I've been trying to understand day count conventions, but the Wikipedia article is confusing. It was my understanding that interest is usually compounded daily - I don't quite understand how it fits in with the formulas.
Interest is, as you say, often compounded daily, but the rate per day is not stated directly. The quoted rate is usually per annum, e.g. 5% per annum. So, how much interest do you pay per day? You would think the daily rate is 5%/365, but the legalese says that they are going to use a 360 day year, and so you pay 5%/360 per day for 365 days. In other words, assuming no payments to clutter up the math, the annual rate of interest (APR as the legalese calls it) is
((1 + 0.05/360)^365 - 1)x100%
I don't know if the answer is different for leap years such as 2012 and whether multi-year contracts such as mortgages take the extra day into account when they inform you of the APR which applies for the length of the contract.
A 30/360 convention in interest calculation means that there are exactly 30 days in a month and there are 12 months [or 360 days in a year].
This convention was used in the early days when computers were not used and most of the calculation were done by hand [remember banking was there before computers].
Now obviously there would be issues as to how to calculate stuff if the investment is on say 31st of the month, the convention says count it as made on 30th. If it was made of 28th or 29th feb, count it as 30th ... etc.
This simplifies the calculation to a large extent and one can do the math mentally without much effort.
This type of calculation was applicable in almost every calculation from Interest on savings, to certifiate [time] deposits to mortgage loans etc.
The example given in wikipedia is on yeilds of bonds
In todays computerized world, this calculation is no longer used and actual/actual calculation is used as all the number crunching is done using computers.
In days leading to computerization, and lesser regulations, there were certain individuals / organization that used the actual/360 calculation; this means that if you take a loan at 5% you are effectively paying 5 days more. This practise is now illegal in most countries and not used anymore except maybe in few stray cases