Aim to get into a target uni: Oxbridge, LSE, Imperial, UCL, Warwick
If not try for a semi-target: Nottingham, Durham, Bristol, Bath, KCL, Edinburgh, St Andrews, Birmingham, Cass
The subject you study isn't really relevant, choose something you're good at and that you will enjoy studying for the 3+ years at university.
Internships don't just 'boost your chances' they are LITERALLY required to get a grad job. The funnel starts in first year (or second year of a four year course) whereby you apply for 'Spring Weeks' - these are a couple of days long placements where you gain a greater insight into the industry, with the potential of being offered a place on the summer internship.
Which brings me on to the next stage, the internship; this is crucial, you'll need to really impress as much as you can in order to land the coveted graduate job offer at the end. These take place in the summer of your penultimate year. Don't worry if you don't land any Spring Weeks, because there will be an even greater selection of banks to choose from for summer - I repeat, it is absolutely crucial you land a summer gig.
Right now, though, focus on your A-level subjects but also get involved around your school or community. You'll need to start building up a strong CV, with content that will impress (but that you also find interesting) if you want to stand a chance. In year 13, apply to some A-level insight programmes - some of which will include a potential fast track (i.e. consideration for a place) for that bank's respective spring week programme.
Some sites you should reference:
Mergers and Inquisitions, the
Unofficial Guide to Banking,
TargetCareers - Banking,
AllAboutFinanceCareers and this is particularly good for a list of A-level programmes:
The Student Ladder.
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