《金融经济学原理》课程教学资源（文献资料）Utility-implied term structures of equity risk premia.pdf_大学文库

Economics Letters 244 (2024)111947 Contents lists available at SclenceDirect Economics Letters ， ELSEVIER journal homepage:www.eisevier.com/locate/ecolet Utility-implied term structures of equity risk premia* Louis R.Piccotti Department of Finance,Spears School of Business,Oklaho a Ste University,Sillwete,OK 74078,United States ARTICLE INFO ABSTRACT risk premium.The tem structure is flat for time 1.Introduction found in Dew-Becker and Giglio (2016)and in van Binsbergen et al. The expected ex m is shown to be flat for stor requires for taking on risks of varyin one-period utility and for uncor cycle frequencies(the Fourie r transform of the autocovariance sloping for interal habit utility,and upward sloping for time preter orm equity risk premia are derived for a range of functi forms.For example.an investor r,w en the repre large tility sure to short-term risks and has a relat exposure to long-run risks.Conversely,for example,an investor vith an equity risk prem ium is asticity of intertemp sut er th I (o run tim aownw arge invest sure to long-nun risks and a relatively small ey term risks ity vields can be manned to a linear time e series model which is These alternative utility functional forms result in different shapes for sistent with a certain utility preference class.Since the shape of the m(D er s changes wi 2 cture quity y In the present paper,commonly used utility preference forms are present paper suggest that the representative investor's utility pre written in more gen l time series t orm and the implied risk premium ences are state-dependent derived.By writi as re alo om the prese onic is relieved.Richer s ts o (2016)analysis and rgen et al.(2012)analysis.The firs than is pe ble in the mo that n ange holding period (or mic range frequency. he fr cu domain The the modef viewed as a generalization of the risk premium term structure results restricted to be monotonic.The second innovation is that the present E-mail address louis.r.piccotti@okstate.edu 0165-1765/2024 BV.All rishts an

Utility-implied term structures of equity risk premia☆ Louis R. Piccotti Department of Finance, Spears School of Business, Oklahoma State University, Stillwater, OK 74078, United States ARTICLE INFO JEL codes: G11 G12 Keywords: Asset pricing Equity risk premium term structure Frequency domain ABSTRACT The fundamental asset pricing problem is extended to the frequency domain for commonly used utility functional forms. For each class of utility functions, closed-form solutions are presented for the term structure of the equity risk premium. The term structure is flat for time separable one-period utility and uncorrelated external habit formation utility, downward sloping for internal habit formation utility, upward sloping for time preference (recursive) utility, and can be upward sloping, downward sloping, or flat for production-based utility. These results suggest how observed risk premium term structures can be mapped to linear time series models consistent with utility preference classes. 1. Introduction The expected excess return on an asset can be viewed as a portfolio of compensations that an investor requires for taking on risks of varying cycle frequencies (the Fourier transform of the autocovariance function between consumption growth and portfolio returns). In this paper, the fundamental asset pricing problem is extended to the frequency domain. Closed-form equity risk premia are derived for a range of commonly used utility functional forms. For example, an investor with internal habit preferences (Abel, 1990; Constantinides, 1990) has a relatively large utility exposure to short-term risks and has a relatively small exposure to long-run risks. Conversely, for example, an investor with an elasticity of intertemporal substitution greater than 1 (long-run time preferences; Epstein and Zin, 1989) has a relatively large utility exposure to long-run risks and a relatively small exposure to short-term risks. These alternative utility functional forms result in different shapes for the term structure of the equity risk premium (Dew-Becker and Giglio, 2016; Lettau and Wachter, 2011, van Binsbergen et al., 2012). In the present paper, commonly used utility preference forms are written in more general time series form and the implied risk premium term structures are