates replacing classical optimal strategies with policies.
- 2.4 Time-consistent strategies: A policy is a Markov strategy that remains optimal against unilateral deviations over an infinitesimal interval while successors follow the policy.The controls are represented by feedback maps of time and wealth.
- 2.4 Time-consistent strategies: Policies are a minimal rationality requirement because any non-policy Markov strategy eventually creates an incentive for a lone decision-maker to deviate.Finite-interval deviations need not be penalized by the policy definition.
3 The Value Function
The value-function framework defines a fixed point for the criterion generated by the Markov strategy associated with the value function. Feedback controls are constructed from this function under regularity and PDE assumptions.
- 3 The Value Function: A value function is a C1,2 function, concave in wealth, that satisfies the paper’s policy-related characterization.Its associated Markov strategy is defined through feedback functions F1, F2, and F3.
- 3 The Value Function: The value function has a fixed-point interpretation: applying its associated Markov strategy yields exactly the investor criterion represented by the function.This connects the function to the corresponding wealth process.
- 3 The Value Function: The feedback controls are obtained from the value function and enter the wealth dynamics through the operator and PDE system.The construction uses regularity and technical assumptions on the relevant PDEs.
- 3 The Value Function: The construction requires a C1,2 solution with exponential growth for the relevant backward problem and prescribed terminal or intermediate functions.The functions include utilities evaluated at consumption, legacy, and terminal wealth.
4 Main Result
The paper’s central result connects a suitable value function to subgame-perfect policy strategies in the time-inconsistent portfolio problem. The value function satisfies a partial differential equation with a non-local term, and the resulting controls form a policy.
- A value function satisfying the paper’s assumptions generates a policy through the specified investment, consumption, and insurance controls.
- The value function is characterized by a partial differential equation containing a non-local term and a terminal boundary condition.
- Concavity of the value function allows the governing equation to be rewritten in a form used to establish the policy property.
5 CRRA Preferences
Under CRRA preferences, the paper reduces the value function and policy construction to an integral equation and establishes existence results under stated assumptions. In the Merton application, hyperbolic discounting permits non-monotone, hump-shaped consumption unlike exponential discounting.
- CRRA Preferences: For CRRA utilities, the value function has the form v(t, x) = a(t)Uγ(x + b(t)), where a(t) satisfies a fixed-point integral equation.
- CRRA Preferences: If Assumption 5.2 holds, the integral equation has a unique global C1 solution, yielding a value function and an associated policy.
- Exponential Discounting: The stock investment amount remains the same as in the standard exponential-discounting Merton problem, attributed to constant stock return and volatility.
- Exponential Discounting: With exponential discounting, the equilibrium policy coincides with the dynamically optimal strategy, whereas consumption and insurance policies generally differ under non-exponential discounting.
- 5.3 The Merton Problem with Hyperbolic Discounting: In the Merton problem with hyperbolic discounting, consumption increases to a satiation point and then decreases, while higher terminal-wealth preference makes that point earlier.
- 5.3 The Merton Problem with Hyperbolic Discounting: A hyperbolic discount function can produce a consumption policy that is neither increasing nor decreasing over time.
- 5.4 The Stationary Case: In the stationary case, the relevant equations have a unique solution, and the resulting functions define a policy.
6 Numerical Results
The numerical section develops approximation schemes for the governing integral equation and examines how discounting and aggregation affect portfolio policies. In the reported experiment, higher aggregation weight increases life-insurance spending.
- Approximation scheme: The numerical procedure discretizes the interval and constructs recursive and explicit approximation schemes for the integral equation.The scheme is presented in multiple steps, including interval discretization and an explicit scheme.
- Approximation scheme: The approximation results are summarized by a theorem for the linearly interpolated function aN(t), with an error bound involving a constant independent of N.The supplied theorem passages state the interpolation setup and the independence of the constant from N.
- Numerical experiment: The experiment uses T = 4, r = 0.05, µ = 0.07, σ = 0.2, p = −1, N = 1000, and ρ = 0.8.These parameters define the illustrative numerical setting.
- Numerical experiment: Higher utility weight m leads to a higher amount spent on life insurance in the numerical experiment.The experiment compares variable and constant aggregation weights while plotting policy maps F2 and F3.
7 Conclusion and future research
The paper formulates time-consistent portfolio management as a subgame-perfect equilibrium problem for investment, consumption, and life insurance. It reports numerical effects of discounting and aggregation while identifying unresolved existence and generality issues.
- Conclusion: Different discount rates for the agent and heirs, or a changing aggregation weight, produce time inconsistency handled through subgame-perfect Nash equilibrium policies.The paper calls these equilibrium strategies policies.
- Conclusion: The model studies an agent investing in a risky asset, consuming, and purchasing life insurance for the agent and heirs.This is the portfolio-management setting analyzed throughout the paper.
- Future research: The introduced integral equation, SDE, and PDE system is established for CRRA utility and bequest functions, with possible extension to concave utilities if the equations admit solutions.The authors present this extension as a direction rather than a completed general result.
- Future research: The paper has no general existence or uniqueness theory for the equations replacing the classical HJB equation and uses an Ansatz to sidestep this difficulty.The authors identify resolving these problems as future work.
- Group portfolios: For groups with two members, even equal constant psychological discount rates can yield time inconsistency after utilities are aggregated with Pareto weights.The model represents group preferences through member-specific utilities, discount factors, and weights.
- Future research: The more general case with unequal member utilities and other heterogeneous-agent macroeconomic problems is left for future research.This defines a stated scope boundary of the current model.
8 Appendix
The appendix supplies proofs for the paper’s analytical results, including PDE representations, integral-equation identities, bounds, and numerical-experiment properties. These arguments support the stated policy and approximation results.
- Proofs: Feynman–Kac representations express the auxiliary functions f1, f2, and f3 as expectations of utility-related quantities.The displayed representations connect the PDE system to expected values.
- Proofs: The appendix uses PDE boundary conditions and regularity assumptions to establish the auxiliary functions needed in the analytical arguments.The supplied passages state C1,2 solutions and list the boundary conditions.
- Proofs: The appendix derives the central integral-equation relation by combining differentiated identities with the model’s preceding equations.The derivation proceeds through equations (8.5)–(8.8) to obtain (4.1).
- Existence arguments: For linear combinations of exponential discount functions, the governing equation becomes an ODE system with local existence, while bounds lead to global existence.The appendix proof describes these steps for Proposition 5.3.
- Numerical properties: Under the numerical-experiment parameters, a(t) increases on part of the interval and decreases near zero, determining non-monotonic behavior of the consumption-rate policy.The appendix attributes the two directional behaviors to different terms in the differential equation.
Appendix E: Proof of Lemma 5.9: Let x ≜a
The appendix proves error and stability bounds for recursive approximation schemes involving a(t) and related sequences. The arguments use induction, Taylor expansion, boundedness, and discrete inequalities.
- Proof of Lemma 5.9: The analysis is split into the cases γ ∈ (0, 1), γ = 0, and γ < 0.The supplied passage states the case distinction without giving its omitted details.
- Proof of Lemma 5.9: The recursive expansion uses a second-order Taylor expansion of a(t_{n+1}) around a(t_n).The remainder coefficient is described as bounded independently of n.
- Proof of Lemma 5.9: The approximation error satisfies a recurrence with linear growth and an O(ε^2) remainder, leading to a bound proportional to |ε|.The displayed recurrence is iterated over the discretization steps.
- Proof of Lemma 5.9: The proof establishes recursive-scheme well-definedness by induction using bounded coefficients and boundedness of a(t).The argument introduces uniform constants before deriving the recursive inequalities.