Active duty members who are stationed overseas (except for Alaska and Hawaii), and are authorized to live off base at government expense, do no receive a Basic Allowance for Housing (BAH). Instead, they receive a different allowance, called Overseas Housing Allowance, or OHA.
What's the difference? Well, BAH (which is paid to military members who live off base in the states), pays a set monthly rate, based on the member's paygrade, location, and whether or not the member has any dependents. In other words, if the set rate for a member is $750 per month, that's what he/she receives, no matter how much the member actually pays for rent and utility costs.
OHA, on the other hand, is based (in part) on the actual amount of rent. For each location, members are assigned a maximum rental cap, which is based on average rental costs for the area, depending on the member's paygrade (the higher the rank, the more expensive housing one is authorized to live in), and whether or not the member is residing with dependents (a member who is living with dependents generally requires larger living quarters than a member who is living alone). In addition to the monthly rental reimbursement (up to the amount of the cap), one's OHA payment also includes an allowance for utilities. This amount is based on random surveys of military members in the area, and is the same for everyone in the area, regardless of paygrade.
Let's look at an example:
Using the rates which were in place on December 1, 2007, an enlisted member in the paygrade of E-6, with dependents, living off base in Ansbach, Germany, would have a maximum rental cap of 830 Euros ($1,245) per month. The assigned utilities rate for Germany is 543 Euros ($814.50) per month. If the member's rent is 830 Euros or greater per month, the member would receive the maximum OHA (1373 Euros, or $2,059.50) per month for rental expenses.
However, if this member lives in a residence where the rent is 730 Euros per month, the member would only receive 1273 Euros ($1,909.50) per month in OHA.
OHA also includes a one-time lump sum allowance, called "MIHA" for moving-in expenses. For Germany (on December 1, 2007), the rate is 550 Euros ($825). So, in the above example, the member would receive an additional $825 in his/her first month's OHA payment.
OHA payments can change, based on the current rate of exchange. Rates are also reviewed at least once every six months.
For current OHA rates, see the Department of Defense's Overseas Housing Allowance Calculator.