derived. By writing the utility preferences as lag equations and as autoregressive equations, the implicit restriction that the risk premium term structure is monotonic is relieved. Richer sets of dynamics can be captured than is possible in the more rigidly defined models of Dew-Becker and Giglio (2016) mapping utility functions to the frequency domain. Therefore, the model of the present paper can be viewed as a generalization of the risk premium term structure results found in Dew-Becker and Giglio (2016) and in van Binsbergen et al. (2012). The term structure of the equity risk premium is shown to be flat for one-period utility and for uncorrelated external habit utility, downward sloping for internal habit utility, and upward sloping for time preference (recursive) utility. When utility is over production-based preferences, the term structure of the equity risk premium is upward sloping in the presence of only mature firms that can easily capture production growth opportunities. However, when the representative investor’s production income is additionally a function of adolescent companies, which require large investments in technology to capture production growth opportunities, then the term structure of the equity risk premium is downward sloping for sufficiently large investment volatilities. Together, these results suggest how the observed term structure of equity yields can be mapped to a linear time series model which is consistent with a certain utility preference class. Since the shape of the term structure of equity yields changes with time (Lettau and Wachter, 2011; Piccotti, 2022; van Binsbergen et al., 2013), the results of the present paper suggest that the representative investor’s utility preferences are state-dependent. Two important additional insights are also gained from the present paper’s model representing an innovation to the Dew-Becker and Giglio (2016) analysis and the van Binsbergen et al. (2012) analysis. The first innovation is that mid-range holding period (or mid-range frequency) dynamics can be modeled more successfully when utility preferences are written in time series form, since the term structure shape is not restricted to be monotonic. The second innovation is that the present ☆ I thank Max Croce (the editor) and an anonymous referee, Hengjie Ai, Paul Calluzzo, and Yangru Wu for their helpful comments and suggestions. All errors are the sole responsibility of the author. E-mail address: louis.r.piccotti@okstate.edu. Contents lists available at ScienceDirect Economics Letters journal homepage: www.elsevier.com/locate/ecolet https://doi.org/10.1016/j.econlet.2024.111947 Received 28 April 2024; Received in revised form 21 August 2024; Accepted 29 August 2024 Economics Letters 244 (2024) 111947 Available online 31 August 2024 0165-1765/© 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies

model suggests how the term structure of the risk premium changes with the correlation structure between the investor’s consumption innovations and the external habit level innovations. When consumption growth sensitivity (beta) with respect to market returns is greater than the external habit growth sensitivity, the term structure is upward sloping. When the consumption growth and external habit growth sensitivities are equal, then the term structure is flat. Finally, when the external habit growth sensitivity to market returns is higher than the consumption growth sensitivity, then the term structure slope is downward sloping. 2. Utility preferences A number of utility specifications have been proposed in the literature characterizing different investor preferences, which include: time separable utility (Merton, 1969), internal habit formation utility (Abel, 1990; Constantinides, 1990; Campbell and Cochrane, 1999), external habit formation utility (Campbell and Cochrane, 1999), utility with time preferences (Epstein and Zin, 1989), and production-based utility preferences (Ai et al., 2018; Bansal and Yaron, 2004). The following example (time separable) preference structures are considered: yt+Δ = ct+Δ one − period utility , (1.i) yt+Δ = ϕp(L)ct+Δ internal habit utility , (1.ii) yt+Δ = ct+Δ + ( ϕp(L) − 1 ) zt+Δ Uncorrelated external habit utility , (1.iii) yt+Δ = ∑q j=1 ψj− 1Et+Δ { ct+jΔ } time preference utility , (1.iv) yt+Δ = ∑q j=1 ψj− 1Et+Δ { ct+j⋅Δ − xt+jΔ } Correlated external habit utility , (1.v) yt+Δ = ∑q j=1 ψj− 1QH,t+jΔ production − based (1.vi) where Δ denotes the time step size, ϕp(L) = 1 − ∑p i=1ϕiLi for ϕi ∈ R+\{∞} is the lag polynomial with coefficients that determine the strength of the habit preference with respect to historic consumption levels, ψj− 1 ∈ R+\{∞} determines the investor’s time preference, QH is household production income (defined below), and p, q ∈ N+. 