Recursive utility with dependence on past consumption; the continuous-time model

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1 Recursive utility with dependence on past consumption; the continuous-time model Knut K. Aase March 20, 2014 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 relax one restriction on dynamic utility, in that we do not assume irrelevance of past consumption for current marginal utility. One motive for this extension is added realism, another is that the state prices depend on past consumption. We use the stochastic maximum principle and forward/backward stochastic differential equations to derive two ordinally equivalent versions. The resulting equilibrium is consistent with reasonable values of the parameters of the utility functions when calibrated to market data, under various assumptions. KEYWORDS: The equity premium puzzle, the risk-free rate puzzle, recursive utility, past dependence, the stochastic maximum principle, forward/backward stochastic differential equations. 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 are 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 The paper was presented at the international conference The Social Discount Rate held in Bergen in May, 2012, and organized by Kåre Petter Hagen in cooperation with the Ministry of Finance, Norway. Special thanks to Thore Johnsen, Steinar Ekern, Gunnar Eskeland, Darrell Duffie, Bjørn Eraker, Bernt Øksendal, and Rajnish Mehra for valuable comments. Any remaining errors are mine. 1

2 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 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 puzzle, but to date there does not seem to be any consensus that the puzzles have been fully resolved by any single of the proposed explanations 1. In the present paper we reconsider recursive utility, where we relax the assumption that past consumption does not matter for marginal utility. We retain the assumption about dynamic consistency. Using the theory of forward/backward stochastic differential equations, this interpretation leads to a new expression for the volatility of the utility process. Aside from this change, we use the basic model 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 test the model is the same as the one used by Mehra and Prescott (1985 in their seminal paper on this subject. Since the state price deflator (the state prices in units of probability depends on past values of consumption and utility, our approach seems like a modest extension. By the properties of recursive model, utility is proportional to the conditional state price deflator in equilibrium. With past dependence it is natural to use a robust method, so we choose to solve for the equilibrium using the stochastic maximum principle. For 1 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 (

3 recursive utility uncertainty is dated by the time of its resolution, and the individual regards uncertainties resolving at different times as being different. With past dependence, we consider two ordinally equivalent utility functions. We derive both equilibrium risk premiums for any risky security, as well as the equilibrium gross return on the riskless asset for both versions of recursive utility. One of the versions turns out to yield the same results as the model with no consumption history in marginal utility, the other version is different. When calibrated to market data we find that the new version is compatible with more stable values of the parameters. Increased stability seems a natural property when marginal utility is allowed to depend on the agent consumption history. In a discrete-time model we obtain similar results. It is based on the works of Epstein and Zin ( While the continuous-time results are exact, the discrete-time version relies on approximations. In the calibrations we have assumed that all income is investment income. Since the goal is to compare two different model, this may be justified. 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 the new asset. In reality the latter is not traded, so the return to the wealth portfolio is not readily observable or estimable from available data. We indicate how the models can be adjusted and still be compared under various assumptions, when the market portfolio is not a proxy for the wealth portfolio. 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 model. One example included in this paper is that we can explain the empirical regularities for Government bills. 2 The paper is organized as follows: In Section 2 we explain the problems with the conventional, time additive model, and also calibrate the standard recursive model. The section contains a preview of some of our results. Section 3 starts with a brief introduction to recursive utility in continuous time, in Section 4 we derive the first order conditions, Section 5 details the financial market, in Section 6 we analyse the main version of recursive utility, 2 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 and Scheinkman (2009, Wacther (2012, Bansal and Yaron (2004, Campbell (1996, Bansal and Yaron (2004, Kocherlakota (1990 b, and Ai (2012 to name some important contributions. Related work is also in Browning et. al. (1999, and on consumption see Attanasio (1999. Bansal and Yaron (2004 study a richer economic environment than we employ. 3

4 where in Section 6.3 we explain our point of departure from the standard recursive utility model. In Section 7 we discuss the main results. Section 8 contains the analysis of the ordinally equivalent version, Section 9 discusses various assumptions when the market portfolio is not a proxy for the wealth portfolio, Section 10 considers optimal portfolio choice, Section 11 points out an extension, and Section 12 concludes. 2 The problems with the standard models 2.1 The additive and separable Eu-model The conventional asset pricing model in financial economics, the consumptionbased capital asset pricing model (CCAPM of Lucas (1978 and Breeden (1979, assumes a representative agent with a utility function of consumption that is the expectation of a sum, or a time integral, of future discounted utility functions. The model has been criticized for several reasons. First, it does not perform well empirically. Second, the usual specification of utility can not separate the risk aversion from the elasticity of intertemporal substitution, while it would clearly be advantageous to disentangle these two conceptually different aspects of preference. Third, while this representation seems to function well in deterministic settings, and for timeless situations, it is not well founded for temporal problems (derived preferences do not in general satisfy the substitution axiom, e.g., Mossin (1969. In the conventional model the utility U(c of a consumption stream c t is given by U(c = E{ T u(c 0 t, t dt}, where the felicity index u has the separable form u(c, t = 1 1 γ c1 γ e δ t. The parameter γ is the representative agent s relative risk aversion and δ is the utility discount rate, or the impatience rate, and T is the time horizon. These parameters are assumed to satisfy γ > 0, δ 0, and T <. In this model the risk premium (µ R r of any risky security labeled R and the equilibrium interest rate are given by µ R (t r t = γ σ Rc (t (1 and r t = δ + γ µ c (t 1 2 γ (γ + 1 σ c(tσ c (t. (2 Here r t is the equilibrium real interest rate at time t, and the term σ Rc (t = d i=1 σ R,i(tσ c,i (t is the covariance rate between returns of the risky asset and the growth rate of aggregate consumption at time t, a measurable and adaptive process satisfying standard conditions. The dimension of the Brownian 4

