Ramsey Rule with Progressive utility and Long Term Affine Yields Curves
Abstract
The purpose of this paper relies on the study of long term affine yield curves modeling. It is inspired by the Ramsey rule of the economic literature, 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, justifying the use of progressive utility. This paper studies, in a framework with affine factors, the yield curve given from the Ramsey rule. It first characterizes consistent progressive utility of investment and consumption, given the optimal wealth and consumption processes. A special attention is paid to utilities associated with linear optimal processes with respect to their initial conditions, which is for example the case of power progressive utilities. Those utilities are the basis point to construct other progressive utilities generating non linear optimal processes but leading yet to still tractable computations. This is of particular interest to study the impact of initial wealth on yield curves.
Keywords: Progressive utility with consumption, market consistency, portfolio optimization, Ramsey rule, affine yields curves.
Introduction
This paper focuses on the modelization of long term affine yield curves. For the financing of ecological project, for the pricing of longevitylinked securities or any other investment with long term impact, modeling long term interest rates is crucial. The answer cannot be find in financial market since for longer maturities, the bond market is highly illiquid and standard financial interest rates models cannot be easily extended. Nevertheless, an abundant literature on the economic aspects of longterm policymaking has been developed. The Ramsey rule, introduced by Ramsey in his seminal work [[26]] and further discussed by numerous economists such as Gollier [[4], [7], [11], [6], [8], [10], [5], [9]] and Weitzman [[28], [29]], is the reference equation to compute discount rate, that allows to evaluate the future value of an investment by giving a current equivalent value. The Ramsey rule links the discount rate with the marginal utility of aggregate consumption at the economic equilibrium. Even if this rule is very simple, there is no consensus among economists about the parameters that should be considered, leading to very different discount rates. But economists agree on the necessity of a sequential decision scheme that allows to revise the first decisions in the light of new knowledge and direct experiences: the utility criterion must be adaptative and adjusted to the information flow. In the classical optimization point of view, this adaptative criteria is called consistency. In that sense, marketconsistent progressive utilities, studied in El Karoui and Mrad [[15], [14], [13]], are the appropriate tools to study long term yield curves.
Indeed, in a dynamic and stochastic environment, the classical notion of utility function is not flexible enough to help us to make good choices in the long run. M. Musiela and T. Zariphopoulou (20032008 [[21], [22], [20], [19]]) were the first to suggest to use instead of the classical criterion the concept of progressive dynamic utility, consistent with respect to a given investment universe in a sense specified in Section 1. The concept of progressive utility gives an adaptative way to model possible changes over the time of individual preferences of an agent. In continuation of the recent works of El Karoui and Mrad [[15], [14], [13]], and motived by the Ramsey rule (in which the consumption rate is a key process), [[16]] extends the notion of marketconsistent progressive utility to the case with consumption: the agent invest in a financial market and consumes a part of her wealth at each instant. As an example, backward classical value function is a progressive utility, the way the classical optimization problem is posed is very different from the progressive utility problem. In the classical approach, the optimal processes are computed through a backward analysis, emphasizing their dependency to the horizon of the optimization problem, while the forward point of view makes clear the monotony of the optimal processes to their initial conditions. A special attention is paid to progressive utilities generating linear optimal processes with respect to their initial conditions, which is for example the case of power progressive utilities.
As the zerocoupon bond market is highly illiquid for long maturity, it is relevant, for small trades, to give utility indifference price (also called Davis price) for zero coupon, using progressive utility with consumption. We study then the dynamics of the marginal utility yield curve, in the framework of progressive and backward power utilities (since power utilities are the most commonly used in the economic literature) and in a model with affine factors, since this model has the advantage to lead to tractable computations while allowing for more stochasticity than the log normal model studied in [[16]]. Nevertheless, using power utilities implies that the impact of the initial economic wealth is avoided, since in this case the optimal processes are linear with respect to the initial conditions. We thus propose a way of constructing, from power utilities, progressive utilities generating non linear optimal processes but leading yet to still tractable computations. The impact of the initial wealth for yield curves is discussed.
The paper is organized as follows. After introducing the investment universe, Section 1 characterizes consistent progressive utility of investment and consumption, given the optimal wealth and consumption processes. Section 2 deals with the computation of the marginal utility yield curve, inspired by the Ramsey rule. Section 3 focuses on the yield curve with affine factors, in such a setting the yield curve does not depend on the initial wealth of the economy. Section 4 provides then a modelization for yield curves dynamics that are nonlinear to initial conditions.