1 It is further assumed that | ∑p j=1ϕj| 0 is the patience parameter which determines the degree of intertemporal substitution (typically, β ≈ 1), dt+1 is the payoff of the asset, u(⋅) is the investor’s utility function, and ct denotes consumption at time t. Solving Eq. (2) and rearranging yields the relationship: 1 = Et{mt+1⋅Rt+1}, (5) mt+1 = β uʹ (ct+1) uʹ (ct) , (6) where mt+1 is the stochastic discount factor (SDF) of the marginal investor pricing the asset and Rt+1 = dt+1/pt is the gross return on the asset. Take the first-order Taylor series expansion of time t + 1 consumption around the point of current consumption (ct), which gives the change in marginal utility to be: uʹ (ct+1) − uʹ (ct) ≈ uʹʹ (ct)(ct+1 − ct). (7) This approximation for the innovation in marginal utility allows the Arrow-Pratt risk-aversion coefficient, θt = − ut uʹ t , to be recovered. Then, noting that Rf = 1 Et {mt+1 } : Et{Rt+1} − Rf = − RfCV{mt+1, Rt+1} (8) ≈ θtRfβCV{ Rc,t+1, Rt+1 } (9) ∝ ∫π − π sc,r(ω)dω, (10) where ∝ denotes proportional to, where the first order approximation for time t + 1 marginal utility is used in Eq. (9), and where the property that the covariance is equal to the integrated cospectrum CV{ Rc,t+1, Rt+1 } = ∫π − π sc,r(ω)dω is used in Eq. (10). θt, Rf , and β are constants with respect to the time t information set. The cospectrum is defined as: 1 R+ and N+, respectively, denote the set of positive real numbers and the set of positive natural numbers. L.R. Piccotti Economics Letters 244 (2024) 111947 2

sc,r(ω) = 1 2π ∑∞ k=− ∞ γ(k) c,r e− iωk , (11) γ(h− 1) c,r = CV{ Rc,t+h, Rt+1 } , (12) where Eq. (12) is the autocovariance function between consumption growth from time t + (h − 1) to time t + h and asset returns (the covariance is not conditional, since the two series are stationary and the covariance is the same for each date). With a sample size of T, the frequencies are ωj = 2πj T for j = 1, ., T 2. Then, the portion of the risk premium in Eq. (8) associated with periodic random components with frequency less than or equal to ωj is proportional to ∫ ωj − ωj sc,r(ω)dω. 4. Implied risk premium term structures In this section, general risk premia term structures that are associated with an example of each of the utility preference forms in Eq. (1) are derived. Assume that one-period asset returns, log consumption changes, and log external habit changes are described by the following discretized vector stochastic differential equation. Assumption 1 Let an asset’s return r, the investor’s consumption growth rc, and correlated external habit growth rx be, respectively, described by the following discretized vector stochastic differential equation. ⎡ ⎣ rt+Δ rc,t+Δ rx,t+Δ ⎤ ⎦ = ⎡ ⎣ αr αc αx ⎤ ⎦Δ + C ⎡ ⎣ ζ1,t+Δ ζ2,t+Δ ζ3,t+Δ ⎤ ⎦ ̅̅̅̅ Δ √ , (13) where (αr, αc, αx) ʹ are the per annum drift terms, ( ζ1,t+Δ, ζ2,t+Δ, ζ3,t+Δ )ʹ are uncorrelated standard normal random variables, and C is the Cholesky decomposition of the (3 × 3) per annum covariance matrix: Σ = ⎡ ⎢ ⎢ ⎣ σ2 r ⋅ ⋅ σcσrρc,r σ2 c ⋅ σxσcρc,x σcσxρc,x σ2 x ⎤ ⎥ ⎥ ⎦, (14) such that Σ = CCʹ . Δ denotes the time step size. Within this model, the following propositions give the expected RP. Proposition 1 gives the expected RP with one-period (OP) preferences over current consumption only. Proposition 1 Assume the one-period utility functional form in Eq. (1.I) and the processes for returns on consumption and the asset in Assumption 1. Then, the expected risk premium is: Et{Rt+Δ} − Rf = θOPβRfσcσrρc,rΔ, (15) where θOP = − uʹʹ (ct) uʹ (ct) (16) is the Arrow-Pratt measure of absolute risk aversion for one-period utility. Proof: See Appendix A. The RP term structure is flat, for one-period utility preferences. This is the result of consumption shocks only affecting contemporaneous utility growth and not affecting utility growth in any other period. The RP term structure level is increasing in the risk aversion coefficient, θOP, and increasing in the covariance between consumption shocks and asset returns. Proposition 