Many sophisticated quantitative investment managers employ systematic carry strategies in their portfolios, yet they remain poorly understood by the investing public due to a tendency to overgeneralize from a limited set of specific examples. Carry strategies are, in fact, a general class of investment opportunities, and can capture a wide array of phenomena in futures and forward markets. In this paper, we will explore three different types of carry trades, including relative value and directional approaches. We will attempt to debunk the perception that carry is, by definition, a highly-levered relative value strategy, and demonstrate that there is not a “one size fits all” approach to the carry trade.

Introduction
In a 2016 Campbell & Company white paper entitled “An Introduction to Global Carry,”1 the basic premise of generalized carry was introduced. “Carry” is an asset’s expected total return, positive or negative, assuming its price is unchanged.2 Regardless of the underlying asset, a carry strategy seeks returns from the net benefit or cost of holding that asset, in excess of the potential for price appreciation/depreciation.

Futures markets, which provide a standardized way to buy or sell a commodity or financial instrument at a specified future date and price, provide an excellent mechanism to harvest the carry risk premium. An asset’s “carrying costs” (or “carrying benefits”) are reflected directly in the futures term structure: the futures contract price at a particular tenor reflects both the current asset price (i.e., spot price) and the net cost or benefit of owning the asset (including both cash flow and non-cash flow items) until contract expiration. As expiration approaches, this carrying cost will diminish and approach zero, leading to a convergence between the futures price and the spot price.3

Extracting the carry risk premium embedded in the futures term structure can be achieved using several different methodologies. However, there is no way to capture carry in a particular market without some sort of price risk. Any attempt to completely hedge out the impact of price changes will also involve ‘hedging’ out any potential profit. An investor that attempts to capture carry in Gold, for example, by shorting Gold futures and buying Gold to hedge spot price risk, would find himself with zero profit at the end of the trade. By the time the investor paid the interest rate (or the opportunity cost) on the money used to buy the gold, paid the transportation costs, hired a security guard, etc., he would have eliminated any premium implied by a positively-sloped futures term structure. This is not a coincidence, but rather a natural consequence of spot-futures parity.4

There are implementations of carry that can mitigate spot price risk, but in doing so, new risks are introduced. In addition, implementations that attempt to directly mitigate spot price risk generally seek to capture some derivative of carry, rather than the level of carry itself.

In this paper, we will explore and contrast several different approaches to harvesting the carry risk premium. We will focus on three specific implementations:

• Relative Value: Neutralizing Sector Risk
• Relative Value: Mitigating Spot Price Risk with Calendar Spreads
• Directional, by Market

We will demonstrate that in each case, the underlying exposure will vary, as will the “hedged” elements, the “unhedged” elements and the expected return.

Carry Implementation
Relative Value: Neutralizing Sector Risk5
In a particular asset class, go long markets with higher-than-average carry and short markets with lower-than-average carry, weighting positions such that the expected net sector exposure is zero. This may involve taking long positions in the highest quartile carry assets and short positions in the bottom quartile carry assets; it might also involve taking long positions in the top 50% of carry assets and short positions in the bottom 50%, or other similar combinations.

What carry exposure is being targeted?
This well-known approach to the carry trade does not seek to capture the absolute level of carry of the underlying assets; instead, it seeks to capture the difference between the carry of the high-carry assets and the low-carry assets within a sector. An asset class with high dispersion in the amount of carry across markets (which often occurs in foreign exchange, for example) may offer a rich opportunity set for this methodology.

The average level of carry within an asset class is generally neutralized by offsetting long and short positions. As such, the trade will not benefit from any tailwind due to a strong overall average carry among markets in the sector. This can, in some cases, be very significant (e.g., fixed income markets).

What price risks are hedged/unhedged?
By weighting positions such that the net sector exposure is minimal, the aggregate portfolio tends to be insulated from a broad asset class re-pricing (i.e., a rally in global rates, for example). The risk from a parallel price movement within a sector has been “hedged.” However, the relative contribution of idiosyncratic risk from particular markets is significant. If prices do not move in a parallel fashion within the sector, the carry premium may be eroded or eliminated (i.e., if the high carry assets fall more than the low carry assets). It may also be enhanced (if the low carry assets fall more than the high carry assets).

