Expanding on what @LWoodyiii had to say. When presenting a case for anything, whether that be hiring more people, or investing in a performance testing tool (or outsourcing your performance testing for that matter), it needs to be presented in terms of money saved. By doing a little leg work, you should be able to back into the $ saved amount.
If you had never had any performance problems, then it would be more difficult to quantify $ saved. But in your case, it should be a little easier to figure this out, as you have already had some significant performance issues. You should be able to put a $ amount to your existing performance problem. You should be able to quantify the revenue lost (lost transactions, lost customers, decreased transaction throughput, etc...) due to the degradation of service. You can also factor in costs associated with fixing and resolving the performance issue. Then it is a matter of comparing the costs of having performance problems vs implementing a performance testing program (tool, training and resource costs).
It also probably would not hurt to spice up the presentation with some anecdotal performance horror stories that were well publicized in the news and how much those outages cost those firms.