5 motion is d > 1. The expression for the risk premium is the continuoustime version of Breeden s consumption-based CAPM. The process µ c (t is the annual growth rate of aggregate consumption and (σ c(tσ c (t is the annual variance rate of the consumption growth rate, both at time t, again dictated by the Ito-isometry. Both these quantities are measurable and adaptive stochastic processes, satisfying usual conditions. The return processes as well as the consumption growth rate process in this paper are also assumed to be ergodic processes, implying that statistical estimation makes sense, in other words, we do not assume these parameters to be constants. Notice that in the model is the instantaneous correlation coefficient between returns and the consumption growth rate given by κ Rc (t = σ Rc (t σ R (t σ c (t = d i=1 σ R,i(tσ c,i (t d i=1 σ d, R,i(t 2 i=1 σ c,i(t 2 and similarly for other correlations given in this model. Here 1 κ Rc (t 1 for all t. Note that with this convention we can equally well write σ R (tσ c(t for σ Rc (t, and the former does not imply that the instantaneous correlation coefficient between returns and the consumption growth rate is equal to one. Prime means transpose. 2.2 The standard recursive model Turning to recursive utility, one more parameter occurs in its most basic form. It is the time preference denoted by ρ. In the form we consider, the parameter ψ = 1/ρ is the elasticity of intertemporal substitution in consumption (EIS, which we refer to as the EIS-parameter. In the conventional Eu-model γ = ρ, but as is well understood, relative risk tolerance (1/γ is something quite different from EIS. What we call the standard recursive model takes the following form: For ρ 1 and with the same notation as above and µ R (t r t = r t = δ + ρµ c (t 1 2 ρ(1 γ 1 ρ σ R(tσ c (t + γ ρ 1 ρ σ R(tσ M (t, (3 ρ(1 ργ 1 ρ σ c(tσ c (t ρ γ 1 ρ σ M(tσ M (t. (4 When ρ = γ this model reduces to the conventional, additive Eu-model 3. 3 The expression (3 for the risk premium can be found in Duffie and Epstein (1992a, then with constant coefficients, derived by dynamic programming. The expression (4 for the interest rate follows from the present paper. 5

6 Expectat. Standard dev. Covariances Consumption growth 1.83% 3.57% cov(m, c = Return S&P % 16.54% cov(m, b = Government bills 0.80% 5.67% cov(c, b = Equity premium 6.18% 16.67% Table 1: Key US-data for the time period Discrete-time compounding. Expectation Standard dev. Covariances Consumption growth 1.81% 3.55% ˆσ Mc = Return S&P % 15.84% ˆσ Mb = Government bills 0.80% 5.74% ˆσ cb = Equity premium 5.98% 15.95% Table 2: Key US-data for the time period Continuous-time compounding. Here σ M (t signifies the volatility of the return on the market portfolio of the risky securities, σ R (tσ M(t = σ RM (t is the instantaneous covariance rate of the returns on any risky asset, with the return of the market portfolio. In the model these quantities are assumed to be measurable, adaptive, ergodic stochastic processes satisfying standard conditions. In Table 1 we present the key summary statistics of the data in Mehra and Prescott (1985, of the real annual return data related to the S&P-500, denoted by M, as well as for the annualized consumption data, denoted c, and the government bills, denoted b 4. Here we have, for example, estimated the covariance between aggregate consumption and the stock index directly from the data set to be This gives the estimate.3770 for the correlation coefficient 5. Since our development is in continuous time, we have carried out standard adjustments for continuous-time compounding, from discrete-time compounding. The results of these operations are presented in Table 2. This gives, e.g., the estimate ˆκ Mc =.4033 for the instantaneous correlation coefficient κ(t. The overall changes are in principle small, and do not influence our comparisons to any significant degree, but are still important. Interpreting the risky asset R as the value weighted market portfolio M corresponding to the S&P-500 index, for the conventional, additive Eu-model 4 There are of course newer data by now, but these retain the same basic features. If our model can explain the data in Table 1, it can explain any of the newer sets as well. 5 The full data set was provided by Professor Rajnish Mehra. 6

7 we have two equation in two unknowns to provide estimates for the preference parameters by the method of moments 6. The result for the Eu-model is γ = 26.3 and δ =.015, i.e., a relative risk aversion of about 26 and an impatience rate of minus 1.5%. If we insist on a nonnegative impatience rate, as we probably should (but see Kocherlakota (1990, this means that the real interest rate explained by the model is larger than 3.3% (when δ =.01, say for the period considered, but it is estimated, as is seen from Table 2, to be less than one per cent. The EIS parameter is calibrated to ψ =.037, which is considered to be too low for the average individual. Similarly, calibrating the standard recursive model we obtain Table 3. Here we have fixed the time impatience rate δ and solved the two equations (3 and (4 in the two remaining unknowns γ and ρ, for values of δ between zero and 35 per cent. From this table we notice that there is a fairly narrow band of values of the impatience rate δ that give reasonable values for the parameters 7, and there is a fair amount of instability in the solution, in that small changes in one of the parameters may lead other parameters to become negative. In applied economics values of the impatience rate between 1 and 2 per cent seem common. One reason for that is of course that the conventional, additive Eu-model is often taken for granted, and from the expression for the interest rate in (2 one simply does not obtain reasonable values for the short rate unless δ is in this range, or smaller. In this connection it may be of interest to consider the study of Andersen et. al. (2008. They use controlled experiments with field subjects in Denmark to elicit the impatience rate and risk preference, ignoring the subject of time preferences. First, an estimate of δ around 25% is reached assuming risk neutrality, second, a new estimate of δ around 10% is obtained assuming risk aversion, with an associated estimate of γ around.74, both based on arithmetic averaging. Notice that a value of about ten per cent does not fit well with the standard recursive model either. 6 Indeed, what we really do here is to use the assumption about ergodicity of the various µ t and σ t processes. This enables us to replace state averages by time averages, the latter being given in Table 2. 7 The recursive model has also another solution where γ varies from to 68.13, and ρ varies from to This is no improvement of the fit for the conventional, additive Eu-model. 7

