Premium Bonds: Are they worth it? Discussion Area

wiggers

Having been skimming through the article and this thread I haven't yet seen a proper calculation of the expected return on investment from PBs or the lottery. The expected return is calculated as the size of the pot divided by total number of prizes times the probablity of winning a prize.

£85.4 million / 1,356,432 x 1/24000 = £0.00263

So for each £1 you can 'expect' to receive 0.26p per month. This works out at 3.2% over a year (compounded monthly) and you would need 1587 years to expect to win a £50 prize for a £1 holding.

Because the prizes are awarded discontinuously (i.e. widely varying amounts) the more chances you have (a 'chance' being £1 in one month) the nearer to the expected return you will realise. Just like an opinion poll has to be conducted with a statistically significant number of people, you will need to have a significant number of chances to get near to the expected return. People with just a few chances may get a higher return than expected, or nothing at all. If you have the maximum holding then you are more likely to get the expected return. You are also more likely to get the the low frequency (higher value) prizes with the maximum holding.

For the National Lottery I calculated a few years ago that the expected return was about -55p for every £1 invested, because you lost your stake. I also calculated that you would need to buy at least £50 worth of Lottery tickets each time to stand a chance of breaking even over time through winning the larger than minimum prizes.

cynic

MODE is the most frequently occuring result, Mean is the arithmetic average.

Some of you who are familar with the noble art of statistics have attacked what Martin is saying. e.g. saying "hey over time the average has gotta be the average dude!" Esssentially though he is correct and the would-be geeks have not got geeky enough I fear!

One can say stuff about the population of premium bonds and I don't think anybody disagrees that the presence of a low number of high value prizes skews the distribution. We have a classic assymetric distribution where the mean and mode do not coincide - the mean return is higher than the MODAL return. Yes indeed when we look at the whole population i.e. if you owned all the premium bonds in the world then the interest rate would come out how it is supposed to.

Now here's the catch: Any single holder of bonds holds a sample of bonds from the entire population. Shouldn't matter say you stattos?
Well sorry but it does
Why?
Because prizes are discrete i.e. lumpy. with 30k holding I am supposed to won £79 every month. There are no £79 prizes.

One can say things about this sample e.g. we can discuss the sample mean and sample mode, but these will not necessarily be the same as the population mean and mode - there will be some variation quite a lot in fact cos the prize values are so dispersed - in fact these sample statistics will have their own distribution (people talk about the distribution of sample means for example)

All you can ever say even if you do some sums (even with say the maximum £30k holding) it takes about X draws for you to be 95% confident that your sample will look like (i.e. share the same characteristics) as the population i.e. get the return you should get. I calculated X to be about 5years once (and even then one poor soul in 20 is still not doing so well)

Martin's table where he excludes some higher prizes has had people up in arms, but because prizes are not continous i.e. are discrete amounts it's not a bad way to look at it - it gives you a flavour of how long you need to wait to get your sample mean up towards the population mean

In particular his line "even if you win a prize the chance of it being bigger than £100 is 160 to one", put another way it takes 13 years of monthly draws to be confident of winning more than a £100 prize, without any of these prizes you are likely to be just earning 2.75%

So a basic point that hasn't been made is that premium bonds are basically a LONG-TERM investment

Hereward

cynic wrote:

Median is the most frequently occuring result, Mean is the arithmetic average...

Sorry to be pedantic, but:

Mode is the most frequently occurring result;
Median is the middle value;
Mean is the arithmetic average.

cynic

The lesson we learn is about "statistics" - nice when looking at populations or when we get to play a game lots of times but if we only get to play a few times or just once then suddenly statistics are not that useful

Tossing a coin & keeping score is good example that everybody intutitively understands:
1. We all know if the coin is fair we expect 50% heads and 50% tails
2. We also know indeed if we tossed the coin loadsaloadsa times this would come out (kinda proving the coin is fair)
3. BUT we also know that at any individual toss we cannot really say what's gonna happen.
These three "rules" are pretty clear to most people, even those people who say "I cannot do maths"

The interesting conclusion is to say that the more times we toss the coin the closer the score will converge to 50% each.

