Introduction / Abstract

The trend following strategy discussed below is based on a landmark paper, - published 2011, by T.Moskowitz et.al and titled Time Series Momentum with data up to 2009. This report brings the data up to December, 20141 and shows how well this simple strategy held up with changing market conditions.

Using only a minimum of parameters it further demonstrates how robust the core of the strategy is. At a time when millions are spent to gain a time advantage of a few milliseconds, this classic strategy adjusts the portfolio on a monthly time frame.

The following paragraphs are extracted from the original paper to outline the strategy. Keep in mind that when expressions like “we or us” are used it is from the perspective of the original authors.

Construction of Return Series

Each day, we compute the daily excess return of the most liquid futures contract (typically the nearest or next nearest to delivery contract), and then compound the daily returns to a cumulative return index from which we can compute returns at any horizon. For the equity indices, our return series are almost perfectly correlated with the corresponding returns of the underlying cash indices in excess of the Treasury bill rate.2

Volatility Estimate

we scale the returns by their volatilities in order to make meaningful comparisons across assets. We estimate each instrument’s ex-ante volatility \(\delta_{t}\) at each point in time using an extremely simple model: the exponentially-weighted lagged squared daily returns (i.e., similar to a simple univariate GARCH model). Specifically, the ex ante annualized variance \(\delta_{t}^{2}\) for each instrument is calculated as follows:

\[ \sigma_{t}^{2}=261\sum_{i=0}^{\infty}(1-\delta)\delta^{i}(r_{t-1-i}-\bar{r}_{t})^{2} \]

where the scalar 261 scales the variance to be annual, the weights \((1-\delta)\delta^{i}\) add up to one, and \(\bar{r}_{t}\) is the exponentially weighted average return computed similarly. The parameter \(\delta\) is chosen so that the center of mass of the weights is \(\sum_{i=0}^{\infty}(1-\delta)\delta^{i}={\delta} / {(1-\delta)}=60\) days. The volatility model is the same for all assets at all times.

Time Series Momentum trading strategies

For each instrument s and month t, we consider whether the excess return over the past k months is positive or negative and go long the contract if positive and short if negative, holding the position for h months. We set the position size to be inversely proportional to the instrument’s ex-ante volatility, \(1/\delta_{t-1}^{s}\) , each month.

For each trading strategy (k,h) we derive a single time series of monthly returns even if the holding period h is more than one month. We derive this single time series of returns following the methodology used by Jegadeesh and Titman (1993): The return at time t represents the average return across all portfolios at that time, namely the return on the portfolio that was constructed last month, the month before that (and still held if the holding period h is greater than two), and so on for all currently “active” portfolios.

Time Series Momentum Factor

For a more in-depth analysis of time series momentum, we focus our attention on a single time series momentum strategy. Following the convention used in the crosssectional momentum literature , we focus on the properties of the 12-month time series momentum strategy with a 1-month holding period (e.g., k = 12 and h = 1), which we refer to simply as TSMOM.

TSMOM by Security and the Diversified TSMOM Factor

We start by looking at each instrument and asset separately and then pool all the assets together in a diversified TSMOM portfolio. We size each position (long or short) so that it has an ex ante annualized volatility of 40%. That is, the position size is chosen to be 40% / σt-1, where σt-1 is the estimate of the ex ante volatility of the contract as described above. The choice of 40% is inconsequential, but it makes it easier to intuitively compare our portfolios to others in the literature. The 40% annual volatility is chosen because it is similar to the risk of an average individual stock, and when we average the return across all securities (equal-weighted) to form the portfolio of securities which represent our TSMOM factor, it has an annualized volatility of 12% per year over the sample period 1985 to 2009, which is roughly the level of volatility exhibited by other factors such as those of Fama and French (1993) and Asness, Moskowitz, and Pedersen (2010).3

The TSMOM return for any instrument s at time t is therefore:

\[ r_{t,t+1}^{TSMOM,s}=sign(r_{t-12,t}^{s})\frac{40\%}{\sigma_{t}^{s}}r_{t,t+1}^{s} \]

We compute this return for each instrument and each available month from January 1985 to December 2009. Panel A of Figure 2 plots the annualized Sharpe ratios of these strategies for each futures contract. As the figure shows, every single futures contract exhibits positive predictability from past one year returns. All 58 futures contracts exhibit positive time series momentum returns and 52 are statistically different from zero at the 5% significance level.

The overall return of the strategy that diversifies across all the \(S_{t}\) securities that are available at time t is:

\[ r_{t,t+1}^{TSMOM}=\frac{1}{S_{t}}\sum_{s=1}^{S_{t}}sign(r_{t-12,t}^{s})\frac{40\%}{\sigma_{t}^{s}}r_{t,t+1}^{s} \] 􏰊􏰏􏰃 We analyze the risk and return of this factor in detail next. We also consider TSMOM strategies by asset class constructed analogously.