1 Progressive Utility and Investment Universe
1.1 The investment universe
We consider an incomplete Itô market, equipped with a standard Brownian motion,
and characterized by an adapted short rate and an adapted
dimensional risk premium vector
. All these processes are defined on a filtered probability space
satisfying usual assumptions; they are progressively processes satisfying minimal integrability assumptions, as .
The agent may invest in this financial market and is allowed to consume a part of his wealth at
the rate .
To be short, we give the mathematical definition of the
class of admissible strategies , without specifying the risky assets. Nevertheless, the incompleteness of
the market is expressed by restrictions on the risky strategies constrained to live in a given progressive vector space
, often obtained as the range of some progressive linear operator .
Definition 1.1 (Test processes).
(i) The selffinancing dynamics of a wealth process with risky portfolio and consumption rate is given by
(1.1) 
where is a positive progressive process, is a progressive dimensional vector in , such that .
(ii) A strategy is said to be admissible if it is stopped with the bankruptcy of the investor (when the wealth process
reaches ).
(iii)The set of the wealth processes with admissible , also called test processes, is denoted by . When portfolios are starting from at time , we use the notation
The following short notations will be used extensively.
Let be a vector subspace of . For any ,
is the orthogonal projection of the vector
onto and is the orthogonal projection
onto .
The existence of a risk premium is a possible
formulation of the absence of arbitrage opportunity. From
Equation (1.1), the minimal state price process , whose the dynamics is , belongs to the convex family of positive Itô’s processes such that is a local martingale for any admissible portfolio. The existence of equivalent martingale measure is obtained by the assumption that the exponential local martingale
is a uniformly integrable martingale. Nevertheless, we are interested into the class of
the socalled state price processes belonging to the family characterized below.
Definition 1.2 (State price process).
(i) An Itô semimartingale
is called a state price process in if for any test process
,
is a local martingale.
(ii) This property is equivalent to the existence of progressive process such that where is the product of by the exponential local martingale , and satisfies
(1.2) 
1.2 consistent Utility and Portfolio optimization with consumption
In long term (wealthconsumption) optimization problems, it is useful to have the choice to adapt utility criteria to deep macroevolution of economic environment. The concept of progressive utility is introduced in this sense. As we are interested in optimizing both the terminal wealth and the consumption rate, we introduce two progressive utilities , for the terminal wealth and for the consumption rate, often called utility system. For sake of completeness, we start refer the reader to [[15]] for a detailled study.
Definition 1.3 (Progressive Utility).
(i) A progressive utility is a  progressive random field on ,
, starting from the deterministic utility function at time , such that for every ,
is a strictly concave, strictly increasing, and
non negative utility function, and satisfying thee Inada conditions:
for every , goes to when goes to
the derivative (also called marginal utility) goes to when goes to ,
the derivative goes to when goes to .
For , the deterministic utilities and are denoted and and in the following small letters and design deterministic utilities while capital letters refer to progressive utilities.
As in statistical learning, the utility criterium is dynamically adjusted to be the best given the market past information. So, market inputs may be viewed as a calibration universe through the testclass of processes on which the utility is chosen to provide the best satisfaction. This motivates the following definition of consistent utility system.
Definition 1.4.
A consistent progressive utility system of investment and consumption
is a pair of progressive utilities and on such that,
(i) Consistency with the testclass: For
any admissible wealth process ,
In other words, the process is a positive supermartingale, stopped at the first time of bankruptcy.
(ii) Existence of optimal strategy:
For any initial wealth , there exists an optimal strategy such that the associated non negative
wealth process issued from satisfies is a
local martingale.
(iii)To summarize, is the value function of optimization problem with optimal strategies, that is for any maturity
(1.3) 
The optimal strategy which is optimal for all these problems, independently of the timehorizon , is called a myopic strategy.
(iv)Strongly consistency
The system is said to be strongly consistent if the optimal process is strictly increasing with respect to the initial condition .
Convex analysis showed the interest to introduce the convex conjugate utilities and defined as the FenchelLegendre random field (similarly for ). Under mild regularity assumption, we have the following results (KaratzasShreve [[12]], Rogers [[27]]).
Proposition 1.5 (Duality).
Let be a pair of stochastic consistent utilities with optimal strategy leading to the non negative
wealth process . Then the convex
conjugate system satisfies :
(i) For any admissible state price density process with ,
is a submartingale, and there exists a unique optimal process with such that is a local martingale.