With any investment strategy, leverage requirements must be considered. An RV Neutralizing Sector Risk carry trade typically requires significant leverage, for two reasons. First, this approach is attempting to capture a derivative of carry, which may be smaller than the level of carry. Second, because the markets within a sector tend to be highly correlated, a sector-neutral carry portfolio requires significant leverage to maintain a reasonable risk target.

Another noteworthy feature of this approach is that it can lead to short positions in positive carry assets (or long positions in negative carry assets), as the portfolio is ‘forced’ to take risk-offsetting positions within a sector. That is the case in the example presented below.

Exhibit C shows the annualized risk-adjusted carry (i.e., ex-ante Sharpe ratio) for a set of futures on 10-year bonds on May 18 2006. At that time, the carry was positive in five of these markets and negative in one. Using the ‘RV Neutralizing Sector Risk’ approach, the investor would enter long positions in the highest-carry markets and short positions in the lowest-carry markets.

In this case, the trade could entail long positions in Japan, Eurozone and Canada, and short positions in US, Australia and UK, with positions sized such that the net exposure to the sector is minimized. For simplicity, we will assume that positions are equal risk-weighted.

• Expected Sharpe: This is proportional to the difference between the ex-ante risk-adjusted average carry of the long positions and the short positions (i.e., 0.4 Sharpe units).6 The proportionality factor is dependent on the volatility and correlation of the contracts and the target portfolio volatility. In this case, the expected Sharpe ratio is approximately 0.7.7
• Leverage: Leverage required to target 1% annualized volatility is 1.4x.8 Every additional 1% of volatility targeted will require an additional 1.4x of leverage9 (i.e., a 5% annualized risk target would require 7x of leverage).

If the positions were held for 1 year, the realized Sharpe ratio (gross) would have been approximately 1.5 (assuming positions are equal risk-weighted at inception with no rebalancing). The outperformance versus the expected Sharpe of 0.7 implies that in this case, the idiosyncratic price movements occurring during the life of the trade were beneficial, on average, adding incremental return to the carry premium.

Relative Value: Mitigating Spot Price Risk with Calendar Spreads
For a particular market, go long the contract in the term structure with the highest carry and short the contract with the lowest carry.

What carry exposure is being targeted?
This approach seeks to capture the difference in carry between two futures contracts in a particular market. The trade will typically involve a long position in the highest carry contract and a short position in the lowest carry contract in the futures term structure.

The trade is designed such that offsetting long and short positions tend to neutralize exposure to the average level of carry across the two contracts, so the trade would generally not benefit from any tailwinds due to a strong average level of carry in a particular market.

What price risks are hedged/unhedged?
The aggregate position would tend to be insulated from parallel shifts in the futures term structure (both contracts appreciate/depreciate to the same degree). A change in the spot price that causes the term structure to shift up or down – but not change shape – will have little impact on the trade.10  

However, the aggregate position will be exposed to a non-parallel shift in the futures term structure (i.e., a change in slope between the two relevant points on the curve). This can have either a positive or a negative impact on performance.

As with the prior implementation, this approach is attempting to capture a derivative of carry rather than the level of carry. As such, the leverage required to generate meaningful returns can be high. Also, because the contracts within the futures term structure of a single market tend to be very highly correlated, significant leverage is needed to maintain a reasonable risk target. Leverage will also be dependent on the volatility of the underlying contracts, as illustrated in Exhibit E.

A consideration somewhat unique to this implementation is the trade-off between the carry opportunity and liquidity. To implement this approach, multiple futures contracts must be traded for each market, preventing the trader from focusing on just the most liquid contract. In general, contracts further from expiration have less depth than the “near” contracts. The largest carry differentials may involve contracts with limited liquidity, restricting the ability to trade in size.

