# Analysis of a Global Futures Trend-Following Strategy

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Stylized Facts

## 2. Materials and Methods

#### 2.1. Data Acquisition and Transformation

_{t,H}and P

_{t,L}are the current daily high and low prices respectively, and P

_{t}

_{−1}is the previous closing price.

_{t}= max[P

_{t,H}− P

_{t,L}, |(P

_{t,H}− P

_{t−1})|, |(P

_{t,L}− P

_{t−1})|]

#### 2.2. Software

- buildGlobalFuturesDataset.R: Extracts data from a MySQL database and builds the R data set.
- establishStylizedFacts.R: Used to explore the stylized facts.
- calibrateMarketModelLRD_log.R: Calibrates the market model.
- selectTradingModelParameters.R: Used to explore the trading model parameters via brute force search.
- simulateMarketModelLRD_rerun_2.R: Simulates the market model and computes the sensitivities.

#### 2.3. Market Model Specifications

_{t}~ N(0,1), σ

_{t}is the time-varying volatility, and μ is the constant drift for the instrument over time period, T.

_{t}= P

_{t}

_{−1}exp(μΔt + σ

_{t}ϵ

_{t})

_{t}(Chou 2005) as shown in Equation (3).

_{t}= (π/8)

^{5}× R

_{t}

_{t}> 0) divided by v, and η

_{t}is an independent and identically distributed (i.i.d) log-normal random variable.

_{t}= vλ

_{t}η

_{t}

_{t}), given by Equation (6). In Equation (6), d is the long memory parameter (0 < d < 0.5); B is the back-shift (or lag operator); and ϕ(z) =1 − ϕ

_{1}z − … ϕ

_{p}z

_{p}, ψ(z) = 1 + ψ1 + … + ψ

_{q}z

_{q}are MA- and AR-polynomials with all roots outside the unit circle. The back-shift operator is used for notational convenience where B

^{m}x

_{t}= x

_{t}

_{=m}. This notation allows (even infinite) distributed lags to be represented concisely.

_{t}= log(R

_{t}) − log(v) = log(λ

_{t}) + e

_{t}

_{t}= ψ(B)e

_{t}

_{t}) as ζ

_{t}and rearranging, we can see that the conditional mean of Zt yields Equation (7), where E(ζ

_{t}) = 0.

_{t}= log(λ

_{t}) = [ϕ

^{−1}(B)ψ(B)(1 − B)−d − 1]e

_{t}

_{t}exhibit a hyperbolic decay (Figure 2), the speed of which depends upon the parameter d as in Equation (8) where c

_{p}

^{Z}> 0 is a constant.

_{p}

^{Z}|k|2d − 1

_{t}are normally distributed, the model is referred to as a Gaussian EFARIMA model. Setting p = 0 and q = 0, the innovations, e = log(η

_{t}), simplifies to Equation (9).

_{t}= e

_{t}

#### 2.4. Market Model Calibration

_{t}and R

_{t}. We assume that R

_{t}(t = 1, 2, …, T) is generated by an EFARIMA process with an unknown parameter vector in Equation (10).

_{e}

^{2}, d, ϕ

_{1}, …, ϕ

_{p}, ψ

_{1}, …, ψ

_{q})T

_{e}

^{2}, d)T

_{t}= log(R

_{t}) − log(v) is a centered Gaussian FARIMA process, maximum likelihood estimation (MLE) can be used to estimate the parameters (Mills 1999; Fox and Taqqu 1986; Giraitis and Surgailis 1990; Beran 1995; Haslett and Raftery 1989). This allows us to use standard, widely-available, estimation software to calibrate the model (McLeod et al. 2007; Veenstra and McLeod 2015). We assume that the price process is a function of the volatility, which is in turn a function of R

_{t}and employ the ARFIMA R package (Veenstra and McLeod 2015) to estimate the parameters of true range for each instrument in the universe under study (Appendix A). For the vast majority of the instruments in the universe under study, the estimated d parameter is between 0.15 and 0.35. All parameters are highly significant.

