Commodity Roll Yield Permutations

First, let us assume that there are three types of market conditions: (i) a hypothetical flat market where there is absolutely no price change for any variable, (ii) a bull market where variables generally trend upwards, and (iii) a bear market where variables generally trend downwards. Again, let S₀ be the current spot price of the asset, and F₀ be the current futures contract price for an illiquid futures contract trading after first notice day and within the delivery period, but prior to the last physical delivery date and contract expiration; let F₁ be the current futures contract price for the liquid nearby future delivery of the underlying asset, and F₂, F₃, F₄ be the current futures contract price for second nearby, third nearby and fourth nearby future delivery of the underlying asset. Additionally, let E(S₁) be the expected future spot price for F₁ future delivery, E(S₂) be the expected future spot price for F₂ future delivery, E(S₃) be the expected future spot price for F₃ future delivery, and E(S₄) be the expected future spot price for F₄ future delivery. Further, we shall assume that each of F₀, F₁, F₂, F₃, and F₄ futures contracts are physically settled in the current year December, and the following year March, June, September and December, respectively.

Permutation Level 1: Assuming one week prior to contract expiration, after first notice day and within the delivery period, but prior to the last physical delivery date and contract expiration, there are three likely scenarios with respect to the relationship between S₀ and F₀. Either: F₀ < S₀, or F₀ = S₀, or F₀ > S₀, with convergence occurring at the time of physical settlement, such that S₀ = F₀ on the contract settlement date. This set of permutations relates to the one percent of open interest futures positions that are settled through delivery, either by bona fide hedgers or by institutional arbitrageurs who still hold the contract into this period of the contract’s life cycle. At this juncture, most speculators have rolled forward and are trading the next liquid futures contract month.

Permutation Level 2: The next set of permutations revolves around the relationship between S₀ and F₁. For this set of permutations we extend the three basic possibilities described in Permutation Level 1 to encompass bullish market conditions, flat market conditions and bearish market conditions. Hence, we note that there are three sets of permutations for each type of market condition. Assuming bullish market conditions, either: F₁ < S₀, or F₁ = S₀, or F₁ > S₀; assuming flat market conditions, either: F₁ < S₀, or F₁ = S₀, or F₁ > S₀; and assuming bearish market conditions, either: F₁ < S₀, or F₁ = S₀, or F₁ > S₀. However, for each of these permutations, given the potential for either backwardation, contango or equilibrium market conditions as such relate to the classic arbitrage model, it is necessary to relate the F₁ future delivery to its corresponding E(S₁) (i.e., expected future spot price). As a result, we can extend the model to include nine possible permutations for each type of market condition. Assuming bullish conditions and we know the value of E(S₁), then either:

F₁ < S₀, where F₁ < E(S₁)

F₁ < S₀, where F₁ = E(S₁)

F₁ < S₀, where F₁ > E(S₁)

F₁ = S₀, where F₁ < E(S₁)

F₁ = S₀, where F₁ = E(S₁)

F₁ = S₀, where F₁ > E(S₁)

F₁ > S₀, where F₁ < E(S₁)

F₁ > S₀, where F₁ = E(S₁)

F₁ > S₀, where F₁ > E(S₁)

As mentioned above, these nine permutations can be applied to flat market conditions and bearish market conditions too. The significance of the underlying market conditions relates to the benefit or detriment the roll yield theoretically provides under either bullish, flat and bearish market scenarios. For example, assuming an established long position, each roll into the forward contract month could result in one of the following theoretical scenarios/outcomes: (1) roll during a bull market in backwardation results in a yield benefit and a positive price change; (2) roll during a bear market in backwardation results in a yield benefit but a negative price change; (3) roll during a flat market but expected future spot market is backwardated results in a yield benefit and zero price change; (4) roll during a neutral market which is neither backwardated or contango results in zero yield benefit and zero price change; (5) roll during a flat market but expected spot market is contango results in a yield detriment and zero price change; (6) roll during a bull market in contango results in a yield detriment but a positive price change; and (7) roll during a bear market in contango results in a yield detriment and a negative price change. Likewise, assuming an established short position, the opposite of these seven theoretical scenarios/outcomes could result.

Additionally, markets do not go straight up in bull markets or straight down in bear markets, and flat markets could go both up then down, down then up, sideways, or any combination thereof. Therefore, assuming that each of F₀, F₁, F₂, F₃, and F₄ futures contracts are physically settled in the current year December, and following year March, June, September and December, respectively; and further, we assume four rolls are transacted in the following year: December to March, March to June, June to September, and September to December; then during each quarterly roll, twelve separate market patterns could theoretically occur: (i) bull, bull, bull, bull; (ii) bull, bear, bull, bull; (iii) bull, bull, bear, bull; (iv) bull, bear, bear, bull; (v) bull, bull, bear, bear; (vi) bull, bear, bull, bear; (vii) bear, bear, bear, bear; (viii) bear, bull, bear, bear; (ix) bear, bear, bull, bear; (x) bear, bull, bull, bear; (xi) bear, bear, bull, bull; and (xii) bear, bull, bear, bull. In order to appreciate the full complexity of permutations, the beneficial or detrimental roll yield scenarios should be overlaid upon each of the quarterly rolls, within the twelve separate market pattern scenarios described above.