(ii) To summarize, is the value function of optimization problem with myopic optimal strategy, that is for any maturity
(1.4) 
(iii)Optimal Processes characterization Under regularity assumption, first order conditions imply some links between optimal processes, including their initial conditions,
(1.5) 
The optimal consumption process is related to the optimal portfolio by the progressive monotonic process defined by
(1.6) 
(iv)By Equation (1.5), strong consistency of implies the monotony of . The system is strongly consistent.
The main consequence of the strong consistency is to provide a closed form for consumption consistent utility system.
Theorem 1.6.
Let be a positive progressive process, increasing in and let be a strictly monotonic solution with inverse of the SDE,
Let be a strictly monotonic solution with inverse of the SDE
Given a deterministic utility system such that , there exists a consistent progressive utility system such that are the associated optimal processes, defined by:
(1.7) 
Observe that the consumption optimization contributes only through the conjugate of the progressive utility . We refer to [[16]] for detailed proofs.
1.3 consistent utilities with linear optimal processes
The simplest example of monotonic process is given by linear processes with positive (negative) stochastic coefficient. It is easy to characterize consumption consistent utility sytems associated with linear optimal processes
Proposition 1.7.
(i) A strongly consistent progressive utility generates linear optimal
wealth and state price processes if and only if it is of the form
The optimal processes are then given by
(ii) Power utilities A consumption consistent progressive power utility (with risk aversion coefficient ) generates necessarily linear optimal processes and is, consequently, of the form .
Proof.
If and , their inverse flows are also linear and
.
(i) a)The linearity with respect to its initial condition of the solution of one dimensional SDE with drift and
diffusion coefficient can be satisfied only when the coefficients and are affine in ,
that is and
, and being one dimensional progressive processes.
Since the only coefficient with some non linearity in the dynamics of is , the previous condition implies that does not depend on . By the same argument, we see that is linear and also does not depend on . For the consumption process, the linear condition becomes
b)We are concerned by strongly consistent progressive utilities, since optimal processes are monotonic by definition.
Then, since
, we see that the marginal utility is given by . By taking the primitive with the condition is given by .
c)We know that and
(from optimality conditions).
Thus
From monotonicity of and , we then conclude that
Integrating yields the desired formula.
(ii) Powertype utilities generate linear optimal processes. So, we only have to consider initial power utilities with the same riskaversion coefficient to characterize the system.
∎
Remark 1.1.
In order to separate the messages and as the risk aversion does not vary in this result, we have deliberately omitted the indexing of the optimal process by , especially in the explicit case of power utilities. Although, optimal process may reflect a part of this risk aversion, therefore in the last section of this work, we take care to make them also dependent on this parameter
1.4 Value function of backward classical utility maximization problem
As for example in the Ramsey rule, utility maximization problems in the economic literature use classical utility functions.
This subsection points out the similarities and the differences between consistent progressive
utilities and backward classical value functions, and their corresponding portfolio/consumption optimization problems.
Classical portfolio/consumption optimization problem and its conjugate problem
The classic problem of optimizing consumption and terminal wealth is determined by a fixed timehorizon and two deterministic utility functions and defined up to this horizon. Using the same notations as previously, the classical optimization problem is formulated as the following maximization problem,
(1.8) 
For any valued stopping and for any positive random variable mesurable , denotes the set of admissible strategies starting at time with an initial positive wealth , stopped when the wealth process reaches 0. The corresponding value system (that is a family of random variables indexed by ) is defined as,
(1.9) 
with terminal condition .
We assume the existence of a progressive utility still denoted that aggregates these system (that is more or less implicit in the literature). When the dynamic programming principle holds true, the utility system is consistent.
Nevertheless, in the backward point of view, it is not easy to show the existence of optimal monotonic processes, or equivalently the strong consistency.
Besides, the optimal strategy in the backward formulation is not myopic and depends on the timehorizon . In the economic literature, is often
taken equal to and the utility function is separable in time with exponential decay at a rate interpreted as the pure time preference parameter: .
It is implicitly assumed that such utility function are equal to zero when tends to infinity.
2 Ramsey rule and Yield Curve Dynamics
As our aim is to study long term affine yields curves, we will focus in the following on affine optimal processes. But let us first recall some results on the Ramsey rule with progressive utility.
