Momentum s Hidden Sensitivity to the Starting Day

Momentum s Hidden Sensitivity to the Starting Day Philip Z. Maymin, Zakhar G. Maymin, Gregg S. Fisher Abstract: We show that the profitability of time- series momentum strategies on commodity futures across their entire history is strongly sensitive to the starting day. Using daily returns with 252- day formation periods and 21- day holding periods, the Sharpe ratio depends on whether one starts on the first day, the second day, and so on, until the twenty first day. This sensitivity is higher for shorter trading periods. The same results also hold in simulation of independent and identically lognormally distributed returns, showing that this is not only an empirical pattern but a fundamental issue with momentum strategies. Portfolio managers should be aware of this latent risk: starting trading the same strategy on the same underlying but one day later could, even after many decades, turn a successful strategy into an unsuccessful one. Keywords: momentum, portfolio management, sensitivity, risk, initial conditions Philip Z. Maymin (corresponding author) NYU- Polytechnic Institute Six MetroTech Center Brooklyn, NY 11201 Telephone: 718-260- 3175 Email: phil@maymin.com Zakhar G. Maymin Quantitative Investment Services, Inc. Email: zak@maymin.com Gregg S. Fisher Gerstein Fisher Research Center Email: gfisher@gersteinfisher.com Electronic copy available at: http://ssrn.com/abstract=1899000

Introduction There are two types of momentum strategies: cross- sectional momentum and time- series momentum. Cross- sectional momentum strategies originate from the seminal study by Jegadeesh and Titman (1993) on equity markets and have been analyzed in commodity futures with research culminating with Miffre and Rallis (2007), Szakmary, Shen, and Sharma (2010), and Fuertes, Miffre, and Rallis (2010). Such strategies form portfolios long winners and short losers. Time- series momentum strategies are applied to each individual time series. The underlying is purchased and held for a period of time if it exceeds the returns to cash over the associated formation period, and is sold for the same holding period otherwise. Thus, in such a time- series momentum strategy, the underlying is always held either long or short, while in a cross- sectional momentum strategy, the middle- ground underlying securities that are neither winners nor losers are not held at all. Time- series momentum has been studied most thoroughly recently in Moskowitz, Ooi, and Pedersen (2010), where they document significant time- series momentum for each of 58 liquid futures when viewed with end- of- month data, with a 12- month formation period and a 1- month holding period. Starting with Jegadeesh and Titman (1993), momentum strategies have often been formed as an average over several starting month- ends, so that the portfolios overlap. However, prices are always taken to be from the last day of the month, rather than from the second- to- last day, third- to- last day, etc., with the exception of Fuertes, Miffre, and Rallis (2010) who compare their month- end results with those from using prices from the 15 th of each month as part of their robustness checks and 1 Electronic copy available at: http://ssrn.com/abstract=1899000

find there is no difference. Rachev, Jasic, Stoyanov, and Fabozzi (2007) use daily data but also form their non- overlapping portfolios based on monthly intervals. We contribute to the literature by exposing a previously unknown risk of momentum strategies relating to their sensitivity not just on the starting month, but, startlingly, on the starting day. Specifically, this paper examines the time- series momentum of liquid futures contracts both on a daily and on a monthly basis and shows that the results of persistent momentum often depend on the exact starting day of the time series, if on a daily basis, or on how many days before the end of the month the strategy is traded, if on a monthly basis. Data Futures data was obtained from CSI. Their dataset includes all prices on all futures traded. They were ranked highest in accuracy by Futures Magazine in 1999 (Sheldon, 1999). Further, they combine electronic trading history with open outcry pit trading history for a comprehensive collection. Every underlying commodity has a sequence of futures contracts with different expiration dates. At any given date, one of those contracts is the most active one, defined as having the highest open interest. The open interest measures the total number of outstanding contracts held by market participants. Creating a single continuous series requires rolling the most active futures contract when a new contract has greater open interest. The adjustment must be made in such a way that the returns to holding the futures contract represent the 2 Electronic copy available at: http://ssrn.com/abstract=1899000