2 Assume the internal habit preferences utility functional form in Eq. (1.Ii) and the processes for consumption growth and returns on the asset in Assumption 1. Then, the expected risk premium is: Et{Rt+Δ} − Rf = θIHP 2π Rfβσcσrρc,rΔ ∫π − π [ 1 − ∑∞ k=1 ϕke− iωk ] dω, (17) where θIHP = − uʹʹ ( ϕp(L)ct ) uʹ ( ϕp(L)ct ) (18) is the Arrow-Pratt measure of absolute risk aversion for utility with internal habit preferences. Proof: See Appendix A. With internal habit preferences (IHP), the RP term structure level is adjusted for consumption shocks having persistent effects on the utility function. Intuitively, the increased portion of the RP attributable to higher frequencies results from current utility being dependent on the level of current consumption, relative to the path that the individual’s consumption has taken in the immediately prior periods, with increased weight being put on the investor’s most recent consumption levels. A similar argument explains why the portion of the RP attributable to low frequencies is lower with IHP utility, relative to the OP utility case. When ϕk = 0 for all k, the IHP case is identical to the OP preferences case and the RP spectrum is flat and equal to that of the OP preferences case. While IHP utility generally assumes that |ϕk| 1, the case where an investor prefers to increase his habit consumption level each period, then the RP term structure level at the lowest frequencies is negative, in order to increase the RP attributable to the highest frequencies. When the investor has utility with uncorrelated external habit preferences (EHP), where the investor’s consumption growth is uncorrelated with the external consumption growth (i.e. household consumption shocks are i.i.d. and uncorrelated across households), the RP term structure level remains constant across all frequencies, as in the oneperiod utility case. Proposition 3 Assume the uncorrelated external habit preferences utility functional form in Eq. (1.Iii) and the processes for consumption growth and returns on the asset in Assumption 1. Then, the expected risk premium is: Et{Rt+Δ} − Rf = θUEHPRf βσcσrρc,rΔ, (19) where θUEHP = − uʹʹ ( ct − ( ϕp(L) − 1 ) zt ) uʹ ( ct − ( ϕp(L) − 1 ) zt ) (20) is the Arrow-Pratt risk aversion coefficient with uncorrelated external habit formation utility. Proof: See Appendix A. The RP with UEHP is flat and the only difference between the RP term structure under UEHP and OP is the risk-aversion coefficient function θ. This flat RP term structure in Proposition 3 is similar to the external habits flat term structure modeled in Dew-Becker and Giglio (2016), but differs from the upward sloping (from high-frequency cycles to low-frequency cycles) external habits term structure calibrated in van Binsbergen et al. (2012) from the Campbell and Cochrane (1999) theoretical model. The flat term structure shape is due to the investor’s consumption innovations being uncorrelated with the external habit innovations in Proposition 3. However, when consumption innovations and external habit innovations are correlated, the term structure is expected to be upward sloping as is shown in Proposition 5 below. There is a large literature showing that investors have a time preference for consumption, which includes the seminal recursive utility case of Epstein and Zin (1989) and the more recent concept of choice aversion, which is axiomatized by Fudenberg and Strzalecki (2015). Utility over consumption with time preferences, in Eqs. (1.iv)-(1.v), depends on expected future consumption choices, which are not known. L.R. Piccotti Economics Letters 244 (2024) 111947 3

To aid with the mathematical tractability for the expectation terms for the time preference utility forms, the consumption level and correlated external habit level are assumed to follow AR(p) processes (note that these processes can be re-written as a vector autoregression model with the appropriate restricted coefficient matrices). Assumption 2 Consumption levels and correlated external habits follow the AR(p) processes: ξt+Δ = Aξt + et+Δ, (21.i) Yt+Δ = AYt + ϵt+Δ, (21.ii) ξt+Δ = [ ct+Δ − μc ct − μc⋯ ct+(2− p)Δ − μc ]ʹ , (22.i) Yt+Δ = [ xt+Δ − μx xt − μx⋯ xt+(2− p)Δ − μx ]ʹ , (22.ii) A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ a1 a2 a3 ⋯ ap− 1 ap 1 0 0 ⋯ 0 0 0 1 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 0 ⋯ 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (23) et+Δ = [ et+Δ 0 ⋯ 0 ] ʹ , (24.i) ϵt+Δ = [ ϵt+Δ 0 ⋯ 0 ] ʹ , (24.ii) where μc is the mean consumption level, μx is the mean habit level, A is the matrix with coefficients that determine the dependence of current consumption and habits on lagged consumption and habit levels, and et+Δ is a zero mean residual term, and ϵt+Δ is a zero mean residual term. Since consumption levels and correlated habit levels are modeled to follow AR(p) processes, the current spread between log consumption and log habit continues to affect future marginal utility growth through the effect that it has on forming expectations about future excess consumption levels over habit levels. The AR parameters in Eqs. (21.I) and (21.ii) are not required to be identical for the results to hold; however, assuming that they are equal simplifies the notation and exposition without a loss of model insight. When Assumption 2 is imposed, the expected consumption level hΔ periods in the future is given by: Et+Δ { ct+(1+h)Δ } = [ Ah− 1 ξt+Δ ] 1,1 + μc, h ≥ 1 (25.i) and the expected excess consumption level over the habit level hΔ periods in the future is given by: Et+Δ { ct+(1+h)Δ − xt+(1+h)Δ } = [ Ah− 1 (ξt+Δ − Yt+Δ) ] 1,1 + (μc − μc), h ≥ 1, (25.ii) where [⋅] m,n is the m, n element of the matrix in [⋅]. Proposition 4 gives the expected risk premium with time preferences utility. Proposition 4 Assume the time preferences utility functional form in Eq. (1.Iv) and the processes for consumption growth and returns on the asset in Assumption 1. Then, the expected risk premium is: Et{Rt+Δ} − Rf = θTP 2πRfβσcσrρc,rΔ × ∫π − π [ 1 +∑q j=2 ∑p k=1 ψj− 1 [ Aj− 1] 1,ke− iω(k− 1) ] dω (26) where θTP = − ut (∑q j=1ψj− 1 ([ Aj− 1 ξt ] 1,1 + μc )) uʹ t (∑q j=1ψj− 1 ([ Aj− 1 ξt ] 1,1 + μc )) (27) is the Arrow-Pratt risk aversion coefficient with time preferences utility. Proof: See Appendix A. The RP term structure level at lower frequencies is increased, relative to the OP preferences case, and the RP term structure level at higher frequencies is decreased, relative to the OP preferences case. In the case where the investor is only concerned with consumption today (ψ0 = 1 and ψj = 0 for j > 0), Proposition 4 reduces to the OP preferences case in Proposition 1. Next, the case of correlated external habits preferences is considered. In this case, both the investor’s consumption and the external habit are contemporaneously correlated with market returns and with each other. The investor’s current spread between their consumption level and the external habit level continues to affect future marginal utility growth through the effect that it has on forming expectations about future relative consumption levels. Proposition 5 presents the RP term structure with correlated external habits. Proposition 5 Assume the correlated external habit utility functional form in Eq. (1.V) and the processes for returns on the asset, consumption growth, and external habit level growth in Assumption 1. Then, the expected risk premium is: Et{Rt+Δ} − Rf = θCEHP 2π Rfβ(bc − bx)σ2 r Δ × ∫π − π [ 1 +∑q j=2 ∑p k=1 ψj− 1 [ Aj− 1] 1,ke− iω(k− 1) ] dω (28) where bl = ( σl σrρl ,r ) /σ2 r for l ∈ {c, x} and where θCEHP = − ut (∑q j=1ψj− 1 ([ Aj− 1 (ξt − Yt) ] 1,1 + (μc − μx) )) uʹ t (∑q j=1ψj− 1 ([ Aj− 1 (ξt − Yt) ] 1,1 + (μc − μx) )) (29) is the Arrow-Pratt risk aversion coefficient with correlated external habit utility. Proof: See Appendix A. Eq. (28) shows that the risk premium term structure shape with correlated external habit preferences depends on the relative sensitivities of consumption and the external habit to market returns. When bc > bx, the term structure is upward sloping from high frequencies to low frequencies, when bc bx > 0. With this condition, consumption is more volatile than the habit and the correlation between consumption and the habit is nonnegative. It is also productive to further reconcile Proposition 5 with the uncorrelated external habit case in Proposition 4. When the external habit level is uncorrelated with market returns and the investor’s consumption growth, bx = 0. Since the preference form in Proposition 4 is backward looking and the preference form in Proposition 5 is forward looking, the intersection of the two is ψ0 = 1 and ψj = 0 for j > 0. Then, the risk premium term structure in this restricted case is flat and Proposition 5 reduces to Proposition 4. Finally, the case with