Seasonal fluctuations are evident for many commodity prices. However, their exact size can be quite uncertain. Hence, seasons affect commodity futures curves in two ways. First, they bias the expected futures price of a specific expiry month relative that of other months. Second, their uncertainty is an independent source of risk that affects the overall risk premia priced into the curve. Integrating seasonal factor uncertainty into an affine (linear) term structure model of commodity futures allows more realistic and granular estimates of various risk premia or ‘cost-of-carry factors’. This can serve as basis for investors to decide whether to receive or pay the risk premia implied in the future curve.

*The post ties in with SRSV’s summary lecture on implicit subsidies, particularly the section on commodity futures markets.*

*The below are quotes from the paper. Emphasis and cursive text have been added. *

### Stochastic seasonal fluctuations

“__Futures prices in many energy and agricultural commodities display seasonal fluctuations.__ Often, those fluctuations are not perfectly predictable… Although it is common to model seasonality as deterministic cycles, in this section we argue that… for a number of energy commodities (gasoil, gasoline, heating oil, and natural gas) and agricultural commodities (corn, soybean, and wheat)…those seasonal fluctuations are in fact stochastic. This distinction is important because __stochastic seasonality implies an additional risk factor that affects risk premia__…From the point of view of market participants, stochastic seasonal fluctuations imply a source of risk that manifests itself in futures prices and risk premia.”

“For most commodities and contract maturities, the tests strongly reject the null hypothesis of deterministic seasonality.”

“We estimate the model of…commodity futures prices….for the period Jan-1984 through Apr-2017. We concentrate our analysis on heating oil [*and*]…on soybeans futures. The model matches the cross-section of futures prices over time, including their seasonal pattern. We __find strong evidence of stochastic seasonality: the peaks and troughs of the seasonal cycle vary over the years, and the amplitude of the seasonal fluctuations decreased over time__, particularly at the end of the sample. Consistent with the theory of storage, the moderation of the estimated seasonal component coincides with a similar mitigation of the seasonal component in stocks of heating oil inventories.”

“Correctly specifying seasonality in futures prices as stochastic is important mostly to avoid erroneously assigning those fluctuations to other risk factors. When we estimate the model imposing deterministic seasonality, the __omitted time variation in the seasonal pattern manifests itself as large fluctuations in the cost-of-carry factors __which, in turn, translate into large and spurious fluctuations in estimated risk premia.”

### A carry curve model

“The purpose of this paper is to develop and estimate an __affine model of futures prices that allows for stochastic variations in seasonality__. We use the model to analyze the implications of stochastic seasonal fluctuations for the pricing of commodity futures and risk premia…Our model features stochastic seasonal fluctuations in both the spot price and the cost-of-carry. By attaching market prices of risk to seasonal factors, we are able to measure the risks associated with stochastic seasonal shocks.”

“[*We*] consider a storable commodity with spot price and with a net cost-of-carry, expressed as a continuously compounded rate of the spot price. The net __cost-of-carry represents the storage and insurance costs of physically holding the commodity net of any convenience yield on inventory__. It is the analogue of the negative of the dividend yield of a stock and can be derived from equilibrium models.”

“To capture seasonality in the spot price and the cost-of-carry, we assume that their loadings… are periodic functions of time… To extract the seasonality of a futures contract with months to maturity, we compute the expected seasonal component at any time conditional on information at that time, and then multiply the resulting expression by a discounting factor.”

“We define the cost-of-carry curve (net of bond yields) as a value that relates to the (log) difference between a futures price and a spot price…[*over and above*] the yield on a zero-coupon bond…To infer the number of factors necessary to capture the variability of the cost-of-carry curve, __we compute principal components of log futures prices net of the contribution of the spot, seasonal, and yield curve factors__.”

“The [*widely used*] basic model of commodity futures assumes that a single factor drives variations in the cost-of-carry… We develop and estimate a multifactor affine model of commodity futures that allows for stochastic seasonality…We show that, in fact, __we need at least three factors to properly account for the dynamics of the cost-of-carry__.”

### Empirical findings

“We perform a principal components analysis on the cost-of-carry curve for each commodity. We conclude that we need at least three factors to appropriately account for the dynamics of the cost-of-carry curve. __The first three principal components account for 90 percent or less of the variability of the cost-of-carry curve__… The __first principal component only accounts for between 16 and 80 percent of the variability__. These results suggest that a model in which the cost-of-carry depends on a single factor misses important features of the data.”

“__We express different notions of risk premia in terms of the components of the affine model__. Since all the strategies that we consider cost zero when they are entered into, ex-ante expected return entirely reflects expected risk premia… The spot premium is the expected return of holding a 1-period futures contract until maturity… The term premium is defined as the 1-period expected holding return of an n-period futures contract in excess of the spot premium.”

“We measure the contribution of the different factors to risk premia. We find that __most high frequency fluctuations in risk premia are due to variations in the spot price factor and other factors associated with the cost-of-carry curve__.”

“We find that [*bond*] __yield curve factors have a significant impact on risk premia, mostly at medium and lower frequencies__… When the slope of the yield curve is positive, long term contracts are relatively more expensive than shorter contracts while the reverse holds when the yield curve is inverted. Thus, changes in the slope of the yield curve over time affect futures prices and risk premia.”