true returns that would have been earned by holding the then- active contract overnight. The standard rolling algorithm of CSI is to wait until a new contract has had the highest open interest for two days in a row, to avoid switching back and forth. In this manner, a single continuous time series of adjusted prices is created for each underlying commodity. These calculated prices may then be used in momentum strategies with the assurance that the returns implied by the prices represent true returns that would have been earned in the market. Different futures had different start dates and open interest. Because some futures trade only thinly or have brief histories, only those futures were used whose maximum open interest over their lifetime exceeded 10,000 contracts, which had at least 20 years of daily data available, and whose lifetime returns exhibited an annualized volatility of at least 5 percent. Further, only the combined floor and electronic series were used. Table 1 lists the descriptions and summaries of the 36 futures contracts studied. (A smaller list of futures that had all been continuously active over the same 25- year period was also explored, and the results were even more extreme, due to the relation, discussed below, between the trading history length and the sensitivity to the starting date.) Overnight daily returns were obtained from the Federal Reserve. The comparison cash return serving as the hurdle to determine if an asset outperformed or underperformed was calculated as the compounded daily Federal Funds Effective rates over the days in the associated period. 3

Methodology The daily results are constructed as follows. On average, there are 252 business days per year and 21 business days per month. On each day after day 252 in each commodity time series, compare the 252- day historical return with the cash return. If the commodity outperformed, record the subsequent 21- day excess return over cash as the profit, and otherwise record the negative of the subsequent 21- day excess return as the profit. This process actually computes 21 different non- overlapping trading strategies: the one starting on the first day and rolling every 21 days, the one starting on the second day and rolling every 21 days, and so on. For each of those 21 strategies, compute the annualized Sharpe ratio, equivalent to the unannualized Sharpe ratio multiplied by the square root of 252 over 21. The monthly results are constructed similarly. First, offset month- end prices are extracted from the time series, with the offset ranging from 0 to - 15. The offset represents how many days before the end of the month to take the prices. It is limited to 15 business days rather than 21 business days to avoid situations where a futures contract may not have had enough trading days in a given month. An offset of zero results in month- end prices, an offset of one results in prices taken always one day before the month end, and so on. This results in sixteen non- overlapping strategies, for each of which we compute the annualized Sharpe ratio, equivalent to the unannualized Sharpe ratio multiplied by the square root of 12. 4

Results Figure 1 shows the Sharpe ratios of each daily 252- day formation period, 21- day holding period momentum strategy, starting on days 1 through 21, for each futures symbol. The dispersion in Sharpe ratios within a single futures contract as a result of different starting days can sometimes be negligible and sometimes extreme. For example, the Sharpe ratio of the Gold momentum strategies is essentially unchanged regardless of the starting day. On the other hand, the Sharpe ratio of the S&P 500 momentum strategy ranged from less than 0.04 to more than 0.30, with much of that gap occurring over the span of just a few days difference as to when the strategy began trading. In some cases the Sharpe ratio can even change signs on nearby days, such as with Orange Juice. Table 2 reports the maximum, minimum, and mean Sharpe ratios across the 21 possible daily strategies, as well as the range, expressed both as the difference between the maximum and the minimum, and as a percentage of the mean. For the majority of the symbols, more than half of their mean Sharpe ratio vanishes if the starting day is chosen poorly. A similar analysis was performed on monthly data, with prices chosen either at the end of the month, or one day before the end of the month, and so on. The results (not shown for space considerations) are similar: the Sharpe ratios of the 12- month formation period, 1- month holding period strategies varied widely depending on how many days before the end of the month the trading was done. 5

Another way of measuring the impact of this effect is to compare the average range of the Sharpe ratios across all futures symbols with the average Sharpe ratio across all futures symbols. To calculate the average range of the Sharpe ratios, we calculate the range for each futures symbol, and then take the average. To calculate the average Sharpe ratio, we calculate the average across all strategies starting on day one, all strategies starting on day two, and so on. It turns out that the average Sharpe ratio is approximately the same regardless of the starting day, and this average Sharpe ratio across all futures symbols is 0.53. Meanwhile, the average range across all futures symbols is 0.20. This means that the choice of starting day across different futures symbols could in principle reduce the expected Sharpe ratio by more than a third. The range in Sharpe ratios within a given futures contract appears to be related to the length of history on which it is tested. Figure 2 shows a scatterplot of the range for each futures contract versus the length in years of its available trading history. Contracts with fewer than 30 years of available trading history have a higher range than those with more history, and therefore an even higher sensitivity to the starting date. Finally, to establish that this result is of fundamental and not only empirical importance, 36 simulations were run with 15, 16,, 50 years worth of daily returns, each distributed independently and identically lognormally with zero mean and 22.68 percent annualized volatility to match the average empirical results of the 36 real- world futures. Figure 3 displays the resulting variety. As with the real world 6

