First thing is that if you are talking about a PhD, we are looking at the world starting 7-8 years from now. It will be different, not only that its not realistic to predict in what ways it will be different. Algotrading may be saturated, or it may be the only game in town.
Remember that your pay is a function of a) the amount of money being made in a market and b) supply / demand for your skills. All are highly variable.
Also BarCap must really think you're something special, $220K is a lot to pay for an option to hire someone N years from now when they're only 18. Well done.
But as a headhunter I see this as an objective evaluation of your capabilities, and one has to assume that you stand out relative to other UCB undergrads. Also BarCap is not a charity, they expect this to pay back, implying that you are not only bright but useful.
That means you will probably earn enough to make good money in your career if optimising for money is your goal, so that leads me to counsel you to put a bit less weight on the 220K because as a % of your career earnings it is not that critical.
If you continue in full time education after your degree, do you have to pay back the bond ?
When you start working for them will you be paid just like any other person who wasn't sponsored ?
The six months attachments are valuable experience whatever you end up doing.
Today Singapore does not pay as well as the US or UK, but we're now talking about a range of time from 8 years to 15 years from now. It may be the case that Sing pays better, or it may be a smoking ruin after raids from Chinese bombers. Over those timescales there are lots of possible outcomes. Barcap may be the largest bank in the world by now, or as recent history shows it may be chopped up by regulators and/or accidents in the market.
Lets assume that algorithmic trading is your goal
You're smart and have the opportunity to talk to lots of people at a major bank about this line of work.
You may not like or have the particular talents for this, so you learn this without too much pain, and easily move to an area that suits you better.
Over the next 3-4 years you can find out about HFT at BarCap, and try to score an assignment there.
Given that they are investing so much time and effort in you, I believe they will try to put you in things you have a passion for.
You should be able to learn which skills are useful to the styles of HFT that BarCap are using, be aware there are several HFT groups there, not all have a name that would allow you to work that out.
That means you can pick course options that improve your chances.
However, they will have plans for you...
This is part of a structure where HR and IT map out what they want you to do, and since they are paying for the music, they get to call the tune. That means it is certain that at least one assignment will be in some shitty part of BarCap. Shitty is of course subjective, but since there are so many parts to a major bank I am certain there are several you will hate a lot.
Lastly, Sam Chan is absolutely right that a PhD is not a choice to be made entire on career enhancement. If you find a particular highly specialist topic interesting, then go for it, but the best way I can characterise a career PhD is like marrying a girl you don't like merely because she has rich parents, and in this case they aren't even all that rich.