Time-Varying Term Structure of Oil Risk Premia.

• Author: Cortazar, Gonzalo

1. INTRODUCTION

   Risk premia represent the annualized price difference between the futures contracts' prices and the markets' expected prices. Therefore, the use of futures prices as the most likely price for a commodity in the future is only valid if the risk premium is equal to zero. Even though the existence of risk premia in futures contracts is not a new finding, there is no consensus on their magnitude, behavior and appropriate estimation procedure (Baumeister and Kilian, 2016; Bianchi and Piana, 2018; De Roon, Nijman, and Veld, 2000; Melolinna, 2011; Palazzo and Nobili, 2010). Moreover, the recent financialization of commodity markets has increased their relevance for investors and strengthened arguments on their time-varying behavior (Hamilton and Wu, 2014; Baker and Routledge, 2017; Ready, 2018; Fattouh, Kilian, & Mahadeva, 2013).

   Understanding the stochastic behavior of commodity risk premia is important for several reasons. First, it provides valuable information on investment returns for agents who treat commodities as an asset class. Second, it helps to relate risk-adjusted expected prices, which are readily available in futures markets, with those of true expected prices, which are preferred by many practitioners for net present value calculations and risk management purposes. Third, it may shed light on some public policy implications by uncovering the macroeconomic determinants of risk premia.

   There have been various attempts in the literature to estimate commodity risk premia. Many practitioners and researchers use futures prices as proxies for market expectations (as discussed in Baumeister and Kilian, 2016; Bianchi and Piana, 2018), implicitly assuming that risk premia are zero. Keynes (1930) and Hicks (1939) had proposed that if producers and other market participants wanted to hedge their risk by selling future contracts, buyers should get a compensation in the form of a risk premia for taking on that risk. Furthermore, there is already evidence on its time-varying nature (De Roon, Nijman, and Veld, 2000; Sadorsky, 2002; Pagano and Pisani, 2009; Achraya, Lochstoer, and Ramadorai, 2013; Etula, 2013; Hamilton and Wu, 2014; Singleton, 2014).

   In the last few years, different methods have been developed to extract risk premia, or equivalently to calculate expected spot prices, from the available data. Even though most of the literature addresses how to get the market's expected interest rates (e.g., Diebold and Li, 2006; Altavilla, Giacomini, and Costantini, 2014; Chun, 2011), some effort has also been oriented to commodities.

   In what follows we present one way of characterizing existing methods for estimating risk premia in commodity markets by classifying them into three approaches: econometric, economic, and market-based.

   In what we call the econometric approach we include Gorton, Hayashi, and Rouwenhorst (2013), Hong and Yogo (2012), Pagano and Pisani (2009) and Baumeister and Kilian (2016) among others. This approach regresses realized spot commodity prices, or a function of them, on different lagged market variables to infer the expected market's spot price. Then the resulting risk premia are obtained by comparing this expected spot price with the futures price for the same maturity. Given that realized future spot prices and current futures prices with same maturity are compared, the required data-sample gets larger as longer-term risk premia are estimated making it difficult to estimate current risk premia for maturities greater than one or two years.

   In what we call the economic approach we include Hamilton and Wu (2014), Bianchi and Piana (2018), and Cortazar, Kovacevic, and Schwartz (2015) among others. These models use no-arbitrage or rational expectation models to infer expected spot prices from past and current market variables, typically futures and spot prices. Even though most of these types of models are successful in fitting futures prices, they do not provide appropriate risk premia estimates. To solve for this issue, asset-pricing models have been extensively applied, obtaining mixed results (Dhume, 2010; Erb and Harvey, 2006; Hong and Yogo, 2012).

   In what we call the market-based approach we include a recent paper by Cortazar, Millard, Ortega, and Schwartz (2019) in which they propose extracting information on expected spot prices directly from market surveys and using them, in addition to spot and futures prices, to calibrate a term structure model. Thus, risk premia are obtained directly from the model as the difference between the expected spot price and the futures price consensus curves. Including survey forecasts in economic models, even though it had not been previously applied to commodities, had been used in various contexts (see Chun (2011), and Altavilla, Giacomini, and Ragusa (2017)). This new approach allows to get risk premia directly from market observations (i.e., analysts' forecasts) as opposed to the traditional methods that usually infer them based on futures prices.