8 γ ρ EIS Conventional Eu-Model δ = Standard recursive model δ = δ = δ = δ = δ = δ = δ = δ = δ = δ = δ = δ = δ = Table 3: Calibrations of the standard models 2.3 Including Government bills There is another problem with the conventional, additive Eu-model. From Table 2 we see that there is a negative correlation between Government bills and the consumption growth rate. Similarly there is a positive correlation between the return on S&P-500 and Government bills. If we interpret Government bills as risk free, the former correlation should be zero for the CCAPM-model to be consistent. Since this correlation is not zero, then γ must be zero, which is inconsistent with the the above (and the model. Since the Government bills used by Mehra and Prescott (1985 have duration one month, and the data are yearly, Government bills are not the same as Sovereign bonds with duration of one year. One month bills in a yearly context will then contain price risk 11 months each year, and hence the real risk free rate should, perhaps, be set strictly lower that 0.80%. Whatever positive value for the risk premium we choose, the resulting value of γ is negative. With bills included, the conventional, Eu-model does not seem to have enough degrees of freedom to match the data, since in this situation the model contains three basic relationships and only two free parameters. The recursive models do better in this regard, and yield more plausible results as they have enough degrees of freedom for this problem. To better understand the problems with contemporary asset pricing, we 8

9 propose to extend the standard recursive model. The analysis we use is of independent interest, where we solve recursive utility models using the stochastic maximum principle, and forward/backward stochastic differential equations. This method works well also with the standard recursive model, and it can incorporate jump dynamics in a natural manner. 2.4 Preview of our results Based on the analysis to be presented later, where we relax the assumption that past consumption does not matter for current marginal utility, the two relationships corresponding to (3 and (4 are and µ R (t r t = ρ 1 + γ ρ σ R(tσ c (t + γ ρ 1 + γ ρ σ R(tσ M (t, (5 r t = δ + ρµ c (t 1 ρ ( 1 + γ + (γ ρ(1 + γ ργ σ 2 (1 + γ ρ c(tσ 2 c (t γρ(ρ γ + (1 + γ ρ 2 σ c(tσ M (t 1 (γ ρ(1 ρ σ 2 (1 + γ ρ M(tσ 2 M (t. (6 We denote this model the unordinal version, or just Model 1. This is the version that gives the most unambiguous separation of risk aversion from consumption substitution of the two versions we consider. As with the standard recursive model, the risk premium of any risky asset in (5 is seen to be a linear combination of the market-based CAPM of Mossin (1966 and the consumption-based CAPM of Breeden (1979. Also here, if γ = ρ risk premiums as well as the real interest rate reduce to those of the conventional Eu-model. As with the standard recursive model, the risk premiums in (5 are endogenously derived and the same is true for the expression for the equilibrium interest rate in (6. Calibrations are presented in Table 4. First notice the stability of the parameter values compared to Table 3. This model explains well relatively high impatience rates, as those reported by Andersen et. al. (2008, without having any difficulty in explaining the observed estimate of the real interest rate of less than one per cent. For the US-data this model calibrates to values of γ < ρ. From Table 3 we notice that the standard recursive model calibrates to γ > ρ when δ =.03, but aside from this, the other values have negative time preference, or γ < ρ. 9

10 Parameters γ ρ EIS The model (5 and (6 δ = δ = δ = δ = δ = δ = δ = δ = δ = δ = δ = δ = δ = Table 4: Calibrations of the model (5 and (6 The issue of preference for early resolution of uncertainty is linked to this, and γ < ρ means that the agent prefers late to early resolution. As we indicate later, the present model may calibrate to preference for early resolution of uncertainty for other sets of data (like for Norwegian data for the period Figure 1 illustrates the the feasible region in (ρ, γ-space. For the conventional model it is the 45 -line shown (ρ = γ. For the recursive utility model it is all of the first quadrant, including the axes. The points above the 45 -line represent preference for late resolution of uncertainty, the points below correspond to early resolution. The point denoted Calibr 1 associated with the present model is in the late resolution part, and corresponds to the US-data set of Table 1 when δ =.040. The point denoted Calibr 2 is associated to the Norwegian data. This may suggest that when consumption history matters, and this history contains enough fluctuations, for the agent to be concerned about the timing of resolution of uncertainty, market volatility must be above a certain level. 3 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 10

11 ρ γ < ρ: Late resolution Calibr 1 ρ = γ γ > ρ: Early resolution Calibr γ Figure 1: Calibration points in the (γ, ρ-space (1994. We are given a probability space (Ω, F, F t, t [0, T ], P satisfying the usual conditions, and a standard model for the stock market with Brownian motion driven uncertainty, N risky securities and one riskless asset (Section 5 provides more details. Consumption processes are chosen form the space L of square integrable progressively measurable processes with values in R +. The stochastic differential utility U : L R is defined as follows by two primitive functions: f : L R R and A : R R, where R are the reals. The function f(t, c t, V t, ω is a felicity index at time t, and A is a measure of absolute risk aversion (of the Arrow-Pratt type for the agent. In addition to current consumption c t, the felicity index also depends on utility V t, and it may also depend on time t as well as the state of the world ω Ω. The utility process V for a given consumption process c, satisfying V T = 0, is given by the representation V t = E t { T t ( f(t, cs, V s 1 2 A(V s Z(s Z(s ds}, t [0, T ] (7 11

