I think you should work out what the Expected Return is for someone who has €100,000 invested.
Hi Brendan, with the changes in interest rates and hikes in DIRT and PRSI, I'm considering putting a considerable chunk in prize bonds. I did some calculations on likely returns and thought I would share them here. Although, as Marc has said, an exact calculation of the odds of a given return is complex, it is easy to simplify roughly to within any given confidence level. (The likely returns are a little less than what Marc calculated). The main thing for people to bear in mind is that for a small investment, your chances of an average return are negligible, and Prize Bonds are
not a good idea in this case. If you want to treat Prize Bonds as an investment and want to have an excellent chance of "average" returns (which I will define below), you would want an absolute minimum of €10k investment, but preferably several times that. I will do a calculation for €100k but I will also talk about the implications for smaller and larger amounts.
Summary.
The prize fund is split between a small number of large prizes which you have essentially no chance of winning, and a large number of small prizes which -- if you invest enough -- you can reasonably rely on an average return from.
Your chances of achieving an average return increase with the amount of money you invest. Note that your return may be more or less than the average no matter how much you invest, but the chance of getting near the average are better for higher amounts and longer periods of investment.
For an investment of €10k to 100k you will probably achieve about 1.18% (equivalent to 2.1% from a bank account assuming 41% DIRT + 4% PRSI). For an investment of 100k+ you may achieve 1.33% (equivalent to 2.4% assuming DIRT and PRSI). Note that these are all probabilistic values with reasonable confidence levels. If you want the excruciating detail, read on.
How the prizes are structured.
The total prize fund is currently calculated at 1.75% of the value of total Prize Bonds held. Latest figures are as of April 2013 and can be viewed on the
State Savings web page. Since the total value is currently €1.7b (made up of 280m individual Prize Bonds), this gives a total annual prize fund of €29.75m, structured as follows:
€1,000,000 x 6 times annually = €6,000,000
€20,000 x 46 times annually = €920,000
€1,000 x 5 times weekly = €260,000
€100 x 500 weekly = €2,600,000
Total annual value: €9,780,000
Now, ironically, the above details can be almost ignored, since you won't be winning any of that. Almost all of your returns will come from the €50 prizes which make up the remainder of the prize fund. The only reason for the above calculation is to subtract the money you won't win (€9.78m) from the total prize fund (€29.75m) to give the total value of €50 prizes: €19.97m. This works out at 399,400 x €50 prizes per year.
What are your personal odds?
It would be messy to calculate odds based on number of prize bonds held. It's easier to calculate your fraction of the total prize bonds by dividing your investment into the total monetary value of all bonds. For a holding of €100,000 you have €100k/€1.7b = 1/17,000 of all bonds. This is very important!
On average you will win one prize out of every 17,000 prizes. (If you invested €10,000 this would change to one prize in 170,000; if you invested €1m it would be one prize in 1,700).
How much will you win?
Unfortunately, few people have an intuitive feel for odds and risk. Think about those €1,000 prizes, 5 times weekly. There are 260 per year. With your investment of €100k you will win one on average every 17,000 / 260 = 65 years. Do you want to bet on a return that might come in every 65 years (or might not)? Clearly not. There's a good chance you won't collect on it in your lifetime. Still, it's worth thinking about this in terms of confidence levels. These are a little complicated to work out so I won't go into details, but will just state that: you have a 50% chance of winning that €1,000 prize within 45 years. If you want a 90% level of confidence, you will have to wait 150 years. And to have a 99% chance of winning, you will wait 300 years!
Now, I'd be prepared to treat a 99% chance of winning as a fairly sure bet, although I'd still want to think about the consequences of the 1% chance that I
wouldn't win, to see if I'm prepared to take that risk. But it's all academic in this case because I'm not prepared to wait 300 years to be 99% sure of a return! The point to illustrate here is that the average time for a return -- 65 years -- is by no means a guarantee that after 65 years you will have won once. You might
never win, or it might take a million years! But there's a 99% chance you'll have won (at least once) after 300 years.
