US will continue to raise interest rates at least until March 2023 until inflation would suddenly drop and pivot then. After that they expect to keep the rate flat and from third quarter of 2023 they will probably lower down the rate slowly and start again with QE (bullish equity, bearish USD)
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The EUR is in a much tougher situation, with inflation exceeding 11pct and the slow action of ECB to increase the rates will keep chasing inflation in the EU for at least 6-24 months longer compared with the US. The EU always is 6-24 months behind the US to take actions and improve conditions of the recession.
Definitely the USD index will drop again, it is at present the USD that dictates the EURUSD rate. How will Europe get their inflation under control with the energy crisis that will be felt next winter, is the big question. Meanwhile more events in Europe going from social unrest, rapid unemployment increase, companies reducing their presence in Western Europe and finally.. risk of war (Russia or internal conflict in the EU?)
So the eurusd will go several times up and and down in the coming period and is quite unpredictable with exception that the trend will be that usd will get weaker