I am thinking of self funding a PhD in Applied Mathematics at either the University of Oxford or Cambridge. Would employers, both for academic (postdoc etc) and industry (research) jobs think less of a PhD holder if he or she is self funded?
E.g. Would they feel that because I failed to secure funding for their PhD, I am inferior to students who did? Although I am planning to self fund to UK schools, it would be nice to hear about views from across the pond (USA) as well.
There's nothing on your CV that needs to indicate exactly how you were supported during your Ph.D. In fact, if you were self-funded, and managed to get an occasional fellowship or other form of support, that's a plus !
All a recruiting committee should (and does) care about post-PhD is the quality of your work (for faculty positions there are additional issues). No one cares about how you were paid to do that work.
Self-funding alone need not be visible in your applications and is unlikely to hurt in any case. That said, getting outside funding is likely to help and you should pursue it if you can.
First, it's not clear how any potential employer would know that you were self funded. Although it's normal to list fellowships or funding that you have received, it's just not normal (in my experience) to list the amount of money you received from a particular fellowship/grant unless it was a very public award.
That said, a list of grants, fellowships, and outside funding will help build a stronger CV. One thing many search committees are looking is a demonstrated ability to bring in grants and funding. Although that can be hard to discern in a graduate student/post-doc, the ability to apply for and get student fellowships can bring a little comfort. Being able to point out in your coverletter that you've thought about grants will inspire a little confidence in your application.
But the standard answer to questions of the form "does x matter for the job market" still applies. The quality of your work, your publications, your letters, and many other things will weigh much more heavily than the fact that you don't have fellowships that others do.
In theory I would say self-funding shouldn't hurt your employment chances, as there's not necessarily a reason to report how you were funded as a PhD student.
That being said, I have found that some people who have an external funding source not tied to a particular project/professor/etc. have two major problems:
Those aren't necessarily reasons not to self-fund, but it's something to consider. None of those will impact employment chances, but they can effect your experience in ways that have downstream effects.
First I have a couple of questions, and suggestions for ways to try to avoid self-funding your PhD:
Do you already have a Masters? If not, it could be a better idea to self-fund a one-year research masters and then try for funding for a PhD after that. That would not leave a 'gap' in your CV, except that probably it's actually too late already to apply for next year.
Have you actually applied anywhere else than Oxford and Cambridge for a PhD? If not, that may be the reason you were unable to obtain funding. There are a lot of other very good universities in the UK (which specific ones are the best will depend on your subject), and doing a PhD with funding at one of those would surely be better than doing one without funding at Oxford or Cambridge.
Or if you were not set on doing your PhD in the UK, you could also choose a country (for example the US or Canada) where PhD students are offered funding as a matter of course.
By the way, if you only applied to Oxford and Cambridge, then Noah Snyder's point, which I think is a good one, doesn't really apply, since it's not the case that you couldn't find anyone willing to fund you, it's just that you didn't try enough possibilities.
It may be possible to apply for funding again once you have already commenced your PhD. I know a few people who self-funded their first year and managed to obtain funding for the rest (these were all in the UK). At least one of them is now a lecturer.
In my opinion you should only try this if you can afford to fund the whole PhD (i.e. don't take out a loan for it!), and even then only if you are very passionate about your subject.
Now to my opinion on your actual question:
You needn't mention how your PhD was funded in job applications, but I suppose that in applications for academic jobs in the UK, employers might notice if no funding source is mentioned. I highly doubt this would count against you at all, as what is really important for your first job after your PhD is the work you have done in your PhD (although an ability to attract funding is highly valued in academia!).