Hopefully it's obvious that there's no point even thinking of winning any of the big Prize Bond prizes, unless you are a gambler rather than an investor. On the other hand, look at the number of €50 prizes annually -- 399,400. With your €100k investment you stand to win 399,400 / 17,000 = 23.5 per year on average, or nearly one per fortnight. Just as importantly, we want to know what the confidence level is that you will achieve this average or close to it. Here we have to use some horrible statistical methods again (the
binomial distribution, for anyone who wants to check it out), so I will just state the result: there is about a two thirds chance that you will win the €50 prize between 20 and 28 times annually, and a 92% chance you will win it between 16 and 32 times. The long and short of this is that "your mileage may vary" but there is a good chance you will get close enough to the average.
Remember: this is based on a 100k investment. The odds of you getting close to the average diminish with a smaller investment. This does not mean you will win a smaller percentage -- you could win
more! But the odds get "lumpier" the less you invest. (If you don't understand this, this investment may not be for you).
So what is the
average return? It is 23.5 x €50 / €100k = 1.175%. But don't forget the confidence levels -- they tell us that there is a 92% chance that your
actual return will be between 0.8% and 1.6% for an investment of €100k within a given year. The
average return doesn't change no matter what you invest. But the confidence level of getting close to the average is higher for a higher investment. For a €10k investment there is much less chance that things will average out in any given year, due to the lumpiness problem. To illustrate, remember that with €10k invested you will win on average 2.35 x €50 / €10k = 1.175%. Still the same average return, but you can only win an integer number of prizes. Here are two graphs for €10k and €100k showing the percentage probability of each number of €50 wins:
The difference in lumpiness should be obvious. For €10k you will win 2.35 times on average, but there's a hefty 9% chance you'll win nothing at all in a given year, and only a 70% total chance that you'll win 1 to 3 times. Things are much smoother with the €100k -- you could still win nothing but the chances are tiny: much less than one in a billion! The chances are less than one in a hundred that you will win less than 14 times (or more than 35 times).
So, an average return of 1.175% would be the final answer except that between the €1000 (and upward) prizes that you will
never win, and the €50 prize that you will win fairly regularly, there is also the €100 prize. Now, there are 15 times fewer of these, so you can only expect to win 1.53 of them every year. When we factor these in the average return goes up to 1.33%.
But -- and this is a big but -- the low average number of €100 wins also gives a much lower confidence level that you will achieve close to the average... that lumpiness factor again. If you only invest €10k, you will probably wait many years before winning a €100 prize, so you should use the lower average return of 1.175% unless your investment is over a timescale of decades. If you invest €1m you are much more likely to quickly average the higher return of 1.33% even over a timescale of a single year. For a €100k investment, you are somewhere in the middle. It might be prudent to bank on 1.175%. But if you leave your money invested for a few years, you are more likely to achieve the higher return of 1.33%.
Remember, this is all statistical.
You are not guaranteed anything. However, for large amounts of money and time, the chances of an average return are good.
EDIT - returns may be higher!
The numbers I used here are based on the
Prize Bond FAQ which gives a total value of outstanding Prize Bonds of €1.7b as of April, and an implied number of €50 prizes of 399,400 per year, or 7,680 per week. However, the
front page of the Prize Bonds website states that there are now over 8,000 €50 prizes per week. And the
results for the latest draw (currently 15-Nov-2013) show 9,346 prizes in that week. Subtracting the 506 prizes greater than €50, that leaves 8,840 €50 prizes. Now,
amtc mentions down page that people use Prize Bonds to accumulate money due for tax. This implies that the number of bonds may reset around end of year. But the 2012 Prize Bonds
annual report says that the value of the fund went up 14% from 2011 to 2012. An increase in fund value doesn't directly change your likely return because your share of the fund, and thus your chances of winning a fraction of a bigger pot, go down accordingly. However, with increasing value, the number of large prizes (which eat up a third of the potential returns) goes down as a fraction of the overall prize fund, leaving more of the small prizes from which your returns primarily come. Basically, as long as the prize structure stays the same, increases in fund value favour investors over gamblers.
EDIT: add graphs
EDIT2: update with finer grained graphs.
EDIT3: add caveat about increase in outstanding bonds.
EDIT4: update with more accurate graph and numbers based on using Stirling's formula