Are premium bonds a good investment?

[Mcr Warrior]

You can get bigger returns in the stock market but you can also get bigger losses. If a bond matures during a dip you’re screwed. It’s not a guaranteed return. I know someone who had fifty grand in a 5-year fixed bond and their profit at maturity was £12.70.

What kind of bond?Sorry, that makes no sense - a stock market dip will have no impact on the redemption value of a bond

Might just have been a structured Stock Exchange-linked bond. There was a spate of these being mis-sold to Joe Public a couple of decades or so ago. Not sure, however, if they have been offered anytime recently. @Tetchytyke. Who was the bond issued by?

P.S. Think we're possibly drifting off topic as regards a discussion on whether or not Premium Bonds are a good investment.

 

[Peter Mugridge]

You are right that the chance of any £1 bond winning is independent of purchase date. But the number of bonds that have been in issue for a long time are now a tiny proportion of the total. In particular the minimum £1 holdings from the early days of premium bonds have a much smaller chance of winning than a minimum holding of £25 invested now. Someone with a minimum holding of £25 invested now has 25 times more chances of winning than someone with a single £1 bond.

An interesting feature of the way the draw is done, with randomly generated numbers, explained to me by a senior professional statistician a few years ago....

Imagine you have a big chart on the wall with, say, a million sequential numbers on it ( bonds are issued in sequence, like with the serial numbers on banknotes ).

Suppose person 1 has the maximum £50k and it's in individual bonds scattered all across that chart.

Suppose person 2 also has the £50k maximum, but it's in a single block within that chart.

Now, act as the random number generator - throw a few darts at the chart. Although each individual bond has a statistically identical chance of winning ( i.e. being hit by one of the darts ), there is a greater chance that something within the big single block of 50k will be hit than one of the loosely scattered numbers will be...

It is better, therefore, to hold single large blocks of bonds rather than to buy them piecemeal on the minimum investment allowed of £25 a time.

Sure, someone with a single bond ( from the time when the minimum investment was £1 ) could win the big £1 million prize, but overall those who hold single large blocks of bonds will do slightly better than the "statistically average luck" that is quoted for bond holding.

 

[Mcr Warrior]

An interesting feature of the way the draw is done, with randomly generated numbers, explained to me by a senior professional statistician a few years ago....

Imagine you have a big chart on the wall with, say, a million sequential numbers on it ( bonds are issued in sequence, like with the serial numbers on banknotes ).

Suppose person 1 has the maximum £50k and it's in individual bonds scattered all across that chart.

Suppose person 2 also has the £50k maximum, but it's in a single block within that chart.

Now, act as the random number generator - throw a few darts at the chart. Although each individual bond has a statistically identical chance of winning ( i.e. being hit by one of the darts ), there is a greater chance that something within the big single block of 50k will be hit than one of the loosely scattered numbers will be...

It is better, therefore, to hold single large blocks of bonds rather than to buy them piecemeal on the minimum investment allowed of £25 a time.

Have to say that I don't quite agree with that "explanation".

 

[Peter Mugridge]

Have to say that I don't quite agree with that "explanation".

I can't reproduce it on here, but he did also demonstrate it on screen with a large selection ( not a million! ) of numbers an a random number generator; the big block in the demonstration did repeatedly get more hits than the scattered numbers.

 

[najaB]

Have to say that I don't quite agree with that "explanation".

I believe it is true. That said it's only a slight increase in probability. The easiest way to visualise it is that the single bond holder has to be an 'exact' match where the holder of the block just has to be 'close enough'.

 

[SuspectUsual]

I believe it is true. That said it's only a slight increase in probability. The easiest way to visualise it is that the single bond holder has to be an 'exact' match where the holder of the block just has to be 'close enough'.

I’m really struggling with this logic

If every number has an identical probability of being selected, it doesn’t matter if they’re sequential or not.

Do sequential numbers have a greater chance of winning the lottery? Of course not

 

[dcbwhaley]

I've done okay out of mine, probably about the same return as if the money was held in the best instant access savings accounts. I've been maxed out at 50k for 4 years, in that time the most I 'won' in a single month was £650, but there have been a few months where I've won nothing. I'm somewhat risk averse, and as with a few other responders, the small possibility I could get a big win is a definite draw.

Same here. For me to get 3.3% tax free return I would need a savings account paying 5.5%, better than any thing available.
And the change of winning a few grand is appealing. Better odds on that than in the lottery

 

[Djgr]

I’m really struggling with this logic

Yes it's pants.

== Doublepost prevention - post automatically merged: 7 Jul 2025 ==

Same here. For me to get 3.3% tax free return I would need a savings account paying 5.5%, better than any thing available.
And the change of winning a few grand is appealing. Better odds on that than in the lottery

And the larger your holding the more likely that your particular return will be very close to the overall mean return for premium bonds as a product. (Statistics 101).