   This paper develops a framework to estimate time-varying risk premia using the market-based approach by extending Cortazar et al. (2019) to allow for a stochastic specification of risk premia following Duffee (2002). Once the term structures for oil risk premia are estimated, we explore their market determinants performing several regressions on different macroeconomic variables and oil market variables that have been previously proposed in the literature (e.g., Bhar and Lee (2011)).

   We find that the risk premia are time varying and are partially explained by market variables, namely inventories, hedging pressure, term premium, default premium and the level of interest rates. In this sense our methodology can estimate significant time-varying ex-ante risk premia directly from futures prices and analysts' forecasts, and search for their relationship with other market variables after having agreed on their levels and structure. It differs from the research that use the economic approach (e.g., Cortazar, Kovacevic, and Schwartz (2015)) on the fact that their risk premia are constant over time. On the other hand, it is fundamentally different to those studies that, following an econometric approach, regress ex-post risk premia (e.g., Baumeister and Kilian (2016)) with market variables due to the different nature of ex-ante and ex-post risk premia, where the first are directly related to market expectations, while the second may differ from the latter by the existence of biases.

   The remainder of this paper is organized as follows. Section 2 presents the theoretical model used to estimate time-varying term structures of risk premia. Section 3 describes the data. Section 4 provides risk premia estimates. Section 5 discusses the market determinants of risk premia and Section 6 concludes.

2. THE MODEL TO ESTIMATE RISK PREMIA

   2.1 Model Definition

   We present an N-factor term structure model which is a non-stationary version of the canonical [A.sub.0](N) Dai and Singleton (2000) model with stochastic risk premia as in Duffee (2002). We propose calibrating this model using both futures prices and analysts' forecasts to obtain a time-varying term structure of risk premia (1).

   Let [S.sub.t] be the spot price of the commodity at time t, then assume that:

   [Please download the PDF to view the mathematical expression] (1)

   [Please download the PDF to view the mathematical expression] (2)

   where h is an nx 1 vector of constants, [x.sub.t] is an nx 1 vector of state variables, [b.sub.1] is a scalar, A is an n*n upper triangular matrix with its first diagonal element being zero and the other diagonal elements all different and strictly positive. Let d[w.sub.t] be an n*1 vector of uncorrected Brownian motions following [Please download the PDF to view the mathematical expression] (3)

   where I is an nxn identity matrix. Dai and Singleton (2000) show that their model has the maximum number of econometrically identifiable parameters and at the same time nests most of the models used in literature.

   To specify time-varying risk premia in our constant-volatility model we resort to Duffee (2002) who shows how to use affine risk premia in all types of Dai and Singleton (2000) canonical models, including the ones with non-stochastic volatility. Let RP, be the commodity risk premia and assume that:

   [Please download the PDF to view the mathematical expression] (4)

   and the risk adjusted version of the model shown in Equations 1 and 2, is [Please download the PDF to view the mathematical expression] (5)

   [Please download the PDF to view the mathematical expression] (6)

   where X is a n+1 vector and A is a nxn matrix which does not need to be diagonal nor triangular. No further restrictions are set for the elements (2) in [lambda] and [LAMBDA].

   Note that in our model the risk-adjusted process differs from the true one not only by a constant risk premium, [lambda], but also by the [LAMBDA] matrix. Thus, futures prices and expected prices depend on different processes for the state variables, the former with the A + [LAMBDA] matrix, while the latter only with matrix A. However, if the [LAMBDA] matrix were set to zero, risk premia would be a constant and not time-varying.

   Futures prices are the expected value of the spot price, [S.sub.t], under the risk-adjusted probability measure, Q (Cox, Ingersoll, and Ross, 1981). Given that the risk-adjusted spot price follows a log-normal distribution, futures prices are given by: [Please download the PDF to view the mathematical expression] (7)

   where the risk-adjusted expected price and variance of [Y.sub.T] can be obtained by replacing Equation 1 into 7: [Please download the PDF to view the mathematical expression] (8)

   with (3) [Please download the PDF to view the mathematical expression] (9)

   [Please download the PDF to view the mathematical expression] (10)

   Analogous to Equations 7, 8, 9 and 10, the expected price should satisfy the following equations: [Please download the PDF to...