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I am considering buying 100 NS&I premium bonds, in the UK. Each $1 bond is automatically entered into a draw with a 1 in 30000 chance to win back what is effectively an interest payment*. My questions are:

A) What is my chance to win with 100 bonds?

B) Assuming I buy 100 more bonds each months for 12 months, how much do my chance to win increase month by month?

Edited to reflect commentary from @CactusCake:

Note that purchasing NS&I Premium Bonds is not really gambling in the traditional sense. This is offered as a savings account in the UK. Any time you would like to make a withdrawal, your bonds can be sold back for 1GBP each (same as purchase price). Instead of paying traditional interest to all account holders, the "pool" of everyone's bonds earns interest each month and that sum is then raffled off in chunks to a randomly selected subset of all account holders. Since your initial stake is never at risk, it isn't really a gamble.

See more here: https://www.nsandi.com/premium-bonds.

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    It is about the ns&i saving bond and as someone investing £100 p.m. it give myself and anyone else investing an idea of return. Especially as there is no interest rate and the returns are speculative. – robin jones Jun 27 '17 at 13:47
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    For practical purposes multiply the number of bonds by the chance of each to win. That's close enough. If this is a statistics test, talk to your teacher. – DJClayworth Jun 27 '17 at 15:00
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    If the number of tickets you have bought is small compared to the probability of winning, DJClayworth's answer is correct. If you get to the point where number of tickets times chance of winning is more than a couple of percent, then you need to get a little more sophisticated. If this is for a math class: Hint: Easiest way is to calculate the probability of NOT winning and subtract from 100%. – Jay Jun 27 '17 at 15:10
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    Here's some information about the account. I think it is on topic for here, since plenty of individuals use these accounts in the UK. – CactusCake Jun 27 '17 at 19:41
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    I would still call this "gambling" as you're trading the interest you would receive for a slim chance at some prize - the more interest you give up (buy buying more interest-free bonds) the better your chances. Your base investment is secure but the fact that you're paying an opportunity cost for a small change at a bigger prize makes it gambling in my book. Would it be any different if you received that interest every month, but then bought lottery tickets with it? – D Stanley Jun 27 '17 at 20:58
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Your odds of winning a prize with 100 bonds are 100 in 30000, which can be reduced to 1/300.

Based on your saving strategy, by month 13 you would have £1,300 in your account, and over that period you would have had 9,100 entries (100 in the first month + 200 in the second month + 300 in the third month and so on) in 13 drawings. Note the odds of winning in any one drawing are not influenced by your odds from any previous drawing, it is always 30,000 to 1. Therefore, by the end of month 13 you will have a 9,100:30,000 chance of winning a prize. This can be reduced to 91/300 or roughly 3 tenths.

If you continue your saving strategy, by month 24 you will have had exactly 30,000 entries, so you can expect to have won a prize at some point within the preceding 2 years (not guaranteed, just "probable").

Now you need to look at what the prizes are. The website estimates that 2,329,721 prizes will be awarded in August 2017, and out of those, a whopping 2,280,757 will be in the amount of £25. Those numbers don't simplify very well, but you can round it off and say that for every 233 winners only 5 of them (2% of the winners pool) are getting a prize larger than £25.

So lets assume you win one prize of £25 at the end of month 24. Your account now has £2,425.00 in it. Good job!

Now let's compare it to a simple interest bearing account at the same 1.15% that this pool earns. Again, you're going to invest £100/month. At the end of 24 months you'll have your investment principal of £2,400 plus (compounded) interest totaling £26.64, for a balance of £2,426.64

So in the end, the two accounts are less than a tenth of a percent different. Many people see this as a plus for premium bonds - "I'm likely to make the same as a low interest bearing account, but there's a chance, a tiny chance that I might win £1,000,000". Of course, the chance really is absolutely minuscule (1 in 34,945,815,000 for each bond, if I did the math right), but we silly humans do so very much love to hope.