This is one reason why those in the fortunate financial situation described above tend to go for the maximum £50,000 holding, so they can predict their return with a high degree of certainty

 

[najaB]

If every number has an identical probability of being selected, it doesn’t matter if they’re sequential or not.

I asked Copilot and it said initially that the probability was the same for sequential or non-sequential tickets:

A probability question for you.


Let's say that there is a raffle where the tickets are issued with sequential ticket numbers. The winning ticket number is chosen lottery style (which is to say by balls pulled randomly). Unlike a lottery, if the chosen number hasn't yet been sold then they will re-draw until they get a number that has been sold.


Is there any difference at all in the probability of winning if I buy individual tickets with non-sequential numbers versus a block of tickets with sequential numbers?

Response:

Great question!
In a raffle where the winning ticket is drawn randomly and unsold tickets are discarded (i.e. redrawn until a sold ticket is selected), every sold ticket has an equal chance of winning, regardless of its number or position in the sequence.
So, whether you buy:

• a block of sequential tickets (e.g. 101, 102, 103), or
• the same number of non-sequential tickets (e.g. 101, 117, 143),

your probability of winning is exactly the same, assuming the total number of tickets you hold is the same.

Why?​

Because the draw is effectively from the set of sold tickets only. The numbering of the tickets is irrelevant to the draw process — it’s just a label. What matters is how many tickets you own, not which ones.
Let me know if you’d like a visual explanation or a simulation to illustrate this!

I then asked for it to run a simulation and this was the response:

Here’s the result of the simulation comparing the probability of winning a raffle when buying a sequential block of tickets versus randomly scattered tickets:

As you can see, the probabilities are virtually identical. This confirms that there is no advantage to choosing sequential or non-sequential ticket numbers — what matters is simply how many tickets you hold.
Let me know if you'd like to explore variations, like unsold tickets or multiple winners!

Note the key phrase there: "the probabilities are virtually identical".

Make of that what you will.

 

[Djgr]

I asked Copilot and it said initially that the probability was the same for sequential or non-sequential tickets:

Response:

I then asked for it to run a simulation and this was the response:

Note the key phrase there: "the probabilities are virtually identical".

Make of that what you will.

Perhaps you can ask it what it means by virtually identical? I think I will rely on a bit of statistical training.

I think simulations based on probability will never give you exactly identical answers. The solution to this problem is based on statistical logic (laws if you prefer)

 

[greatkingrat]

NS&I mention this myth in their FAQs

Does buying Premium Bonds in a sequence increase the chances of winning?
It’s a popular myth that holding Premium Bonds in a sequence can improve your chances of winning, but this simply isn’t true. The machine we use to generate the numbers for each prize draw—ERNIE—generates numbers completely at random, which are then matched to our database of eligible Bonds to determine the winners. Since no Bonds are actually entered into ERNIE, it doesn’t ‘know’ anything about the Bonds themselves, such as whether the Bond number is part of a sequence of numbers or not. The only thing that can increase your chances of winning is holding more Premium Bonds – but holding them in sequence won’t make any difference.

FAQs | NS&I Adviser

Home page

www.nsandi-adviser.com

 

[SuspectUsual]

it said initially that the probability was the same for sequential or non-sequential tickets

Note the key phrase there: "the probabilities are virtually identical".

Make of that what you wil

Is it that the first answer is a technical answer based on the laws of probability and the second is the outcome of a simulation using randomised data?

(Simple comparison - toss a fair coin and the odds of head and tail are both [almost - exactly if we exclude the coin landing on its edge] 50%, but do a simulation with an odd number of attempts and you'll get an outcome that at the very least makes one outcome a little more likely than the other)

 

[Magdalia]

An interesting feature of the way the draw is done, with randomly generated numbers, explained to me by a senior professional statistician a few years ago....

Imagine you have a big chart on the wall with, say, a million sequential numbers on it ( bonds are issued in sequence, like with the serial numbers on banknotes ).

Suppose person 1 has the maximum £50k and it's in individual bonds scattered all across that chart.

Suppose person 2 also has the £50k maximum, but it's in a single block within that chart.

Now, act as the random number generator - throw a few darts at the chart. Although each individual bond has a statistically identical chance of winning ( i.e. being hit by one of the darts ), there is a greater chance that something within the big single block of 50k will be hit than one of the loosely scattered numbers will be...

It is better, therefore, to hold single large blocks of bonds rather than to buy them piecemeal on the minimum investment allowed of £25 a time.

NS&I mention this myth in their FAQs

As it says on the NS&I website, it is a myth.

If every number has an identical probability of being selected, it doesn’t matter if they’re sequential or not.