Recalling the term structure for RBOB Gasoline presented in Exhibit A, we can construct a Calendar Spread carry trade for that market at that particular point in time.11

The annualized risk-adjusted carry (ex-ante Sharpe) for each contract is shown in Exhibit E. In this example, the high-carry contract is Oct-12 and the low-carry contract is Dec-12, so the calendar spread could involve a long position in Oct-12 and a short position in Dec-12. 12

• Expected Sharpe: This is proportional to the difference between the ex-ante risk-adjusted carry of the long position and the short position (i.e., 1 Sharpe unit). In this case, the ex-ante Sharpe ratio is approximately 7. 13 It’s worth noting that this is not realistically what one would expect – especially for commodity markets, the assumption that the futures term structure remains unchanged tends to be violated. Market expectations about future spot price movements tend to be reflected in the term structure more so in commodities than in other markets. In fixed income, for example, the ex-ante Sharpe expectation tends to be much more realistic.
• Leverage: The leverage required to target 1% annualized volatility is approximately 0.5x.14 Every additional 1% of volatility targeted will require an additional 0.5x of leverage. While this may seem low, it is a result of the extremely high volatility of the underlying RBOB contracts (27% annualized, compared to 4.7% annualized, on average, for the fixed income contracts in Fig.3). To illustrate the incremental leverage required for a calendar spread, we can compare 0.5x to the leverage required to trade just the October contract in a directional carry strategy. In that case, the leverage requirement would be 0.03x for each 1% of volatility targeted. Employing a calendar spread raises the leverage required by nearly 20 times.

Directional, by Market:
Take long positions in positive carry futures contracts and short positions in negative carry futures contracts, focusing on the most attractive opportunities. Trade the most liquid contract in each market.

What carry exposure is being targeted?
This implementation seeks to capture the level of carry in each market. This is in contrast to the relative value approaches, which generally seek to capture some derivative of the carry in each market.

The position entered will always be in the direction of the carry (i.e., long positions in positive-carry contracts and short positions in negative-carry contracts). Because there may be no offsetting long/short trades, this implementation may not neutralize sector exposure; therefore, the trade will tend to benefit from any tailwind due to a high average level of carry in the sector.

What price risks are hedged/unhedged?
This approach includes exposure to the direction of spot price moves. For a particular trade, an adverse spot price movement has the potential to offset the carry premium. For markets with significant carry (either positive or negative), it will take relatively larger adverse spot price movements to erode the premium.

A key difference between the directional and relative value implementations is that a directional approach has little direct exposure to changes in the slope of the futures term structure. There is also little direct exposure to non-parallel price movements in an asset class.

Let’s revisit the example presented in Exhibit C and consider the application of a directional, rather than a relative value, strategy. Recall that on May 18 2006, the carry was positive in five of the 10Y bonds and negative in one. A directional carry strategy would therefore take a long position in each of the positive carry markets (Japan, Eurozone, Canada, US, Australia) and a short position in the single negative carry market (UK).

• Expected Sharpe: The expected Sharpe ratio is proportional to the average of the absolute value of the ex-ante risk-adjusted carry in each market (i.e., 0.3 Sharpe units). In this case, the expected Sharpe ratio is approximately 0.5.15 This is somewhat lower than the expected Sharpe for the RV approach (0.7). The lower Sharpe ratio is a result of the reduced diversification in the underlying trades (5 longs and 1 short for directional, versus 3 longs and 3 shorts for relative value; the latter case involves more risk-offset), and the corresponding higher volatility of returns: prior to scaling, the trailing volatility of the directional portfolio is nearly twice that of the relative value portfolio.
• Leverage: The leverage required to target 1% annualized volatility is 0.8x.16 Every additional 1% of volatility targeted will required an additional 0.8x of leverage. This is just over half of the leverage required for the RV approach (1.4x).

If the position was held for 1 year (entered on May 22 2006, assuming a 1-day lag from the carry signal to the trade date), the realized Sharpe ratio (gross) would have been approximately 0.7 (assuming positions are equal risk-weighted at inception with no rebalancing). The moderate outperformance versus the expected Sharpe (0.5) suggests that in this case, spot price movements during the life of the trade were beneficial, adding incremental return.

