Sizing futures bets

[Ganchrow]

How do those of you who bet futures tend to size your bets?

I've put together a modified version of Kelly (what else?) to account for time-value-of money, but am interested in knowing if anyone else has any ideas.

[Doug]

I usually bet the limit on futures.

[Wheell]

Futures are problematic for two reasons: The first is that you have time/value of money issues. The second is that it is difficult to truly know your edge with the large increase in variables a futures bet provides. I took a team totals bet in baseball that iIfelt was seriously off but I had to consider that bet a "fun" bet even though it was for max. Let me put it this way: You will have a very hard time generating strong returns relying on futures bets. On the other hand, in terms of creating a long term rooting interest for a cheap amount futures are great.

[Ganchrow]

I usually bet the limit on futures.

Most books offer rather large limits on season totals futures, which may not always be a reasonable bet quantity, especially when betting at longer odds.

Back in October of 2005 (can't get there from here) I had posted what I had considered (and in retrospect still do consider) a long-term value play. The bet was at the Greek and the wording was "Independent candidate to win White House in 2008." This bet was offered at +50,000 which I believed to be quite generous. InTrade was at the time offering the same bet at +4,445 for and -5,782 against.

So even if we assumed that all the market vig on the InTrade bet is on the the dog side (resulting in a break-even fave bet), this would imply a win probability of about 1.7001%, correspondsing to an incredible edge of a little over 750%! For a quarter-Kelly bettor, this would imply a stake of 0.3759% of bankroll (assuming no hedge). Considering that the maximum bet was $1,000 (I believe -- although memory could be failing me again), this means that those quarter-Kelly bettors with bankrolls of approximately $266,000 or more (which, somewhat coincidentally, I'd expect to roughly represent the median bankroll among pro and semi-pro sportsbettors) would bet the max, and those with smaller bankrolls would bet less than the max.

So this has all been straight forward Kelly up to this point. But here's the question ... let's assume for the moment that you could bet the InTrade market at zero slippage (meaning that one could transact infinitely large quantities without impacting the market price -- a rather specious assumption to be sure). In this situation, how much would one bet on the +50,000 instrument and how much would one bet on the hedge instrument?

It should quickly become apparent that Kelly is effectively useless on this situation. Because the market contains a riskless arbitrage, the sum of bets across the two instruments, assuming the max bet size constraint on neither leg binds, is 100%.

To best see why this probably won't accurately reflect most bettor's preferences, we should look at the specifics. The quarter-Kelly stake on this market would theoretically be 0.5939% on the primary bet (Independent wins) and 99.4061% on the hedge (Independent does NOT win). This means that if the Independent candidate won (a 1.7001% likelihood event) the player's bankroll would increase by 197.5%, and if the Independent candidate lost (a 98.2999% likelihood event) the player's bankroll would increase by 1.125%, an expected return of 4.4643% over 3 years, and an expected annualized return of 1.4665%.

So does this seem attractive? Would you really want to tie up ALL your money for 3 years just to earn an expected return of 1.4665% per year? (This corresponds to a certainty-equivalent of 2.0944% for 3 years and 0.6933% annualized. Pretty poor -- especially when you consider credit rating uncertainty for the Greek). I know I'd never make this bet.

So this is the general problem with long odds futures bets ... you can't really bet all that much on the underdog as you're very likely going to lose, and if you hedge, you could be tying up a lot of money for a long period of time, with a greatly diminished expectation.

Conversely, on a short odds future, while the hedge would be very cheap, the expected return on the primary bet would be quite low. Furthermore for sufficiently short odds bets, chances are that even for the smallest bettors the max bet size would constrain. (This is part of the reason why short odds futures bets often trade at low or no vig. Justin had pointed out this phenomenon a month or so ago regarding Super Bowl futures. And if this were the case for 2 or 3-day futures, imagine how much more likely it would be to hold for 2 or 3-month or longer futures -- this could be quite relevant to bettors sufficiently with low hurdle rates.).

So what's the solution? How does one optimize in the face of a pure arbitrage and a relatively long period to contract expiry? What about if you were able to buy on margin?

Does anyone have any thoughts on this? I'll post what I consider to be the most direct solution if people show interest.

[Ganchrow]

Futures are problematic for two reasons: The first is that you have time/value of money issues.

Very true, indeed.

The second is that it is difficult to truly know your edge with the large increase in variables a futures bet provides.

I think it often can be fairly easy to identify prop bets that are likely to be profitable using a Poisson valuation methodology, an issue discussed by Stanford Wong in Sharp Sports Betting.

On the other hand, in terms of creating a long term rooting interest for a cheap amount futures are great.

A fact that certainly hold true for the recreational gambler.

[ShamsWoof10]

Buying on margin at a sportsbook..? What book let's you do that and what would you have to do to qualify..?

I think futures are relative.. Super Bowl futures are much ore difficult then regular season win totals although the odds are better too... I think it is much easier to correctly pick a regular season win total as opposed to a regular game... A good team will be a good team throughout the year but can lose on "Any Given Sunday"... For example If you put your bankroll on four team's season wins in the NFL you are more likely to hit those then picking games week to week over a three month course... I had a friend who use to bet regular season wins and would hit most of them, in comparison he would bet throughout the season and be up some weeks and down others... At the end of the year he profited more off of the futures...

[Ganchrow]

Buying on margin at a sportsbook..? What book let's you do that and what would you have to do to qualify...

No book of which I'm aware, although it certainly wouldn't surprise me if this were available in the UK.