This is correct. The winning numbers are generated by ERNIE, which is an acronym for Electronic Random Number Indicator Equipment. The winning numbers are random and it makes no difference whether the numbers held are scattered or in a block.

I asked Copilot and it said initially that the probability was the same for sequential or non-sequential tickets:

Response:

I then asked for it to run a simulation and this was the response:

Note the key phrase there: "the probabilities are virtually identical".

Make of that what you will.

What I make of it is that Copilot's simulation of randomness won't be as random as ERNIE.

And the larger your holding the more likely that your particular return will be very close to the overall mean return for premium bonds as a product.

This only applies if all prizes are the same value. In practice, a big holding will get closer to the median return, which isn't quite the same thing, because of the distorting effect of the large prizes.

 

[styles]

An interesting feature of the way the draw is done, with randomly generated numbers, explained to me by a senior professional statistician a few years ago....

Imagine you have a big chart on the wall with, say, a million sequential numbers on it ( bonds are issued in sequence, like with the serial numbers on banknotes ).

Suppose person 1 has the maximum £50k and it's in individual bonds scattered all across that chart.

Suppose person 2 also has the £50k maximum, but it's in a single block within that chart.

Now, act as the random number generator - throw a few darts at the chart. Although each individual bond has a statistically identical chance of winning ( i.e. being hit by one of the darts ), there is a greater chance that something within the big single block of 50k will be hit than one of the loosely scattered numbers will be...

It is better, therefore, to hold single large blocks of bonds rather than to buy them piecemeal on the minimum investment allowed of £25 a time.

Sure, someone with a single bond ( from the time when the minimum investment was £1 ) could win the big £1 million prize, but overall those who hold single large blocks of bonds will do slightly better than the "statistically average luck" that is quoted for bond holding.

That's absolute garbage.

 

[Djgr]

As it says on the NS&I website, it is a myth.



This is correct. The winning numbers are generated by ERNIE, which is an acronym for Electronic Random Number Indicator Equipment. The winning numbers are random and it makes no difference whether the numbers held are scattered or in a block.


What I make of it is that Copilot's simulation of randomness won't be as random as ERNIE.


This only applies if all prizes are the same value. In practice, a big holding will get closer to the median return, which isn't quite the same thing, because of the distorting effect of the large prizes.

Whilst true, only 10% of the prize winnings is allocated to the "large prizes", meaning that the median and mean are probably not that far apart.

 

[AM9]

That's absolute garbage.

Yes, the fallacy is that 50k out of a total 1m (i.e. 5%) is not representative. There are claimed to be over 128bn individual bond number currently held, so even a 50k block will be less than 0.000039%. Add to that the randomness in the draw is not necessarily anything to do with spatial distribution meaning that a number form a geometrical block is no more likely to be chosen than one from 50k numbers across the total field.

 

[najaB]

Perhaps you can ask it what it means by virtually identical? I think I will rely on a bit of statistical training.

I think simulations based on probability will never give you exactly identical answers. The solution to this problem is based on statistical logic (laws if you prefer)

For what it's worth, I have run the simulation myself in Python several times. Each time the result has slightly favoured sequentially numbered tickets. But only very slightly:

python3 simulation.py
Simulation Results over 100000 draws:
Number of your tickets: 100
Total number of tickets sold: 10000
------------------------------
Scenario: Sequential Tickets (e.g., 1 to 100)
Wins: 1018
Estimated Probability of Winning: 0.0102

Scenario: Randomly Selected Tickets (e.g., 543, 9876, ...)
Wins: 965
Estimated Probability of Winning: 0.0097

--- Theoretical Expectation ---
The theoretical probability of winning is 100 / 10000 = 0.0100.
As you can see, both scenarios yield very similar estimated probabilities,
confirming that the type of tickets (sequential vs. random) does not matter.

Very similar, but so far in favour of sequentially numbered tickets in each run.

 

[jfollows]

My guess would be that this is showing up whatever random number generator you used, which is unlikely to be totally random. Unless you used tealeaves or something ……

 

[AM9]

For what it's worth, I have run the simulation myself in Python several times. Each time the result has slightly favoured sequentially numbered tickets. But only very slightly:

Very similar, but so far in favour of sequentially numbered tickets in each run.

At the moment, the ratio of a maximum holding to the total number of bonds is 1:2,560.000, so as the total number of bonds continues to increase, any difference between the two cases tends further towards zero.

 

[Djgr]

For what it's worth, I have run the simulation myself in Python several times. Each time the result has slightly favoured sequentially numbered tickets. But only very slightly:

Very similar, but so far in favour of sequentially numbered tickets in each run.

It (Python?) does clearly say the following:

Theoretical Expectation ---
As you can see, both scenarios yield very similar estimated probabilities,
confirming that the type of tickets (sequential vs. random) does not matter.

