Macroeconomic Fluctuations, Oil Supply Shocks, and Equilibrium Oil Futures Prices

Transcription

1 Macroeconomic Fluctuations, Oil Supply Shocks, and Equilibrium Oil Futures Prices Steffen Hitzemann November 215 Abstract What is the role of macroeconomic fluctuations and of oil supply shocks for oil prices, volatilities, and risk premia? I analyze this question within a general equilibrium asset pricing framework with an oil sector. The benchmark calibration shows that short-run shocks to macroeconomic growth and oil productivity shocks account for more than 9% of the volatility in the oil market and are responsible for the mean-reversion behavior of oil prices. On the other hand, long-run macroeconomic growth risks are the main driver of risk premia on oil futures and their upward-sloping term structure, which is observed in the data. The model consistently explains quantity and price dynamics in the oil sector and in the general macroeconomy, and furthermore sheds light on the intricate relationship between oil and equity returns. This paper was previously circulated under the title Production-Based Asset Pricing and the Oil Market. I thank Max Croce, Erik Gilje, Vincent Glode, Itay Goldstein, Urban Jermann, Marcel Prokopczuk, Nick Roussanov, Philipp Schuster, Krista Schwarz, Ivan Shaliastovich, Luke Taylor, Marliese Uhrig- Homburg, Amir Yaron, as well as participants of the 214 Finance, Probability, and Statistics Workshop of the Institute of Mathematical Statistics, the Workshop on Commodity Markets and their Financialization at UCLA, the 215 Annual Meeting of the German Finance Association, the 215 SAFE Asset Pricing Workshop at Goethe University Frankfurt, and seminar participants at Duke University (Fuqua), Humboldt University Berlin, Syracuse University (Whitman), University of British Columbia (Sauder), and the University of Pennsylvania (Wharton) for valuable discussions and helpful comments and suggestions. Department of Finance, Fisher College of Business, The Ohio State University, Columbus, OH 4321, hitzemann.6@osu.edu. 1

2 1 Introduction Fueled by the high volatility of oil prices in the last decade and the tremendous rise of commodity index investments, the interest of economic agents in understanding the price behavior in oil markets could not be higher. A substantial contribution to this topic is the development of partial equilibrium models, be it for commodities in general (e.g., Routledge, Seppi, and Spatt 2; Carlson, Khokher, and Titman 27) or the oil market in particular (e.g., Litzenberger and Rabinowitz 1995; Kogan, Livdan, and Yaron 29). The typical setup of partial equilibrium models, however, precludes the analysis of at least two important issues. First, taking oil demand as exogenously given, it is not possible to investigate how macroeconomic fluctuations translate to the oil market and drive spot and futures prices. Second, a thorough quantitative analysis of risk premia is not possible in partial equilibrium since the aggregate risk in the economy is not considered. In fact, many related models assume risk-neutrality or work with an exogenously specified pricing kernel. The aim of this paper is to study the oil market in general equilibrium. To do so, I consider a production economy with a general productive sector and an oil sector, in which households derive utility from consuming a bundle of oil and a general consumption good. Firms engaged in the oil sector endogenously decide about the drilling of new oil wells to increase oil production. Incorporating real-world evidence provided by Anderson, Kellogg, and Salant (214), oil is extracted from active wells at the highest possible rate but is subject to production disruptions, which I capture by means of oil productivity shocks. Firms furthermore manage oil inventories, and they decide endogenously about the amount of oil that is released for consumption. On the other side, the general productive sector which I consider as the macroeconomy is modeled in line with the production-based asset pricing framework with long-run productivity risk proposed by Croce (214). The model reproduces the empirical behavior of oil-related quantities and prices considerably well and maintains the qualities of the production-based asset pricing framework in explaining the general macroeconomy. Regarding the oil sector, the benchmark calibration of my model closely matches first and second moments related to oil investment and production and also generates realistic dynamics of oil consumption and inventories. Furthermore, the benchmark calibration produces a backwardated oil futures price curve on average and 2

3 a considerable variation of the slope, a downward-sloping volatility term structure, and a positive and upward-sloping term structure of oil futures risk premia, in line with what is observed empirically. I characterize the role of short- and long-run macroeconomic productivity shocks and oil productivity shocks for the dynamics of important quantities and prices within the model. As the analysis reveals, even a short-run macroeconomic productivity shock has a persistent impact on the oil sector according to the following mechanism. By driving up general consumption, a positive short-run macro shock also leads to an increased oil demand due to the complementarity of the two goods. Since oil production is very rigid in the short run, firms find it optimal to release a certain amount of oil from existing inventories and to increase production gradually in the longer run by drilling new wells. In consequence, oil consumption continues to rise and inventories start filling up again at the same time. The continued gradual increase of oil consumption translates to a mean-reversion behavior of the oil price, which is given by the marginal rate of substitution between oil and the general good. While the effect of oil productivity shocks is somewhat comparable to that of short-run macro shocks, the impact of long-run macroeconomic productivity shocks differs in several ways. First, oil prices gain momentum in response to long-run macro shocks and do not exhibit mean-reversion. Second, positive long-run macro shocks yield higher oil prices and a more contangoed futures price curve at the same time, while high spot prices are associated with a more backwardated futures price curve for the other kinds of shocks. Third, long-run macro shocks explain only a small amount of the volatility in the oil market less than 2% for short maturities and up to about 1% for longer-term contracts. In addition to that, I show that both the long-run component of macroeconomic productivity risk and the existence of inventories in the model are critical ingredients to generate a sizeable positive risk premium for short-term oil futures and an upward-sloping term structure of risk premia. While long-run productivity risk is required to obtain sizeable risk premia at all, the existence of inventories determines the short-term behavior of oil prices to different kinds of shocks. Since positive long-run macro shocks lead to a reduction of general consumption in favor of additional investments due to the substitution effect (see Croce 214), oil prices fall if oil consumption cannot be adjusted immediately, which is the case in a model without 3

