A few thoughts on money management

If you only read one thing by this handicapper, let it be this: An eloquently written piece on handling your bankroll.

If imitation is indeed the sincerest form of flattery, then castigation has to finish a close second. So I have to say that it’s a perverse pleasure to see my name in the paper, so to speak. Even if it’s just a handicapper who vehemently, vociferously, and uncategorically believes my money management strategies are wrong and fundamentally unsound. For the un-initiated, my bet-sizing strategy is quite simple: make a flat percentage bet (I recommend 1%, but anywhere up to 2% is okay) no matter what happened yesterday or the day before, and your advantage will come through over time. Then, re-evaluate your bankroll and re-figure that percentage at a pre-determined point, either in time (6 months, 12 months) or volume (50% increase in bankroll, 100%, etc.). The choices to be made within the above parentheses really depend on your own investment needs. There’s no set right answer for those questions. But what is a bit more black and white is the first part: flat percentage betting. My most vocal detractor (to date), whose picks and commentary are also found on TDS, is quite adamant about the flat-bet “plateau” method’s inadequacy. His thinking, the nearest I can figure it, is that because you’ve got a positive expectation on each bet, you should expect to win money every day. Because you’ll win money every day, you should steadily increase your bets directly proportionately to your bankroll, in order to maximize profit. One’s bankroll, it would figure then, would climb upwards and upwards ad infinitum since we have a positive expectation. Piece of cake. End of story. Right?

Wrong. The problem with our so-called positive expectation in the above scenario is that it doesn’t manifest itself every  day we bet. Just because we expect to win 60% (for the sake of round numbers; actually, I expect to win the equivalent [including moneylines] of between 56-58%) doesn’t mean that we’ll go 3-2 every day. If we went 3-2 every day, this would be an easy business! Instead, there’s this thing called binomial distribution that I had explained to me by people who know math much better than I do. What it means is despite our example of a 60% winning expectation, we can go 1-4 in a small enough sample size. Only when the sample size (the number of games) is much, much larger do we expect to see that 60% edge manifest itself. Imagine that you’ve got a weighted coin that you bought after being told that it would land on heads 60% of the time. Flipping that coin ten times wouldn’t be enough to find out if the guy who sold you the coin ripped you off. It could be 8 tails, 2 heads, or 5 and 5, or just about anything. Only after about a thousand flips would the edge start to emerge, shrinking the standard deviation. So what has this got to do with betting a rolling percentage of your bankroll based on a “current balance” rather than a “flat percentage”? Simply put: we have no idea (at least I don’t!) of what I’m going to do in the span of a day, betting-wise. I could go 0-6 or 6-0. In a year, I know that I won’t go the equivalent of 6-0 or 0-6, but not in a day.

So adjusting daily doesn’t make sense – the daily fluctuations are too great. Another example: In a span of 46 decisions, I’d be delighted to go 26-20, but there’s no way I could predict in what order those wins and losses would come. And in the rolling percentage scheme, it’s the order – which is essentially a randomly decided-upon factor – that is the main ingredient. I will illustrate the above example and do the math that my counterpart urged me to. Here goes.

Last year in the NBA, I had the following 10 days (assume that all bets are 11-10):

Day One: 4-5

Day Two: 6-4

Day Three: 0-2

Day Four: 3-1

Day Five: 4-0

Day Six: 2-4

Day Seven: 0-2

Day Eight: 3-1

Day Nine: 2-0

Day Ten: 2-1

TOTAL: 26-20, +4 units

I think we’d all agree that this order of wins and losses on a daily basis is no less ordinary than any other. So let’s examine what would happen with a $1,000 bankroll: $10.00 to win $9.10 as our first-day bet, and adjusting to the nearest penny thereafter, assuming a 1% bet (if you want to do 2%, fine, it’s simply a matter of scale):

Day 1: -13.65 986.35

Day 2: +14.38 1000.73

Day 3: -20.00 980.73

Day 4: +16.98 997.71

Day 5: +36.32 1034.03

Day 6: -22.54 1011.49

Day 7: -20.22 991.27

Day 8: +17.15 1008.42

Day 9: +18.34 1026.76

Day 10: +8.43 1035.19

Not too shabby: a $35.19 cent profit on a $1,000 bankroll, good for 3.52%. The flat-bettor’s result, on the other hand, is easier to figure out:

26 wins x +$9.10 = 236.60

20 losses x -$10.00 = 200.00

A little bit better: a $36.60 profit on that same $1,000 bankroll, good for 3.66%.

It’s important to note that there are most definitely orders of wins and losses that would compute larger gains for the first method. And likewise, there are examples that I could have cooked up that would make it seem even worse for the rolling-percentage player. But the important thing to emphasize is that the fluctuations are completely unpredictable and random. I’m currently in the process of finding a computer science friend of mine construct a model that will give us the rough probabilities of the rolling percentage method actually benefitting the long-term investor. I’m not saying it can’t ever happen. I am saying that if it does happen, it’s because of chance rather than any statistical principle.
thedailyspread.com | March 17th, 2000

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