#### 2.5. Trading Model Specifications

_{t}. Filters are commonly used to smooth price series. We use exponentially weighted moving averages (EMAs) to smooth both price and the true range time series. The core rules of our simple trading model are detailed briefly in the next two sub-sections.

#### 2.5.1. Long Position

_{t}

_{−1,F}is above the slow EMA

_{t}

_{−1,S}and we have no position, we enter a long position of p

_{t}units (Equation (12)).

_{t}= floor[f A

_{t}

_{−1}÷ max[ATR

_{t}

_{−1}× M, L]]

_{t}

_{−1}is the EMA of the true range for the previous time step, M is the risk multiplier, and L is the ATR floor.

_{t}. For each subsequent time, t, we update our stop level as in Equation (13).

_{t}= max[P

_{t}− ATR

_{t−1}× M, s

_{t−1}]

_{t}moves below the stop loss level, s

_{t}

_{−1}.

#### 2.5.2. Short Position

_{t−}

_{1,F}is below the slow EMA

_{t}

_{−1,S}and we have no position, we enter a short position of p

_{t}units (Equation (14)).

_{t}= −floor[f A

_{t}

_{−1}÷ max[ATR

_{t}

_{−1}× M, L]]

_{t}. For each subsequent time, t, we update our stop level as in Equation (15).

_{t}= min[P

_{t}+ ATR

_{t−1}× M, s

_{t−1}]

## 3. Results

#### 3.1. Parameter Selection

#### 3.2. Sensitivity Analysis

#### 3.2.1. Trend Sensitivity

#### 3.2.2. Serial Dependence in True Range Sensitivity

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Item # | Instrument Name | d | log(V) | var(e) | mu |