12 where E t ( denotes conditional expectation given F t and Z(t is an R d - valued square-integrable progressively measurable volatility process, to be determined in our analysis. Here d is the dimension of the Brownian motion B t. We think of V t as the remaining utility for c at time t, conditional on current information F t, and A(V t is penalizing for risk. If, for each consumption process c t, there is a well-defined utility process V, the stochastic differential utility U is defined by U(c = V 0, the initial utility. The pair (f, A generating V is called an aggregator. Since V T = 0 and Z(tdB t is a martingale, (7 has the stochastic differential equation representation ( dv t = f(t, c t, V t A(V t Z(t Z(t dt + Z(tdB t. (8 If terminal utility different from zero is of interest, like for applications to e.g., life insurance, then V T may be different from zero. We think of A as associated with a function h : R R such that A(v = h (v, where h h (v is two times continuously differentiable. U is monotonic and risk averse if A( 0 and f is jointly concave and increasing in consumption. A may also depend on time t. The preference ordering represented by recursive utility is usually assumed to satisfy A1: Dynamic consistency, in the sense of Johnsen and Donaldson (1985, A2: Independence of past consumption, and A3: State independence of time preference (see Skiadas (2009a. One of the advantages with the recursive model is that utility may depend on the past. This we make use of in the present paper. Below we relax assumption A2 related to marginal utility at any time t > 0: In the recursion (7, if V s depends on past consumption for s t > 0, so will V t. In this paper we consider two ordinally equivalent specifications: The first has the Kreps-Porteus utility representation, which corresponds to the aggregator with a CES specification f 1 (c, v = δ c 1 ρ v 1 ρ and A 1 ρ v ρ 1 (v = γ v corresponding to two functions u(c = c1 ρ v1 γ and h(v =, say. If, for 1 ρ 1 γ example, A 1 (v = 0 for all v, this means that the recursive utility agent is risk neutral. The version (9 is our main one. Here ρ 0, ρ 1, δ 0, γ 0, γ 1 (when ρ = 1 or γ = 1 it is the logarithms that apply. The elasticity of intertemporal substitution in consumption ψ = 1/ρ. The parameter ρ is the time preference parameter. Here u( and h( are different functions, resulting in the desired disentangling of γ from ρ. 12 (9

13 An ordinally equivalent specification can be derived as follows. When the aggregator (f 1, A 1 is given corresponding to the utility function U 1, there exists a strictly increasing and smooth function ϕ( such that the ordinally equivalent U 2 = ϕ U 1 has the aggregator (f 2, A 2 where f 2 (c, v = ((1 γv γ 1 γ f1 (c, ((1 γv 1 1 γ, A2 = 0. The function ϕ is given by U 2 = 1 1 γ U 1 γ 1, (10 for the Kreps-Porteus specification. It has has the CES-form f 2 (c, v = δ 1 ρ c 1 ρ ((1 γv 1 ρ 1 γ ((1 γv 1 ρ 1 γ 1, A 2 (v = 0. (11 Is should be emphasized that the reduction to a normalized aggregator (f 2, 0 does not mean that intertemporal utility is risk neutral, or that the representation has lost the ability to separate risk aversion from substitution (see Duffie and Epstein(1992a. The corresponding utility U 2 retains the essential features, namely that of (partly disentangling intertemporal elasticity of substitution from risk aversion. This is the version we shall call the standard one, analyzed by Duffie and Epstein (1992a. It is instructive to recall the that the conventional additive and separable utility has aggregator f(c, v = u(c δv, A = 0. (12 in the present framework (an ordinally equivalent one. As can be seen, even if A = 0, the agent of the conventional model is not risk neutral. 3.1 Homogeniety The following result will be made use of in Section 6.3 and 8.3 (U = U 1. For a given consumption process c t we let (V (c t, Z (c t be the solution of the BSDE { dv (c t = ( f(t, c t, V (c t + 1 (c A(V 2 t Z(t (c Z(t (c dt + Z(t (c db t V (c T = 0 (13 13

14 Theorem 1 Assume that, for all λ > 0, (i λ f(t, c, v = f(t, λc, λv; t, c, v, ω (ii A(λv = 1 A(v; v λ Then V (λc t = λv (c t and Z (λc t = λz (c t, t [0, T ]. (14 Proof By uniqueness of the solution of the BSDEs of the type (13, all we need to do is to verify that the triple (λv (c t, λz (c t, λk t ( (c is a solution of the BSDE (13 with c t replaced by λc t, i.e. that ( d(λv (c t = f(t, λc t, λv (c t + 1 (c A(λV 2 t λz(t (c λz(t (c dt +λz(t (c db t ; 0 t T (15 λv (c T = 0 By (i and (ii the BSDE (15 can be written ( λdv (c t = λf(t, c t, V (c t (c A(V 2 λ t λ 2 Z(t (c Z(t (c dt +λz(t (c db t ; 0 t T λv (c T = 0 (16 But this is exactly the equation (13 multiplied by the constant λ. Hence (16 holds and the proof is complete. Remarks 1 Note that the system need not be Markovian in general, since we allow f(t, c, v, ω; (t, ω [0, T ] Ω to be an adapted process, for each fixed c, v. 2 Similarly, we can allow A to depend on t as well 8. Corollary 1 Define U(c = V (c 0.Then U(λc = λu(c for all λ > 0. Notice that the aggregator in (9 satisfies the assumptions of the theorem. 4 The First Order Conditions In the following we solve the consumer s optimization problem, where the assumption A2 plays no role, using the stochastic maximum principle and forward/backward stochastic differential equations. We return to the issue 8 although not standard in Economics. 14

