Futures and spot returns on the same underlying asset often diverge, and the magnitude of this divergence is known as the futures “roll yield.” The cumulative impact of roll yield can be quite significant, in some cases being similar in magnitude to the entire gain or loss an investor experiences over the lifetime of a trade. In spite of the importance of roll yield in futures markets, misconceptions abound regarding its nature, measurement, and relevance.

This paper aims to demystify roll yield by connecting it to ideas familiar to investors and addressing common misunderstandings. The roll yield is not the result of “rolling” positions from one contract to the next. We demonstrate this by walking through the life cycle of a futures trade. We also demonstrate that there is a direct relationship between roll yield and the slope of the futures term structure. Finally, we show that the decomposition of a futures return into its spot return and roll yield components provides insights that have implications for both carry and trend-following strategies. At Campbell, many of these ideas are used in the ongoing development of new trend-following strategies. They also serve as the foundation for some of the strategies included in Campbell PRISM, a quantitative trading programme which utilises a suite of non-trend-based alpha strategies to exploit structural market opportunities.

**Relating roll yield to the roll transaction**

Futures-spot divergence is known as the futures “roll yield,” which can be defined as:

**EQUATION 1:** *Roll Yield = Futures Return – Spot Return*

A common misconception is that roll yield represents the P&L generated on the day of the contract roll.

For example, if rolling a futures position requires you to sell the current contract and buy the next one at a higher price, it may appear that the transaction will cause a loss (since you are selling low and buying high). However, this is not the case; the roll yield accumulates over the life of the trade as the futures price converges to spot at expiration. [1]

In order to observe the impact of the roll event, let’s follow along with an investor who is long a futures contract.

*On 8 March, 2013, an investor (Lois) goes long one contract* [2] *of the March 2013 e-Mini (ESH3) at the closing price of 1549.50. *[3] *As the 15 March expiration approaches, Lois wants to maintain her exposure, so on 14 March she sells her position at the closing price of 1562.25, and goes long one June 2013 contract (ESM3) at the closing price of 1556.00. On 18 March, Lois closes her position at 1546.75.*

There are a few questions to consider. What is Lois’s cumulative P&L? How much of it is directly attributable to the roll? And to what extent is her P&L different from simply comparing the front futures prices on 8 March and 18 March?

**CLICK IMAGE TO ENLARGE**

Fig.1 shows the daily P&L in each contract. We see that nothing unusual happens on the roll day. There is, however, a 6.25 point difference between ESH3 and ESM3 on the roll day, which some mistakenly believe is a roll ‘cost’ (or in this case, a roll gain). The price difference does not directly impact P&L, but it does make any type of analysis involving prices problematic. Simply splicing the front contract prices together does not properly account for the P&L. To correct this, we need to shift the closing prices of the “rolled-into” contract by the difference in prices on the roll date (i.e., 6.25). This “joined” price series does not represent an actual contract, but it accurately accounts for the P&L experienced by an investor.

**What drives the difference between the joined price and front price?**

The difference between the joined and spliced prices over time is called the cumulative roll adjustment, which is the sum of roll adjustments at each roll date.

**EQUATION 2: ***Joined Price = Front Price + Cumulative Roll Adjustment*

Since each roll adjustment is directly related to the slope of the futures term structure, we see that the cumulative roll adjustment reflects an average of the term structure on the roll dates. For example, if the next contract has a lower (/higher) price than the front contract, then the cumulative roll adjustment will be positive (/negative). This type of market is in backwardation (/contango). Suppose we assume that rolls occur an instant before expiry, where the spot price is approximately equal to the front contract price. Then Equation 2 becomes

**EQUATION 3:*** Joined Price (Roll Date) ≈ Spot Price + Cumulative Roll Adjustment*

If we use Equation 3 to calculate the profit of a long futures position that is put on and unwound on a roll date an instant before the roll, we get:

**EQUATION 4: ***Futures Return (Roll Date to Roll Date) = Spot Return + Cumulative Roll Adjustment*

And comparing this to Equation 1:

**EQUATION 5: ***Roll Yield Between Roll Dates = Cumulative Roll Adjustment Between Roll Dates*

**Why is cumulative roll adjustment a useful concept?**

Since the roll adjustment does not represent the return of any instrument, there has been some debate about the usefulness of such a decomposition. The reason it is useful is that the two terms on the right side of Equation 2 can behave in dramatically different ways.

**CLICK IMAGE TO ENLARGE**

Fig.2 depicts what Lois would have experienced had she held a long position in the front contract since March 1998 (“Joined Price” curve). It also shows the spliced price (“Front Price” curve). The cumulative roll adjustment is simply the difference between them. It is non-zero on roll dates, and has extended periods where it is upwardly sloping, downwardly sloping, or flat. The long-term persistence across multiple roll days suggests that roll yield is more predictable than spot price. We have essentially decomposed the futures return into a part that looks very noisy and a part that looks very smooth.