production-based utility preferences is considered. Within this class of models, the interplay between production growth shocks and investment in capital continues to affect future utility over production output through the effect that it has on the law of motion of the household’s production income. Assume that the representative investor’s production income QH,t follows a structure similar in the spirit to that of Ai et al. (2018). Let the investor’s production income QH,t be a linear combination of the outputs of mature firms (QM,t) and of adolescent firms (QA,t) with the following structure: QH,t = QM,t + QA,t, (30) Ql ,t = λQl ,t− 1 + vl ,t, j ∈ {M,A}, (31) L.R. Piccotti Economics Letters 244 (2024) 111947 4

vl ,t = Gl ,t − Il ,t, (32) where λ ∈ (0, 1) is an autoregressive parameter, and where vl is a normally distributed innovation term with variance σ2 l and contemporaneous correlation with market returns of ρl ,r for l ∈ {M, A}. The innovation terms vM and vA are assumed to be serially and cross-serially uncorrelated; however, can be contemporaneously correlated. Gl ,t and Il ,t are, respectively, productivity growth and investment growth shocks. In order to be consistent with Ai et al. (2018), mature firms have sufficient capital in place to benefit from positive news shocks, however, adolescent firms require making a substantial investment in new capital in order to take advantage of growth opportunities. The following correlation structures between growth shocks and investment shocks with market returns hold: ρ(M) G,r > 0, ρ(M) I,r ≈ 0, ρ(A) G,r > 0, ρ(A) I,r > 0, and the variance of investment is much larger than the variance of productivity growth for adolescent firms, σ2 A,I≫σ2 A,G. As a result, ρM,r > 0 and ρA,r can be either positive or negative. The autoregressive parameter λ is assumed to be the same for mature and adolescent firms for notational simplicity of exposition, but this is not a required assumption for the results to hold. The structure in Eqs. (30)–(32) fits within the production-based model of Ai et al. (2018), where the outputs QM,t and QA,t could further be subordinated on Cobb-Douglas parameters. Alternatively, if there is no adolescent firm (i.e. QA,t = 0 ∀t) then the model fits within the long-run risks model of Bansal and Yaron (2004), where the firm’s aggregate dividend is equal to the mature firm’s output. Therefore, the implied term-structures of equity can be derived from the presently proposed time series production-based model for both the Ai et al. (2018) and Bansal and Yaron (2004) models. Proposition 6 presents the term structure of expected equity returns. Proposition 6 Assume the production-based (PB) utility functional form in Eq. (1.Vi) and the production output process in Eqs. (30)–(32). Then, the expected risk premium is: Et{Rt+Δ} − Rf = θPBRfβ(bM + bA)σ2 r Δ 2π × ∫π − π [ 1 + ∑q j=2 × ∑p k=1 ψj− 1λk+j− 1 e− (k− 1)iω ] (33) where the autocovariances have been arbitrarily truncated at a lag length of p and where bl = ( σl σrρl ,r ) /σ2 r for l ∈ {M,A}. Proof: See Appendix A. Since in the Bansal and Yaron (2004) model, QA,t = 0, it follows that bA = 0 in Proposition 6 and the term structure of equity is upward sloping. Within the model of Ai et al. (2018), Proposition 6 suggests that the equity risk premium term structure can be either upward sloping, downward sloping, or flat. If the adolescent firm’s interaction between investment volatility and correlation with news innovations (σA,IρI,r) is not sufficiently large in magnitude, then the term structure of equity returns is upward sloping, since ρA,r > 0 and bM + bA > 0. Conversely, if the interaction σA,IρI,r is sufficiently large in magnitude, then ρA,r < 0 which leads to bA < 0 and bM + bA < 0 and the term structure of equity returns is downward sloping. In the special case where bM + bA = 0, the term structure of the risk premium is flat. Data availability No data was used for the research described in the article. Appendix proofs Proof of Proposition 1: From Eqs. (8)–(12), the expected risk premium is: Et{Rt+Δ} − Rf = θOPRfβ 2π ∫π − π ( ∑∞ k=− ∞ γ (k) t,dy,re− iωk ) dω, (A. 1) where θOP = − uʹʹ (ct) uʹ(ct) is the Arrow-Pratt measure of absolute risk aversion for time separable utility. The autocovariance terms for OP utility are: γ (h− 1) t,dy,r = CVt {( ct+hΔ − ct+(h− 1)Δ ) ,rt+Δ } = CVt { rc,t+hΔ,rt+Δ } = { σcσrρc,rΔ for h = 1 0 for h ∕= 1 . (A. 2) Substituting the autocovariances from Eq. (A. 2) into Eq. (A. 1) gives: Et{Rt+Δ} − Rf = θOP 2π Rfβσcσrρc,rΔ ∫π − π dω = θOPRfβσcσrρc,rΔ, (A. 3) Q.E.D. Proof of Proposition 2: From Eqs. (8)–(12), the expected risk premium is: Et{Rt+Δ} − Rf = θIHPRfβ 2π ∫π − π ( ∑∞ k=− ∞ γ (k) t,dy,re− iωk ) dω. (A. 4) The autocovariances between marginal utility growth and asset returns with IHP utility are: L.R. Piccotti Economics Letters 244 (2024) 111947 5

