How to evaluate / calculate the Brand Return on investment (ROI) mathematically?

What is te best way to ensure Profitable Return-on-Investment (ROI) in Marketing?

Research and experimentation - whatever medium you are advertising in, you must have some way of determining whether or not the advertising campaign was successful.

For example, if you are placing an ad in the local newspaper, be sure to include a coupon or special offer for those who mention the ad - you can then create a record of how many people mentioned the ad or brought in the coupon and compare the value of the sales which your ad generated versus the cost of running the ad in the newspaper.

(If the ad did not include a coupon or special offer, there is a good chance that no one would mention seeing the ad and you would not be able to determine whether or not your ad campaign was worthwhile)

Edit: Sorry for missing the main part of the question on mathematic evaluation. The Return on Investment can be described as follows:

Revenue from advertising minus Cost of advertising equals Return on Investment for advertising

(You could also add the cost of developing a new product or service to the cost of advertising if you are adding the new product or service for the sake of getting new business)

You will need to research and experiment to find a profitable ROI. This may mean talking to other marketers, doing some research on the Google Adwords bid tool if you are running a pay-per-click campaign (find the keywords which are very relevant and *aren't* expensive!), or running that newspaper ad for a few weeks to see what happens.