# «Abstract Motivated by the problems of the conventional model in rationalizing market data, we derive the equilibrium interest rate and risk premiums ...»

Recursive utility using the stochastic

maximum principle

∗

Knut K. Aase

November 3, 2015

Abstract

Motivated by the problems of the conventional model in rationalizing market data, we derive the equilibrium interest rate and risk premiums using recursive utility in a continuous time model. We use the

stochastic maximum principle to analyze the model. This method uses

forward/backward stochastic diﬀerential equations, and works when

the economy is not Markovian, which can be the case with recursive utility. With existence granted, the wealth portfolio is characterized in equilibrium in terms of utility and aggregate consumption. The equilibrium real interest rate is derived, and the resulting model is shown to be consistent with reasonable values of the parameters of the utility function when calibrated to market data, under various assumptions.

KEYWORDS: The equity premium puzzle, recursive utility, the stochastic maximum principle.

JEL-Code: G10, G12, D9, D51, D53, D90, E21.

1 Introduction Rational expectations, a cornerstone of modern economics and ﬁnance, has been under attack for quite some time. Questions like the following were sometimes asked: Are asset prices too volatile relative to the information arriving in the market? Is the mean risk premium on equities over the riskless ∗ Norwegian School of Economics, Bergen, Norway. The ﬁrst version of the paper was presented at the international conference ”The Social Discount Rate” held in Bergen in May, 2012, and organized by K˚ Petter Hagen in cooperation with the Ministry of are Finance, Norway. Special thanks to Thore Johnsen, Steinar Ekern, Gunnar Eskeland, Bernt Øksendal, Darrell Duﬃe, Rajnish Mehra and two anonymous referees for valuable comments. Any remaining errors are mine.

rate too large? Is the real interest rate too low? Is the market’s risk aversion too high?

Mehra and Prescott (1985) raised some of these questions in their wellknown paper, using a variation of Lucas’s (1978) pure exchange economy with a Kydland and Prescott (1982) ”calibration” exercise. They chose the parameters of the endowment process to match the sample mean, variance and the annual growth rate of per capita consumption in the years 1889 The puzzle is that they were unable to ﬁnd a plausible parameter pair of the utility discount rate and the relative risk aversion to match the sample mean of the annual real rate of interest and of the equity premium over the 90-year period.

The puzzle has been veriﬁed by many others, e.g., Hansen and Singleton (1983),Ferson (1983), Grossman, Melino, and Shiller (1987). Many theories have been suggested during the years to explain the puzzle1.

We reconsider recursive utility in continuous time along the lines of Duﬃe and Epstein (1992a-b). In their papers two ordinally equivalent versions of recursive utility were established, and one version was analyzed by the use of dynamic programming. The version left out is analyzed in the present paper.

Our method is the stochastic maximum principle, which gives explicit results for both risk premiums and the short rate. This method does not require the underlying processes to be Markov. This may be important in applications.

For example, in Bollerslev, Engle, and Wooldridge (1988) it is indicated that the conditional variance of the market return ﬂuctuates across time. When the conditional variance is random, the state price deﬂator is not a Markov process but still our approach is valid. With random conditional moments, dynamic programming may not be appropriate, which follows from the nature of the Bellman equation.

When evaluating utility of consumption, the recursive utility maximizer is not myopic, but rather takes into account more than just the present. As a consequence, when calculating the conditional probabilities of the future state prices of the economy, not only the present, but also the past values of the basic economic variables matter, i.e., the Markov property can fail2. The Constantinides (1990) introduced habit persistence in the preferences of the agents.

Also Campbell and Cochrane (1999) used habit formation. Rietz (1988) introduced ﬁnancial catastrophes, Barro (2005) developed this further, Weil (1992) introduced nondiversiﬁable background risk, and Heaton and Lucas (1996) introduce transaction costs.

There is a rather long list of other approaches aimed to solve the puzzles, among them are borrowing constraints (Constantinides et al. (2001)), taxes (Mc Grattan and Prescott (2003)), loss aversion (Benartzi and Thaler (1995)), survivorship bias (Brown, Goetzmann and Ross (1995)), and heavy tails and parameter uncertainty (Weitzmann (2007)).

