Prediction markets and lotteries (and my one simple trick for winning the lottery)

 A short and slightly mad post today, inspired by weird thoughts I was having when I awoke from my sleep this morning (yes, you are all thinking, a completely unwanted insight into the life of Rob). Clickbait is at the end.

There has recently been an uptick in the world of prediction markets. Although not new, and certainly in the UK political betting in a market based environment has been around for yonks (Betfair exchange*, take a bow) the recent entrance of Kalshi** and Polymarket** has spiced things up somewhat. 

* I'm a sometime customer but otherwise unconnected.

** No connection, not a customer, and god knows certainly no endorsement.

Now, as Matt Levine will tell you, to an extent these markets are mostly regulatory arbitrages around the fact that gambling on sports is heavily constrained in the US; Kalshi has taken advantage of an onshore loophole during a friendly political environment, whilst Polymarket is offshore crypto and hence can do whatever it wants. But at least in theory they do help to achieve the economists wet dream of a complete market which would allow you to bet on any outcome and therefore perfectly hedge yourself.

Anyway my mad thought was this; what if I could bet on whether I will win the lottery? 

There is one thing AI is good for. One thing.

In many ways that is a straightforward bet; I could specify the numbers I was going to pick and the conditions of the bet. For example, does only the jackpot count, or subsidiary prizes? For the purposes of this discussion it's simpler to assume a lottery with just one jackpot prize which is shared if multiple numbers win. And unlike certain other bets in these markets there is no debate about whether the bet will pay off.

But this is also quite mad; as there is a much easier way to make this bet, which is to actually play the lottery! Furthermore, it would extremely difficult to find someone to take the other side of this bet. Lottery ticket type bets are massively positively skewed; the person taking the other side must be willing to accept a huge negative skew of a tiny profit if they win or a huge loss if they lose. 

Of course they could hedge this risk by buying a lottery ticket. But then for this to make sense they would have to give the buyer of the bet worse odds than the lottery itself, effectively selling the ticket at a small mark up. This can make sense, again as a regulatory arbitrage if you want to 'play' the gigantic US lotteries but don't actually live there; and some bookmakers do offer this service, but not as far as I know in a market environment where you can choose to sell the bet, only as a standard bookie:client arrangement.

Lotteries only work because there is one counterparty who is willing to take all the bets - the 'promoter'. Because they are taking all the bets, and these bets are by nature perfectly uncorrelated (in the example of the simple lottery jackpot only lottery with prize sharing we are discussing), and they set the size of the prize fund based on ticket sales, they are in a no lose situation. In fact they normally have a substantial edge; even ignoring the possibility of nobody winning. 

(In many lotteries if there are no jackpot winners the prize fund 'rolls over' to the following draw and isn't taken by the promoter. We will return to this possibility later.)

In the UK national lottery this edge is set by law and reduces the prize fund by half before it's paid out, but lottery winnings are tax free. In US state lotteries where the top line reduction is lower the prize is also subject to income tax. Broadly speaking in both cases the net prize fund is around half of the total. Buying lottery tickets, as is well known, is a subsantial -ve EV bet. It is gambling, in the pejorative sense of the word. And nonsense like this, ought to be banned.

However let's imagine a world in which there are people who are willing to take the other side of this bet. One way of doing this is to split the bets into tiny chunks. For example I could take on 0.01 pennies of risk that a given set of numbers will not win a $1 stake lottery with a $1 million dollar prize; if the market is priced perfectly then if I lose I will lose 0.01*1 million = $10,000 if the numbers come up. And I could make 1 million of these bets to cover all possible outcomes and stand to win $10,000. As the market is perfectly priced then assuming everyone buys every combination of numbers I will not make or lose anything. With a million people like me betting against lottery wins, and a million people betting on lottery wins, we'd effectively have a lottery.

Importantly though, this lottery would have no house edge. Instead of 50% of the winnings being drained away by the promoter, 0% of the winnings would be. And of course those selling lottery 'tickets' by betting against possible winning combinations would have the option of doing so at a slightly lower probability. For example, in a 1:million odds lottery offering 900,000:1 on winning. The buyers of lottery tickets are facing a 10% house edge here, but that is still much better than the 50% loss of a normal lottery. There would probably be some cost to the prediction market operating, but even if it was 5% that is still a gain for both buyers and sellers.

Note: It seems likely that buyers will overpay for long odds tickets; as per this research which was highlighted by FT Alphaville the day after I originally published this: "Contracts trading at 5 cents win only 4.18% of the time, implying mispricing of -16.36%" 

Note: The presence of multiple levels of jackpots in normal lotteries makes the house edge much less transparent than it would be for a single prize with known odds. I guess this is deliberate.

Note: Just as in a normal lottery people bet on 'lucky' numbers it might be that sellers of lottery 'tickets' do not bet against all possible numbers coming up, or vary the price they offer on certain numbers. Absurd numbers of people buy birthdays or sequences of numbers. If this happened here then the 'price' of those 'tickets' would be bid up versus totally random numbers, allowing the sellers of those 'tickets' to make higher profits. A risk premium in lotteries!

In any normal world this wouldn't be allowed to happen. Effectively the parallel lottery is free riding on the actual lottery. Governments tend to be very protective of their gambling monopolies! Of course there is nothing to stop the prediction market setting up their own 'lottery', a trusted counterparty who will draw some numbers and post them on the internet every week. But perhaps this is too complicated. After all, lottery type payoffs are already widely available on the internet in the form of dubious crypto coins masquerading as investments. Why overcomplicate things? 