1 | Brent Crude Oil | 0.2355 | −6.2819 | 0.1489 | 0.0092 |

2 | Crude Oil | 0.2209 | −6.3405 | 0.17 | 0.0122 |

3 | Ethanol | 0.2809 | −6.7159 | 0.4139 | 0.0571 |

4 | Gas Oil | 0.2198 | −6.3044 | 0.1542 | 0.0096 |

5 | Gas-RBOB | 0.2011 | −6.1794 | 0.1672 | 0.0022 |

6 | Heating Oil | 0.2449 | −6.2499 | 0.1439 | 0.0099 |

7 | Nat Gas | 0.2886 | −6.14 | 0.1594 | −0.0022 |

8 | WTI Crude Oil | 0.2177 | −6.2201 | 0.3278 | −0.0024 |

9 | ECX EUA Emissions | 0.3822 | −5.7152 | 0.1031 | −0.0142 |

10 | Nat Gas | 0.372 | −5.8487 | 0.4072 | −0.0386 |

11 | AUDUSD | 0.2638 | −6.185 | 0.14 | 0.0067 |

12 | CADUSD | 0.2694 | −6.1393 | 0.1334 | 0.002 |

13 | CHFUSD | 0.2222 | −6.1448 | 0.1587 | 0.0034 |

14 | EURUSD | 0.2132 | −6.1638 | 0.1455 | 0.0014 |

15 | GBPUSD | 0.2373 | −6.1258 | 0.1397 | 0.0004 |

16 | JPYUSD | 0.2847 | −6.0951 | 0.1754 | −0.0058 |

17 | NZDUSD | 0.183 | −6.2719 | 0.3233 | 0.0123 |

18 | US Dollar Index | 0.183 | −6.0317 | 0.1807 | −0.0018 |

19 | EURCHF | 0.3317 | −6.1226 | 0.5683 | −0.0181 |

20 | EURGBP | 0.203 | −5.9885 | 0.4755 | −0.0013 |

21 | EURJPY | 0.2507 | −6.35 | 0.4758 | 0.0084 |

22 | BRLUSD | 0.2252 | −6.5292 | 0.7088 | 0.0223 |

23 | CZKUSD | 0.0844 | −6.5137 | 1.1831 | −0.0043 |

24 | HUFUSD | 0.0897 | −6.4835 | 1.1712 | −0.0073 |

25 | MXNUSD | 0.3382 | −6.1907 | 0.1611 | 0.005 |

26 | PLNUSD | 0.1315 | −6.5222 | 1.1359 | 0.0062 |

27 | RUBUSD | 0.2953 | −6.163 | 0.3264 | 0.0046 |

28 | ZARUSD | 0.1748 | −6.2179 | 0.6859 | −0.0049 |

29 | USDKRW | 0.273 | −6.0789 | 0.2116 | −0.0022 |

30 | Corn | 0.3198 | −6.0545 | 0.172 | −0.0009 |

31 | Oats | 0.3307 | −6.2312 | 0.2129 | 0.0118 |

32 | Rough Rice | 0.286 | −6.0389 | 0.208 | −0.0084 |

33 | Soybean Meal | 0.3132 | −6.322 | 0.1647 | 0.0203 |

34 | Soybean Oil | 0.244 | −6.1338 | 0.1459 | 0.0053 |

35 | Soybeans | 0.2783 | −6.2305 | 0.1561 | 0.0164 |

36 | Wheat | 0.2748 | −6.0158 | 0.1416 | −0.0066 |

37 | Corn | 0.286 | −6.5743 | 0.6159 | 0.0274 |

38 | Milling Wheat | 0.3447 | −6.5546 | 0.6575 | 0.0339 |

39 | Rapeseed | 0.2804 | −6.5859 | 0.5275 | 0.0166 |

40 | Wheat | 0.3195 | −6.2202 | 0.4898 | 0.0065 |

41 | Dow Jones Industrial (mini) | 0.3304 | −6.1172 | 0.1621 | 0.0105 |

42 | MSCI EAFE (mini) | 0.3136 | −6.1508 | 0.2195 | 0.01 |

43 | Nasdaq 100 (e-mini) | 0.3325 | −6.1532 | 0.1491 | 0.0136 |

44 | Russell 2000 (mini) | 0.3029 | −6.1539 | 0.1477 | 0.0125 |

45 | SP 500 (e-mini) | 0.3359 | −6.1039 | 0.1596 | 0.0089 |

46 | Belgian 20 | 0.239 | −6.0888 | 0.2724 | 0.0123 |

47 | CAC 40 | 0.293 | −6.1944 | 0.1899 | 0.007 |

48 | DAX | 0.2998 | −6.2708 | 0.2117 | 0.0081 |

49 | DJ Euro STOXX 50 | 0.309 | −6.0845 | 0.1636 | 0.0058 |

50 | EOE (Amsterdam) | 0.2994 | −5.9945 | 0.1449 | 0.0028 |

51 | FTSE 100 | 0.3368 | −5.9297 | 0.1181 | 0.004 |

52 | IBEX 35 | 0.2771 | −6.2074 | 0.1749 | 0.0072 |

53 | MIB FTSE | 0.3057 | −6.2135 | 0.1974 | 0.0011 |

54 | Nikkei 225 | 0.2941 | −6.3303 | 0.3148 | 0.0055 |

55 | OMX | 0.386 | −6.9382 | 0.165 | 0.0124 |

56 | SPTSE 60 | 0.2485 | −6.0678 | 0.1485 | 0.0113 |

57 | SMI | 0.3235 | −5.9756 | 0.1364 | 0.008 |

58 | SPI 200 | 0.2745 | −6.3048 | 0.2375 | 0.0099 |

