Correlations do exist, but whether they are between two currencies, or a currency and a commodity(or even interest rates) they are dynamic relationships and change constantly. Sometimes a currency pair is the flavour of the month, whilst at other times, for no apparent reason, the pair loses its lustre. Hence you need to be closely monitoring your trading strategy to reflect this change.
Arguably the most discernible correlation over the past 12 months has been the flow into high yielding currencies, for example the AUD/JPY and NZD/JPY. This correlation between interest rates and currencies has been the stand out trade as the market chases returns while currency volatility remains low. However, this is a very mature trading strategy and numerous central bankers and traders are now questioning how much longer the trend will last.
The relationship between currencies is interwoven to a large extent with the relationship to commodities. As a simplistic example, a strong gold price usually (not always) has a positive correlation to the AUD and the ZAR (Sth African Rand) as these countries have extensive gold resources. A stronger oil price usually helps the CAD (Canadian dollar) and the NOK (Norwegian Krona) and hurts the Yen. The reasoning is that Canada and Norway have vast oil reserves so a price rise is good for their companies while Japan imports virtually all of their oil so a rise in price will impose a higher cost on Japan.
Currencies are not only quoted against the USD, for example AUD/USD, USD/JPY and EUR/USD but also against each other as a cross rate, for example AUD/NZD, AUD/JPY and EUR/JPY. This adds another layer of complexity to correlations.
Obviously if the USD is weak then you would expect the other currencies to maintain or gain value against the USD. However there may be specific events in a particular country that changes this view. For instance over the past year the Asian currencies have generally been rising in value against the USD as the market has bought the view that these Asian countries will continue to grow more strongly than the more established western economies. This trend has not been welcomed by certain Asian central banks and they have tried to limit their currencies strength by firstly “jawboning” and at times intervening in the market. These actions have in turn influenced the relative values of Asian currencies against one another where one central bank is more laissez-faire than another.
In summary, correlations do exist but the relationships can vary from day to day, week to week and month to month. How you trade these constantly moving relationships will depend in part on your trading time frame, your level of risk aversion, the capital available, time available to monitor the markets and a numerous other considerations.