It should be noted that these nine permutations can be extended to include the relationships between S₀ and F₂, S₀ and F₃, or S₀ and F₄, resulting in twenty-seven additional permutations assuming the following four potential roll scenarios in any one year: December to March, December to June, December to September, and December to December. It is also noted that these permutations in the real world are a difficult set of relationships for the average speculator to arbitrage since these permutations involve the cash spot market (i.e., S₀) and thus require actual outlay for storage costs.

Permutation Level 3: The next set of permutations builds upon ideas presented in Permutation Level 1 and 2, but extends established concepts around the relationship between F₁ and F₂ (and by extrapolation, the relationships between F₁ and F₂, F₃, F₄; and F₂ and F₃, F₄; and F₃ and F₄). Therefore, if one assumes bullish market conditions, then either: F₁ > F₂, or F₁ = F₂, or F₁ < F₂; assuming flat market conditions, then either: F₁ > F₂, or F₁ = F₂, or F₁ < F₂; and assuming bearish market conditions, then either: F₁ > F₂, or F₁ = F₂, or F₁ < F₂. However, for each of these permutations, given the potential for either backwardation, contango or equilibrium market conditions as such relate to the classic arbitrage model, it is necessary to relate the F₁ future delivery to its corresponding E(S₁), and also relate the F₂ future delivery to its corresponding E(S₂). As a result, the model is extended to include twenty-seven possible permutations for each type of market condition. Therefore, for each underlying market condition, and assuming we know the value of E(S₁) as well as E(S₂), then either:

F₁ > F₂, where F₁ > E(S₁) and F₂ > E(S₂)

F₁ > F₂, where F₁ = E(S₁) and F₂ > E(S₂)

F₁ > F₂, where F₁ < E(S₁) and F₂ > E(S₂)

F₁ > F₂, where F₁ > E(S₁) and F₂ = E(S₂)

F₁ > F₂, where F₁ = E(S₁) and F₂ = E(S₂)

F₁ > F₂, where F₁ < E(S₁) and F₂ = E(S₂)

F₁ > F₂, where F₁ > E(S₁) and F₂ < E(S₂)

F₁ > F₂, where F₁ = E(S₁) and F₂ < E(S₂)

F₁ > F₂, where F₁ < E(S₁) and F₂ < E(S₂)

F₁ = F₂, where F₁ > E(S₁) and F₂ > E(S₂)

F₁ = F₂, where F₁ = E(S₁) and F₂ > E(S₂)

F₁ = F₂, where F₁ < E(S₁) and F₂ > E(S₂)

F₁ = F₂, where F₁ > E(S₁) and F₂ = E(S₂)

F₁ = F₂, where F₁ = E(S₁) and F₂ = E(S₂)

F₁ = F₂, where F₁ < E(S₁) and F₂ = E(S₂)

F₁ = F₂, where F₁ > E(S₁) and F₂ < E(S₂)

F₁ = F₂, where F₁ = E(S₁) and F₂ < E(S₂)

F₁ = F₂, where F₁ < E(S₁) and F₂ < E(S₂)

F₁ < F₂, where F₁ > E(S₁) and F₂ > E(S₂)

F₁ < F₂, where F₁ = E(S₁) and F₂ > E(S₂)

F₁ < F₂, where F₁ < E(S₁) and F₂ > E(S₂)

F₁ < F₂, where F₁ > E(S₁) and F₂ = E(S₂)

F₁ < F₂, where F₁ = E(S₁) and F₂ = E(S₂)

F₁ < F₂, where F₁ < E(S₁) and F₂ = E(S₂)

F₁ < F₂, where F₁ > E(S₁) and F₂ < E(S₂)

F₁ < F₂, where F₁ = E(S₁) and F₂ < E(S₂)

F₁ < F₂, where F₁ < E(S₁) and F₂ < E(S₂)

As can be inferred by the variety of roll yield permutations above, the number of potential manifestations can be overwhelming when one also considers variations as a result of timing rolls.

Economic modeling is often premised on the idea of distilling complex data sets into axioms that can be robustly applied to the intricacies of the real world. However, the purpose of our roll yield permutations model is to cause the opposite reasoning: take self-evident precepts about the commodities futures market and reveal how truly complex the trade decision-making process is. We do not portend that these permutations allow for “natural” economic constraints or elucidate the underlying dynamics of the arbitrageur, hedger, speculator paradigm in the legacy of Keynes, Hicks, Kaldor, Working, Brennan, Cootner, etc. Rather, these permutations demonstrate the range of possibilities when one clarifies the simplified arbitrage model by incorporating the pivotal relationship between F_t and its corresponding E(S_t), as elucidated by the classic arbitrage model. Accordingly, we admit that if F₂ is backwardated relative to F₁ [i.e., F₁ > F₂], then it is a likely indication that E(S₂) is also backwardated relative to E(S₁) [i.e., E(S₁) > E(S₂)], but this may not always be the case, and researchers and practitioners should not presume as much.