4 inventories. A contemporaneous fall in the stochastic discount factor results in a negative risk premium for short-term oil futures in this case. To the contrary, oil consumption is adjusted very flexibly when firms maintain inventories, resulting in increasing oil prices and a positive risk premium. The fact that oil prices gain momentum after long-run macro shocks leads to an even stronger increase of longer-term futures in the benchmark calibration, which therefore carry a higher risk premium than short-term contracts. An interesting aspect of the mechanism behind the risk premia of oil futures in general equilibrium is that it highlights the importance of inventories even when investment in new oil wells is incorporated into the model, while investment- and inventory-based models are sometimes understood as alternative approaches in partial equilibrium. Although this paper does not particularly focus on the dramatic price movements in the oil market during the period from 23 to 28, 1 my findings may still contribute to understanding specific features of oil prices during this time. In particular, the result that positive long-run macroeconomic productivity shocks lead to a less pronounced mean-reversion behavior of oil prices and a more contangoed futures price curve might provide an explanation for this pattern to be found empirically in the recent boom and bust period (see Ready 212). Furthermore, the model helps to quantify the impact of speculators on oil prices and risk premia by providing a benchmark for the part of risk premia that is unrelated to speculative activity. Quantitatively, more than half of the oil futures risk premium is explained by the model across all maturities, leaving a remaining premium of about 3% per year to be potentially driven by speculation-related factors. Finally, the general equilibrium approach of this paper enables me to analyze the sometimes puzzling relationship between oil and stock market returns in a theoretical framework. I find that oil and equity returns are positively related after macroeconomic productivity shocks, which are responsible for a considerable amount of the variation in both markets. On the other hand, oil productivity shocks are followed by a strong reaction of oil prices, but have almost no influence on equity returns. The model thus confirms the empirical findings of Kilian and Park (29), especially that the relationship between oil and equity returns 1 There is a fast growing literature concerned with the boom and bust in oil prices during this period and related topics, such as the financialization of commodity markets and the role of speculators, e.g., Stoll and Whaley (21), Ready (212, 214), Tang and Xiong (212), Cheng, Kirilenko, and Xiong (214), Kilian and Murphy (214), Singleton (214), and Sockin and Xiong (214). 4

5 depends on the type of the preceding shock. The remainder of this paper is organized as follows. Section 2 reviews the related literature. Section 3 presents a production-based asset pricing framework with an oil sector which enables me to study the oil market in general equilibrium. In Section 4, I specify the equilibrium of the model and characterize the model solution. I calibrate the model and provide an overview of important quantities and prices compared to their empirical counterparts in Section 5. Section 6 investigates the dynamics of quantities and prices in detail and analyzes the futures price curve, the volatility term structure, and risk premia in the oil market in general equilibrium. Section 7 discusses the relationship of oil and equity returns within my model. Section 8 concludes and outlines directions for future research. 2 Related Literature The approach of this paper to analyze the oil market within a general equilibrium production-based asset pricing framework is naturally related to two strands of literature. First, there is a long tradition of theoretical models for commodity prices, of which the vast majority work in partial equilibrium by taking commodity demand and the pricing kernel as exogenously given. Several papers in this literature emanate from the classical theory of storage (see Kaldor 1939; Working 1948, 1949; Telser 1958) and emphasize the role of inventories as a cushion against unexpected demand shocks. Williams and Wright (1991) and Deaton and Laroque (1992) formalize this theory in terms of a utility maximization problem of rational inventory holders and elaborate on the asymmetry arising from the non-negativity constraint on inventories. Routledge, Seppi, and Spatt (2) introduce forward contracts into this framework and investigate the slope of the forward price curve and its relation to the state of inventories as well as the term structure of volatilities. Gorton, Hayashi, and Rouwenhorst (212) connect the theory of storage with the theory of normal backwardation to make qualitative predictions about the relation between inventories and risk premia. In contrast to these papers, a few studies abstract from inventories and focus alternatively on optimal investment and production decisions (Litzenberger and Rabinowitz 1995; Carlson, Khokher, and Titman 27; Kogan, Livdan, and Yaron 29). In these models, adjustment costs or physical constraints on investment and production correspond to the non-negativity 5