Now I could use lotsa jargon & symbols and put a lot of you off, but using statistics gives you ability to discuss how quickly this score "should" converge. Note we cannot say with certainty (but that would feel wrong if we could) but we can say things like "after 100 tosses I am 95% confident that there will be between 45 and 55 heads"

This thing about statistics pops up in lotsa financial situations, so it's quite useful to understand.
In premium bonds if you get to play lots i.e. own them for 5-10years if you have max holding then the statistics start coming out as they "should"

Hereward

cynic wrote:

Tossing a coin & keeping score is good example that everybody intutitively understands:
1. We all know if the coin is fair we expect 50% heads and 50% tails
2. We also know indeed if we tossed the coin loadsaloadsa times this would come out (kinda proving the coin is fair)...

This isn't quite true, if you flip a coin there are at least three possible outcomes: Heads, Tails or it lands on its side. The probability of a coin landing on its side is so small it can be said to be statistically insignificant, but in the real world that possibility still exists, albeit extremely unlikely.

cynic

there's also a probability that the tossed coin quantum-tunnels through the catching hand, not to mention the probabilty of getting hit by a meteorite on the way down but I think these can be safely disregarded for the example
thanks for the earlier correction tho!

nrsql

Geoff_W wrote:

Back in 2000 I bought £10,000 of Premium Bonds (at the suggestion of a Financial Advisor!) at a time when the 'interest rate' was probably nearer 4.5%. To be fair, I did have some small wins, but when I finally gave up on them in 2003 and cashed them in, my return worked out at about 2.1% per year; very below average.

The money was put into Cash ISA's which have performed better than the Premium Bonds.

Geoff

Your FA advised to to invest in premium bonds rather than a cash isa?

As to a lot of the discussions here I would recommend
A Mathematician Plays the Stock Market by John Allen Paulos.

It gives a lot of insight into the reasons why people do odd things with investments and why superficially sensible plans might be destined to fail. How odds can be misleading.
It gets a bit bogged down in places and doesn't really convince in some places but especially the first half is a good read.
It stems from his loss on enron which is all entertaining and mirrors how a lot of people lose on the stock market - I would would doubt if anyone has not been through the same situation (hopefully to a lesser extent but compare with pension funds).

Hereward

cynic wrote:

there's also a probability that the tossed coin quantum-tunnels through the catching hand, not to mention the probabilty of getting hit by a meteorite on the way down but I think these can be safely disregarded for the example
thanks for the earlier correction tho!

True, but these outcomes are independent of the coin, as these rely on some external factor (space, meteor); however, the edge of the coin is integral to it and, therefore, it should not be discounted. I agree for your example that it can be safely ignored, but it should have been acknowledged.

cynic

Setting phasers to pedantic:
The quantum characteristics of the coin are actually integral to the coin and the speed at which it is tossed so if we include your "lands on its edge" outcome we should include this "quantum-tunnelling" outcome too, but suddenly this sounds a bit silly.

I think we could discuss for ages and acknowledge all sorts of outcomes that as you concede can be safely ignored!

That might however obscure the point I was trying to illustrate which I think was something about the use of statistics in relation to premium bonds
You clearly understand that point which is great, but our verbal jousting may put off someone who doesn't which is a shame.

But thanks again for the earlier correction

Hereward

cynic wrote:

Setting phasers to pedantic:
The quantum characteristics of the coin are actually integral to the coin and the speed at which it is tossed so if we include your "lands on its edge" outcome we should include this "quantum-tunnelling" outcome too, but suddenly this sounds a bit silly.

I think we could discuss for ages and acknowledge all sorts of outcomes that as you concede can be safely ignored!

That might however obscure the point I was trying to illustrate which I think was something about the use of statistics in relation to premium bonds
You clearly understand that point which is great, but our verbal jousting may put off someone who doesn't which is a shame.

But thanks again for the earlier correction

Well said, I had no intention of detracting from a very good post.