Percent Change

A rise of 10 means something different on a series sitting at 20 than on one sitting at 20,000. Percent change removes the scale, reporting each step as a fraction of where it started:

Task: write percent_change(series) returning one value per step, each rounded to 4 decimal places.

• The output is one shorter than the input — the first value has nothing before it to compare against.
• Return the result as a fraction, not a percentage: a rise from 100 to 110 is 0.1, not 10.
• Divide by the earlier value, never the later one. Dividing by the wrong end gives a number that looks nearly right for small moves and is badly wrong for large ones.
• No value in the series is zero, so the division is always safe.
• A flat step gives exactly 0.0; a fall gives a negative number.

The thing to notice is that these don't add up. Go up 50% and then down 50% and you are not back where you started — 1.5 × 0.5 = 0.75, a 25% loss. Percent changes multiply, which is exactly why cumulative returns are computed as a product of (1 + r) rather than a sum, and why a stock that halves needs to double to recover.