This example is a good reminder that spot price movements will not always be adverse; they can also be additive to the trade. By applying a directional approach to a wide range of markets across the major asset classes, the resulting portfolio will typically include some trades where the spot move is additive and some where it detracts from returns, reducing the overall net effect of spot price moves. A portfolio construction technique that incorporates the correlation among the underlying markets has the potential to further enhance this effect.

Recall:

• Futures Return = Spot price change + Carry
• Portfolio Return = ∑i (Spot price changei + Carryi )
• Over a diversified portfolio of positions across all asset classes, the impact of the spot price change will tend to be low compared to the contribution from carry, on average, over the long-term.

That last point is very important. Much like trend following, directional carry is best expressed through a multi-asset, highly diversified portfolio. Over time, the diversification across markets, sectors and geographies will tend to mitigate the impact of adverse spot price movements in any one market.

To quantify the historical impact of spot price movements on a directional carry strategy, we can examine the performance of a Hypothetical Global Carry Strategy,17 designed as a benchmark for the performance of a diversified multi-asset carry portfolio using a directional implementation. This hypothetical rules-based strategy was originally introduced in Campbell’s 2016 paper, “An Introduction to Global Carry,” and was constructed using actual futures data since 1992, with a universe of 61 markets spanning Equity, Fixed Income, FX and Commodities. Assumptions include equal risk weighting by asset class, constant capital, a 1-day lag between signal generation and execution, and a 10% annualized volatility target.18 We can use this simulated track record to evaluate the “carry capture” ratio of such a portfolio: how much of the ex-ante carry has historically been eroded by adverse spot price changes? By decomposing the returns into spot and carry components, we can determine the long-term average carry capture rate.19

Exhibit H shows the cumulative performance of the Hypothetical Global Carry Strategy from Jan-92 to May-17, including the carry and spot contribution to total return, along with the quarterly return distribution (bottom chart) for the spot/carry components. Since the spot price is in many cases unobservable, and since the positions change day to day, we have to make certain assumptions and approximations to decompose the return into carry and spot contributions. As such, the carry return represents, over the long term, the approximate ex-ante carry opportunity identified by the Strategy, summed across markets. In other words, if the spot price (or relevant market condition) remained unchanged during the life of the trade, the total return would approximately equal the carry return, when measured over a long time period.

During the 25-year simulated track record, the cumulative performance of the Hypothetical Global Carry Strategy reflected a 79% carry capture ratio (total return/carry = 79%),20 meaning that adverse spot price movements offset approximately 21% of the ex-ante carry premium. This demonstrates the efficacy of this statement, which bears repeating: By applying a directional approach to a wide range of markets across the major asset classes, the resulting portfolio will include some trades where the spot move is additive and some where it detracts from returns, reducing the overall net effect of spot price moves.

While the overall portfolio had a carry capture rate of 79%, there was a significant variation by asset class. In fixed income and foreign exchange, spot price moves have historically been favorable, on average, adding incremental gains to the ex-ante carry premium (carry capture greater than 100%). However, in commodities and equity indices, spot price movements have tended to offset the carry premium by approximately 60%, on average.21

 
Multi-Asset Carry: As the prior discussion suggests, there are many reasons why a directional carry implementation may be well-suited to a stand-alone multi-asset carry portfolio. At Campbell, we rely on both directional and relative value methodologies in our various investment programs, but believe that a directional approach may be more appropriate for a stand-alone multi-asset carry portfolio for the following reasons:

• The potential to capture the average level of the carry premium in a particular market, rather than a derivative of the carry premium (i.e., the difference between carry of two contracts/markets).
• The leverage required for this implementation tends to be more moderate than for an RV approach: because you are attempting to capture the level of carry, rather than the difference between the carry of two instruments, less leverage tends to be required to potentially generate meaningful returns.
• The ability to use portfolio construction techniques to mitigate the risk from an adverse spot price move in a particular market. As mentioned above, when implemented with a highly diversified portfolio (across markets, asset classes and geographies – with the most important dimension being asset class), the Hypothetical Global Carry Strategy captured 70% – 80% of the carry opportunity, on average.
• This “capture ratio” will vary over short periods of time. As the observation period is lengthened, spot price moves have less of an impact (to a point), with more of the performance stemming from the carry premium.