I am sure that a simulation of tossing a coin will not give an exact 50% heads and 50% tails. Simulation doesn't really override logic.

 

[najaB]

It (Python?) does clearly say the following:

Theoretical Expectation ---
As you can see, both scenarios yield very similar estimated probabilities,
confirming that the type of tickets (sequential vs. random) does not matter.

I am sure that a simulation of tossing a coin will not give an exact 50% heads and 50% tails. Simulation doesn't really override logic.

It's only a slight effect, but repeated runs have produced the same result - it's always a very slight bias in favour of sequentially numbered tickets.

Funny that you mention a coin toss isn't 50/50 because a fair coin toss *isn't* 50/50. There is a slight bias towards it landing on the side that was facing up when it was tossed.

 

[Magdalia]

Whilst true, only 10% of the prize winnings is allocated to the "large prizes", meaning that the median and mean are probably not that far apart.

10% is big enough to make a difference.

For example, if the payout rate is 4%, that means that the small prizes are paying out 3.6%, and the median winnings will be much nearer to 3.6% than 4%.

For what it's worth, I have run the simulation myself in Python several times. Each time the result has slightly favoured sequentially numbered tickets.

That suggests to me that the simulation is not random.

 

[Tetchytyke]

Might just have been a structured Stock Exchange-linked bond. There was a spate of these being mis-sold to Joe Public a couple of decades or so ago. Not sure, however, if they have been offered anytime recently. @Tetchytyke. Who was the bond issued by?

That sounds like a familiar product, it was sold about 15 years ago. No idea beyond that.

 

[sprunt]

My guess would be that this is showing up whatever random number generator you used, which is unlikely to be totally random. Unless you used tealeaves or something ……

Probably, but ERNIE won't be totally random either - there's not really any such thing as a perfect random number generator.

 

[Tetchytyke]

For what it's worth, I have run the simulation myself in Python several times. Each time the result has slightly favoured sequentially numbered tickets. But only very slightly

100,000 draws isn’t enough to demonstrate a distortion. For comparison, slots games are tested to about 10,000,000 spins to demonstrate that the RTP (return to player, I.e. the chance of winning) is statistically accurate.

 

[Killingworth]

I've had a £5 block since they were first launched in 1956, thanks Dad. That block has won twice, a £25 and a £1,000.

I've added more since and netted another £1,000 recently. I used to take note of where the wins were coming from but now have better things to do. There was one month I got about 5 or 6 £25 wins and noted several were in one large block, more than I'd have expected.

No tax to consider and one day I might get a really big win, but only £150 this month! A large holding held over the long term is a safe bet. I'm getting better interest on other savings and some of my shares have done very well - but others haven't!

Horses for courses.

 

[Magdalia]

Probably, but ERNIE won't be totally random either - there's not really any such thing as a perfect random number generator.

ERNIE will be nearer to being totally random than anything easily available to people like us. And NS&I are now on ERNIE 5, which will be nearer to being totally random than ERNIE 1-4.

 

[styles]

They address this on their website:

But how can you prove it’s all random?​

The test of proving that ERNIE’s outputs are robust has long been discussed by experts in the field of randomness.
However, from the outset, engineers working in conjunction with – but exclusive of – the ERNIE programme have proven that the numbers created by the machine are unpredictable and follow no set patterns.
The original system to test ERNIE’s randomness was called Pegasus, designed by Dame Stephanie Shirley, who went on to be one of Britain’s greatest computer engineers.
Since then, as with ERNIE’s evolution, testing randomness has developed and is now managed by the Government Actuary’s Department (GAD).
Each month after the Premium Bonds numbers have been generated, they are sent securely to GAD who run a number of tests to identify whether the outputs are truly random:

• The frequency test – whether every possible character in each position of the Bond number appears as often as it should.
• The serial test – looking at the number of times one digit follows another (for example the number of 3s coming directly after 7s).
• The poker test – looking at the number of times that a group of characters generated consecutively contain four identical characters, three of a kind, two pairs, one pair and all different.
• The correlation test – looking for correlation between characters in two different Bond positions over a series of Bond numbers.

ERNIE has never failed to be anything but random in every test carried out by GAD.

ERNIE | NS&I Corporate Site

nsandi-corporate.com

Sounds fairly robust to me!

 

[dcbwhaley]

Probably, but ERNIE won't be totally random either - there's not really any such thing as a perfect random number generator.

Yes there is. There are lots of unpredictable physical events such as electronic noise, atmospheric noise, radioactive decay which can be used to generate genuinely random numbers

 

[bleeder4]

For 23 years I had two £500 Premium Bonds which never won anything at all. Two years ago I sold them and took out ten £100 Premium Bonds instead. Same amount of money, but using more lower value bonds rather than higher value bonds. In the two years since doing that, I have had two £50 wins. Coincidence or not? I have no idea.