This was very interesting thanks! I was especially surprised that the effective interest rate of Premium Bonds is 1.4%, which is pretty good compared to today's savings rates in the UK. When I last checked (quite a few years ago) Premium Bonds were quite a poor deal.
It's a little unfair that you said "the Lottery prize fund is weighted very heavily towards the larger sums that you'll realistically never win" but don't qualify that the effective interest rate of Premium Bonds also includes some high value but unlikely prizes. I wonder what the interest rate is for 50th percentile prize winners.
Edit: This page[1] on MoneySavingExpert shows the median (50th percentile) Premium Bond winnings are 1.16% over a year if you have £15,000 invested, and are 0%(!) if you have £1,000 invested. They also have a winnings calculator[1], which shows that median probabilities don't compound over time in the way you'd expect: for £1,000 over 5 years the median return is £50, which is equivalent to 0.98% compound annual interest. (It may actually be fractionally higher because I don't think the calculator assumes you re-invest the winnings in Premium Bonds.)
That sounds right. Using the latest figures I can find online, the lowest Premium Bond prize (£25) was £75,665,750 of the total £88,099,800 prizes[0], i.e. over 85%, whereas lowest National Lottery prize (free ticket worth £2) was £2,306,736 of the total £19,891,729 prizes[1], i.e. under 12%. So if you calculate on only ever winning the lowest prizes, the Premium Bond's not too bad notional return is likely to be not that much worse in practice, but the unfavourable distribution towards the headline grabbing large jackpots means the National Lottery's very bad notional return is likely to be significantly worse in practice.
It's a little unfair that you said "the Lottery prize fund is weighted very heavily towards the larger sums that you'll realistically never win" but don't qualify that the effective interest rate of Premium Bonds also includes some high value but unlikely prizes. I wonder what the interest rate is for 50th percentile prize winners.
Edit: This page[1] on MoneySavingExpert shows the median (50th percentile) Premium Bond winnings are 1.16% over a year if you have £15,000 invested, and are 0%(!) if you have £1,000 invested. They also have a winnings calculator[1], which shows that median probabilities don't compound over time in the way you'd expect: for £1,000 over 5 years the median return is £50, which is equivalent to 0.98% compound annual interest. (It may actually be fractionally higher because I don't think the calculator assumes you re-invest the winnings in Premium Bonds.)
[1] https://www.moneysavingexpert.com/savings/premium-bonds/
[2] https://www.moneysavingexpert.com/savings/premium-bonds-calc...