in the risky asset is held, while on a sell signal the position in the risky asset is sold and the proceeds are invested against the risk-free interest rate. If a technical trading rule has forecasting power, then it should beat the buy-and-hold strategy consistently and persistently. It should advise to buy when prices rise and it should advise to sell when prices fall. Therefore its performance, i.e. mean return or Sharpe ratio, will be compared to the buy-and-hold performance to examine whether the trading strategy generates valuable signals. The advantage of this procedure is that it circumvents the question whether it is possible to hold an actual short3 position in an asset. We define Pt as the price of the risky asset, It as the investment in the risky asset and St as the investment in the risk free asset at the end of period t. The percentage/100 costs of initializing or liquidating a trading position is denoted by c. The real profit during a certain trading position including the costs of initializing and liquidating the trading position is determined as follows:
|
|
| |
It-1 |
St-1 |
It |
St |
costs |
| Initiate a double position |
|
|
|
|
|
| Post-1 ≠ 1 ∧ Post=1 |
2 Pt-1 |
-Pt-1 |
It-1+2(Pt-Pt-1) |
(1+rf)St-1 |
c Pt-1 |
| Liquidate a double position |
|
|
|
|
|
| Post=1 ∧ Post+1 ≠ 1 |
|
|
Pt |
(1+rf)St-1+Pt |
c Pt |
| Initiate a risk free position |
|
|
|
|
|
| Post-1 ≠ -1 ∧ Post=-1 |
0 |
Pt-1 |
0 |
(1+rf)St-1 |
c Pt-1 |
| Liquidate a risk free position |
|
|
|
|
|
| Post=-1 ∧ Post+1 ≠ -1 |
|
|
Pt |
(1+rf)St-1-Pt |
c Pt |
| Initiate a long position |
|
|
|
|
|
| Post-1 ≠ 0 ∧ Post=0 |
Pt-1 |
0 |
It-1+(Pt-Pt-1) |
0 |
0 |
| Liquidate a long position |
|
|
|
|
|
| Post=0 ∧ Post+1 ≠ 0 |
|
|
Pt |
0 |
0 |
| Position not changed |
|
|
It-1+ |
|
|
| Post-1=Post |
|
|
(1+Post)(Pt-Pt-1) |
(1+rf)St-1 |
0 |
|
The profit at day
t is equal to (
It+
St)-(
It-1+
St-1)-
costs. The net return of a technical trading strategy during a trading position is then equal to
Note that because a continuous long position in the risky asset is the benchmark the trading signals are superimposed upon, liquidating the double or risk free position means a return back to the long position. Furthermore, costs are defined to be paid only when a double or risk free position is initialized or liquidated. For example, if a risk free position is held until the end of day
t is turned into a double position from the beginning of day
101