6 constraint in the inventory-based approach, restricting producers in their flexibility to adjust commodity supply. Interestingly, the analysis of my paper reveals that both investment and inventories are critical to producing key characteristics of oil futures prices and risk premia and should not be understood as alternative concepts in general equilibrium. There exist a few papers that take steps towards a quantitative general equilibrium analysis of the oil market. Casassus, Liu, and Tang (213) consider a general equilibrium setting with two commodities that is tailored to an analysis of convenience yields dependent on the relative scarcity of a commodity. Ready (212) integrates oil into an endowment economy with long-run risk and studies structural changes of oil futures price features during the recent boom and bust period. In a paper developed in parallel to mine, Ready (214) extends his analysis by taking the hybrid approach of adding exogenous oil supply to a production economy setting. Arseneau and Leduc (213) study the aspect of commodity storage within a general equilibrium framework with an explicit focus on food price dynamics. Closest to this paper is the work by Casassus, Collin-Dufresne, and Routledge (29), who analyze oil prices within a general equilibrium production economy with irreversible investment. While the authors present a number of interesting theoretical results such as a state-dependent futures risk premium, several critical modeling choices lead to obvious inconsistencies with the data and therefore preclude reliable quantitative predictions. 2 Second, this paper contributes to the production-based asset pricing literature. Following the idea of Cochrane (1991, 1996), this literature jointly studies macroeconomic quantities and asset prices within real business cycle models. While the classical setting fails to reproduce key features of asset prices (see Rouwenhorst 1995), Jermann (1998) proposes a framework that generates an equity premium as sizeable as in the data and a low risk-free rate. His approach introduces habit-based preferences and adjustment costs, which can be considered as a shortcut to modeling two separate productive sectors for investment and consumption goods (see Boldrin, Christiano, and Fisher 21). In analogy to the developments in the 2 Among other issues, Casassus, Collin-Dufresne, and Routledge (29) find that investment into new oil wells (i.e., oil drilling) is infrequent and lumpy, in contradiction to the empirical observation that oil drilling takes place all the time and responds strongly to oil prices (see Anderson, Kellogg, and Salant 214). The model is furthermore based on the classical real business cycle framework with CRRA preferences, which has proven to be unsuccessful in reproducing asset prices and risk premia. Finally, the authors report an upper bound for the oil price of USD 48.4 within their calibrated model, which is obviously inconsistent with the data. 6

7 consumption-based asset pricing literature, Croce (214) proposes an alternative setup with long-run productivity risk and Epstein and Zin (1991) preferences. This framework improves on habit-based models by generating a less volatile risk-free rate and a more realistic correlation of equity returns and consumption growth. Recent work extends this literature into several directions. Several papers consider settings with multiple productive sectors to study the cross-section of stock returns (e.g., Zhang 25; Gomes, Kogan, and Yogo 29). Other works investigate the capabilities of the production-based asset pricing approach in explaining the term structures of interest rates and equity returns (Ai, Croce, Diercks, and Li 213; Jermann 213). This paper has elements of both, in that it extends the production-based asset pricing framework by an explicitly modeled oil sector and studies the term structure of oil futures prices, volatilities, and risk premia. 3 Model In this section, I present a production-based asset pricing model that enables me to study oil prices in general equilibrium. I consider an economy consisting of households that derive utility from consuming a bundle of oil and a general consumption good, and firms that operate in the oil sector and in a general productive sector. The general productive sector stands for the macroeconomy and is modeled in line with the production-based asset pricing framework with long-run productivity risk proposed by Croce (214). The firms activities in the oil sector include the drilling of new oil wells, the extraction of oil from active wells, and the active management of oil inventories. 3.1 Household The representative household consumes a bundle v(c, B) = [ ] (1 θ)c 1 1 ρ + θb ρ ρ 1 (1) composed of a general consumption good C and of oil barrels B, with θ describing the relative importance of oil consumption in the economy, and ρ is the constant elasticity of substitution. Oil and the general consumption good are complements for ρ close to zero, 7

8 while they are substitutes for large ρ. The household maximizes utility according to Epstein and Zin (1991) preferences V t = [ (1 β)v(c t, B t ) 1 1 ψ + βet [ V 1 γ t+1 ] 1 ] 1 ψ 1 1 γ 1 ψ 1, (2) which allow to separate the intertemporal elasticity of substitution ψ from the relative risk aversion γ. The parameter β is the subjective discount factor. The household s utility maximization is subject to the intertemporal budget constraint W t+1 = (W t C t P t B t )R W t+1, (3) where W t denotes the overall wealth of the household, R W t+1 is the return on wealth, and P t is the oil price Firms and Production The representative firm manages the capital and oil stock of the economy and generates cash-flows by selling the general consumption good and oil to the market. More precisely, the amount of the general consumption good sold is equal to the production output Y t minus the new investment I t in the general productive sector and the investment H t for the drilling of new oil wells. On the other hand, the firm decides about the amount of oil S t of the current oil stock O t that is stored until the next period, while the remaining amount of oil is sold to the market at price P t. Overall, the representative firm s goal is to maximize firm value, which is equivalent to maximizing the expected sum of discounted cash-flows E t s= M t+s (Y t+s I t+s H t+s + P t+s (O t+s S t+s )), (4) where M t+s is the s-period stochastic discount factor. The capital stock of the economy evolves according to K t+1 = (1 δ)k t + I t G t K t, (5) 3 The price of the general consumption good is always 1 since it is chosen to be the numeraire. 8