15 of relaxing A2 in Section 6.3. For any of the versions i = 1, 2 formulated in the previous section, the representative agent s problem is to solve subject to sup c L U( c { T } { T } E c t π t dt E c t π t dt. 0 0 Here V t = Vt c and (V t, Z t is the solution of the backward stochastic differential equation (BSDE { dv t = f(t, c t, V t, Z(t dt + Z(t db t (17 V T = 0. Notice that (17 covers both the versions (9 and (11 that we intend to analyze, where f(t, c t, V t, Z(t = f i ( c t, V t 1 2 A i(v t Z(t Z(t, i = 1, 2. For α > 0 we define the Lagrangian ( T L( c; λ = U( c α E π t ( c t c t dt. 0 Important is here that the quantity Z(t is internalized. Market clearing will finally determine this quantity. In order to find the first order condition for the representative consumer s problem, we use Kuhn-Tucker and either directional 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. Because of the generality of the problem, let us instead utilize the stochastic maximum principle (see Pontryagin (1972, Bismut (1978, Kushner (1972, Bensoussan (1983, Øksendal and Sulem (2013, or Peng (1990: We then have a forward/backward stochastic differential equation (FBSDE system 15

16 consisting of the simple FSDE dx(t = 0; X(0 = 0 and the BSDE (17. The Hamiltonian for this problem is H(t, c, v, z, y = y t f(t, ct, v t, z t α π t ( c t c t (18 and the adjoint equation is { dy t = Y (t ( f v (t, c t, V t, Z(t dt + f z (t, c t, V t, Z(t db t Y 0 = 1. (19 If c is optimal we therefore have ( t { Y t = exp f 0 v (s, c s, V s, Z(s 1 ( f 2 z (s, c s, V s, Z(s 2} ds t + f z (s, c s, V s, Z(s db(s 0 a.s. (20 Maximizing the Hamiltonian with respect to c gives the first order equation or y f c (t, c, v, z α π = 0 α π t = Y (t f c (t, c t, V (t, Z(t a.s. for all t [0, T ]. (21 Notice that the state price deflator π t at time t depends, through the adjoint variable Y t, on the entire optimal paths (c s, V s, Z s for 0 s t, which means that the economy does not display the usual Markovian structure. When γ = ρ then Y t = e δt for the aggregator (12 of the conventional model, so the state price deflator is a Markov process, the utility is additive and dynamic programming works well. For the representative agent equilibrium the optimal consumption process is the given aggregate consumption c in society, and for this consumption process the utility V t at time t is optimal. We now have the first order conditions for both the versions of recursive utility outlined in Section 3. We start with the nonordinal version denoted Model 1, with aggregator given by (9. First we specify the financial market model. 5 The financial market Having established the general recursive utility form of interest, in his section we specify our model for the financial market. The model is much like the 16

17 one used by Duffie and Epstein (1992a, except that we do not assume any unspecified factors in our model. Let ν(t R N denote the vector of expected rates of return of the N given risky securities in excess of the riskless instantaneous return r t, and let σ(t denote the matrix of diffusion coefficients of the risky asset prices, normalized by the asset prices, so that σ(tσ(t is the instantaneous covariance matrix for asset returns. Both ν(t and σ(t are progressively measurable, ergodic processes. The representative consumer s problem is, for each initial level w of wealth to solve sup U(c (22 (c,ϕ subject to the intertemporal budget constraint dw t = ( W t (ϕ t ν(t + r t c t dt + Wt ϕ t σ(tdb t. (23 Here ϕ t = (ϕ (1 t, ϕ (2 t,, ϕ (N t are the fractions of total wealth W t held in the risky securities. Market clearing requires that ϕ tσ(t = (δt M σ(t = σ M (t in equilibrium, where σ M (t is the volatility of the return on the market portfolio, and δt M are the fractions of the different securities, j = 1,, N held in the valueweighted market portfolio. That is, the representative agent must hold the market portfolio in equilibrium, by construction. 6 The analysis for the nonordinal model For our main model, Model 1, we now turn our attention to pricing restrictions relative to the given optimal consumption plan. The first order conditions are given by α π t = Y t f 1 c (c t, V t a.s. for all t [0, T ] (24 where f 1 is given in (9. The volatility Z(t and the utility process V t satisfiy the following dynamics ( dv t = δ c 1 ρ t V 1 ρ t 1 ρ V ρ t + 1 γ Z (tz(t dt + Z(tdB t (25 2 V t where V (T = 0. This is the backward equation for the ordinal model. Aggregate consumption is exogenous, with dynamics on of the form dc t c t = µ c (t dt + σ c (t db t, (26 17