Our model does not violate history independence in the sense of Section 6 of Kreps and Porteus (1978).

conditional distribution of of future consumption may depend on history in complicated ways. The stochastic maximum principle allows us to derive the optimality conditions without explicitly specifying the dependence.

We base our treatment on the basic framework developed by Duﬃe and Epstein (1992a-b) and Duﬃe and Skiadas (1994), which elaborate the foundational work by Kreps and Porteus (1978) and Epstein and Zin (1989) of recursive utility in dynamic models. The data set we use to calibrate the model is the same as the one used by Mehra and Prescott (1985) in their seminal paper on this subject.

Generally not all income is investment income. We assume that one can view exogenous income streams as dividends of some shadow asset, in which case our model is valid if the market portfolio is expanded to include new assets. In reality the latter are not traded, so the return to the wealth portfolio is not readily observable or estimable from available data. We indicate how the model may be adjusted to account for this under various assumptions, when the market portfolio is not a proxy for the wealth portfolio. Here we also present an example using Norwegian data from the period 1971-2014, in which case we do have the summary statistics related to the wealth portfolio.

The present model calibrates very well to these data.

Besides giving new insights about these interconnected puzzles, the recursive model is likely to lead to many other results that are diﬃcult, or impossible, to obtain using, for example, the conventional, time additive and separable expected utility model. One example included in the paper is related to empirical regularities for Government bills.3 Some of extant literature contributes to more realistic, but also more complex models, often based on approximations. An example is Bansal and Yaron (2004) exploring ’long run consumption risk’. For a relative risk aversion of 10 and EIS of 1.5, they are able to replicate the stylized facts quite well. They use the Campbell and Shiller (1988) approximation for the log interest rate. Their work is based on the Epstein and Zin (1989) discrete time approach, in which they employ a richer economic environment. Not surprisingly, this paper comes a long way in explaining several asset pricing anomalies. In contrast, our expression for the equilibrium short rate is exact, There is by now a long standing literature that has been utilizing recursive preferences.

We mention Avramov and Hore (2007), Avramov et al. (2010), Eraker and Shaliastovich (2009), Hansen, Heaton, Lee, Roussanov (2007), Hansen, Heaton, Lee (2008), Hansen and Scheinkman (2009), Wacther (2012), Campbell (1996), Bansal and Yaron (2004), Kocherlakota (1990b), and Ai (2010) to name some important contributions. Related work is also in Browning et. al. (1999), on consumption see Attanasio (1999), on climate risk see Cai, Judd, and Lontzek (2013, 2015), and Pindyck and Wang (2013). Bansal and Yaron (2004) study a richer economic environment than we employ.

and so is the expression for the risk premiums. Using our approach with a less elaborate model, we are able to explain many of the same features, for more plausible values of the preference parameters.

In particular, also our model predicts lower asset prices as a result of a rise in consumption volatility. Furthermore, when the EIS is larger than 1, agents demand a large equity premium because they fear that a reduction in economic growth prospects or a rise in economic uncertainty will lower asset prices. As noticed by Bansal and Yaron (2004), this can justify many of the observed features of asset market data from a quantitative point of view.

In order to address the particular puzzle at hand, it is a clear advantage to deviate as little as possible from the basic framework in which it was discovered. This way one obtains a laboratory eﬀect, where it is possible to learn what really makes the diﬀerence. Otherwise it is easy to get lost in an ever increasing and complex model framework. From our approach it follows that the solution is simply the new preferences. We do not even need unspeciﬁed ”factors” in the model of the ﬁnancial market (as Duﬃe Epstein (1992a) use).

It has been a goal in the modern theory of asset pricing to internalize probability distributions of ﬁnancial assets. To a large extent this has been achieved in our approach. Consider the logic of the Lucas-style models. Aggregate consumption is a given diﬀusion process. The solution of a system of forward/backward stochastic diﬀerential equations (FBSDE) provide the main characteristics in the probability distributions of future utility. With existence of a solution to the FBSDE granted, market clearing ﬁnally determines the characteristics in the wealth portfolio from the corresponding characteristics of the utility and aggregate consumption processes.