9 where δ is the capital depreciation rate and G t is a Jermann (1998) adjustment cost function of the form ( ) ( It G t = I t a + a 1 K t K t 1 1 ξ ( It K t ) ) 1 1 ξ. (6) Following the literature, the adjustment cost function captures the costly reallocation of goods between the general productive sector and the related consumption sector in reduced form. 4 For high values of the parameter ξ >, capital can be reallocated very flexibly between the consumption and productive sector, while reallocation is very costly for low values of ξ. The parameters a and a 1 are chosen in such way that adjustment costs G t and the first derivative G t are zero at the deterministic steady state of the model. Output is produced according to a Cobb-Douglas production function where A t is the stochastic productivity of labor. Y t = (A t L Y t ) 1 α K α t, (7) I assume an exogenous constant labor supply of L Y t = l Y. Following Croce (214), the stochastic productivity A t consists of a short-run and a long-run component and overall follows the dynamics A t+1 = A t exp{µ + x t + ε A t+1}, (8) x t+1 = φx t + ε x t+1, (9) where ε A t+1 and ε x t+1 are mutually independent and i.i.d. normal shocks with mean zero and standard deviations σ A and σ x. In analogy with the long-run risk literature on consumptionbased asset pricing originating from Bansal and Yaron (24), the long-run component x t of productivity risk makes it possible to generate sizeable risk premia while keeping the volatility of consumption growth which arises endogenously in the production-based asset pricing framework at a realistic level. 4 In the oil sector, I model the limited flexibility of firms to adjust oil production explicitly in the next section. 9

10 3.3 Oil Sector In the oil sector of the economy, the firm maintains an amount of oil wells U t. More precisely, I consider U t as the amount of oil that is contained in the active oil wells of the economy and extracted over time. The oil production at time t is given by E t = κ t U t, (1) where κ t is a stochastic extraction rate specified below. Accordingly, the law of motion for oil wells is U t+1 = (1 κ t )U t + Z t, (11) with Z t being the amount of new oil wells drilled. Drilling oil wells requires capital investment H t and labor L Z t according to the investment function Z t = ζ(a t L Z t ) 1 τ H τ t, (12) where τ is the capital share of oil drilling and ζ is a scaling factor. Again, I assume an exogenous constant labor supply L Z t = l Z and I summarize the constant factors of the function as ζ = ζ l Z 1 τ. The particular specification of oil investment and production appears intuitive, but contains two non-trivial modeling choices that reflect the characteristic structure of the oil industry, as pointed out by Anderson, Kellogg, and Salant (214). First, the daily rental cost for a drilling rig (the rig rate) increases in a convex manner in the drilling activity, leading to the concave investment function specified. Second, and more importantly, production from existing oil wells is subject to a physical capacity constraint. Up to this constraint, however, it is virtually always optimal to produce at the highest possible extraction rate since marginal production costs are very low. Following this characteristic feature of the real-world oil industry, I refrain from endogenizing the oil production decision and I assume that the maximum extraction rate κ t follows an AR process κ t+1 = η(1 χ) + χκ t + ε κ t+1 (13) 1

11 with mean η, where ε κ t+1 is an i.i.d. normal shock with mean zero and standard deviation η σ κ that is independent of the macroeconomic shocks defined above. The stochastic nature of the extraction rate captures production disruptions and other productivity fluctuations in the oil extraction process. The representative firm maintains oil inventories and decides in each period explicitly about the amount S t of the available oil that is stored and not released to the market. The oil stock O t of the economy, consisting of oil inventories from the previous period and new oil production, accordingly evolves as O t+1 = (1 ω)s t Π t A t + E t+1, (14) where ω is the storage cost and Π t captures the opportunity cost of possible stock-outs arising from the non-negativity constraint on inventories. 5 I follow the standard approach in the macroeconomics literature to define stock-out costs exogenously in order to ensure tractability of the model. 6 inventories In particular, I specify Π t as an inverse-quadratic function of ( ) St Π t = π A t 2 ( St A t ) 2 (15) with parameter π, similar to that proposed by Maccini and Pagan (213). 4 Equilibrium and Model Solution This section establishes the solution of the outlined model. For that, I specify the equilibrium conditions and formulate the related problem of a central planner. I further derive the prices of oil futures contracts as well as the risk-free rate and the market return in the context of the model. 5 I define stock-out costs relative to the size of the economy by normalizing with productivity A t. 6 See Wen (211) for a discussion of the related literature. Partial equilibrium models for commodity markets with an explicitly modeled non-negativity constraint are studied by Routledge, Seppi, and Spatt (2) and by the classical theory of storage literature discussed in Section 2. 11

