How to convince Management to move from Manual to Automation testing?

I was asked this question. 

Here are my thoughts: 

All the automation testers earlier would have faced the issue of convincing the Management at some part of their career, mainly when automation was new to the market. 

The question itself, shows, Management is used to either legacy methods, or they don't want to invest for monetary reasons thereby may not be open to change. 

The question can be dealt at 2 levels: 

1) How to technically convince the management

2) How to mentally convince the management

For point 1: 

• Best approach is to create a Proof of Concept & highlight the benefits. 

Now one may or may not have "paid" time to create PoC. So, it’s your call, when and how would you create it, many times one has to spend time after work hours. 

• You can create a Return on Investment document - highlighting, how much execution time is saved - not always this statistic would look good, as the time spent on Automation script development can be reasonably high; but you can highlight other benefits of automation like ,

• unattended execution ,
• over-night execution ,
• parallel execution,
• multi-browser execution ,
• multi-device execution ,
• automation can takeover repetitive tasks, while manual testers can focus on other key tasks,  

 So the overall benefits of automation add up. Project these benefits into the future, and see how the statistics look like. 

Hope that, once you have your automation framework in place, you see fast execution; you show all the other benefits, stats & eventually give a nice presentation to Management.
But still, they don't buy-in then? : A very generic solution to point 2 is: be the change you wish to see in the World. In this case the change is in the mindset of the management to move from manual to automated testing, World would be your company. 
You continue & hope for someone eventually to see/acknowledge the benefits. 

Manual testing can't be made redundant , unless there is Artificial Intelligence :-). Automated testing is a supportive activity to make Testing cycle efficient & effective; it is not a sole alternative to manual testing. 

Lastly, even if one succeeds in convincing the management, it is really important to have a proper Test Automation Strategy in place - which at times can be very intricate, to come up with a good ROI.