Perhaps a more interesting bet with some chance of being implemented is betting on nobody winning the lottery jackpot. Remember earlier I said many lotteries have a rollover function. Take the Euromillions for example. Odds of winning the jackpot are 1 in 139 million. But typically only 20 to 30 million tickets are purchased. Hence in any given draw the chances are the jackpot will not be won. For example in October and November last year there was no winner for 13 draws and the jackpot reached 156 million Euros from a starting value of 14.8 million. 

If you calculate the exact probability assuming everyone picks their numbers randomly then you get what I would call 'reasonable' odds, so the skew on the bet isn't absurd and it isn't mad to think people would want to bet on this. 

The maths here isn't simple as the longer these stretches of no jackpot go on, the more tickets are sold as the jackpot gets more attractive (in the aforementioned period of 13 draws average sales nearly doubled from 20 million to almost 40 million by the end of the run). So you would need to be able to predict ticket sales and have a good understanding of both the rules of the lottery and betting patterns. Ironically then there will be more skill required here than just betting on random numbers.

I will finish with - as promised- my own lottery winning strategy: If you are going go bet on the lottery, normally a -EV bet as discussed, then doing so when the jackpot is very high after lots of rollovers isn't a bad idea. The retained money from earlier draws makes it a slightly +ve EV bet. Whilst this doesn't guarantee a win, it does mean that in all possible lifetimes you will come out ahead. So that's nice.

See this paper which Andre Loeffler pointed me towards after I published this post.

Are markets that are good for trend good just because they have also gone up a lot, or because carry, or...

I have a friend, ex-colleague and TTU co-host who runs a fixed income focused CTA. We have regular coffees (he pays, so his fund is doing ok) and one of our favourite topics for arguing debating making polite conversation about is why fixed income is so much better for trend than anything else. He's biased, but then arguably so am I; I started my career trading rates options, and we both managed AHL's fixed income portfolio (though not at the same time).

Certainly if you looked at the performance of fixed income it looks good! Consider this extract from my fourth book, tables 37 and 38:

Only vol is better, and that is just two markets. But then, bonds have been a great investment on a long only basis. The data in that book covers a period when bond prices could very easily be summarised as "number go up!". Here are tables 10 and 11 from the same book:

Bonds have the highest Sharpe Ratio on a long only basis (actually it's long only with position sizes adjusted daily for vol: a constant forecast of +10). 

But still, there must be something else going on here. Equities have also done awfully well long only (SR 0.46), but they have the worst trend following performance of all. FX has had crap SR long only, but did pretty well trend following. What's more, it will probably be useful to distinguish between different kinds of tailwind from secular up trends: 

• Price go up (or down, since if we're trend following long/short we are cool with that, c.f. vol)
• High Positive carry (or negative, same reasons)

Of course there are fancy ways of dealing with this, like CAPM style regressions (of which there are plenty in the book); but let's keep things simple here with some scatter plots.

What is being scattered? Well on the y-axis let's do SR of trend following, and on the x-axis the SR of the long only adjusted price return, spot price return, or cumulated carry return (the difference between spot and adjusted price). Each scattered point will represent the returns of one market over a five year period, and I colour the point to indicate the asset class. And we'll also need to do different graphs for each speed of trend following.

Messy python (requires psystemtrade)

Fast trend (momentum 8)

We would expect for faster momentum to have a relatively weak relationship. Let's see:

Yup, nothing there.

Medium trend (momentum 16)

Perhaps the start of a slope there, but not for carry.

Slow trend (momentum 64)

So adjusted prices look good, spot prices less so, and carry not much going on. Don't forget that adjusted is mechanically spot+carry, but over time that addition will make sense if they are uncorrelated. It looks like adjusted prices show clearer trends than eithier of the components do.

But is there anything special?

This doesn't tell us whether there is something special about bonds, equities or FX. Let's repeat the plots for momentum64 but without the scatter points; instead we fit a regression line to each asset class and plot that.

Now that is the money shot for this blog! We can see a clear split:

• Bonds, FX, Metals, Vol: We can convert adjusted price drift into trend SR, with a slope of around 0.9. Note: The vol line in green is the one almost perfectly aligned to the orange line going up to SR 1.0 on the x-axis.
• Ags, Energies, Equities: The relationship is much weaker 

What about the other two types of prices?

We can see there is a consistent effect in spot prices, but carry is much weaker. The exception is vol; much of the adjusted price downtrend in vol comes from the vol premium which is basically carry on the futures curve.

I won't do everything, but here are the adjusted price plots for the other two speeds.

Although the vol line has a worse intercept, the same pattern of the relationship is present.

Summary

To an extent, secular up(down) trends in asset prices make it easier to make profits from (slower) momentum. The effect is roughly 0.7 to 0.9 units of trend following SR for every unit of secular drift depending on the speed of trading. And it's remarkably consistent across asset classes that fall into this category: they are not energies, equities or ags. So there is nothing special about bonds; but there are two distinct types of asset classes. 

Here is another view: a distribution of trend SR returns in five year blocks by asset class for momentum64:

And combining with this plot, showing the distribution of adjusted price returns (SR) in five year blocks by asset class:

One last scatter plot of the medians SR from the above two plots:

We can say:

• Bonds, FX and Vol did as expected, converting trends of various strengths to momentum trend SR (they lie on an imaginary line running diagonally upwards)
• Metals, which we know is also a strong trend convertor, did even better than expected 
• Equities is definitely an underperformer as we'd expect from a poor trend convertor.
• Energies and Ags seems to have got lucky; though they are not great trend convertors they did better than expected (lying just above the imaginary line).

Is this trend conversion a property of the asset class, or just a fluke? I dunno. But it doesn't look like there is anything special about bonds.