futures data, some simulations display constancy while others vary wildly depending on the starting day. Further, as shown in Figure 4, the Sharpe ratios vary more when the trading history is shorter, again as with the real world futures data. Of course, these results all hinge on the momentum strategy. A buy- and- hold strategy in any market with relatively low autocorrelation should not exhibit such sensitivity to the initial day. Conclusion Momentum strategies are typically regarded as robust and exhibiting universality. However, they bring with themselves a hidden risk: their profitability can vary greatly simply as a result of the particular day the strategies begin trading. An analysis of both daily and monthly momentum strategies on the entire histories of various liquid futures contracts shows that the particular days chosen as trading days, a decision made once at the start of trading, can have substantial long- term consequences on profitability. These discrepancies become even wider when the trading strategies are applied to shorter periods of data. Further, simulated data yields similar results, suggesting that this is a not only an empirical risk but a fundamental one as well. Portfolio managers ought to be aware of this latent risk. The range in profitability can render what appear to be winning strategies completely unprofitable, or even make them losers. 7

We propose two possible solutions for portfolio managers trading momentum based on our findings. One is to roll every single day, thus averaging out the results and reducing the risk of choosing the wrong day. Another is to employ multiple futures contracts as underlyings for their momentum strategies. Future research could determine what characteristics or patterns of futures prices, in addition to the effect from the length of the trading history documented here, are associated with an increased sensitivity of the momentum strategy to the starting day, or what aspects of a strategy such as the momentum strategy make it likely to be sensitive to the starting day. References Fuertes, A- M., Miffre, J., Rallis, G. (2010). Tactical allocation in commodity futures markets: Combining momentum and term structure signals. Journal of Banking and Finance 34, 2530-2548. Jegadeesh, N., Titman, S. (1993). Returns to buying winners and selling losers: implications for stock market efficiency. Journal of Finance 56, 699-720. Knight, S. (1999). How clean is your end- of- day data? Futures Magazine 28:9, 64-69. Miffre, J., Rallis, G. (2007). Momentum strategies in commodity futures markets. Journal of Banking and Finance 31, 1863-1886. Moskowitz, T., Ooi, Y.H., Pedersen, L.H. (2010). Time series momentum. Working paper retrieved on July 7, 2011 from http://pages.stern.nyu.edu/~lpederse/papers/timeseriesmomentum.pdf. Rachev, S., Jasic, T., Stoyanov, S., Fabozzi, F.J. (2007). Momentum strategies based on reward- risk stock selection criteria. Journal of Banking and Finance 31, 2325-2346. Szakmary, A., Shen, Q., Sharma, S. (2010). Trend- following trading strategies in commodity futures: A re- examination. Journal of Banking and Finance 34, 409-426. 8

Table 1: Description of Futures Contracts This table lists the name, exchange, start date, and end date (in our sample) for each of the 36 futures symbols with more than 20 years of daily data, at least 10,000 lifetime maximum open interest, and at least 5 percent annualized volatility. Symbol Exchange Short Name Start Date End Date AD CME Australian Dollar 19861125 20110628 BO2 CBT Soybean Oil 19500717 20110628 BP CME British Pound 19700101 20110628 C2 CBT Corn 19490103 20110628 CC2 NYCE Cocoa 19651230 20110628 CD CME Canadian Dollar 19720516 20110628 CL2 NYMEX Crude Oil- Light 19830330 20110628 CT2 NYCE Cotton #2 19670322 20110628 CU CME Euro 19720516 20110628 DX2 FINEX U.S. Dollar Index 19710104 20110628 FC CME Cattle- Feeder 19711130 20110628 GC2 COMEX Gold 19750102 20110628 HG2 COMEX CopperHG 19660103 20110628 HO2 NYMEX Heating Oil #2 19781115 20110628 JY CME Japanese Yen 19720516 20110628 KC2 NYCE Coffee 19720816 20110628 LB CME Lumber 19691001 20110628 LC CME Cattle- Live 19641130 20110628 LH CME Hogs- Lean 19660228 20110628 NG2 NYMEX Natural Gas- Henry Hub 19900403 20110628 NK CME Nikkei 225 Index 19900925 20110628 O2 CBT Oats 19490103 20110628 OJ2 NYCE Orange Juice- Frozen 19661026 20110628 PA2 NYMEX Palladium 19770103 20110628 PL2 NYMEX Platinum 19640114 20110628 RB2 NYMEX Gasoline- Reformulated Blendstock 19841203 20110628 RR2 CBT Rice- Rough 19860820 20110628 S2 CBT Soybeans 19490103 20110628 SB2 NYCE Sugar #11 19651230 20110628 SF CME Swiss Franc 19720516 20110628 SI2 COMEX Silver 19630612 20110628 SM2 CBT Soybean Meal 19510829 20110628 SP2 CME S&P 500 19500103 20110628 TY CME T- Note- US 10 Yr w/prj A- CBT 19820503 20110628 US CBT T- Bond- US 19770822 20110628 W2 CBT Wheat 19220103 20110628 9

