First prizes won five draws in. 2 x €50 in today's draw.
Prize Bonds are looking attractive alternatives to deposits
- Thread starter Raging Bull
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First prizes won five draws in. 2 x €50 in today's draw.
postman pat
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€50 today ..ends a bad streak
Well done Marc, nice piece.
One minor quibble -- you say Prize Bonds might work out better than cash deposits if you are in the top tax bracket. But DIRT isn't dependent on your tax bracket, so I wasn't sure what you were getting at there. I suppose there's a dependence on how much you earn from deposit interest, since if you are over the "chargeable" limit you additionally pay 4% PRSI, but this is dependent on the amount of your unearned income rather than your tax bracket.
A second even more minor quibble is about the statement that "the graph is more lumpy toward the left hand side indicating that you are more likely to do badly than to do well". That's not wrong, but it could be caveatted by saying "you are more likely to do badly than to do well in any given year". In the long run, of course, you're likely to do "averagely". First we have to agree that by "doing well" we mean getting your average return or better. Let's revisit the €5k graph:
Clearly, the most likely number of €50 wins (called the mode of the distribution) is 1, since that number has the highest percentage chance at 36.28% (based on latest figures). The average (or mean) number of wins over a long period is actually 1.176 -- higher than the mode. (We get the average by just dividing the total number of €50 prizes by our fraction of the total Prize Bonds held). Now, you're right that in any given year you're more likely to do "badly" (i.e. get less than the mean) because if we sum the first two columns to give the chance of getting either zero or one wins -- both of which are less than the mean -- we get 67.13%, i.e. a 2/3 chance of doing worse than average. On the other hand, if we subtract that number from 100% we get 32.87%, which is the chance of getting any number greater than one. Now, clearly, each year we win more than one prize, it makes up for a year when we win zero. And we're more likely to win more than one (32.87%) than we are to win zero (30.85%). Not only that, but some of those "more than one" years will be more than two! So it's easy to see why the mean is higher than the mode.
Intuitively, the reason for this phenomenon of the mean (i.e. average) being higher than the mode is because of the skew caused by not being able to win less than zero times, but having a possibility of winning many times more than the mean. (In theory, with a single prize bond, you could win all 470,400 x €50 prizes in a year, but that would take much longer than the lifetime of the universe on average
).
Now for the science bit. Technically, the type of distribution in our €5k graph above is called a binomial distribution. You can imagine a smoother version of it where we're allowed to win fractions of prizes. A smoothly continuous version of the binomial distribution is called a Poisson distribution. That's as distinct from a normal or Gaussian distribution -- the well-known bell curve -- of a truly randomly distributed variable. A Gaussian is a symmetrical curve where the mode and the mean are always the same, i.e. your most likely number of wins is identical to your average number of wins. It is to be noted that as the average number of wins increases, the Poisson distribution more and more closely approximates a Gaussian distribution, which is another way of saying what we already concluded: the more you invest the more likely you are to hit the average in any given year. For smaller investments you'll still win the average but over a longer (perhaps very much longer) number of years.
I'll end with some graphs showing how the Poisson distribution (in blue) differs from the Gaussian (in purple). Here they both are with a mean (vertical grey line) of 1.17, exactly the average number of €50 wins we expect for our €5k Prize Bond investment. The mode (which, remember, is the most likely outcome) is the position along the horizontal axis of the highest point on each curve. The mean, on the other hand, is the position on the horizontal axis where the area under the curve to the left of the mean is equal to the area under the curve to its right. For the Gaussian, which is symmetrical, the mean and the mode are always the same -- but we can only achieve this by allowing for the impossible case of negative numbers of wins as can be seen where the purple curve goes negative. But for the Poisson distribution the mode is less than the mean:
Here is another case, where the Poisson and the Gaussian once again have the same mean, this time equal to 23.52 -- the average number of €50 wins for our €100k investment. See how much more closely the Poisson distribution approximates the Gaussian and the mode is much closer to the mean:
Finally, here are some graphs plucked from [broken link removed] showing a binomial distribution (quite similar to our Prize Bond one) overlaid with a Gaussian with the same mean, for different values of the mean. We see how the binomial distribution, like the Poisson (its continuous version) more nearly approximates the Gaussian as the mean goes up:
[broken link removed]
You can also see that more of the binomial distribution hangs out the left hand side of the area enclosed by the Gaussian because its mode is less than the mean, as we expect. (As Marc puts it, it is "more lumpy to the left", which actually technically is called a right skew, because the tail to the right is less steep).