To this point, for simplicity, our transformation has focused primarily on roll dates, when spot price approximates futures price. Let’s now introduce a more precise decomposition, which is valid on non-roll dates. Let’s rewrite Equation 2 as:

We have now introduced the concept of spot/futures basis, which is the difference between the front futures price and the spot price. For markets in backwardation (/contango) the basis is generally negative (/positive). We introduce a fictitious futures market which is perpetually in backwardation, with the deferred contract always three points lower than the front contract. An investor decides to go long this market on the day the front month expires, when the spot price is 17.5 and the new front contract is 14.5.

**CLICK IMAGE TO ENLARGE**

Fig.3 shows the spot price, basis, roll adjustments, and the joined price over time. Note that if we sum the spot, basis, and roll adjustment curves, we get the joined prices. The basis starts at -3 immediately after each roll, and slowly accretes to zero immediately prior to the next roll, as the front futures price converges to the spot price. Now, let’s go back to Equation 6:

This looks similar to Equation 3, which combined the spot and basis together. Conceptually, the return due to backwardation/contango is represented by the sum of the basis and roll adjustment. Furthermore, unless the spot price has a strong drift, the cumulative roll adjustment will be the dominant factor driving the futures return. Comparing Equation 7 to the defining equation for roll yield, we get:

**EQUATION 8: ***Roll Yield = Basis Return + Cumulative Roll Adjustment*

Note that this differs from Equation 5 in that we are no longer restricted to roll dates. Since the basis is bounded by the shape of the term structure, the basis return is also bounded. Thus over long periods the cumulative roll adjustment will dominate, and we have:

**EQUATION 9: ***Roll Yield ≈ Cumulative Roll Adjustment*

where equality holds if the basis return is zero.

**Implications for trend following**

The central idea behind trend following is that returns have some degree of persistence. Thus, if total returns have been positive (/negative), a trend-following strategy would go long (/short). Since for futures markets, the total return is reflected in the joined price series, we use joined prices in the calculation of the trend signal.

If we put “persistence” in front of the terms in Equation 4, we get:

**EQUATION 10: ***Persistence in Futures Return = Persistence in Spot Return + Persistence in Cumulative Roll Adjustment*

This shows that all else equal, the stronger the persistence in the roll adjustment, the stronger the persistence in the futures return. Therefore, Equation 10 makes clear that trend following has a strong connection to the behaviour of the roll adjustment. In fact, our previous white paper [4] showed that nearly half of the cumulative performance of a trend-following strategy was attributable to roll yield.

**Implications for carry strategies**

It would be nice if we could get direct exposure to the roll adjustment due to its high persistence, but it does not represent the return of any instrument. However, what if the price change of the front contract (the “noisy” part) averaged out to zero over time? Then, Equation 2 gives:

**EQUATION 11:** *Futures Return Over Life of Trade ≈ Roll Adjustment over life of Trade*

We want to emphasise that this holds only if the front contract price change is small compared to the roll adjustment. Nevertheless, it immediately suggests an actionable trading strategy, called the “Carry” trade:

Under certain conditions, the roll adjustment contains information about expected profit. If the sign and magnitude of the roll adjustment are not expected to change quickly, then we can use today’s roll adjustment as an estimate going forward. Thus, using Equation 11, we can write:

**EQUATION 12: ***Futures Return Over Life of Trade ≈ Today’s Roll Adjustment * Time in Trade*

We can exploit this by going long markets for which the roll adjustment is positive (i.e., backwardated) and short markets for which the roll adjustment is negative (i.e., in contango).

We are not claiming that we are obtaining direct exposure to the roll adjustment. But under a specific set of assumptions, the roll adjustment can behave like a ‘real’ P&L in a sense captured by Equation 11.

The carry trade is not riskless, however. Let’s revisit Equation 12, putting back in the spot return:

**EQUATION 13: ***Futures Return ≈ Spot Return + Today’s Roll Adjustment * Time*

As long as the roll adjustment exceeds any adverse moves in the spot, the carry trade will be profitable. Thus we can think of the roll adjustment as a “buffer” against adverse spot price moves.

Furthermore, Equation 13 shows that risk-adjusted performance of a carry strategy will depend on the relative magnitude of the roll adjustment versus volatility of spot moves. Therefore, performance can be improved by reducing the impact of adverse spot price movements in a way that does not significantly degrade the roll adjustment.

Equation 13 and the subsequent discussion provide a framework to develop ‘enhanced’ carry strategies through the prediction and mitigation of adverse spot price moves.

**Notes**

- In saying that the futures price converges to the spot price at expiration, we are somewhat idealising things; each sector has its own idiosyncrasies that make this statement inexact.
- For simplicity, we say that one contract corresponds to one unit of the S&P 500 index. The actual e-Mini contract gives exposure to 50 units of the underlying index.
- To simplify the discussion, we assume that all trades are done without commissions, slippage, or other transaction costs.
- ‘Prospects for CTAs in a Rising Rate Environment’, Campbell White Paper Series, January 2013.

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## Commentary

## Issue 139

## Man FRM Early View

## February 2019

## Man FRM