Figure 1: Sharpe Ratios of Daily Momentum Strategies This figure shows the Sharpe ratio of each daily momentum strategy with a 252- day formation period and 21- day holding period for each futures symbol. The different bars represent the Sharpe ratios of strategies starting on different days, from the first day to the 21 st day. 10

Table 2: Range of Sharpe Ratios of Daily Strategies This table reports the minimum, mean, maximum, range (difference between maximum and minimum), and range as a percentage of the mean of the Sharpe ratios of the 21 daily momentum strategies with a 252- day formation period and a 21- day holding period for each futures symbol. Symbol Min Mean Max Range Range/Mean AD 0.19 0.39 0.52 0.32 82% BO2 0.20 0.27 0.36 0.16 60% BP 0.28 0.34 0.42 0.14 41% C2 0.22 0.29 0.37 0.15 53% CC2 0.07 0.20 0.34 0.27 138% CD 0.85 0.94 1.05 0.20 21% CL2 0.08 0.17 0.27 0.20 118% CT2 0.34 0.43 0.51 0.17 39% CU 0.36 0.46 0.59 0.22 48% DX2 0.37 0.44 0.56 0.20 45% EM 3.20 3.44 3.67 0.48 14% FC 0.38 0.44 0.50 0.12 27% FF 4.18 4.25 4.30 0.12 3% FV 0.57 0.64 0.71 0.14 21% GC2 0.41 0.51 0.57 0.16 30% HG2 0.32 0.41 0.46 0.14 34% HO2 0.14 0.25 0.36 0.22 85% JY 0.42 0.49 0.62 0.20 41% KC2 0.12 0.23 0.33 0.21 91% LB 0.46 0.54 0.59 0.12 23% LC 0.24 0.32 0.43 0.19 59% LH 0.14 0.24 0.35 0.21 90% NG2 0.07 0.25 0.41 0.34 135% NK 0.06 0.19 0.29 0.23 121% O2 0.20 0.29 0.39 0.19 66% OJ2-0.13-0.04 0.08 0.21-598% PA2 0.35 0.50 0.60 0.25 51% PL2 0.24 0.33 0.45 0.21 62% RB2 0.09 0.20 0.32 0.23 117% RR2-0.00 0.06 0.15 0.16 246% S2 0.02 0.08 0.16 0.14 166% SB2 0.09 0.18 0.26 0.17 94% SF 0.35 0.44 0.54 0.19 42% SI2 0.26 0.36 0.43 0.16 45% SM2 0.07 0.15 0.21 0.14 97% SP2 0.04 0.19 0.30 0.27 144% TU 1.34 1.48 1.63 0.29 19% TY 0.30 0.39 0.47 0.17 43% US 0.19 0.33 0.43 0.23 71% W2 0.24 0.33 0.41 0.17 53% 11

Figure 2: Range of Daily Sharpe Ratios vs. Length of Trading History This scatterplot shows the negative relation between the range of the Sharpe ratios for the various daily momentum strategies on each futures contract plotted against the length of the trading history of that futures contract. 12

Figure 3: Sharpe Ratios of Simulated Daily Momentum Strategies This figure shows the Sharpe ratio of each daily momentum strategy with a 252- day formation period and 21- day holding period for 36 different simulated time series having respectively 15, 16, 50 years of trading history, with all simulated returns independently and identically distributed lognormally with a zero mean and a 22.68 percent annualized standard deviation. The different bars represent the Sharpe ratios of strategies starting on different days, from the first day to the 21 st day. 13

Figure 4: Range of Simulated Daily Sharpe Ratios vs. Length of Trading History This scatterplot shows the negative relation between the range of the Sharpe ratios for the various daily momentum strategies on each simulated futures contract plotted against the length of the trading history of that simulated futures contract. 14