18 where µ c (t and σ c (t are measurable, F t adapted stochastic processes, satisfying appropriate integrability properties. We assume these processes to be ergodic, so that we may replace (estimate time averages by state averages. The function f of Section 4 is given by f(t, c, v, z = f 1 (c, v 1 2 A(vz z, and since A(v = γ/v, from (19 the adjoint variable Y has dynamics ({ dy t = Y t v f 1(c t, V t + 1 γ 2 V 2 t Z (tz(t } dt A(V t Z(t db t, (27 where Y (0 = 1. From the FOC in (49 we get the dynamics of the state price deflator. In this section use the notation f for f 1 for simplicity. We also use the notation Z(t/V (t = σ V (t, valid for V 0. By Theorem 1 the term σ V (t is homogeneous of order zero in c. We then seek the joint determination of V t and σ V (t. Notice that Y is not a bounded variation process, and by Ito s lemma dπ t = f c (c t, V t dy t + Y t df c (c t, V t + dy t df c (c t, V t. (28 By the adjoint and the backward equations this is dπ t = Y t f c (c t, V t ({f v (c t, V t γσ V (tσ V (t}dt γσ V (tdb t + Y t f c c (c t, V t dc t + Y t f c + Y t ( 1 2 v (c t, V t dv t + dy t df c (c t, V t 2 f c c (c t, V 2 t (dc t f c c v (c t, V t (dc t (dv t f c v (c t, V 2 t (dv t 2. (29 Here f c (c, v := f(c, v c = δc ρ v ρ, f v (c, v := f(c, v v = δ 1 ρ (1 ρc1 ρ v ρ 1, f c (c, v c = δρc (1+ρ v ρ, f c (c, v v = δρv ρ 1 c ρ, and 2 f c c 2 (c, v = δρ(ρ + 1vρ c (ρ+2, 2 f c v 2 (c, v = δρ(ρ 1vρ 2 c ρ. 2 f c c v (c, v = δρ2 v ρ 1 c (ρ+1, 18

19 6.1 The risk premiums Denoting the dynamics of the state price deflator by dπ t = µ π (t dt + σ π (t db t, (30 from (29 and the above expressions we obtain the drift and the diffusion terms of π t as and µ π (t = π t ( δ ρµc (t ρ(ρ + 1σ c(tσ c (t + ρ(γ ρσ c(tσ V (t (γ ρ(1 ρσ V (tσ V (t (31 σ π (t = π t ( ρσc (t + (γ ρσ V (t (32 respectively. Notice in particular that π t is not a Markov process since µ π (t and σ π (t depend on π t, and the latter variable depends on consumption and utility from time zero to time t. Interpreting π t as the price of the consumption good at time t, by the first order condition it is a decreasing function of consumption c since f cc < 0. The risk premium of any risky security with return process R is given by µ R (t r t = 1 π t σ π (t σ R (t. (33 It follows immediately from (32 and (33 that the formula for the risk premium of any risky security R is µ R (t r t = ρ σ c (t σ R (t + (γ ρσ V (t σ R (t. (34 This is our basic result for risk premiums. It remains to determine σ V (t, which we do below. Before that we turn to the interest rate. 6.2 The equilibrium interest rate The equilibrium short-term, real interest rate r t is given by the formula r t = µ π(t π t. (35 The real interest rate at time t can be thought of as the expected exponential rate of decline of the representative agent s marginal utility, which is π t in equilibrium. 19

20 In order to find an expression for r t in terms of the primitives of the model, we use (31. We then obtain the following r t = δ + ρµ c (t 1 2 ρ(ρ + 1σ c(tσ c (t ρ(γ ρσ cv (t 1 2 (γ ρ(1 ρσ V (tσ V (t. (36 This is our basic result for the equilibrium short rate. The potential for these two relationships to solve the puzzles should be apparent. We return to a discussion later. We proceed to link the volatility term σ V (t to observable quantities in the market that can be estimated from market data. 6.3 The determination of the volatility of utility The standard approach In order to determine Z(t, i.e., to solve the adjoint equation, first notice that the wealth at any time t is given by W t = 1 π t E t ( T t π s c s ds. (37 From Theorem 1 it follows that the nonordinal utility function U is homogenous of degree one. By the definition of directional derivatives we have that U(c ; c = lim α 0 U(c + αc U(c α = lim α 0 (1 + αu(c U(c α = lim α 0 U(c (1 + α U(c α = lim α 0 αu(c α = U(c, where the third equality uses that U is homogeneous of degree one. By the Riesz representation theorem it follows from the linearity and continuity of the directional derivative that ( T U(c ; c = E π t c t dt = W 0 π 0 (38 0 where W 0 is the wealth of the representative agent at time zero, and the last equality follows from (37 for t = 0. Thus U(c = π 0 W 0. Let V t = V (c t denote future utility at the optimal consumption for our representation. Since also V t is homogeneous of degree one and continuously differentiable, by Riesz representation theorem and the dominated 20

21 convergence theorem, the same type of basic relationship holds here for the associated directional derivatives at any time t, i.e., V t (c ; c = E t ( T t π s (t c s ds = V t where π s (t for s t is the state price deflator at time s t, conditional on time t information. As for the discrete time model, it follows that with assumption A2, the consumption history in the adjoint variable Y t is simply removed from the state price deflator π t, so that π s (t = π s /Y t for all t s T. It is then the case that V t = 1 π t W t. (39 Y t By this result, market clearing and Ito s lemma it follows that σ V (t = 1 1 ρ (σ M(t ρσ c (t. (40 Inserting this expression into (34 and (36 the version of recursive utility presented in Section 2.2 results. The version derived by Duffie and Epstein (1992a is the ordinally equivalent one, based on the aggregator in (11. This was claimed to be better suited for dynamic programming, the solution method used by them. As it happens, these ordinally equivalent versions turn out to yield identical risk premiums and interest rates under assumption A Dependence on consumption history It seems reasonable that an individual s current marginal utility is affected by the individual s consumption history, not only of current consumption. For additional utility models in which past consumption plays a role in determining utility, see Sundaresan (1989. Recursive utility is especially well suited to deal with past dependence. In our model one natural way to include this is as follows. We keep the first order conditions in (24 at time t, also seen from this time on. This amounts to letting marginal utility depend on the past consumption history. By relaxing assumption A2, it no longer follows that π s (t has the form given above. In doing this, we make sure that dynamic consistency holds. In addition homogeneity in c must be satisfied so that Theorem 1 applies, and finally the relationship V 0 = π 0 W 0 in (38 must result at t = 0, for any such extension. To this end, consider π (t s = π s for all t s T. (41 21