The paper is organized as follows: Section 2 starts with a brief introduction to recursive utility in continuous time, in Section 3 we derive the ﬁrst order conditions, Section 4 details the ﬁnancial market, in Section 5 we analyze our chosen version of recursive utility. In Section 6 we summarize the main results, and present some calibrations. Section 7 explores various alternatives when the market portfolio is not a proxy for the wealth portfolio, Section 8 presents the calibration to Norwegian data, Section 9 points out some extensions, and Section 10 concludes.

2 Recursive Stochastic Diﬀerentiable Utility In this section we recall the essentials of recursive, stochastic, diﬀerentiable utility along the lines of Duﬃe and Epstein (1992a-b) and Duﬃe and Skiadas (1994).

when d 1, and similarly for other correlations given in this model. Here −1 ≤ κRc (t) ≤ 1 for all t. With this convention we can equally well write σR (t)σc (t) for σRc (t), and the former does not imply that the instantaneous correlation coeﬃcient between returns and the consumption growth rate is equal to one.

Notice that the aggregator in (3) satisﬁes the assumptions of the theorem.

3 The First Order Conditions In the following we solve the consumer’s optimization problem. The consumer is characterized by a utility function U and an endowment process e. For any of the versions i = 1, 2 formulated in the previous section, the representative agent’s problem is to solve

Important is here that the quantity Zt is part of the solution of the BSDE.

Later we show how market clearing will ﬁnally determine the corresponding quantity in the market portfolio as a function of Z and the volatility σc of the growth rate of aggregate consumption. This internalizes prices in equilibrium.

In order to ﬁnd the ﬁrst order condition for the representative consumer’s problem, we use Kuhn-Tucker and either directional (Frechet) derivatives in function space, or the stochastic maximum principle. Neither of these principles require any Markovian structure of the economy. The problem is well posed since U is increasing and concave and the constraint is convex. In maximizing the Lagrangian of the problem, we can calculate the directional derivative U (c; h), alternatively denoted by( U (c))(h), where U (c) is the gradient of U at c. Since U is continuously diﬀerentiable, this gradient is a linear and continuous functional, and thus, by the Riesz representation theorem, it is given by an inner product. This we return to in Section 5.3.

Because of the generality of the problem, let us here utilize the stochastic maximum principle (see Pontryagin (1972), Bismut (1978), Kushner (1972), Bensoussan (1983), Øksendal and Sulem (2013), Hu and Peng (1995), or Peng (1990)): We then have a forward/backward stochastic diﬀerential equation (FBSDE) system consisting of the simple FSDE dX(t) = 0; X(0) = 0 and the BSDE (10)7. The Hamiltonian for this problem is ˜ H(t, c, v, z, y) = yt f (t, ct, vt, zt ) − α πt (ct − et ), (11)

4 The ﬁnancial market Having established the general recursive utility of interest, in this section we specify our model for the ﬁnancial market. The model is much like the one used by Duﬃe and Epstein (1992a), except that we do not assume any unspeciﬁed factors in our model.

Let ν(t) ∈ RN denote the vector of expected rates of return of the N given risky securities in excess of the risk-less instantaneous return rt, and let σ(t) denote the matrix of diﬀusion coeﬃcients of the risky asset prices, normalized by the asset prices, so that σ(t)σ(t) is the instantaneous covariance matrix for asset returns. Both ν(t) and σ(t) are progressively measurable, ergodic processes. For simplicity we assume that N = d, the dimension of the Brownian motion B.

The representative consumer’s problem is, for each initial level w of wealth to solve sup U (c) (15) (c,ϕ) subject to the intertemporal budget constraint

As can be seen, πt depends on past consumption and utility from time zero to the present time t. Unless the terms µc (t), σc (t), and σV (t) are all deterministic, the state price process πt is not a Markov process. If the parameters are deterministic in the conventional model, this implies that σM (t) = σc (t) which is not supported by data (see Table 1 below). Hence these quantities must then be stochastic. If we allow this in the recursive model, our method still works, while dynamic programming is then ruled out - the stochastic maximum principle allows us to derive some optimality conditions without explicitly specifying the dependence.