12 4.1 Equilibrium Conditions I impose a standard competitive rational expectations equilibrium to solve the model. In particular, an equilibrium is given by a set of consumption, investment, and inventory strategies such that the household s and the firm s utility is maximized and that markets clear. In the following I derive the first order conditions of the household and the firm and impose the market clearing conditions Household s First Order Conditions The household s first order conditions are obtained by maximizing its utility (2) with respect to the intertemporal budget constraint (3). Technically, the household s problem accords to the problem in a two-good endowment economy, and I refer to Yogo (26) for detailed derivations. First, the intratemporal condition holds that the price of oil is equal to the marginal rate of substitution between oil and the general consumption good, i.e., Second, define the pricing kernel as M t+1 = [ β P t = ( ) 1 Ct+1 C t with ϑ ( ) ( B C = (1 θ + θ B ) 1 1 C ρ ) 1 θ 1 θ ( Bt C t ψ ( ϑ(b t+1 /C t+1 ) ϑ(b t /C t ) ) 1 ρ. (16) ) ] 1 ρ 1 ψ R W 1 1 ψ 1 1 γ 1 ψ 1 1 γ t+1, (17) 1 1 ρ. 7 For this pricing kernel, the standard Euler equation E t [M t+1 R t+1 ] = 1 (19) 7 An equivalent expression for the pricing kernel in terms of the value function is given by M t+1 = β ( ) 1 Ct+1 C t ψ ( ϑ(b t+1 /C t+1 ) ϑ(b t /C t ) ) 1 ρ 1 ψ E t [ V t+1 V 1 γ t+1 ] 1 1 γ 1 ψ γ. (18) 12

13 holds for the returns R t+1 of all assets traded in the economy Firm s First Order Conditions I derive the firm s first order conditions along the lines of Cochrane (1991, 1996). For that, the firm s utility (4) is maximized with respect to the related laws of motion. Detailed derivations are provided in Appendix A. The resulting first order conditions can be expressed in terms of Euler equations E t [ Mt+1 R i t+1] = 1, i = I, H, S (2) for the return on investment in the general productive sector, R I t+1, the return on drilling new oil wells, R H t+1, and the return on holding inventories, R S t+1. In particular, these returns are Rt+1 I = α Y t+1 I K t+1 + ((1 δ) + G t+1 t+1 K t+1 G t+1 )Q I t+1, (21) Q I t Rt+1 H = P t+1κ t+1 + (1 κ t+1 )Q H t+1, (22) Q H t Rt+1 S = (1 ω Π t)q S t+1, (23) Q S t where Q i t, i = I, H, S are the marginal rates of transformation between new capital, oil wells, or inventories, and the general consumption good, as given by Q I t = 1, Q H 1 G t = t 1 τ Z t /H t, Q S t = P t. (24) In other words Q i t, i = I, H, S is the value of an additional unit of new capital, oil wells, or inventories in terms of the general consumption good. Therefore, the intuition for the return on investment in new oil wells given by (22) is that investing in an additional unit of oil wells today yields the production of κ t+1 units of oil with price P t+1 tomorrow, plus the remaining 1 κ t+1 units of the oil well with value Q H t+1. The returns (21) and (23) correspond to an analogous intuition, where especially the former one is well understood in the literature. 13

14 4.1.3 Market Clearing The markets for both the general consumption good and for oil clear in equilibrium. Therefore, general consumption, general investment, and oil investment add up to the output of the general productive sector, i.e., C t + I t + H t = Y t. (25) On the other hand, oil consumption equals the overall oil stock available minus the amount of oil stored to the next period, i.e., B t + S t = O t. (26) 4.2 Model Solution According to the welfare theorems, the model can be reformulated as the problem of a central planner. Precisely, the central planner maximizes the household s utility (2) with respect to the laws of motion (5), (8), (9), (11), (13), (14) and the feasibility conditions (7), (12), (25), (26). I solve the problem numerically by applying perturbation techniques. In particular, I use a second-order local approximation as provided by the dynare package. 8 For technical details I refer to Appendix B. Relevant prices and returns are obtained based on the allocation of the central planner. The time-t price of a futures contract on oil FT,t O with maturity at t + T is given by = E t [M t+1 (F O T 1,t+1 F O T,t)], (27) where F O,t = P t and I assume marking to market at each period of time. The one-period return of oil futures with maturity at t + T is defined as RT,t+1 O = F T O 1,t+1. (28) FT,t O 8 See The dynare package is widely applied in macroeconomics and macrofinance, see for example Ai, Croce, and Li (213) or Croce (214). 14

15 The expected return is interpreted as the risk premium of T -period oil futures. Furthermore, the risk-free rate of the economy is given by R f t = 1 E t [M t+1 ]. (29) Finally, the market return is the weighted average of the different investment returns, Rt+1 M = K tq I t Rt+1 I + U t Q H t Rt+1 H + O t Q S t Rt+1 S. (3) K t Q I t + U t Q H t + O t Q S t Consistent with the literature, I calculate the equity risk premium based on levered excess returns, i.e., I consider R LEV ex,t = (1 + DE)(R M t R f t 1), (31) with an average debt-to-equity ratio DE that is set to 1 in line with Rauh and Sufi (212). 5 Calibration In this section, I present two calibrated versions of the model. The first version which I call the benchmark calibration is a calibration of the full model as outlined above. The second version does not allow for inventory holdings, such that the households oil consumption is always equal to the amount of oil that is produced. For both models, I compare several statistics of general macroeconomic and oil-related quantities and prices to their empirical counterparts. To do so, I combine several sources of data. Macroeconomic quantities are calculated based on the National Income and Product Accounts (NIPA) tables published by the Bureau of Economic Analysis (BEA). While the NIPA tables also provide adequate proxies for oil-related investment and consumption, I use data from the Energy Information Administration (EIA) for oil production and inventories. Finally, I obtain data on equity returns, interest rates, and oil futures prices from the standard financial databases. 9 I consider 3 years of data from 1984 to 213, with the launch of crude oil futures trading determining the beginning of the sample period. 9 A detailed description of the data and the construction of the different time series is provided in Appendix C. 15

