[Paper Review] Ramsey Rule with Progressive Utility in Long Term Yield Curves Modeling

Nicole El Karoui, Caroline Hillairet+1 more|arXiv (Cornell University)|Apr 7, 2014

Economic theories and models33 references3 citations

TL;DR

This paper extends the Ramsey rule to long-term yield curve modeling by introducing progressive utility for both investment and consumption, solving a second-order stochastic PDE of HJB type to ensure intertemporal consistency. It shows that the marginal utility yield curve depends on risk aversion and volatility structure, with decreasing yield curves possible under low risk aversion and persistent volatility.

ABSTRACT

The purpose of this paper relies on the study of long term yield curves modeling. Inspired by the economic litterature, it provides a financial interpretation of the Ramsey rule that links discount rate and marginal utility of aggregate optimal consumption. For such a long maturity modelization, the possibility of adjusting preferences to new economic information is crucial. Thus, after recalling some important properties on progressive utility, this paper first provides an extension of the notion of a consistent progressive utility to a consistent pair of progressive utilities of investment and consumption. An optimality condition is that the utility from the wealth satisfies a second order SPDE of HJB type involving the Fenchel-Legendre transform of the utility from consumption. This SPDE is solved in order to give a full characterization of this class of consistent progressive pair of utilities. An application of this results is to revisit the classical backward optimization problem in the light of progressive utility theory, emphasizing intertemporal-consistency issue. Then we study the dynamics of the marginal utility yield curve, and give example with backward and progressive power utilities.

Motivation & Objective

• To model long-term yield curves in illiquid markets where standard financial models fail.
• To incorporate adaptive preferences via progressive utility that adjust to new economic information over time.
• To establish a financial interpretation of the Ramsey rule using marginal utility and consumption dynamics.
• To resolve intertemporal consistency issues in backward optimization by embedding progressive utility theory.
• To analyze the dynamics of the marginal utility yield curve under power utility functions.

Proposed method

• Introduces a consistent pair of progressive utilities for investment and consumption, extending classical utility theory to dynamic, information-adaptive frameworks.
• Derives and solves a second-order stochastic PDE of HJB type involving the Fenchel-Legendre transform of consumption utility.
• Applies the Davis price (marginal utility indifference pricing) to price non-replicable zero-coupon bonds in incomplete markets.
• Uses affine factor models and Vasicek-type short rate dynamics to derive explicit expressions for bond volatility and yield curve behavior.
• Distinguishes between forward and backward optimization frameworks, showing how maturity and risk aversion affect yield curve shape.
• Analyzes asymptotic yield curve behavior as maturity tends to infinity, particularly under long-run volatility conditions.

Experimental results

Research questions

• RQ1 How can the Ramsey rule be financially interpreted in the context of long-term yield curve modeling?
• RQ2 What conditions ensure intertemporal consistency in long-horizon portfolio optimization under progressive utility?
• RQ3 How does the marginal utility yield curve behave as maturity approaches infinity under progressive power utilities?
• RQ4 What is the impact of risk aversion and volatility structure on the shape of the yield curve in backward optimization?
• RQ5 How do forward and backward progressive utility frameworks differ in their yield curve implications?

Key findings

• The marginal utility yield curve coincides with the equilibrium interest rate under the Ramsey rule, but this robustness is limited to small trades.
• For infinite maturity, the yield curve is non-decreasing in risk aversion α if α ≥ 1/2; it may decrease if α < 1/2, depending on the long-run volatility structure.
• The optimal investment strategy depends on the maturity only through the risk premium component Γᴿ, while consumption volatility ν* is proportional to the orthogonal volatility Γ⊥.
• In the Vasicek model with orthogonal noise, ν* depends on time to maturity (T−t), while κ* is independent of maturity.
• When lim_{T→∞} ||Γ(T)||²/(T−t) > 0 and α < 1/2, the yield curve for infinite maturity can be decreasing, even in a log-normal market.
• The Davis price provides a linear pricing rule that enables consistent valuation of non-replicable zero-coupon bonds under progressive utility.