It turns out I miscalculated anyway and the return needed to turn a decent differential is higher (4% ish), so you're probably right in that there's not a huge difference between them. The flexibility that £1m gives you would still sway me though, especially because predicting your life circumstances 10 years into the future is hard.
You guys seem to just be figuring out what NPV is... Obvs the value of 10k this month is different to the value now of the promise of 10k sometime next year or 5 years from now and so on etc..
You can just use the NPV formula in excel, open office etc.. and get the present value of a bunch of future monthly or yearly cashflows given some discount rate (usually a near risk-free rate).
Since you're talking about annual returns if you want to simplify things, just to take a look at the principle of this, then look at 120k per year instead of 10k per month. Keep in mind that interest rates are v.low at the moment so cashflows won't be discounted too much.
Essentially you've got 1.2 million of future cashflows vs a 1 million lump sum, if you want to present value of those 1.2 million in cashflows and interest rates are low, we're looking at less than 1% for bond yields etc.. then the NPV of the 1.2 million in future cashflows ends up being worth
more than the 1 million lump sum. So with no other considerations, the monthly payments are "better".
This is a slightly silly situation as generally, they should be about equal, in fact if you win a US lottery and turn down the lump sum in favour of future payments then the future payments option essentially comes from the same lump sum being used to purchase an annuity (which is in turn invested in safe govt bonds), the NPV of those future payments should therefore be the same as the lump sum you could choose to claim. There shouldn't be a "free lunch" here.
That begs the question of where are these cashflows coming from? If an annuity then can you sell it - you've got something (an annuity) that is maybe currently worth say 1.15 million now can you sell it now for say 1.1 million? Ergo taking the lump sum is leaving 100k on the table, taking the annuity and then immediately selling it gets you instantly more than the lump sum.
If instead, the future cash flows come from the company offering the prize then they're not risk-free and the discount rate being assumed for them to be equal to 1 million now (around 3.75% if we assume annual payments to keep it simple) might be a bad offer in terms of the risk re those cashflows being maintained - will the company still be around etc.. What returns could you get elsewhere for taking similar risk?
If you were going to invest either in say the stock market, property etc.. then (assuming annual cashflows for simplicity) anything over 3.75% makes taking the lump sum better.
this is all assuming that we're in the UK and there is no tax to pay - with US lotteries you have extra potential headaches with taking the cashflows, often they're across 30 years not just 10 - what if federal or state taxes increase in future, you have some minor benefit from your personal allowance or offsets, basic rate etc.. but when we're talking about a high 8 or 9 figure lump sum vs annual 7 figure payments that becomes moot, vast majority of it is getting taxed at the highest rate either way and if that increases in future then... Of course, if you win at a time of very high taxes and you believe some future administration will significantly lower them then... Also, 30 years is a long time, whether the annuity can be sold in general (or upon the death of the winner) is a key question - the beneficiaries of your estate could be left with an awkward inheritance tax situation otherwise if they inherit an annuity (or share of) with years of large payments left they'd be facing a big tax bill they can't necessarily afford to pay, would require some rather less than efficient borrowing against those future payments.