16 5.1 Parameter Choices The preference parameters are chosen identically for both model variants and are reported in Table 1. In line with the long-run risk literature (see Bansal and Yaron 24), I set the relative risk aversion γ to 1 and the intertemporal elasticity of substitution ψ to 2. The annualized subjective discount factor β is fixed at.96. The oil share in the economy, expressed by the parameter θ, is quantified by Wei (23) to be.4. Furthermore, I set the elasticity of substitution ρ between oil and capital to.9, in line with Backus and Crucini (2). Table 1: Preference parameters. All preference parameters are chosen in line with the literature and are identical for both model calibrations. Parameter Value Subjective discount factor β.96 Risk aversion γ 1 Intertemporal elasticity of substitution ψ 2 Importance of oil θ.4 Constant elasticity of substitution ρ.9 Table 2 presents parameters that describe the structure of the general macroeconomy and of the oil sector. For the parameters related to the general macroeconomy, I exactly follow Croce (214) and set the capital share α to.34, the annual depreciation rate of capital δ to.6, and the annual average growth rate of the economy µ to 1.8%. Moreover, Croce (214) identifies a short-run component of productivity with volatility σ A of 3.35% and a long-run component with volatility σ x of.1σ A and autocorrelation φ of.8. For the constant labor supply, I follow Tallarini (2) and fix l Y at.18. Standard values for parameters describing the industrial structure of the oil sector are much less established in the literature. At least, Casassus, Collin-Dufresne, and Routledge (29) provide an estimate of.1 for the oil inventory costs ω. Furthermore, I infer the average rate of oil production η from Anderson, Kellogg, and Salant (214) who present micro-level data on the oil production of active wells in Texas. In their data, the production from pre-existing oil wells halves in about four years, 1 which corresponds to an annual decline rate of 16%. Finally, I estimate the mean- 1 Since oil wells in Texas have a relatively low production volume compared to average oil wells in the US, 16

17 Table 2: Calibrated parameters. This table reports parameters describing the general macroeconomy and the oil sector for the benchmark model and the model variant without inventories. Parameters related to the general macroeconomy, except the capital adjustment costs, are fixed in line with the calibration presented by Croce (214). Oil inventory costs and the average oil production rate are chosen in line with the literature, and the parameters of the oil productivity process are estimated using data from the EIA. The capital adjustment cost parameter, the parameters of the oil investment function, and the oil stock-out costs are calibrated by matching the moments reported in Table 3. The models are calibrated at an annual frequency. Parameter Benchmark No Inventories General macroeconomy Capital share α Depreciation rate of capital δ.6.6 Average growth rate µ 1.8% 1.8% Capital adjustment costs ξ Autocorrelation of expected growth φ.8.8 Volatility of short-run risk σ A 3.35% 3.35% Volatility of long-run risk σ x.1σ A.1σ A Oil sector Oil investment factor ζ Capital share of oil drilling τ Oil inventory costs ω.1 Oil stock-out costs π.53 Average oil production rate η Mean-reversion of oil productivity χ Volatility of oil productivity risk σ κ 5.26% 5.26% reversion parameter and the standard deviation of the oil productivity process (13) by using annual oil well productivity data from the EIA. I obtain a mean-reversion rate χ of.87 and a volatility of oil productivity shocks σ ω of 5.26%. Four parameters are not specified yet: The economy s capital adjustment costs ξ, the parameters of the investment function for oil wells, τ and ζ, and the stock-out costs for oil I consider the top 1% producing oil wells in Texas presented in Figure 9.b of Anderson, Kellogg, and Salant (214) as representative. 17

18 inventories, π. I calibrate these parameters by matching four moments: the equity premium, the first and second moment of oil-specific investment relative to general investment, and the ratio of oil inventories and oil production. The capital adjustment cost parameter determines the degree of capital flexibility to smooth the consumption stream by investments and is therefore responsible for the endogenous volatility of consumption growth within the model. By matching the equity premium, I ensure a fair analysis of oil futures risk premia later in the paper based on a level of endogenous consumption volatility that is also needed to generate the level of equity returns. The parameters τ and ζ specify the link of the oil sector and the general macroeconomy on the investment side of the model. I match both the level and the volatility of oil-related investment relative to general investment to make sure that both the magnitude and the fluctuation of oil investments are at realistic levels. These three parameters have naturally different values in the benchmark model and the model without oil inventories (see Table 2). Finally, I calibrate the oil stock-out cost parameter π for the model with inventories to match the empirical level of the oil inventory-production ratio. 5.2 Aggregate Prices and Quantities This section provides an overview of prices and quantities generated by the models compared to their empirical counterparts, and I inspect the underlying mechanisms in Section 6. As reported in Table 3, both the model with oil inventories and the model without inventories are able to match the equity risk premium, the ratio of oil-related and general investment, and the relative volatility of oil-related investment and general investment. In addition, the benchmark model with inventories also matches the amount of inventories held by the economy relative to the annual oil production. Table 4 shows important quantities related to the general macroeconomy and the oil sector that the models are not calibrated to. The quantity-related statistics are very similar for the model with and without inventories, except for the volatility of oil consumption relative to production. The investment-output ratio is higher than in the data, but in a similar order of magnitude. Stronger deviations are observable for the relative volatility of consumption growth, which is too high relative to the data, and the volatility of investment growth, which is too low. While this pattern is qualitatively also observable in the production-based 18