22 We must examine the expression for π t = Y t f 1 c (c t, V t. By the above results this can be written { t ( δ π t = Y 0 exp 0 1 ρ + δρ 1 ρ c1 ρ s Vs ρ 1 γ γ(1 γσ V (sσ V (s ds t 0 σ V (sdb s } δc ρ t V ρ t (42 From (42 we notice that π t is homogeneous of degree zero in c for all t by inspection of Y t, since V is homogeneous of degree one. With this choice we obtain the same homogeneity results as the standard solution. Dynamic consistency holds by symmetry, when the observer stands at time t and looks back at the consumption history. With this choice π s (t it follows that From (37 we get that V t (c ; c = E t ( T t π s c s ds = V t (c. V t = π t W t (43 at the optimal consumption path c, so that for this version of recursive utility with dependence on history, the optimal utility at time t is the deflated wealth at this time. This is our departure from the standard model, as is seen when comparing with (39. Since the Ito process V t is a function of the agent s wealth and the state price deflator, it is a consequence of Ito s lemma that its diffusion term is Z(t = σ π (tw t + σ W (tπ t. (44 We now use (32 and (23, and observe that in equilibrium ϕ t σ(t = σ M (t, so that by (23, σ W (t = W t σ M (t. This gives Z(t = π t W t (ρσ c (t + (γ ρσ V (t + W t σ M (t π t. Since Z(t = V t σ V (t, we get the following equation for σ V (t from which it follows that σ V (t = σ M (t ρσ c (t (γ ρσ V (t, σ V (t = γ ρ ( σ M (t ρσ c (t. (45 22

23 By comparing with (40, this shows our point of departure from the standard recursive model. From the new interpretation, we obtain a different value for the quantity Z in the forward/backward system of equations. This gives new results for Model 1. It seems reasonable that the agent s evaluation of an additional unit of consumption at time t depends on the agent s consumption history 9. 7 Discussion of the model depending on consumption history In the expressions for the equilibrium risk premiums and the real interest rate σ V (t was the only undetermined quantity. Inserting (45 into (34 and (36 we obtain the expressions and µ R (t r t = r t = δ + ρµ c (t ρ 1 + γ ρ σ R(tσ c (t + γ ρ 1 + γ ρ σ R(tσ M (t, (46 ρ ( 1 + γ + (γ ρ(1 + γ ργ γρ(ρ γ (1 + γ ρ 2 σ c(tσ M (t 1 2 (1 + γ ρ 2 σ c (t σ c (t (γ ρ(1 ρ (1 + γ ρ 2 σ M(tσ M (t. (47 Taking existence of equilibrium as given, the main results in this section are then summarized as Theorem 2 For the model specified in Sections 3-6, with the consumption history dependence of Section 6.3.2, in equilibrium the risk premium of any risky asset is given by (46 and the real interest rate by (47. The resulting risk premiums are linear combinations of the consumptionbased CAPM and the market-based CAPM at each time t as for the standard version, only the coefficients are different. The last two terms in the short rate has the potential to explain the low, observed values of the real rate. The covariance term between consumption and the market index cancels out in the ordinary version. The model was calibrated in Section 2.3. In Table 5 we illustrate a few additional parameter values of interest to the ones of Table 4. The Kelly criterion means that γ = 1, and corresponds 9 Notice that we have not imposed any exogenous history on marginal utility not inherent in the model of recursive utility. 23

24 γ ρ EIS δ Model (46-(47: γ = γ = ρ = Table 5: Various Calibrations Consistent with Table 2. to logarithmic utility for the certainty equivalent. It gives a plausible fit to the data. Risk neutrality seems less natural, but does not give contradictory values for the other parameters. The situation with ρ = 1 is allowed here, but gives an implausible value for γ, a risk loving representative consumer. 7.1 Government bills In the above discussion we have interpreted Government bills as risk free. As mentioned in Section 2.3, this may not be entirely correct. Exactly what risk premium bills command we can here only stipulate. With the same assumption as in Section 2.2, for a risk premium of.0050 for the bills we have a third equation, namely µ b (t r t = ρ 1 + γ ρ σ b,c(t + γ ρ 1 + γ ρ σ b,m (48 to solve together with the equations (46 and (47. With the covariance estimates provided in Table 2, we have three equations in three unknowns, giving the following values δ = 0.036, γ =.63 and ρ = This risk premium of the bills may seem a bit high, but these results are far better than for the conventional, additive Eu-model. 7.2 Early resolution The calibration points reported in Table 4 correspond to otherwise plausible values of the various parameters, except that they are located in the late resolution part of the (ρ, γ-plane where γ < ρ. Consider the following hypothetical question. How large must the volatility of the market portfolio be in order for the agent to be concerned about early resolution? It can be seen from the expressions for the equity premium and the interest rate that when the volatility of the market portfolio increases, this may tend to change the calibration points to the region where γ > ρ. 24