Performance by sector

Stats

risk/perf stats
                           TSMOM TSMOM_CM TSMOM_EQ TSMOM_FI TSMOM_FX
Annualized Return         0.1608   0.1223   0.1878   0.1832   0.1201
Annualized Std Dev        0.1205   0.1432   0.2766   0.2946   0.1829
Annualized Sharpe (Rf=0%) 1.3340   0.8542   0.6791   0.6220   0.6567
max.Drawdown
                   TSMOM  TSMOM_CM  TSMOM_EQ  TSMOM_FI  TSMOM_FX
Worst Drawdown 0.1621032 0.3135071 0.5364262 0.5504765 0.2865146
Performance by month
      Jan  Feb  Mar  Apr  May  Jun  Jul  Aug  Sep   Oct  Nov  Dec TSMOM
1985  4.3  3.8 -5.3  4.0  6.4  1.1 -3.1  5.3  0.7   0.8  4.8  2.2  27.3
1986 -2.6 10.3  8.8  0.8 -4.3  3.7  4.0  2.5 -4.1  -1.3  4.2 -0.3  22.6
1987  6.6  4.0  4.4 -0.7  1.7 -0.1  4.1  1.0  4.0 -10.7  4.7  1.8  21.4
1988 -0.2  0.5  4.2  0.2  2.1 -0.4 -1.9  1.4  2.1   0.5  0.2  0.5   9.5
1989  6.3 -5.1  6.0  1.0  2.3 -0.3  4.0  3.3  1.1  -2.4  2.2  5.9  26.4
1990 -0.8  0.9  4.9 -1.6 -4.9  1.0  3.4 11.0  5.1  -4.4 -0.5  0.8  14.9
1991 -0.3 -2.2 -1.9  3.4 -0.8  0.9 -2.6  3.4  2.6   0.7  0.0  7.5  10.7
1992 -1.6 -0.2 -1.9  1.9  1.2  2.5  5.5  5.2  3.3  -0.4  1.6  1.9  20.3
1993  2.5  5.5 -0.5 -1.1  2.9  1.6  3.2  6.9 -2.2   4.5  0.9  6.2  34.5
1994  1.1 -4.8 -6.5 -0.3  0.5 -1.2 -2.0  2.0  2.8   1.7  2.3  1.9  -3.1
1995  0.8 -2.5  4.2 -1.0  1.1  1.0 -1.5 -1.6  4.5   1.7  3.3  1.6  12.0
1996 -2.8 -2.4  2.5  4.1  0.3  1.8 -4.6  4.3  8.6   2.2  7.6  3.1  26.7
1997  3.9 -1.6  2.1  3.0  0.8  3.4  7.5 -5.6  4.0  -0.4  1.7  5.0  25.8
1998  3.0  2.7  4.9  0.2  4.8  3.0  4.5  6.4  0.8  -1.2  3.2  3.9  42.6
1999 -0.3 -2.6 -3.9 -0.8  0.4 -2.3  0.5 -0.2  0.4   1.9  6.0  6.7   5.3
2000  0.3  1.9 -3.5  1.8  0.9  1.1  0.0  5.5 -1.4   2.8  4.0 -1.1  12.6
2001  0.2  3.0  6.2 -6.3  0.0  2.2  0.6  4.0  9.4   5.0 -5.9 -0.4  18.1
2002  1.6  0.4 -5.0  1.7  2.9  6.2  5.7  3.3  6.9  -3.1 -0.9  9.3  32.0
2003  4.9  5.0 -2.7  0.0  6.5 -2.3 -7.2 -2.1  6.0   3.6  1.9  7.4  21.9
2004  0.8  5.9  0.4 -4.5 -1.5 -1.6 -0.7  0.3  5.9   1.1  5.3  0.9  12.5
2005  2.3 -1.4  0.6 -3.7  2.6  5.1  0.3  2.2  1.6  -2.3  3.8  1.9  13.6
2006  4.0 -0.4  5.1  2.6 -2.9  0.4 -0.7 -1.4  1.4   1.6  0.9  3.8  15.0
2007  2.7 -3.5  1.9  5.1  2.2  2.2 -5.5 -2.3  4.1   5.0 -2.4  2.8  12.3
2008 -3.3  9.5 -2.9 -2.2  0.1  8.0 -3.6 -3.6 -1.5   9.0  8.0  6.2  24.6
2009  0.2  2.5 -2.2 -4.5 -7.4  1.5 -3.4  2.1  0.9  -2.2  4.4 -1.5  -9.6
2010 -4.7  2.1  5.2  2.1 -5.2  1.2 -0.9  5.8  1.3   2.2 -0.6  4.4  12.8
2011 -1.0  4.1 -1.9  5.2 -1.3 -2.4  1.1  2.7 -9.5  -1.8  2.7  1.4  -1.5
2012 -0.6 -2.2  0.2  3.6 10.8 -7.1  1.9 -1.7 -3.6  -1.2 -0.4  1.1  -0.1
2013  0.9  1.7  1.3  6.0 -2.2  0.5  3.4 -0.9 -1.3   2.3  3.8  1.9  18.6
2014 -5.0 -0.8  1.1  0.0 -0.1  2.3 -0.7  3.5  3.9   0.8  8.6  4.7  19.2
     TSMOM_CM TSMOM_EQ TSMOM_FI TSMOM_FX
1985     17.7    106.7     28.4     -8.5
1986     -7.9     89.6     29.2     24.6
1987     29.7      8.7     -5.5     25.3
1988      4.2    -19.1      4.4     45.4
1989     29.3     18.8     20.1     20.4
1990     20.8      5.1     38.4     -3.3
1991     15.1    -17.9     52.7     -1.9
1992      8.2     24.3     35.5     27.9
1993     23.7      3.3    131.6     21.0
1994     19.2    -24.3    -43.2      8.3
1995      3.3      5.9     24.5     19.1
1996     25.7     40.7     -0.3     36.9
1997     -0.2     74.9     70.0     10.5
1998     48.3     21.7     78.5      7.1
1999      8.1     33.1    -15.7     -3.0
2000     27.3    -12.7     12.5     -1.6
2001     14.4     30.2      6.8     21.6
2002      6.9     61.1     69.4     24.6
2003     22.6      0.0     -3.4     68.1
2004      7.9     40.3     13.4     -0.5
2005      5.3     81.7      1.7     -3.3
2006     26.6     50.9    -20.9      9.1
2007     10.0      7.2     16.6     13.0
2008      8.9     43.8     32.9     23.5
2009     -5.7    -19.1     -6.3    -16.2
2010      9.5     -5.0     24.5     13.5
2011    -11.0    -23.8     49.3    -14.6
2012    -13.3     -0.1     19.4      6.1
2013     21.7     76.1     -9.7      3.3
2014     14.3      7.7     26.8     23.6