19 Table 3: Calibrated moments. Empirical moments are calculated based on annual US data from 1984 to 213. Both model calibrations match the equity risk premium, the ratio of oil-related and general investment, and the relative volatility of oil-related investment and general investment. The benchmark model also matches the ratio of oil inventory and production. Lowercase letters refer to log variables, and is the first difference operator. Statistic Data Benchmark No Inventories Equity risk premium E[rex,t+1] LEV [%] Ratio of oil-related and general investment E[H/I] [%] Relative volatility of oil-related and general investment σ( h)/σ( i) Oil inventory-production ratio E[S/E] asset pricing model without an oil sector studied by Croce (214), it is more pronounced in the model calibrations presented in here. The obvious reason for this deviation is that the equity risk premium in the data from 1984 to 213 which the model is calibrated to is very high. Precisely, my calibrations match an equity risk premium of almost 6.5%, while the equity risk premium in the otherwise considered sample starting in 1929 is about 5%. Therefore the endogenous volatility of consumption growth has to be relatively high, which is achieved through considerable adjustment costs that limit the possibility of consumption smoothing by adjustments of the capital invested. As a result, the volatility of investment growth is relatively low. It can be concluded that the deviations from the data observed for macroeconomic quantities are typical of the production-based asset pricing framework in general and do not specifically occur due to the existence of an oil sector in the economy. The quantities related to the oil sector are reasonably close to those observed empirically. Especially, the relative volatilities of oil production and investment almost exactly match the data. The volatility of oil consumption and oil inventories both measured relative to the volatility of oil production are in a reasonable order of magnitude in the benchmark calibration, but higher than in the data. Both statistics suggest that the inventory technology described by the model is extraordinarily flexible compared to reality. While not introduced 19

20 Table 4: Quantities. This table presents important macroeconomic and oil-specific quantities that the models are not calibrated to. Empirical moments are calculated based on annual US data from 1984 to 213. Oil consumption is expressed in monetary units in the data and therefore corresponds to P t B t in the model. Lowercase letters refer to log variables, and is the first difference operator. Statistic Data Benchmark No Inventories Investment-output ratio E[I/Y ][%] Relative volatility of general consumption and output σ( c)/σ( y) Relative volatility of general investment and output σ( i)/σ( y) Relative volatility of oil production and oil investment σ( e)/σ( h) Relative volatility of oil consumption and oil production σ( p + b)/σ( e) Relative volatility of oil inventories and oil production σ( s)/σ( e) in here, the presence of inventory adjustment costs as considered by Gomes, Kogan, and Yogo (29) could help to reduce these fluctuations. Table 5 summarizes relevant price statistics generated by the model calibrations in comparison to the data. On the macroeconomic side, the level and volatility of the real risk-free rate are fit very well by both model variants considered. The clear superiority of the benchmark model compared to the model without inventories comes to light when considering oil futures prices, volatilities, and related risk premia. The benchmark calibration successfully reproduces a futures price curve that is in backwardation on average, but whose slope considerably varies over time. Although the variation of the slope is slightly lower than in the data, the model still generates a contangoed futures price curve 38% of the time. The volatility of short-term oil futures is explained less well by the benchmark calibration, being only about half of that observed in the data. A possible explanation is that factors that are unrelated to fundamental variables account for a considerable amount of asset price volatilities, in line 2

21 Table 5: Prices. This table presents important price variables that the models are not calibrated to. Empirical moments are calculated based on annual US data from 1984 to 213. Annual oil futures returns are constructed according to a monthly roll strategy of the related contracts. Lowercase letters refer to log variables, and is the first difference operator. Statistic Data Benchmark No Inventories Risk-free rate E[r f t ] [%] Volatility of risk-free rate σ(r f t ) [%] Slope of oil futures price curve E[F1,t/F O 12,t] O Variation of futures price curve σ(f1,t/f O 12,t) O [%] Volatility of short-term oil futures σ(r O 2,t+1) [%] Slope of volatility term structure σ(r O 2,t+1)/σ(r O 12,t+1) Short-term oil risk premium E[r2,t+1] O [%] Slope of term structure of oil futures risk premia E[r O 12,t+1 r O 2,t+1] [%] with the excess volatility puzzle (LeRoy and Porter 1981; Shiller 1981). In fact, the model can also explain only a bit more than half of the equity volatility, even when idiosyncratic cash-flow shocks that are not priced by the market are incorporated (see Croce 214). The volatility term structure of oil futures produced by the model is clearly downward-sloping, in accordance with the data. Finally, a sizeable risk premium on short-term oil futures and a clearly upward-sloping term structure of futures risk premia is generated by the benchmark calibration. The risk premia on both short-term and long-term futures are smaller than in the data, but in a considerable order of magnitude. In sharp contrast to that, the model without inventories generates a volatility of short-term 21