Conclusion
In this paper, we sought to clarify that there is not a “one size fits all” approach to carry. The perception that carry is, by definition, a highly-levered relative value strategy is not necessarily accurate. There are a number of equally valid ways to implement a carry trade, three of which were presented here. Exhibit J provides a comparison of the three approaches, highlighting the different carry exposure and hedged/unhedged risks.

Regardless of the specific implementation used, the carry risk premium has historically offered an attractive opportunity set, and warrants consideration as part of a broader allocation scheme.

Footnotes

1. Roberts, Susan 2016. “Introduction to Global Carry”, Campbell White Paper Series.
2. A more precise way to express this concept is that carry is an asset’s expected total return assuming that the relevant “market conditions” remain unchanged. This will differ by asset type. For fixed income, the relevant market condition may be the term structure of interest rates, while for foreign exchange it may be the spot price of the underlying.
3. For a discussion of spot-futures parity, which governs the relationship between the current asset price and the futures price, please see Hull, John 2012. Options, Futures and Other Derivatives (8th edition). Boston: Prentice Hall.
4. There are times when spot-futures parity does not hold exactly, in which case there may be an arbitrage opportunity in capture deviations from spot-futures parity, but this is separate from the carry premium and beyond the scope of this paper.
5. Throughout this paper, the term “neutralizing” is used to represent a direct attempt to reduce/minimize exposure, and is not intended to imply that exposure has been completely eliminated.
6. For simplicity, please note that this assumes positions are equally-risk weighted, with an equal number of longs and shorts. This assumption is made throughout this paper. There are also other equally valid ways to approach position sizing (e.g., sizing positions proportional to the carry in each market).
7. Calculation available upon request. All results are before transactions costs and fees.
8. Calculation available upon request. All results are before transactions costs and fees.
9. Leverage is defined as the sum of the absolute value of notional positions divided by capital.
10. Technically, this would be true if the trade were put on with an equal number of contracts on each leg of the spread. However, it can sometimes be more appropriate to size the trade such that expected volatilities are equal for each leg. In our example, it happens that the approaches lead to the same result.
11. Please note that more complicated, multi-leg term structure trades are also possible, such as a so-called “butterfly,” which in the RBOB example might involve going long one contract of Oct-12, short two contracts of Nov-12 and long one contract of Dec-12.
12. In this example, we consider only the first several contracts in the futures term structure. While there may be opportunities further out on the curve, there tends to be a trade-off between carry opportunity and liquidity as contracts get further from expiration. In our example, we define the carry to be the difference between the “near” contract (i.e., next to expire) and the contract of interest. An alternative and equally valid approach would be to use the difference between consecutive-maturity contracts to calculate the carry. Generally, the closer the points are together, the more highly correlated they tend to be, leading to more risk offset and less relative alpha differences (i.e., there is also a trade-off between more risk offset and the relative alpha opportunity).
13. Calculation available upon request. All results are before transactions costs and fees.
14. Calculation available upon request. All results are before transactions costs and fees.
15. Calculation available upon request. All results are before transactions costs and fees.
16. Calculation available upon request. All results are before transactions costs and fees.
17. Please see disclosures for a full description of the Hypothetical Global Carry Strategy construction methodology.
18. Please note that we do not include the interest on cash or trading costs, which is somewhat conservative, as the interest income would on average have exceeded the trading costs over the period considered.
19. Please note that the construction of the Hypothetical Global Carry Strategy was intended to be as generic as possible, and does not rely on methodologies used by Campbell. Please see the disclosures for a detailed description of Strategy construction and some of the inherent limitations of hypothetical data.
20. These calculations do not include any haircut to the simulated returns.
21. For markets in which it is difficult/impossible to store and/or short the underlying asset, the usual cash and carry (and reverse cash and carry) arbitrage considerations do not apply, and thus the futures term structure can embed some market expectations about future spot price movements. This is the case with commodity markets, and thus spot price movements tend to strongly erode profits to simple carry strategies in that sector. In our traded commodity carry models, we adopt commodity-specific methodologies designed to mitigate this spot price drag.