Data Sources

Country Equity Indices

Universe consists of equity index futures from nine developed equity markets:

  • SPI 200 (Australia), CAC 40 (France), DAX (Germany), FTSE/MIB (Italy), TOPIX (Japan), AEX (Netherlands), IBEX 35 (Spain), FTSE 100 (U.K.), S&P 500 (U.S.)

Futures prices are obtained from Bloomberg and Datastream

MSCI country level index returns are used prior to the availability of futures returns

Country Bond Indices

Universe consists of bond index futures from 13 developed bond markets:

  • Australia 3-year Bond, Australia 10-year Bond, Euro Schatz, Euro Bobl, Euro Bund, Euro Buxl, Canada 10-year Bond, Japan 10-year Bond (TSE), Long Gilt, U.S. 2-year Note, U.S. 5-year Note, U.S. 10-year Note, U.S. Long Bond

Futures prices are obtained from Bloomberg, Datastream and Morgan Markets

JP Morgan country level bond index returns are used prior to the availability of futures returns

Daily returns are scaled to a constant duration of 2 years for 2 and 3-year bond futures, 4 years for 5-year bond futures, 7-years for 10-year bond futures and 20 years for 30-year bond futures

Foreign Exchange

Contains 12 cross-currency pairs:

  • AUD-NZD; AUD-USD; EUR-JPY; EUR-NOK; EUR-SEK; EUR-CHF; EUR-GBP; JPY-AUD; GBP-USD; USD-EUR; USD-CAD; USD-JPY

Universe of currency forwards covers 10 exchange rates:

  • Australia, Canada, Germany spliced with the Euro, Japan, New Zealand, Norway, Sweden, Switzerland, U.K., U.S.
  • We look at forwards for the first nine currencies vs. USD; these nine underlie the 12 cross-currency pairs

Spot and forward interest rates from Citigroup are used to calculate currency returns going back to 1989 for all currencies except CAD and NZD, which go back to 1992 and 1996, respectively

Prior to that, spot exchange rates from Datastream and IBOR short rates from Bloomberg are used to calculate returns

Commodities

Universe covers 24 commodity futures

  • Aluminum, Copper, Nickel, Zinc
    • From London Metal Exchange (LME)
  • Brent Crude, Gas Oil, Cotton, Coffee, Cocoa, Sugar
    • From Intercontinental Exchange (ICE)
  • Live Cattle, Lean Hogs
    • From Chicago Mercantile Exchange (CME)
  • Corn, Soybeans, Soy Meal, Soy Oil, Wheat
    • From Chicago Board of Trade (CBOT)
  • WTI Crude, RBOB Gasoline spliced with Unleaded Gasoline, Heating Oil, Natural Gas
    • From New York Mercantile Exchange (NYMEX)
  • Gold, Silver"
    • From New York Commodities Exchange (COMEX)
  • Platinum
    • From Tokyo Commodity Exchange (TOCOM)

Futures prices are obtained from Bloomberg

  1. AQR Capital Management, LLC.dataset↩

  2. Bessembinder (1992) and de Roon, Nijman, and Veld (2000) compute returns on futures contracts similarly and also find that futures returns are highly correlated with spot returns on the same underlying asset.↩

  3. Also, this portfolio construction implies a use of margin capital of about 5-20%, which is well within what is feasible to implement in a real-world portfolio.↩