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A bear market is a sustained period of declining prices, typically defined as a drop of 20% or more from recent highs. In forex, the term is applied more loosely to describe a currency or asset in a prolonged downtrend. Bear markets are driven by pessimistic sentiment, weakening economic data, or risk-off conditions. Traders can profit in bear markets by selling (going short).
If EUR/USD falls steadily from 1.1200 to 1.0600 over three months due to deteriorating eurozone economic data, that is a fall of about 5%. That is far short of the 20% equity-market definition, but forex traders often still call a sustained decline like this a bear market for the euro. Short sellers profit from this decline.