A stock falls 50% and then rises 50%. Where does it end up?

Say it ends down 25%: $100 falls to $50, and a 50% rise on $50 only gets back to $75. Explain the changing base, give the rule of one minus x squared, and note that a 50% loss needs a 100% gain.

The numbers, step by step

A stock that starts at $100, falls 50% and then rises 50%.

After the fall$100 × (1 − 50%)

$50

After the rise$50 × (1 + 50%)

$75

General rule1 − 50% squared

75% of the start

Gain needed to recover$100 ÷ $50 − 1

100%

Up 10% then down 10%$100 × 1.1 × 0.9

$99

What the interviewer listens for

• Do you land on $75, down 25%?
• Do you explain that the base changes?
• Do you know the order doesn't matter?
• Can you say what gain recovers a loss?

Common mistakes

• Saying it's back where it started because the two percentages match.
• Adding the percentages instead of multiplying the factors, 0.5 times 1.5.
• Thinking the order matters.Up then down and down then up both end at 75% of the start.

Likely follow-ups

• What about down 20% then up 20%?One minus 0.2 squared is 0.96, so it ends down 4%.
• What gain do you need to recover a 20% loss?25%, since $80 has to grow by $20, and $20 over $80 is 25%.

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