γ (h− 1) t,dy,r = CVt{(ϕp(L)ct+hΔ − ϕp(L)ct+(h− 1)Δ,rt+Δ = CVt { ϕp(L)rc,t+hΔ,rt+Δ } = ⎧ ⎪⎨ ⎪⎩ − ϕh− 1σcσrρc,rΔ for h > 1 σcσrρc,rΔ for h = 1 0 for h < 1 , (A. 5) where θIHP = − uʹʹ (ϕp(L)ct) uʹ (ϕp(L)ct) is the Arrow-Pratt coefficient of risk aversion. Substituting the autocovariances from Eq. (A. 5) into Eq. (A. 4) gives: Et{Rt+Δ} − Rf = θIHP 2π Rfβσcσrρc,rΔ ∫π − π [ 1 − ∑∞ k=1 ϕke− iωk ] dω. (A. 6) Q.E.D. Proof of Proposition 3: From Eqs. (8)–(12), the expected excess return is: Et{Rt+Δ} − Rf = θUEHPRfβ 2π ∫π − π ( ∑∞ k=− ∞ γ (k) t,dy,re− iωk ) dω. (A. 7) where θUEHP = − uʹʹ (ct+hΔ+(ϕp (L)− 1)zt+hΔ ) uʹ (ct+hΔ+(ϕp (L)− 1)zt+hΔ ) is the Arrow-Pratt measure of absolute risk aversion for UEHP utility. Under external habit preferences, investors’ excess consumption with respect to aggregate consumption is what matters. With i.i.d. atomistic investors that have uncorrelated consumption shocks, aggregate consumption growth is: rz,t+Δ = ∫ r (h) c,t+Δdh = α(h) c , (A. 8) where h denotes the h’th household. Since investors are atomistic with uncorrelated shocks, in the aggregate, their consumption shocks net out to 0 and aggregate consumption growth is a constant that is equal to the mean consumption drift term of investors. Therefore, there is only a non-zero covariance between contemporaneous marginal utility growth and returns. This is shown in Eq. (A. 9): γ (h− 1) t,dy,r = CVt {( st+hΔ − st+(h− 1)Δ ) ,rt+Δ } = CVt { rc,t+Δ,rt+Δ } = { σcσrρc,rΔ for h = 1 0 for h ∕= 1 , (A. 9) where st+hΔ = ct+hΔ + ( ϕp(L) − 1 ) zt+hΔ. Substituting the autocovariances from Eq. (A. 9) into Eq. (A. 7) gives: Et{Rt+Δ} − Rf = θUEHP 2π Rfβσcσrρc,rΔ ∫π − π dω = θUEHPRfβσcσrρc,rΔ, (A. 10) Q.E.D. Proof of Proposition 4: For notational convenience, ut (∑q j=1 ψj− 1 ([ Aj− 1 ξt ] 1,1 + μc ) ) is abbreviated to ut(⋅). From Eqs. (8)–(12), the expected excess return is: Et{Rt+Δ} − Rf = θTPRfβ 2π ∫π − π ( ∑∞ k=− ∞ γ (k) t,dy,re− iωk ) dω, (A. 11) where θTP = − ut(⋅) uʹ t(⋅) is the Arrow-Pratt measure of absolute risk aversion for TP utility. The autocovariances between marginal utility growth with time preferences utility and asset returns are: L.R. Piccotti Economics Letters 244 (2024) 111947 6

γ (h− 1) t,dy,r = CVt {(∑q j=1 ψj− 1Et+hΔ { ct+(h− 1+j)Δ } − ∑q j=1 ψj− 1Et+(h− 1)Δ { c(t+(h− 2+j)Δ) } ) ,rt+Δ } = CVt {∑q j=1 ψj− 1 [ Et+hΔ { ct+(h− 1+j)Δ } − Et+(h− 1)Δ { ct+(h− 2+j)Δ }],rt+Δ } = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 0 for h > p ∑q j=2 ψj− 1 [ Aj− 1] 1,hσcσrρc,rΔ for 1 < h ≤ p ∑q j=1 ψj− 1 [ Aj− 1] 1,1σcσrρc,rΔ for h = 1 0 for h < 1 . (A. 12) Substitute the autocovariances from Eq. (A.12) into Eq. (A.11) and collect the σcσrρc,rΔ terms: Et{Rt+Δ} − Rf = θTPRfβσcσrρc,rΔ 2π × ∫π − π [ ∑q j=1 ψj− 1 [ Aj− 1] 1,1 + ∑q j=2 ψj− 1 [ Aj− 1] 1,2e− iω + . +∑q j=2 ψj− 1 [ Aj− 1] 1,pe− (p− 1)iω]dω. (A. 13) Separate out the first sum so that all summations will be from 2 to q and note that ψ0 = 1 and that [ A0] 1,1 = 1: Et{Rt+Δ} − Rf = θTPRfβσcσrρc,rΔ 2π ∫π − π dω + θTPRfβσcσrρc,rΔ 2π × ∫π − π [ ∑q j=2 ψj− 1 [ Aj− 1] 1,1 +∑q j=2 ψj− 1 [ Aj− 1] 1,2e− iω + ⋯ + ∑q j=2 ψj− 1 [ Aj− 1] 1,pe− (p− 1)iω ] dω. Collect the summations of ψ and collect the summation of terms from 1 to k to yield: Et{Rt+Δ} − Rf = θTP 2π Rfβσcσrρc,rΔ × ∫π − π [ 1 +∑q j=2 ∑p k=1 ψj− 1 [ Aj− 1] 1,ke− (k− 1)iω ] dω. ∫π − π [ 1 +∑q j=2 ∑p k=1 ψj− 1 [ Aj− 1] 1,ke− (k− 1)iω ] dω. (A. 14) Q.E.D. Proof of Proposition 5: The proof follows closely to the proof of Proposition 4. For notational convenience, ut (∑q j=1 ψj− 1 ([ Aj− 1 (ξt − Yt) ] 1,1 + (μc − μx) )) is abbreviated to ut(⋅). From Eqs. (8)–(12), the expected excess return is: Et{Rt+Δ} − Rf = θCEHPRfβ 2π ∫π − π ( ∑∞ k=− ∞ γ (k) t,dy,re− iωk ) dω, (A. 15) where θCEHP = − ut(⋅) uʹ t(⋅) is the Arrow-Pratt measure of absolute risk aversion for CEHP utility. The autocovariances between marginal utility growth with discounted external habit preferences and asset returns are: L.R. Piccotti Economics Letters 244 (2024) 111947 7