In theory, there's no reason why books shouldn't provide this service. Margin rates would be another way for books to make money. Betting credits are essentially free to books and so the only real issue is risk.

On reconsideration, I believe that Tradesports has or had something like this.

[Justin7]

Futures are often easy to beat - so easy, that you tend to just fire away at the limits even with a moderate bankroll. In most futures though, you're done with your bet in 6 months tops.

I assume that most futures wagers have at least a 10% EV. If my money is tied up for 6 moths, I'm still getting about a 20% annualized return.

On your presidential future play, you are tieing up money for at least 18 months. You could buy a risk-free treasury bond returning over 5% per year... so to tie up money for 18 months, you need better than a 7.5% EV. I wouldn't play the arbitrage you identified, because the EV is worse than conservative investing.

Now, if you have credit, you can be much more aggressive with futures (as long as the bet limits are still the constraining factor).

[Ganchrow]

I think futures are relative.. Super Bowl futures are much ore difficult then regular season win totals

Here's where Justin talked about Super Bowl props. It's a good read, and Justin was right-on in his analysis.

Super Bowl props. while technically only veryshort-term futures, may have more in common with longer-term futures than one might think at first glance. This would be the case insofar as the cost of capital may be considerably higher around Super Bowl time.

[Ganchrow]

I wouldn't play the arbitrage you identified, because the EV is worse than conservative investing.

Right. That's the whole point. While the "solution" I gave is indeed Kelly-optimal, no one is really going to play. The question is, what is the correct way to optimize?

Now, if you have credit, you can be much more aggressive with futures (as long as the bet limits are still the constraining factor).

Possibly. Although there's still an opportunity cost issue when playing at a one of a bettor's regular credit shops.

An interesting question would be how to optimize if on top of your $X 0% interest you could also borrow from the book at some specified interest rate. For example, if you could borrow at 1% per annum (remember that the book isn't really floating cash, but rather only betting credits --hence it could afford to offer very low rate) and can make 1.5% per annum guaranteed on your futures position, then that's pure arbitrage profit (credit risk aside).

[Justin7]

Right. That's the whole point. While the "solution" I gave is indeed Kelly-optimal, no one is really going to play. The question is, what is the correct way to optimize?

Possibly. Although there's still an opportunity cost issue when playing at a one of a bettor's regular credit shops.

An interesting question would be how to optimize if on top of your $X 0% interest you could also borrow from the book at some specified interest rate. For example, if you could borrow at 1% per annum (remember that the book isn't really floating cash, but rather only betting credits --hence it could afford to offer very low rate) and can make 1.5% per annum guaranteed on your futures position, then that's pure arbitrage profit (credit risk aside).

Optimizing... if you are post-up, decide what you think a reasonable return on your investment is through other means. Look at your betting history for the last 3 years. Smaller plays can realize amazing returns hitting props and overnight lines. It might even be 50% per year. It wouldn't be lower than 5% a year.

Whatever your expected alternative return, deduct that from your theoretical EV. If you think your future has a 10% edge, but you could make 8% annualized for 18 months, that futures play is actually -2% compared to your alternative investment.

For most credit players, their credit is much higher than the limits for futures such that there isnt' really an opportunity cost of using it. If it were though, I'd use the same methodology.

[Ganchrow]

Optimizing... if you are post-up, decide what you think a reasonable return on your investment is through other means. Look at your betting history for the last 3 years. Smaller plays can realize amazing returns hitting props and overnight lines. It might even be 50% per year. It wouldn't be lower than 5% a year.

Whatever your expected alternative return, deduct that from your theoretical EV. If you think your future has a 10% edge, but you could make 8% annualized for 18 months, that futures play is actually -2% compared to your alternative investment.

This will only hold in the case of a risk-neutral player. In the case of a risk-averse player, it's actually a bit more involved.

Assuming Kelly preferences, and
o = odds
p = win prob
r = risk-free interest rate over relevant time period
k = Kelly multiplier (1 => full-Kelly, ½ => half-Kelly, ¼ => quarter-Kelly)

the optimal stake, S* is given by

     (op-r-1)   (k+r)
S* = -------- × -----
     (o-r-1)    (1+r)

[SIZE="1"][I](proof available upon request)[/I][/SIZE]

So to give an example, assuming a quarter-Kelly bettor, payout odds of +500, a win probability of 20%, and a risk-free interest rate of 4%, the correct stake would be (6×20%-4%-1) / (6-4%-1) × (0.25+4%) / (1+4%) = .675%. (Using the linear method would yield 0.600%, which to be fair is only off by about 12½%)

But this is all fairly trivial. What we're really looking to do is determine how to jointly optimize the primary and hedge positions subject to both individual and aggregate budget constraints.

For most credit players, their credit is much higher than the limits for futures such that there isnt' really an opportunity cost of using it. If it were though, I'd use the same methodology.

The issue at hand is that of hedging a very large underdog. In such a case the opportunity cost could be substantial (and as mentioned earlier, this could be an especially powerful effect when liquidity is at a premium -- say right around Super Bowl or NCAA Tourney time).

A case in point would be the 2008 election example I gave above. This is made even more explicit when one considers the hedge. On a $1,000 bet, hedging completely would require a $493,482.49 bet on the fave. Freezing up this credit line over a 3-year period may well be substantial. Now one might certainly respond that the expectation on the bet would only be $7,517.51 -- certainly nothing to write home about for a bettor with a half-million dollar line of credit, but possibly not completely irrelevant either when considering it's a pure arbitrage.

So the issue returns by commodius vicus of recirculation to opportunity cost and environs.