  • "I'm likely to make the same as a low interest bearing account" - no you're likely to make nothing. A few people will make millions. The average gain for the millions or participants is about the same as a low-interest savings account. Roughly the same as buying lottery tickets with the earned interest,. – D Stanley Jun 27 '17 at 21:45
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    "Roughly the same as buying lottery tickets with the earned interest" is very accurate (I considered putting that in my answer when writing it), but "you're likely to make nothing" isn't. Even if you only spend invest the minimum £100, if you leave it in there for 300 months then you are indeed probably going to win a £25 prize (note - "probably" and "likely" are not guarantees). The 1.15% interest account meanwhile would have earned £32.90, enough for 7 extra lottery tickets. – CactusCake Jun 27 '17 at 21:53
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    Bear in mind that Premium Bonds are tax-free, so for some that have maxed out other tax-free investments they are a good deal for that reason. As the ISA allowance has risen this has become less significant over the years. – GS - Apologise to Monica Jun 28 '17 at 5:57
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The real thing you should look at is the "expected" return, i.e. the average amount you can expect to win per bond. The current return is 1.15%, i.e. for each £1 bond you'll get an average of 1.15p per year. With 100 bonds you'll get an average of £1.15 per year, or just under 10p/month. If you buy 100 more bonds per month the average return will rise by the same amount each month, getting to £1.15/month after 12 months.

If you want to know the chances of winning anything at all, it's quite small for the volume of bonds you're looking at. Since each bond has a 1/30,000 chance of winning, the chance of 100 bonds winning something is just under 1/300, and will increase by about that amount each month to reach 1/25 after a year. More precisely, it starts off at about 1/300.5, and finishes at about 1/25.5.

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    Although note that the minimum prize is now something quite large (£50?) so with £100 of bonds you wait a long time to get anything. I've had £200 of bonds for almost 15 years and only just got my first win. – Vicky Jun 27 '17 at 22:05
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    @Vicky Yes, it's "biased" (though that's not really the right word) towards higher-value holdings because of the £25-or-nothing quantisation of the lowest payout. A small-value regular savings account will "trickle-in" modest interest payments; with PBs, you (more often than not) get nothing (per bond) and occasionally get something (usually £25). For a low-value holding, the "lots of nothing" still add up to nothing. A higher-value holder has enough bonds that at least one of them wins £25 fairly often and so gets a more "interest-like" steady(ish) stream of money. – TripeHound Jun 29 '17 at 8:06
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    Yes the problem with the "expected return" number is that, while it's entirely mathematically sound, it's bumped up by the very small chance of a very large win. Somewhere I've seen an analysis of what happens if you exclude the large prizes (which realistically you're just not going to win) and then PBs don't look so good. It's like saying "I expect giant meteors to kill 10000 people a year" because once every 100000 years something hits us that could kill a billion. – timday Jun 29 '17 at 15:03
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A) What is my chance to win with 100 bonds?

Your odds of winning at least once in the month with 100 bonds, each having a 1 in 30000 chance of winning each month is:

one minus the chance of winning nothing, or put in numbers

1 - (29999/30000)^100 = 1 - 0.99667 = 0.0033

So about a third of a percent a month, with 100 bonds.

Your odds are not simply 100/30000 (although with so few tickets, it's very close to that); if it was that simple, it would imply that buying 30000 bonds would guarantee you a prize, and that's clearly not the case (the chance of winning something in a month if you have 30000 bonds is actually about 63 percent).

B) Assuming I buy 100 more bonds each months for 12 months, how much do my chance to win increase month by month?

The same calculation can be used to calculate your chance of winning at least once each month.

Month one:    1 - (29999/30000)^100  = 0.003328
Month two:    1 - (29999/30000)^200  = 0.006645
Month three:  1 - (29999/30000)^300  = 0.009950
Month four:   1 - (29999/30000)^400  = 0.013245  (crossing the 1% barrier!)
Month five:   1 - (29999/30000)^500  = 0.016529
Month six:    1 - (29999/30000)^600  = 0.019802
Month seven:  1 - (29999/30000)^700  = 0.023063
Month eight:  1 - (29999/30000)^800  = 0.026314
Month nine:   1 - (29999/30000)^900  = 0.029555
Month ten:    1 - (29999/30000)^1000 = 0.032784
Month eleven: 1 - (29999/30000)^1100 = 0.036003
Month twelve: 1 - (29999/30000)^1200 = 0.039211 (almost 4% chance of winning something this month)

As an aside, if you're looking for somewhere to put your money to work, low cost index funds with dividends reinvested are a good way to go. On a personal note, I steer away from the FTSE100 (not diverse enough for my liking), but fifty quid a month into the S&P500 and a broad Japan index is a pretty solid way to get started.

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