Convergence Rates of Turnpike Theorems for Portfolio Choice in Stochastic Factor Models
Abstract
Turnpike theorems state that if an investor's utility is asymptotically equivalent to a power utility, then the optimal investment strategy converges to the CRRA strategy as the investment horizon tends to infinity. This paper aims to derive the convergence rates of the turnpike theorem for optimal feedback functions in stochastic factor models. In these models, optimal feedback functions can be decomposed into two terms: myopic portfolios and excess hedging demands. We obtain convergence rates for myopic portfolios in nonlinear stochastic factor models and for excess hedging demands in quadratic term structure models, where the interest rate is a quadratic function of a multivariate Ornstein-Uhlenbeck process. We show that the convergence rates are determined by (i) the decay speed of the price of a zero-coupon bond and (ii) how quickly the investor's utility becomes power-like at high levels of wealth. As an application, we consider optimal collective investment problems and show that sharing rules for terminal wealth affect convergence rates.
Summary
This paper investigates the convergence rates of turnpike theorems in portfolio choice within stochastic factor models. The core idea behind turnpike theorems is that, given a sufficiently long investment horizon, an investor's optimal strategy will closely resemble the constant relative risk aversion (CRRA) strategy if their utility function is asymptotically similar to a power utility. The paper focuses on determining how quickly this convergence occurs for both myopic portfolios (those disregarding future investment opportunities) and excess hedging demands (adjustments made to account for future changes in the investment environment). The methodology involves a probabilistic approach leveraging martingale duality methods and Malliavin calculus. The analysis is conducted in two main settings: general stochastic factor models and quadratic term structure models. The paper derives convergence rates for myopic portfolios in the former and for excess hedging demands in the latter. The authors show that these rates are influenced by two key factors: the decay speed of zero-coupon bond prices and the rate at which the investor's utility function becomes power-like at high wealth levels. As an application, the paper explores optimal collective investment problems, demonstrating that the specific rules used to share terminal wealth among investors impact the convergence rates. The main contributions are the derivation of convergence rates for both myopic portfolios and excess hedging demands, a novel application of Malliavin calculus with assumptions independent of the investment horizon, and an analysis of the impact of sharing rules on convergence in collective investment problems. This research is significant because it provides a quantitative understanding of how quickly optimal investment strategies converge to the CRRA strategy under general utility functions and stochastic environments, offering practical implications for portfolio management and financial modeling.
Key Insights
- •The convergence rate of the turnpike theorem in stochastic factor models is determined by the decay speed of zero-coupon bond prices (E[H_T]) and the similarity between the investor's utility function and a power utility (captured by parameters α and q).
- •The paper extends previous results on convergence rates from the Black-Scholes model to more general stochastic factor models, showing that the rate `E[H_T]^(1 - α/q-1)` is a natural generalization of the rate in Bian and Zheng [3]. If the interest rate is a positive constant, r(Y_t) = r > 0, then the convergence rate is e^(-r(1 - α/q-1)T), which is the same rate as in Bian and Zheng [3].
- •Uniform convergence in the wealth variable for portfolio proportions is proven (Theorem 2.5), a result not previously documented.
- •The paper provides the first derivation of convergence rates for excess hedging demands, showing they are the same as those of myopic portfolios and uniform convergence in wealth also holds.
- •In optimal collective investment problems, the sharing rules for terminal wealth significantly affect the convergence rates. A linear sharing rule can lead to faster convergence than a Pareto-optimal sharing rule when the least risk-averse investor is no less risk-averse than the log investor.
- •The paper assumes conditions (Assumptions 2.1–2.3) that are independent of the investment horizon, allowing for the application of Malliavin calculus techniques as the investment horizon tends to infinity.
- •The convergence rate of the turnpike theorem under quadratic term structure models is given by O(E[H_T]^(γ_(n-1) - γ_n)/γ_(n-1)), where γ_i represents the relative risk aversion level of the i-th investor.
Practical Implications
- •Portfolio managers can use the derived convergence rates to assess how long they need to maintain strategies tailored to specific utility functions before they can safely transition to simpler CRRA strategies, optimizing computational efficiency and reducing model complexity.
- •Financial modelers can incorporate the factors influencing convergence rates (bond price decay, utility function similarity) into their models to improve the accuracy of long-term portfolio projections and risk assessments.
- •The analysis of collective investment problems provides insights for fund managers designing wealth-sharing rules, highlighting the trade-offs between different rules and their impact on the convergence of optimal strategies.
- •Future research could extend the analysis to nonlinear stochastic factor models, potentially requiring more advanced techniques due to the complexity of solving semilinear PDEs for optimal portfolios in such models.
- •The methodology can be applied to other areas of financial economics where turnpike theorems are relevant, such as consumption-investment problems and dynamic asset pricing.