25 The critical value of σ M (t for this data set turns out to be about.245. For example, for a value of σ M (t corresponding to 27.4%, we get preference for early resolution of uncertainty, ceteris paribus, for the values δ =.00, γ = 4.48, and ρ =.64. In some countries, this order of magnitude for the volatility of the market portfolio is not uncommon. An example is Norway where this volatility has been estimated to 35% for the period When the market volatility is high, normally the risk premium is also high. Consider the following example. Example 1. The data for Norway for the period are: In real terms σ M (t = 0.35, risk premium = 8.71% and the risk-free rate is 2.25%. Assuming the rest of the summary statistic in real terms for Norway for the period indicated (σ M,c (t =.00048, σ c (t =.024, µ c (t =.0311, µ M (t =.11, this gives the calibrated values for the nonordinal model of this section δ =.01, γ = 3.03 and ρ =.58 (EIS = 1.72 i.e., preference for early resolution. In Figure 1 this is denoted Calibr 2. For the ordinary recursive model we find the following possibility: δ =.01, γ = 1.21, and ρ = 1.73, i.e., preference for late resolution at low parameter values. This magnitude of the volatility of the market portfolio is common in many countries, but Norway has the highest stock market volatility in the following group: France, Germany, The Netherlands, Sweden, Switzerland, UK, Japan and US. This example demonstrates that the representative agent may be of the type of this paper s Model 1, and still calibrate to data consistent with preference for early resolution. 8 The ordinally equivalent model For Model 2 the first order conditions are given by α π t = Y t f c (c t, V t a.s. for all t [0, T ] (49 where f(t, c, v, σ v = f 2 (c, v is given in (11, and where the adjoint variable Y (t is ( t f Y t = exp 0 v (c s, V s ds a.s. (50 10 Data made available by Thore Johnsen, based on 25

26 As can be noted, for this version the adjoint process is of bounded variation 11. The model for the aggregate consumption is the same as before, and the process V t is assumed to follow the dynamics where dv t (1 γv t = µ V (t dt + σ V (t db t (51 σ V (t = (1 γv t σ V (t, and µ V (t = δ ( c 1 ρ t 1 ρ ((1 γv t 1 ρ 1 γ ((1 γv t 1 ρ 1 ρ Here we have called Z 2 (t = σ V (t, and f 2 = f for simplicity of notation. When γ > 1 utility V is negative so the product (1 γv > 0 a.s., which gives us a positive volatility of V provided σ V (t > 0 a.e. From the FOC (49 we then get the dynamics of the state price deflator: Using Ito s lemma this becomes dπ t = f c (c t, V t dy t + Y t df c (c t, V t. (52 f c dπ t = Y t f c (c t, V t f v (c t, V t dt + Y t c (c f c t, V t dc t + Y t v (c t, V t dv t ( 1 2 f c + Y t 2 c (c t, V 2 t (dc t f c c v (c t, V t (dc t (dv t f c 2 v (c t, V 2 t (dv t 2. (53 Here f v (c, v := f c (c, v c f c (c, v := f(c, v v = δ 1 ρ = ( δ ρ c ρ 1 γ ρ, (1 γv 1 γ f(c, v c = δ c ρ ( (1 γv 1 ρ 1 γ 1, (c 1 ρ( (1 γv 1 ρ 1 γ (ρ γ + (γ 1, f c (c, v v = δ(ρ γ c ρ ( (1 γv 1 ρ 1 γ,. 2 f c δ ρ (1 + ρ c ρ 2 (c, v = c2 ( 1 ρ (1 γv 1 γ 1, 2 f c ρ δ (γ ρ c ρ 1 (c, v = c v ( 1 ρ, (1 γ 1 γ and 2 f c δ (γ ρ (1 ρ c ρ (c, v = v2 ( 1 ρ (1 γv 1 γ Originally the author derived this FOC using utility gradients based on a result of Duffie and Skiadas (

27 Denoting the dynamics of the state price deflator by dπ t = µ π (t dt + σ π (t db t, (54 from (53 and the above expressions we now have that the drift and the diffusion terms of π t are given by µ π (t = Y t ( δ 2 1 ρ (1 γδ2 1 ρ (ρ γ c2(1 ρ 1 t ((1 γv t 2(1 ρ 1 γ +1 c ρ t ((1 γv t 1 ρ 1 γ +1 δ ρ c ρ δ c ρ t (ρ γ ((1 γv t 1 ρ 1 γ f(ct, V t and 1 2 δ ρc ρ t (ρ γ ((1 γv t 1 ρ 1 γ +1 σ cv (t t ((1 γv t 1 ρ 1 γ +1 µ c (t δ ρ (1+ρ c ρ t ((1 γv t 1 ρ 1 γ +1 σ c(tσ c (t δ (ρ γ (1 ρ c ρ t ((1 γv t 1 ρ 1 γ +1 σ V (tσ V (t, (55 σ π (t = Y t δ c ρ t (( ρ σ c (t ((1 γv t 1 ρ 1 γ +1 + (ρ γ ((1 γv t 1 ρ 1 γ ((1 γvt σ V (t (56 respectively. 8.1 The risk premium for the ordinally equivalent version The risk premium is as before given by µ R (t r t = 1 π t σ Rπ (t, (57 where σ Rπ (t is the instantaneous covariance of the increments of R and π. Combining the FOC with the result in (56, the formula for the risk premium in terms of the primitives of the model is accordingly given by µ R (t r t = ρ σ Rc (t + (γ ρ σ RV (t. (58 This is a basic result of our analysis with recursive utility, and is seen to be the same as for the nonordinal version based on (9 in terms of σ V (t. We return to the equilibrium determination of this term. Before we do that, we give an expression for the equilibrium interest rate r t. 27