Premium Bonds

You're right in that the number of small prizes will be similar, but the value of the smaller prizes will be higher i.e. there will be more £50 and £100 prizes to replace the number of £25 prizes:

"There's also a considerable change to the distribution of smaller prizes – it's estimated that the number of £50 and £100 prizes will both rise by nearly 400,000 each, though this is mostly offset by the number of £25 prizes falling by around 900,000."

Therefore, most of us will be quids in.

I would have preferred more increase in £25 prizes rather than £100k and £50k ones ("it's estimated that the number of £100,000 prizes will triple from 18 to 56, the number of £50,000 prizes will rise from 36 to 112"), which hardly any of us are likely to win, but I'm aware that most punters/savers in these Premium Bonds don't think logically so the increase in higher value prizes will get them salivating. 

When these prizes are set there is a balance struck by having some very high prizes so this is covered in the media and encourage people to buy these in the hope of fulfilling a dream (gambling) and those who want a regular tax free income (investing). If too many high prizes a lot of people get nothing and walk away feeling unlucky and go elsewhere as can been seen on this board.

The problem is as you say for punters/gamblers they often lack logic, so many people playing the lottery for years with the same numbers and losing money say they can't give up now otherwise they have lost all of their past bets and if theses numbers have not come up in the past they believe it increase the chances of coming up in the future. I don't play the lottery but my only advice to those that do is chose the least popular numbers so less chance of sharing a big win i.e 13, numbers higher than 31 as people love to use birthdays.

I like everyone would like to dream of the big win but like you I am more realistic and prefer more money in smaller prizes so I get a better return with less chance of the big win, for me this is more about investing tax free than gambling.

The one thing that is never mentioned in the stats is the loss income before you go into the first draw so check out the time of month you invest so you have the least time the money is sitting around waiting for the first draw.

10% was a good rule-of-thumb prior to the latest increase, but it's more like 20% now. This is because the total of the middle and higher tier prizes used to be 10% of the prize fund (5% in each tier), but it's now 10% in each of those tiers, so 20% total.

EDIT: This 20% theory is discredited in the posts below. The reason is that with 20% of the prize fund in middle and higher tier prizes it's no longer valid to assume the the average investor won't win any of those prizes. 

Interesting, because this isn't how the guy who designed that calculator above sees it.  Taking a £30,000 holding, he discards (claims to, I haven't checked) the outlying top and bottom 1% of prizes, resulting in his 'trimmed mean' of 2.7% - exactly 10% off NSI's rate.  He regards that as the expected return on £30k (hover over the question mark for his explanation).

I think I prefer your analysis, at least for a discrete year or less.  Having said that, the number of mid-tier prizes has increased from 1/3rd of a percent to 1% of the total number of prizes.  So we should be seeing far more £500 and even £1000 wins creeping in now... Over the longer term, I think knocking off 20% will turn out to be too punitive.    

The true figure is likely to be somewhere between the two of course, and it will be interesting to check back after a few months.  

Doing these analyses isn't made any easier when you realise

• NS&I can't add up, and
• Martin Lewis is using completely different figures for the Dec '22 draw

NS&I's total number of prizes doesn't equal their column sum - their figure of 4,992k is out by a mere 2,000 prizes!  I suspect they've published the wrong figure for the number of £50 prizes paid out.  This isn't the first time they've screwed up and I'm convinced they have the same dimwit listing my prizes each month...  

Martin claims a total of 4,977k prizes - for the same month .  His column sum adds up correctly at least, but his individual figures per prize band don't match those published by NS&I and nor does his total.

Other than that, everything's fine and things are looking up. 

Your right, both methods of estimation are far from perfect. In reality the average return is likely to be a bit higher than simply assuming you never win any middle or higher tier prizes. I agree that knocking off 20% of the prize rate is likely to be an under estimate. Something just over 2.6% is probably about right. 

I can't see any justification for picking 1% as the number of outliers to ignore. However, it just so happens that using 1% means that all middle and higher tier prizes are rejected in both the pre and post January prize funds. 

Here's another (imperfect, but possibly slightly better) way of estimating the likely average return from the new prizefund: 

52.50% ( 2,621,112 /  4,992,880 * 100 ) of the prizes will be £25.
23.25% ( 1,160,883 / 4,992,880 * 100 ) of the prizes will be £50.
23.25% ( 1,160,883 / 4,992,880 * 100 ) of the prizes will be £100.
1% (100% - 52.50% - 23.25% - 23.25% ) of the prizes will be higher than £100.

Someone with the max invested (£50,000) with average luck would expect to win 25 prizes per year ( 50,000 / 24,000 * 12 ).
13.125 (25 * 52.50%) of those prizes will most likely be £25. I.e. £328.125 (13.125 * £25).
5.8125 (25 * 23.25%) of those prizes will most likely be £50. I.e. £290.625 (5.8125 * £50).
5.8125 (25 * 23.25%) of those prizes will most likely be £100. I.e. £581.25 (5.8125 * £100).
0.25 (25 * 1%) of those prizes will most likely be > £100.

So, each year, with average luck, someone with the max invested would expect to win £1,200 (£328.125 + £290.625 + £581.25) = 2.4% plus one higher value prize every 4 years. If we assume that the higher value prize is the most likely £500, then that adds an average £125 per year, which raises the total to £1,325 per year, or 2.65%.