22 oil futures that is considerably larger than in the data. Additionally, this high volatility is compensated by a highly negative risk premium, which is also contradicted by the data. Finally, the large negative risk premium at the short end leads to a futures price curve in this model calibration that is in contango on average. In the next section, I inspect the underlying mechanisms that are responsible for the particular outcomes of the two model variants. 6 Oil Prices in General Equilibrium This section analyzes oil spot and futures prices and their driving forces within the general equilibrium context of this paper. First, I inspect the dynamics of general macroeconomic and oil-specific quantities as well as the resulting oil price fluctuations within my model. Second, I investigate the precise implications of the model for oil futures contracts in detail. In particular, the model sheds light on the role of short- and long-run macroeconomic as well as oil-specific productivity shocks for the slope of the futures price curve, the volatility term structure, and the term structure of risk premia. Finally, I highlight why a model that does not incorporate oil inventories fails to explain the term structure of oil futures risk premia. 6.1 Quantity Dynamics The behavior of general macroeconomic and oil-specific quantities in response to the different kinds of shocks is illustrated by Figure 1 and Figure 2 in terms of impulse response functions. While I discuss the prevailing effects for the benchmark model in here, I point out the differences to a model without oil inventories in Section 6.6. A positive short-run macroeconomic productivity shock leads to a temporarily increasing output of the general production technology, resulting in a higher supply of the general consumption good. As in classical real business cycle models, the representative agent uses parts of this additional output to increase present consumption, but also invests more in the general productive sector in order to smooth consumption over time. These effects translate to the oil sector. Due to the complementarity of of oil and the general consumption good, the marginal utility of oil consumption rises and it is optimal to increase the consumption of oil as well. 22

23 Since oil production is rigid in the short run, the resulting demand is satisfied by existing inventories. In contrast, oil production can well be adjusted in the longer run by additional investments, and it is optimal for the agent to expand oil drilling activities. As a result, the amount of active oil wells and therefore also the oil production of the economy increases slowly by slowly, such that inventories start filling up again after some time. Overall, oil inventories cushion the arising short-term demand shock for oil until the production is finally adjusted, in line with intuition. Through this mechanism, even a short-run macroeconomic productivity shock has a persistent impact on the oil supply of the economy. For long-run shocks, already the dynamics of general macroeconomic quantities is more complex, as pointed out by Croce (214). In counteraction to the resulting income effect that stimulates consumption in the same way as for a short-run shock, the persistent increase in productivity growth incentivizes investments in order to generate an even higher output in the future, at the expense of present consumption. In my benchmark calibration, the latter effect dominates such that consumption decreases initially and rises again gradually in the subsequent time as a consequence of the increased output. The oil sector responds to the long-run macroeconomic productivity shock in a similar fashion. Again, the complementarity of the two consumption goods makes oil consumption run parallel to the consumption of the general good, such that inventories rise due to the initial reduction in oil consumption. The steadily increasing consumption of oil in the sequel is facilitated by additional oil investments that enhance the level of production little by little. Altogether, positive long-run macro shocks lead to decreasing present consumption levels and increased investments both for the general macroeconomy and the oil sector, and a gradual subsequent surge of outputs and consumption. Finally, let us consider the quantity dynamics in response to an exogenous increase in the amount of oil produced from existing oil wells. In a similar way that a short-run macro shock stimulates consumption of the general good, an oil-specific productivity shock leads to a higher oil consumption and due to the complementarity also to an increasing general consumption. In this case, the additional consumption of the general good is financed by a reduction of investment into new oil wells, which is feasible because of the increased productivity of the existing wells. In fact, the enhanced oil production even leads to a rise in oil inventories on top of the higher consumption levels. 23

24 h t a t c t i t s t b t e t x Macroeconomic Productivity Shocks Short Run Benchmark Model No Inventories x x x x x Months 5 x 1 4 Long Run x x x x x Months Figure 1: Quantities and macroeconomic productivity shocks. This figure shows impulse response functions of general macroeconomic as well as oil-related quantities for positive short- and long-run macroeconomic productivity shocks materializing at t = 1. The blue lines stand for the benchmark calibration, the red dashed lines for the model without inventories. Parameters are chosen according to Table 1 and Table 2. Lowercase letters refer to log variables, and is the first difference operator. 24

25 Oil Productivity Shocks h t κ t i t e t 2 x Benchmark Model No Inventories x x Months u t c t b t s t 15 x x x x Months Figure 2: Quantities and oil productivity shocks. This figure shows impulse response functions of general macroeconomic as well as oil-related quantities for positive oil productivity shocks materializing at t = 1. The blue lines stand for the benchmark calibration, the red dashed lines for the model without inventories. Parameters are chosen according to Table 1 and Table 2. Lowercase letters refer to log variables, and is the first difference operator. 6.2 Spot and Futures Price Dynamics Understanding the dynamics of general economic and oil-specific quantities in response to the different kinds of shocks provides direct insights into the underlying mechanisms for oil price fluctuations. More precisely, recall that the spot price of oil is given by the marginal rate of substitution between oil and the general consumption good, which is monotonically increasing in the ratio of general consumption and oil consumption according to (16). Since oil and general consumption respond in the same direction to all three kinds of shocks considered, the precise impact on prices is determined by the relative magnitude of consumption 25