One minor quibble -- you say Prize Bonds might work out better than cash deposits if you are in the top tax bracket. But DIRT isn't dependent on your tax bracket, so I wasn't sure what you were getting at there. I suppose there's a dependence on how much you earn from deposit interest, since if you are over the "chargeable" limit you additionally pay 4% PRSI, but this is dependent on the amount of your unearned income rather than your tax bracket.
A second even more minor quibble is about the statement that "the graph is more lumpy toward the left hand side indicating that you are more likely to do badly than to do well". That's not wrong, but it could be caveatted by saying "you are more likely to do badly than to do well in any given year". In the long run, of course, you're likely to do "averagely". First we have to agree that by "doing well" we mean getting your average return or better. Let's revisit the €5k graph:
Clearly, the most likely number of €50 wins (called the mode of the distribution) is 1, since that number has the highest percentage chance at 36.28% (based on latest figures). The average (or mean) number of wins over a long period is actually 1.176 -- higher than the mode. (We get the average by just dividing the total number of €50 prizes by our fraction of the total Prize Bonds held). Now, you're right that in any given year you're more likely to do "badly" (i.e. get less than the mean) because if we sum the first two columns to give the chance of getting either zero or one wins -- both of which are less than the mean -- we get 67.13%, i.e. a 2/3 chance of doing worse than average. On the other hand, if we subtract that number from 100% we get 32.87%, which is the chance of getting any number greater than one. Now, clearly, each year we win more than one prize, it makes up for a year when we win zero. And we're more likely to win more than one (32.87%) than we are to win zero (30.85%). Not only that, but some of those "more than one" years will be more than two! So it's easy to see why the mean is higher than the mode.
Intuitively, the reason for this phenomenon of the mean (i.e. average) being higher than the mode is because of the skew caused by not being able to win less than zero times, but having a possibility of winning many times more than the mean. (In theory, with a single prize bond, you could win all 470,400 x €50 prizes in a year, but that would take much longer than the lifetime of the universe on average
).Now for the science bit. Technically, the type of distribution in our €5k graph above is called a binomial distribution. You can imagine a smoother version of it where we're allowed to win fractions of prizes. A smoothly continuous version of the binomial distribution is called a Poisson distribution. That's as distinct from a normal or Gaussian distribution -- the well-known bell curve -- of a truly randomly distributed variable. A Gaussian is a symmetrical curve where the mode and the mean are always the same, i.e. your most likely number of wins is identical to your average number of wins. It is to be noted that as the average number of wins increases, the Poisson distribution more and more closely approximates a Gaussian distribution, which is another way of saying what we already concluded: the more you invest the more likely you are to hit the average in any given year. For smaller investments you'll still win the average but over a longer (perhaps very much longer) number of years.
I'll end with some graphs showing how the Poisson distribution (in blue) differs from the Gaussian (in purple). Here they both are with a mean (vertical grey line) of 1.17, exactly the average number of €50 wins we expect for our €5k Prize Bond investment. The mode (which, remember, is the most likely outcome) is the position along the horizontal axis of the highest point on each curve. The mean, on the other hand, is the position on the horizontal axis where the area under the curve to the left of the mean is equal to the area under the curve to its right. For the Gaussian, which is symmetrical, the mean and the mode are always the same -- but we can only achieve this by allowing for the impossible case of negative numbers of wins as can be seen where the purple curve goes negative. But for the Poisson distribution the mode is less than the mean:
Here is another case, where the Poisson and the Gaussian once again have the same mean, this time equal to 23.52 -- the average number of €50 wins for our €100k investment. See how much more closely the Poisson distribution approximates the Gaussian and the mode is much closer to the mean:
Finally, here are some graphs plucked from [broken link removed] showing a binomial distribution (quite similar to our Prize Bond one) overlaid with a Gaussian with the same mean, for different values of the mean. We see how the binomial distribution, like the Poisson (its continuous version) more nearly approximates the Gaussian as the mean goes up:
[broken link removed]
You can also see that more of the binomial distribution hangs out the left hand side of the area enclosed by the Gaussian because its mode is less than the mean, as we expect. (As Marc puts it, it is "more lumpy to the left", which actually technically is called a right skew, because the tail to the right is less steep).
Very interesting thread, like seeing the the Ghoul's returns. Presumably Marc was referring to Dub Nerd in his article. Great analysis Dub Nerd. Not sure if it was factored in, but what about inflation eroding the 100K investment of the Ghoul?
Further up the thread it was mentioned that there is a rumour that newer bonds win more prizes, but that doesn't logically make sense does it. If it did, than one ought to sell and buy once a year.
Further up the thread it was mentioned that there is a rumour that newer bonds win more prizes, but that doesn't logically make sense does it. If it did, than one ought to sell and buy once a year.