59 | TOPIX | 0.2845 | −6.2746 | 0.2545 | 0.0074 |

60 | MSCI EM (mini) | 0.2801 | −6.0982 | 0.1943 | −0.0027 |

61 | MSCI Taiwan | 0.2195 | −6.1515 | 0.1817 | 0.0092 |

62 | SP CNX Nifty | 0.1867 | −6.3298 | 0.4275 | 0.0175 |

63 | Hang Seng | 0.1864 | −6.2192 | 0.1769 | 0.0128 |

64 | Hang Seng (mini) | 0.1814 | −6.1844 | 0.1782 | 0.0128 |

65 | Hang Seng China Enterprises | 0.2633 | −6.2258 | 0.153 | 0.0158 |

66 | IPC | 0.3258 | −5.8855 | 0.1262 | 0.0064 |

67 | KOSPI 200 | 0.2357 | −6.5445 | 0.4424 | 0.0126 |

68 | Feeder Cattle | 0.2522 | −6.1331 | 0.1549 | 0.0109 |

69 | Lean Hogs | 0.2451 | −5.9264 | 0.1445 | −0.0064 |

70 | Live Cattle | 0.2083 | −6.1276 | 0.1491 | 0.0088 |

71 | Copper | 0.2635 | −6.1975 | 0.1522 | 0.0091 |

72 | Gold | 0.3005 | −6.18 | 0.1894 | 0.0108 |

73 | Palladium | 0.3813 | −6.2387 | 0.2212 | 0.0111 |

74 | Platinum | 0.3442 | −6.2491 | 0.1652 | 0.011 |

75 | Silver | 0.3047 | −6.1876 | 0.1795 | 0.0066 |

76 | USD Deliverable Swap 10 year | 0.299 | −6.0867 | 0.208 | 0.0123 |

77 | USD Deliverable Swap 5 year | 0.2849 | −6.0717 | 0.2412 | 0.007 |

78 | USD Govt 10yr | 0.2797 | −6.2127 | 0.1532 | 0.0173 |

79 | USD Govt 15–30yr | 0.2606 | −6.1806 | 0.1457 | 0.0141 |

80 | USD Govt 2yr | 0.2951 | −6.2539 | 0.1923 | 0.0209 |

81 | USD Govt 30yr | 0.3198 | −6.1727 | 0.1455 | 0.0156 |

82 | USD Govt 5yr | 0.2751 | −6.2401 | 0.1649 | 0.0176 |

83 | AUD Govt 10yr | 0.1748 | −6.3502 | 0.3921 | 0.0076 |

84 | AUD Govt 3yr | 0.1689 | −6.3336 | 0.4123 | 0.0084 |

85 | CAD Govt 10yr | 0.2198 | −6.1027 | 0.2192 | 0.0182 |

86 | CHF Govt 10yr | 0.2652 | −6.3925 | 0.3511 | 0.02 |

87 | DEM Govt 10yr | 0.2364 | −6.4194 | 0.2785 | 0.0172 |

88 | DEM Govt 2yr | 0.2896 | −6.3881 | 0.3155 | 0.0179 |

89 | DEM Govt 5yr | 0.2472 | −6.4036 | 0.2799 | 0.0186 |

90 | FRF Govt 10yr | 0.3146 | −6.098 | 0.1477 | 0.0307 |

91 | GBP Govt 10yr | 0.2338 | −6.0316 | 0.1308 | 0.0132 |

92 | ITL Govt 10yr | 0.4065 | −5.9488 | 0.1303 | 0.0201 |

93 | ITL Govt 2yr | 0.4559 | −6.019 | 0.2678 | 0.0294 |

94 | JPY Govt 10yr (mini) | 0.3342 | −6.3402 | 0.2904 | 0.0182 |

95 | KRW Govt 10yr | 0.295 | −6.3422 | 0.2746 | 0.0275 |

96 | Butter | 0.2603 | −6.0707 | 0.8538 | 0.0164 |

97 | Cocoa | 0.2437 | −6.11 | 0.1751 | 0.0011 |

98 | Coffee | 0.2895 | −6.0532 | 0.1854 | −0.0044 |

99 | Cotton 2 | 0.3031 | −6.1075 | 0.181 | −0.0023 |

100 | Lumber | 0.2773 | −5.9161 | 0.1449 | −0.0138 |

101 | Milk-Class III Fluid | 0.3337 | −6.1052 | 0.3525 | 0.0072 |

102 | Orange Juice | 0.2462 | −6.1736 | 0.2549 | 0.0069 |

103 | Robusta Coffee | 0.3565 | −6.0909 | 0.2177 | 0.0004 |

104 | Sugar 11 | 0.3112 | −6.2256 | 0.1673 | 0.0063 |

105 | Sugar 5 | 0.2975 | −6.2787 | 0.2071 | 0.0126 |

106 | TSR20 Rubber | 0.3321 | −6.1874 | 0.3907 | 0.0106 |

107 | Cocoa | 0.3086 | −5.9108 | 0.1417 | 0.0039 |

108 | USD STIR | 0.4512 | −6.1231 | 0.2431 | 0.0303 |

109 | AUD STIR | 0.2282 | −6.0767 | 0.3378 | 0.0081 |

110 | CAD STIR | 0.3843 | −5.9982 | 0.2806 | 0.0265 |

111 | CHF STIR | 0.4136 | −6.0595 | 0.2454 | 0.0332 |

112 | EUR STIR | 0.4227 | −5.7489 | 0.2022 | 0.0162 |

113 | GBP STIR | 0.33 | −5.9621 | 0.2374 | 0.0244 |

114 | VIX | 0.4345 | −5.9535 | 0.237 | −0.0443 |