2.1 Ramsey rule
Financial market cannot give a satisfactory answer for the modeling of long term yield curves, since for longer maturities, the bond market is highly illiquid and standard financial interest rates models cannot be easily extended.
Economic point of view of Ramsey rule
Nevertheless, an abundant literature on the economic aspects of longterm policymaking has been developed. The Ramsey rule is the reference equation in the macroeconomics literature for the computation of long term discount factor. The Ramsey rule comes back to the seminal paper of Ramsey [[26]] in 1928 where economic interest rates are linked with the marginal utility of the aggregate consumption at the economic equilibrium. More precisely, the economy is represented by the strategy of a riskaverse representative agent, whose utility function on consumption rate at date is the deterministic function . Using an equilibrium point of view with infinite horizon, the Ramsey rule connects at time the equilibrium rate for maturity with the marginal utility of the random exogenous optimal consumption rate by
(2.1) 
Remark that the Ramsey rule in the economic literature relies on a backward formulation with infinite horizon, an usual setting is to assume separable in time utility function with exponential decay at rate and constant risk aversion , that is . is the pure time preference parameter, i.e. quantifies the agent preference of immediate goods versus future ones. is exogenous and is often modeled as a geometric Brownian motion.
In the financial point of view we adopt here the agent may invest in a financial market in addition to the money market.
We consider an arbitrage approach with exogenously given interest rate, instead of an equilibrium approach that determines them endogenously.
It seems also essential for such maturity to adopt a sequential decision scheme that
allows to revise the first decisions in the light of new knowledge and direct experiences:
the utility criterion must be adaptative and adjusted to the information flow. That is why we consider consistent progressive utility.
The financial market is
an incomplete Itô financial market: notations are the one described in Section 1.1, with a standard Brownian
motion , a (exogenous) financial short term interest rate and a dimensional risk premium . In
the following,
we adopt a financial point of view and consider either the progressive or the backward formulation for the optimization problem.
Marginal utility of consumption and state price density process
(i) The forward dynamic utility problem
Proposition 1.5 gives a pathwise relation between the marginal utility of the optimal consumption and the optimal state price density process, where the parameterization is done through the initial wealth , or equivalently or since
,
(2.2) 
The forward point of view emphasizes the key rule played by the monotony of with respect to the initial condition ,
under
regularity conditions of the progressive utilities (cf [[15]]).
Then as function of , is decreasing, and is an increasing function of .
This question of monotony is frequently avoided, maybe because with power utility functions (the example often used in the
literature)
is linear in as does not dependent on . We shall come back to that issue in Section 4.
(ii) The backward classical optimization problem
In the classical optimization problem,
both utility functions for terminal wealth and consumption rate are deterministic, and a given horizon is fixed. In this backward point of view,
optimal processes are depending on the time horizon : in particular the optimal consumption rate depends on the time horizon through the optimal state price density process leading to the same pathwise relation (2.2) as in the forward case,
(2.3) 
So, in general the notation of the forward case are used, but
with the additional symbol () to address the dependency on in the classical backward
problem.
Conclusion: Thanks to the pathwise relation (2.2), the Ramsey rule yields to a description of the equilibrium interest rate as a function of the optimal state price density process , , that allows to give a financial interpretation in terms of zero coupon bonds. More dynamically in time,
(2.4) 
2.2 Financial yield curve dynamics
Based on the foregoing,
it is now proposed to make the connection between the economic and the financial point of view through the state price
densities processes and the pricing.
Let , ( for market), be the market price at time of a zerocoupon bond paying one unit of cash at maturity . Then, the market yield curve is defined as usual by the actuarial relation,
Thus our aim is to give a financial interpretation of for in terms of price of zerocoupon bonds.
Remark that is solution of an optimization problem whose criteria depend on the utility functions, yet the utilities do not intervene in the dynamics of .
In term of pricing, the terminal wealth at maturity represents the payoff at maturity of the financial product, whereas the consumption may be interpreted
as the dividend distributed by the financial product before .
Replicable bond
For admissible portfolio without consumption , it is straightforward that for any state price process is a local martingale, and so under additional integrability assumption, . So the price of does not depend on . This property holds true for any derivative whose the terminal value is replicable by an admissible portfolio without consumption, for example a replicable bond,
Besides, for any state price density process
with goods integrability property.
Non hedgeable bond
For non hedgeable zerocoupon bond, the pricing by indifference is a way (among others) to evaluate the risk coming from the unhedgeable part.