Hi Marc
I spotted a small error in the article. The graph title says that it's for a 50k investment but your article states that the graph shows the return for a 5k investment. Not a big thing but just in case someone invests 5k and then turns up at your door complaining they haven't done as well as your graph ...
Anyway, I can't even pretend to follow the sums on this really. It's fascinating to me that people can but statistics was the only subject in college that I was worse at than accounting. So glad it was only for one semester! But pedantry? I'm good at that. LOL.
This is exactly the point of Marc's article. People with smaller amounts (who are in the majority) have much less of a chance of winning than they think.....
I'm probably getting another batch of PBs today, 33,000 worth. Have a tax return to pay in October - I'll park the money in PBs, leave it in for at least 3 months and withdraw it in time to pay the tax having hopefully won a few 50 euro prizes.
I presume I will get batch of bonds in sequence so it will be easy to track how much the new bonds win and how much my existing bonds win.
If the new bonds do better then I might start thinking about conspiracy theories again!
I presume I will get batch of bonds in sequence so it will be easy to track how much the new bonds win and how much my existing bonds win.
If the new bonds do better then I might start thinking about conspiracy theories again!
People always sound surprised when they say that.
Don't be. With a couple of hundred (i.e. €200) you will win a €50 prize on average every 21.26 years at current rates. But, just as The Ghoul on this thread has gone seven and eight weeks without a win when he is due to win every couple of weeks on average, you could easily go one or two centuries without a win and could take many centuries to achieve an average return. I can't say this often enough -- for small amounts of money you should view Prize Bonds as a place to park your money and be able to cash it in when you want. To a first approximation you are not going to win, EVER!
EDIT: username123 got there before me.
As with any investment, your returns may be more or less than inflation. With the likely returns for a substantial PB investment being 1.2% - 1.3% at present, you don't need very high inflation to wipe them out. (However, they're still better than instant access bank deposits).
Yes, there is such a perception, but it's not true unless the PB company are committing fraud or have programmed the computer wrong. All the calculations here assume that each draw is independent and you have the same odds in each one. If I was to hazard a complete guess as to why the erroneous perception arises, it would be as follows...
If you write down your sequence of wins and losses as has been done by some contributors here, you'd have something like:
Week 2: | 1 x €50
Week 3: | 1 x €50
Week 4: | 0 x €50
The average is going to depend on how many PBs you hold. However, as with any random process, you will get excursions from the mean. The longer you go on, the more extreme will be the excursions, even though you will always return to the average in the long run. This is a very difficult concept to grasp. Let's take The Ghoul as our guinea pig.
He's had runs of seven or eight weeks without a win, whereas with his investment he would expect to win just a fraction less than once every two weeks. But over the course of a year, such a void period is not really very surprising. If you waited long enough, you'd eventually have a whole year of no winnings (although my quick calculation suggests that could take a trillion years to happen!). Anyway, void periods of random lengths are to be expected.
But equally, periods of excess wins are to be expected. The Ghoul's had some of those too. But did he write exuberant posts expressing surprise at his good fortune and noting that he was well ahead of expectations? Oh god, no! Our innate optimism means that we expect to be rewarded! But eaten bread is soon forgotten. So we notice the voids and complain about them bitterly, whereas we treat the wins as our just deserts.
As I said, over time the length of the longest voids and greatest excesses both increase. But because of a greater propensity to notice the voids, we think our luck is diminishing.
Could be total poppycock
... I'd be interested if anyone has a different theory.Now that's jammy. The €1000 prize should be a once in 140 year event for you on average. The good news is that it in no way affects your independent chance of winning an average number of €50 prizes (13 or so) for an overall 3% tax free return (equivalent to 5.1% gross assuming 41% DIRT).
Thanks for the comments everyone.
To be fair to Niall, the editing actually improved the article but when you are required to be concise to start with some of the nuances can be lost.
Dub Nerd:
The reference to top rates of tax was an attempt to capture a high earner with considerable passive investment income who would be subject to a punitively high tax rate on their investment income due to PRSI.
Similarly, the "lumpy graph" was obviously meant to reference any one year rather than forever, but I think this is implied in the overall context.
Finally I noticed the graph was mislabeled but again hopefully the meaning wasn't lost in the overall context.
To be fair to Niall, the editing actually improved the article but when you are required to be concise to start with some of the nuances can be lost.
Dub Nerd:
The reference to top rates of tax was an attempt to capture a high earner with considerable passive investment income who would be subject to a punitively high tax rate on their investment income due to PRSI.
Similarly, the "lumpy graph" was obviously meant to reference any one year rather than forever, but I think this is implied in the overall context.
Finally I noticed the graph was mislabeled but again hopefully the meaning wasn't lost in the overall context.