Asset Management Job Tasks

Not sure exactly what you are asking, but yes, asset management is often involved in underwriting of acquisitions (usually in due diligence period), this is not universal at all firms, but I'd say it is increasingly seen as a "best practice". And yes, AM has to do forecasts and budgets annually/quarterly, etc. A lot of that depends on what type of reporting is required depending on deal type/firm type/regulatory requirements, etc. 

Can't say for AMs in the world outside of the public sector but if you are asking do AMs do underwriting, I've yet to meet one who doesn't have input into the underwriting of projects. I think the questions a bit confusing, but I also don't know a single AM who isn't heavily involved in the budgeting process, It's a core responsibility in asset management as far as I believe regardless of structure

Asset management will mostly never do a first underwriting. Normally the transactions team will pull them in at a certain point in the underwriting process in order to get their input. Asset management is close to the properties in the markets they cover so they know best about rent growth, concessions, etc. They also (at my firm) oversee the third party property managers along with budgeting and forecasting.