AppliedMultiple choiceFree solution

A backtest computes pnl = signal * returns, where both are indexed by day. What is wrong?

  1. ANothing – that is the standard vectorised form
  2. BThe signal earns the same day’s return, which it could not have known
  3. CThe returns should be log returns rather than simple ones
  4. DTransaction costs are missing from the expression
Show the answer and worked solution

Answer: B – The signal earns the same day’s return, which it could not have known

The expression pays the signal at time tt the return realised at time tt, which is a return you could not have known when you formed the position. Every backtest written this way is trading on information from the future, and the effect is usually large enough to turn noise into a Sharpe ratio of several. The fix is signal.shift(1) * returns, and more than one period of lag if execution happens later than the close the signal was computed from. Missing costs and the choice between log and simple returns are real issues and both are worth far less than this one.

Worked solution

  1. Formula
    pnlt=signalt−1×returnst\text{pnl}_t = \text{signal}_{t-1} \times \text{returns}_t
  2. Substitute
    the code uses signalt\text{the code uses } \text{signal}_t
  3. Solve
    signalt is formed after rt is known\text{signal}_t \text{ is formed after } r_t \text{ is known}
  4. fix: signal.shift(1)\text{fix: signal.shift(1)}
  5. Answer
    One period of look-ahead\text{One period of look-ahead}

Sanity check. Lag further when execution happens later than the close the signal was computed from.

  • A. It is the standard bug. The signal must be lagged relative to the return it earns.
  • B. Correct. Use signal.shift(1) * returns, and lag further if execution happens later than the close.
  • C. A reasonable preference, but not the error here.
  • D. True but secondary; the alignment error is worth far more than the costs.

Takeaway: Shift the signal, not the return.

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A backtest computes pnl = signal * returns, where both are indexed by day. What is wrong? · QuantMax