Krugman argues the euro will stay strong, no matter what happens to Greece. He suggests thinking of "the euro-dollar exchange rate as if it were an exchange rate between the solid euro core — Germany, France, and a few others — and the United States." NYT (6-1-2011) The Strength of a Failing Euro (Wonkish) by Paul Krugman
According to Krugman, investors expect the European Central Bank (ECB) to keep inflation in Europe lower than in the U.S. This means that even if nominal interest rates are the same in the EU as in the U.S., investors expect higher real interest rates in the EU.
Example:
real interest rate = nominal interest rate - inflation rate (approximately)
Suppose nominal interest rate = 3.5% in both places, but inflation = 2% in EU and 3% in U.S., then the real interest rate in the EU = 1.5% and in the U.S. = .5%.
European investments look better.
As long as the real rate of interest is higher in the EU, investors will want to park their money in Europe instead of the U.S.. This will require buying euros (and possibly selling dollars) which drives up the price of the euro (in terms of dollars).
Other experts caution that the euro could be set for a fall. Once the full extent of European banks' exposure to Greek debt is revealed, the markets may conclude that Europe's core economies are not as solid as previously thought. If so investors would pull their money out of Europe and into the U.S., especially into U.S. Treasuries. This would weaken the euro and strengthen the dollar.
Let's look at the strength of the euro in terms of lunch room trades. In this parable let Cheetos = euros and peanuts = dollars. The European Cheeto Bank (ECB) sets the interest rates for Cheetos - if you save one Cheeto today you get two Cheetos next week. Initially a child can trade 5 peanuts for 1 Cheeto.
Suppose on Tuesday, a teen rock idol announces a love for Cheetos (or the ECB raises the Cheeto interest rate), then Cheetos become even more popular. Those unlucky kids whose moms' packed them peanuts will have to give up even more peanuts to get a Cheeto. They might have to give up 10 peanuts for 1 Cheeto. The cost of Cheetos in terms of peanuts has increased from 5 to 10. The Cheeto (euro) is strong against the peanut (dollar).
However, suppose on Thursday, a popular starlet announces that she thinks Cheetos are gross. She further exclaims that anyone who wants to be thin and cool should only eat natural foods, like peanuts (e.g. European banks' balance sheets are revealed to be gross). Peanut packers are suddenly looking good. The cost of Cheetos in terms of peanuts might fall to 4 as students flock to the safety of food without artificial colors. The Cheeto (euro) is weak against the peanut (dollar).
And yes, I did trade peanuts for Cheetos at school. Fortunately, I had a nice friend who gave me Cheetos even though there was no market for peanuts. While this does suggest that economists need to model altruism more carefully, we can probably safely ignore it in the foreign currency markets.