The utility indifference price is the cash amount for which the investor is indifferent between selling (or buying) a given quantity of the claim or not.
This pricing rule is non linear and provides a bidask spread. If the investor is aware of its sensitivity to the unhedgeable risk, they can try to transact
for a little amount. In this case, the “fair price” is the marginal utility indifference price (also called Davis price
[[2]]), it
corresponds to the zero marginal rate of substitution.
We denote by ( for utility) the marginal utility price at time of a zerocoupon bond paying one cash unit at maturity , that is . Based on the link between optimal state price density and optimal consumption, we see that
(2.5) 
Remark that is also equal to . Nevertheless, besides the economic interpretation, the formulation through the
optimal consumption is more relevant than the formulation through the
optimal wealth : indeed the utility from consumption is given, while the utility from wealth is more constrained.
According to the Ramsey rule (2.4), equilibrium interest rates and marginal utility interest rates are the same. Nevertheless, for marginal utility price, this last curve is robust only for small trades.
The martingale property of yields to the following dynamics for the zero coupon bond maturing at time with volatility vector
(2.6) 
Using the classical notation for exponential martingale, , the martingale can written as an exponential martingale with volatility . Using that , we have two characterisations of
Taking the logarithm gives
(2.7) 
2.3 Yield curve for infinite maturity and progressive utilities
The computation of the marginal utility price of zero coupon bond is then straightforward using (2.5) leading to the yield curve dynamics
for finite maturity, and for infinite maturity.
As showed in Dybvig [[24]] and in El Karoui and alii. [[23]] the long maturity rate behaves differently
according to the long term behavior of the volatility when ,

If , then a.s and is infinite.

Otherwise, , and is a non decreasing process, constant in and if .
In this last case, which is the situation considered by the economists, all past, present or future yield curves have the same asymptote.
3 Progressive utilities and yield curves in affine factor model
Recently, affine factor models have been intensively developed with some success to capture under the physical probability measure both financial and macroeconomics effects, from the seminal paper of Ang and Piazzesi (2003). As explained in Bolder&Liu (2007) [[1]], Affine termstructure models have a number of theoretical and practical advantages. One of the principal advantages is the explicit description of market participants aggregate attitude towards risk. This concept, captured by the market price of risk in particular, provides a clean and intuitive way to understand deviations from the expectations hypothesis and simultaneously ensure the absence of arbitrage.
3.1 Definition of affine market
The affine factor model makes it possible to compute tractable pricing formulas, it extends the lognormal model (studied in [[16]]) to a more stochastic model. Affine model, which generalizes the CIR one, was first introduced by D. Duffie and R. Kan (1996) [[3]], where the authors assume that the yields are affine function of stochastic factors, which implies an affine structure of the factors. Among many others, M. Piazzesi reports in [[25]] some recent successes in the study of affine term structure models. Several constraints must be fulfilled to define an affine model in mutidimentional framework, but we will not discuss the details here and refer to the works of Teichmann and coauthors [[17]], [[18]].
Properties of affine processes and their exponential
We adopt the framework of the example in Piazzesi ([[25]], p 704). The factor is a Ndimensionnal vector process denoted by and is assumed to be an affine diffusion process, that is the drift coefficient and the variancecovariance matrix are affine function of :
(3.1) 
The affine constraint is expressed as:

, where and are deterministic.

, where is deterministic, and the matrix is a diagonal matrix, with eigenvalues The affine property concerns the variance covariance matrix or equivalently (since is diagonal) the positive eigenvalues of : that must be positive
^{1} with deterministic , where denotes the transposition of a vector or a matrix.
Characterization of market with affine optimal processes
To be coherent with the previous market model (Section 1.1), we have to define the set of admissible strategies , at date , and its orthogonal . Let us first to point out that the volatility vector of any process ( is deterministic) is given by . Thus if (linear space) is the set of admissible strategies, then at time it depends on and is necessarily given by,
The deterministic space and its orthogonal are assumed to be stable by , or equivalently the matrix
is commutative with the orthogonal projection on . A block matrix (up to an orthogonal
transformation) satisfies this property. Furthermore, and are stable by .
The set of admissibles strategies being well defined, we denote by and by the elements of the
linear
spaces and .
We consider two types of assumptions:
(i) The spot rate and the consumption rate are affine positive processes
and .
(ii) The volatilities of the optimal processes and have affine structure,