This is a pretty complex one to work through, but you can do it with some data diving as well as well as the information you've given.
- Time and Labor in your group - assuming you have your automation suite in place, each time it runs, you are running (assuming that the suite runs in under a day) with 3 person-days for a total of 24 hours. The cost to run manual regression is, from your post, 100 person-days (10 people x 10 days) or 800 hours. So each run without modification saves 776 hours.
- Time and Labor external to your group - This will take some data diving. You will need to work out how many regression issues, on average, escape to production per release, and how much time is spent dealing with them post-release. Once you've got that number, you can compare against the number of regression issues, on average, that escape after your automation is implemented, and figure out a time saving. If you've targeted your regression automation properly, it could be a substantial amount.
- Regression bugs not released to production - you'll need to do some digging here, too - this time to locate all the regression issues that are caught by your automation and work out how both how likely it was that the issue would have been caught in manual regression, and how much time was saved. If you're running the automated regression on a daily or more frequent basis, you should see a lot of savings here, via an average time per issue.
To use some numbers pulled from air:
- Every run of your automation saves 776 hours.
- If it takes on average 500 hours to update the automation per deployment, then your time savings are (776 * number of runs per deployment) - 500. If you run the automation 10 times per deployment that's still over 7000 person-hours available for other tasks.
If you design your automation well enough, you should be able to reduce your refactoring time significantly, increasing your time savings.
And if your management requires a money amount on estimated savings, you'll need to include the development time to create the automation as a sunk cost (because it will happen no matter what kind of automation is used). Say, 5 times the time cost of a full manual regression run to do it properly. That means that by the sixth run of your automation, you'll be into positive ROI - but as you implement automated runs on a smaller scale, you will start seeing benefits).