γ (h− 1) t,dy,r = CVt {(∑q j=1 ψj− 1Et+hΔ { ct+(h− 1+j)Δ − xt+(h− 1+j)Δ } − ∑q j=1 ψj− 1Et+(h− 1)Δ { c(t+(h− 2+j)Δ) − xt+(h− 2+j)Δ } ) ,rt+Δ } = CVt {∑q j=1 ψj− 1 [ Et+hΔ { ct+(h− 1+j)Δ } − Et+(h− 1)Δ { ct+(h− 2+j)Δ }],rt+Δ } +CVt {∑q j=1 ψj− 1 [ Et+hΔ { − xt+(h− 1+j)Δ } − Et+(h− 1)Δ { − xt+(h− 2+j)Δ }],rt+Δ } = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 0 for h > p ∑q j=2 ψj− 1 [ Aj− 1] 1,h(bc − bx)σ2 r Δ for 1 < h ≤ p ∑q j=1 ψj− 1 [ Aj− 1] 1,1(bc − bx)σ2 r Δ for h = 1 0 for h < 1 , (A. 16) where bl = ( σl σrρl ,r ) /σ2 r for l ∈ {c, x} is the sensitivity of investor’s consumption and external habit to market returns. Substitute the autocovariances from Eq. (A.16) into Eq. (A.15) and collect the (bc − bx)σ2 r Δ terms: Et{Rt+Δ} − Rf = θCEHPRfβ(bc − bx)σ2 r Δ 2π × ∫π − π [ ∑q j=1 ψj− 1 [ Aj− 1] 1,1 + ∑q j=2 ψj− 1 [ Aj− 1] 1,2e− iω + . +∑q j=2 ψj− 1 [ Aj− 1] 1,pe− (p− 1)iω ] dω. (A. 17) Separate out the first sum so that all summations will be from 2 to q and note that ψ0 = 1 and that [ A0] 1,1 = 1: Et{Rt+Δ} − Rf = θCEHPRfβ(bc − bx)σ2 r Δ 2π ∫π − π dω + θCEHPRfβ(bc − bx)σ2 r Δ 2π × ∫π − π [ ∑q j=2 ψj− 1 [ Aj− 1] 1,1 +∑q j=2 ψj− 1 [ Aj− 1] 1,2e− iω + ⋯ + ∑q j=2 ψj− 1 [ Aj− 1] 1,pe− (p− 1)iω ] dω. Collect the summations of ψ and collect the summation of terms from 1 to k to yield: Et{Rt+Δ} − Rf = θCEHP 2π Rfβ(bc − bx)σ2 r Δ × ∫π − π [ 1 +∑q j=2 ∑p k=1 ψj− 1 [ Aj− 1] 1,ke− (k− 1)iω ] dω. (A. 18) Q.E.D. Proof of Proposition 6: The proof follows closely to the proofs of Proposition 4 and Proposition 5. For notational convenience, ut (∑q j=1 ψj− 1QH,t+jΔ ) is abbreviated to ut(⋅). From Eqs. (8)–(12), the expected excess return is: Et{Rt+Δ} − Rf = θPBRfβ 2π ∫π − π ( ∑∞ k=− ∞ γ (k) t,dy,re− iωk ) dω, (A. 19) where θPB = − ut(⋅) uʹ t(⋅) is the Arrow-Pratt measure of absolute risk aversion for PB utility. The autocovariances between marginal utility growth with discounted external habit preferences and asset returns are: L.R. Piccotti Economics Letters 244 (2024) 111947 8

γ (h− 1) t,dy,r = CVt {(∑q j=1 ψj− 1QH,t+(h− 1+j)Δ − ∑q j=1 ψj− 1QH,t+(h− 2+j)Δ ) ,rt+Δ } = ⎧ ⎪⎪⎨ ⎪⎪⎩ ∑q j=2 ψj− 1 ( λ(h− 1)+(j− 1) (bM + bA) ) σ2 r Δ for h ≥ 1 0 for h < 1 . (A. 20) where bl = ( σl σrρl ,r ) /σ2 r for l ∈ {M,A} is the sensitivity of investor’s production income attributable to firm type to market returns. Substitute the autocovariances from Eq. (A.20) into Eq. (A.19) and collect the (bH +bA)σrΔ terms: Et{Rt+Δ} − Rf = θPBRfβ(bM + bA)σ2 r Δ 2π × ∫π − π [ 1 + ∑q j=2 ∑p k=1 ψj− 1λk+j− 1e− (k− 1)iω ] , (A. 21) where the autocovariance order has been arbitrarily truncated at a lag length of p and where bl = ( σl σrρl ,r ) /σ2 r for l ∈ {M,A}. Q.E.D. References Abel, AB., 1990. Asset pricing under habit formation and catching up with the joneses. Am. Econ. Rev. 80, 38–42. Ai, H., Croce, M.M., Diercks, AM., Li, K., 2018. News shocks and the production-based term structure of equity returns. Rev. Financ. Stud. 31, 2423–2467. Bansal, R., Yaron, A., 2004. Risks for the long run: a potential resolution of asset pricing puzzles. J. Financ. 59, 1481–1509. Campbell, JY., Cochrane, JH., 1999. By force of habit: a consumption-based explanation of aggregate stock market behavior. J. Political Econ. 107, 205–251. Constantinides, G.M., 1990. Habit formation: a resolution of the equity premium puzzle. J. Political Econ. 98, 519–543. Dew-Becker, I., Giglio, S., 2016. Asset pricing in the frequency domain: theory and empirics. Rev. Financ. Stud. 29, 2029–2068. Epstein, L.G., Zin, S.E., 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretic framework. Econometrica 57, 937–969. Fudenberg, D., Strzalecki, T., 2015. Dynamic logit with choice aversion. Econometrica 83, 651–691. Lettau, M., Wachter, J.A., 2011. The term structures of equity and interest rates. J. Financ. Econ. 101, 90–113. Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: a continuous-time case. Rev. Econ. Stat. 51, 247–257. Piccotti, L.R., 2022. Portfolio returns and consumption growth covariation in the frequency domain, real economic activity, and expected returns. J. Financ. Res. 45, 513–549. van Binsbergen, J., Brandt, M., Koijen, R., 2012. On the timing and pricing of dividends. Am. Econ. Rev. 102, 1596–1618. van Binsbergen, J., Hueskes, W., Koijen, R., Vrugt, E., 2013. Equity yields. J. Financ. Econ. 110, 503–519. L.R. Piccotti Economics Letters 244 (2024) 111947 9