where Pt1 and Pt are futures prices at time t 1 and t, respectively, and Sj,t is an indicator variable that takes one of three values: 1 for a long position, 0 for a neutral position (i.e., out of the market), and 1 for a short position.[3]Measuring trading returns on a daily basis is consistent with the process of the daily settlement (mark-to-market) in the futures market. The daily net trading return is then given by:
nj
rj,t1 rjg,t1 dt1 
Ninj b ln(1 c) (2)
where n is the number of round-turn trades for a contract, Nin is the number of days “in” the market (e.g., Nin N Nout where Nout is the number of days “out” of the market), dt1 is an indicator variable having a value of 1 for in-days and 0 for out-days, and c is the value for round-turn proportional transaction costs. This article considers three scenarios for transaction costs: (1) $100 transaction costs per contract per round-turn trade for each sample period (1985–1994, 1995–2004, and 1985–2004); (2) $100 for the in-sample period, 1985–1994, and $50 for the out-of-sample period, 1995–2004; and (3) $50 for each sample period. Transaction costs of $100 per round-turn trade are quite conservative when compared with the sum of the bid-ask spread and commissions estimated in other studies of technical trading rules (e.g., Levich & Thomas, 1993; Silber, 1994; Szakmary & Mathur, 1997).
The Sharpe ratio measures the excess return per unit of total risk. In futures markets, there is no need to sacrifice the risk-free return in order to participate in an alternative investment because traders can deposit treasurybills for margin requirement. The Sharpe ratio (SR) of a trading rule j can then be calculated by:
SRj rjsˆ j (3)
where rj denotes the annualized mean net
return during a given sample period and sˆ j denotes
the standard deviation. Technical trading rules are optimized based on each
performance criterion. That is, for a given sample period a trading rule
showing the highest mean net return or Sharpe ratio among the full set of
trading rules is chosen as the best rule.
Other assumptions included in the trading model are: (1) all trading is on a one contract basis, i.e., only one contract is used for each transaction; (2) no pyramiding of positions or reinvestments of profits are allowed; and (3) sufficient funds are assumed available to meet the margin requirement that may occur due to trading losses.
White’s (2000) and Hansen’s (2005) tests evaluate the performance of the best rule in the context of the full universe of technical trading rules. Thus, it is critical to the analysis to construct an appropriate universe of technical trading rules, because it directly influences test results. For example, considering only currently popular trading rules, which are likely to have subtle “survivorship bias” over a long time period, may bring about spurious results (Neely et al., 1997; Sullivan et al., 1999). This study approximates the full “universe” of technical trading rules based on Lukac et al.’s (1988) 12 trading systems and three additional trading systems.[4] These trading systems cover the major groups of technical trading systems such as moving averages, channels, momentum oscillators, filters, and combinations. According to Lukac, Brorsen, and Irwin, their trading systems were selected to be representative of the various types of systems that had been suggested by actual traders, previous studies and books.
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