115 | VSTOXX (mini) | 0.4162 | −5.9015 | 0.1454 | −0.0248 |

## Appendix B

#### Evaluation Measures

_{t}is our return over period t, HPR

_{t}is our holding period return or one plus our return over the tth period, and TWR

_{T}is our terminal wealth relative or one plus our total return over T periods. We can approximate the return with Equation (A2).

_{T}is the approximate terminal wealth relative (i.e., one plus the approximate total return over the T periods), and HPRt is the holding period return (i.e., the return over the tth period).

_{T}is arithmetic average of the holding period returns over the T periods (Equation (A3)).

- If AHPR
_{T}is less than or equal to 1, then regardless of the other two variables, SDHPR_{T}and T, our result can be no greater than 1 (i.e., our total return will be less than or equal to zero). - If AHPR
_{T}is less than 1, then as T approaches infinity, TWR_{T}approaches zero. This means that if AHPR_{T}is less than 1, we will eventually go broke. - If AHPR
_{T}is greater than 1, increasing T increases our TWR_{T}. - If we reduce our SDHPR
_{T}more than we reduce our AHPR_{T}our TWR_{T}will rise.

^{2}, and SDHPR

^{2}.

_{P}

_{,t}is a function of the weights and the returns of portfolio investment components. We define the portfolio return for I component investments for the period t given the period returns r

_{i,t}and portfolio weights w

_{i,t}for each component investment i in Equation (A6).

_{t}be a vector of portfolio component weights for period t, ’ denote the transpose operator, and R

_{t}be a vector of the period t component returns, we can use matrix notation to define the portfolio return as shown in Equation (A7).

_{t}is a vector of portfolio component weights for period t, ’ denotes the transpose operator, R

_{t}is avector of the period t component returns, and Σ is the return covariance matrix. In the portfolio context, EGM

_{T}is also a function of the return covariance. Reducing the return covariance-keeping all other return properties the same—thus reduces portfolio variation, increasing EGM

_{T}.

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Parameter | Value |
---|---|

atrLookback | 20 |

atrMultiplier | 4 |

fastLookback | 120 |

slowLookback | 180 |

longOnly | 0 |

commissionPerShare | 0 |

accountSize | 100,000 |

fPercent | 0.01 |

minRisk | 0.001 |

stopTWR | 0.7 |

nRowsScenario | 1250 |

nPathsScenario | 1000 |

S0 | 1 |

T | 5 |

xMean | −6.2146 |

xSigma2 | 0.15 |

randomSeed | 1,234,567 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nokes, D.; Fulton, L.
Analysis of a Global Futures Trend-Following Strategy. *J. Risk Financial Manag.* **2019**, *12*, 111.
https://doi.org/10.3390/jrfm12030111

**AMA Style**

Nokes D, Fulton L.
Analysis of a Global Futures Trend-Following Strategy. *Journal of Risk and Financial Management*. 2019; 12(3):111.
https://doi.org/10.3390/jrfm12030111

**Chicago/Turabian Style**

Nokes, Derek, and Lawrence Fulton.
2019. "Analysis of a Global Futures Trend-Following Strategy" *Journal of Risk and Financial Management* 12, no. 3: 111.
https://doi.org/10.3390/jrfm12030111