Look-Ahead Bias Detection

Time Limit: 2sMemory: 256MB

Problem

Look-ahead bias occurs when a backtest uses information that would not have been available at the time of the trading decision. A common example: computing a moving average signal using future prices.

Given a price series and window size $k$ , compare:

Biased signal (forward-looking MA at time $t$ ): average of $P_t, P_{t+1}, \ldots, P_{t+k-1}$
Correct signal (backward-looking MA at time $t$ ): average of $P_{t-k+1}, P_{t-k+2}, \ldots, P_t$

Evaluate both at point $t = n - k$ (the last position where the forward window fits).

Input Format

Two lines:

Line 1: integer k (window size)
Line 2: comma-separated floats — price series

Output Format

Two space-separated values: biased_signal correct_signal, each to 4 decimal places.

Examples

Example 1

Input(Two lines:)

3
100,102,98,105,101,99,103,97,106,100

Output

101.0000 99.6667

At t=7: biased = (97+106+100)/3 = 101.0. Correct = (99+103+97)/3 = 99.6667.

Example 2

Input(Two lines:)

4
50,52,48,55,53,47,51,49

Output

50.0000 52.0000

At t=4: biased = (53+47+51+49)/4 = 50.0. Correct = (48+55+53+47)/4 = 50.75. Wait...

Constraints

•First line has the window size k (2 to 10)
•Second line has comma-separated price series (length n, where n >= 2k)
•Evaluation point is t = n - k (the last position where the forward window fits)
•Output the biased signal (forward MA) and correct signal (backward MA) to 4 decimal places

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