Monday, June 4, 2012

The Trader Who Made $710,000 at the Blackjack Table: Why it is Better to Be Lucky Than Good Sometimes


There was a popular article in the Wall Street Journal on a hedge fund manager that racked up $710,000 in profit at the blackjack table.  According to the report, he started with a $10,000 line of credit and turned in his big profit over a couple of sessions at the tables in the Bellagio.  He summed up his strategy as follows:

[He] follow[s] the instructions on a blackjack strategy card he lays out on the table. “The only thing I do is I vary my bet sizes based on how well I’m doing,” he said. Each time he wins, he increases the size of his wager; similarly, he decreases the size of his wager each time he loses. “The concept is, if I lose 10 hands in a row, I’m going to lose my minimum bet at least nine of those times…but if I win 10 hands in a row by pressing my bet every time—not doubling it, just making it bigger—I’m going to win a lot more than when I lose 10 hands in a row,” he said.

Setting aside the ins and outs of “proper” blackjack strategy for a second (do you always double down on 10?), I thought it would be interesting to simulate his risk management and position sizing strategy. 

Based on the above quote, I extrapolated the following rules:

-  Break down your starting bankroll into units.  For instance, if you have a $10,000 bankroll to start with, divide your bankroll into 100 units for a basic bet of $100 per hand.

-  Your first bet will always be 1 unit.

-  If the last hand was a win, the next bet size will be increased by some multiplier.  As the article stated, you don’t double your bet but increase by some set amount.

-  If the last hand was a loss, the next bet size will be one unit times some multiplier.

-  Expectancy and edge play a large role in blackjack strategy.  In general, the house or casino has about a 5% edge over blackjack players.  Using perfect blackjack strategy and a 6 card deck, the house advantage can be reduced to 0.44%.  According to one resource, a card counter who performs flawlessly can expect a 1% edge over the casino. 

So in the best case scenario and assuming the hedge fund manager was not counting cards, his edge would be -0.44%.  This is the definition of a negative expectancy outcome – there is no way that he could win over the long run.  Your winning percentage would then be defined as 49.56%.

Building the Spreadsheet

If you have read any of the recent posts, you know the ins and outs of a Monte Carlo simulation.  For this simulation, I again assumed we would run 1000 simulations of 500 hands and determine the statistics on the entire run.

I assumed we would start with a bankroll of $10,000 and that a unit would be $100 giving us 100 units to start with.

If the previous hand was a loss, our Losing Multiplier would be 1.  Stated otherwise, if we lose a hand then our next bet will be equal to 1 unit times the Losing Multiplier.  This is the “cut your losses” mentality.

If the previous hand was a win, our Multiplier would be 1.1.  If we win a hand then our next bet will be equal to our profit on the previous hand times the multiplier.  This is the “let your winners ride” reasoning. 

Here is a screenshot of the spreadsheet:


For Hand No. 0, we reference $F$2 as our starting bankroll.

For Hand No. 1, the formula is:

=IF(RAND()<=Winning Percentage, Previous Account Balance + Unit, Previous Account Balance – Unit)

Or

=IF(RAND()<=$I$3,E7+$F$1,E7-$F$1)

This tells Excel to generate a random number.  If the random number is less than or equal to our winning percentage, then you add 1 unit to the starting bankroll.  If the random number is greater than our winning percentage, then you subtract 1 unit from the starting bankroll.  We don’t worry about the Losing Multiplier or Winning Multiplier because we don’t have a previous hand history.

For Hand No. 2, we apply the “let winners ride” and “cut our losses” mentality.  The formula is:

=IF(RAND()<=$I$3,IF((F7-E7)>0,F7+((F7-E7)*$I$2),F7+$F$1*$I$1),IF((F7-E7)>0,F7-((F7-E7)*$I$2),F7-$F$1*$I$1))

Again, we tell Excel to compute a random number to determine a “win” or a “loss” for the current hand. 

Then you have to look at the bankroll for the previous hand to determine our bet size.  Remember, if we had a winner last hand we would increase our bet size to an amount equal to the profit times the winning multiplier.  If we had a loser last hand, our bet size is equal to one unit times the losing multiplier. 

So if Hand No. 2 is a win AND your bankroll from the last hand increased (last hand was a winner), then take the profit from the last hand times winning multiplier and add it to your bankroll.  Conversely, if Hand No. 2 is a loss AND your bankroll from the last hand decreased (last hand was a loser), then subtract one unit times the losing multiplier to the previous bankroll.

The process described above is reversed if Hand No. 2 is a loser.

Applying these rules to Simulation 1, here is a chart of our bankroll over 500 hands:


Running the Statistics

Next, we can look at the statistics after 1000 runs of 500 hands of blackjack to determine our expectations for what this system gives us as far as profit and loss.

Profit/Loss
Minimum
(8737)
Maximum
8763
Average
(527)
Median
(508)
Standard Deviation
2516
Low
High
1-SD (68% Confidence)
(3043)
1988
2-SD (95% Confidence)
(5559)
4504

Here is the histogram for our system:

Profit/Loss
Bins
Occurrences
Ind %
Cum %
(8,737)
1
0.10%
0.10%
(6,987)
3
0.30%
0.40%
(5,237)
22
2.21%
2.61%
(3,487)
92
9.24%
11.85%
(1,737)
194
19.48%
31.33%
13
277
27.81%
59.14%
1,763
225
22.59%
81.73%
3,513
129
12.95%
94.68%
5,263
38
3.82%
98.49%
7,013
12
1.20%
99.70%
8,763
3
0.30%
100.00%

Yikes, we didn’t get anywhere close to $710,000!  In fact, this wasn’t even in the realm of possibility!  As expected, you can easily expect to be down $527 on average if you bet 110% of your profits after every winning hand.  At best, you would be up more than $8,700 bucks 3 times out of 1000!

Changing Our Parameters to Make $710,000

If we are to accept that the house edge is .44% even playing perfectly, then what winning multiplier would we have to use to even have a chance of making $710,000?

By using a Winning Multiplier of 1.6 (betting 160% of your profits after a winning hand), then we do have an outside chance of making the type of money that we want.  Check out the stats below:

Profit/Loss
Minimum
(15375)
Maximum
819518
Average
(490)
Median
(3788)
Standard Deviation
32294
Low
High
1-SD (68% Confidence)
(32785)
31804
2-SD (95% Confidence)
(65079)
64098

Profit/Loss
Bins
Occurrences
Ind %
Cum %
(15,375)
1
0.10%
0.10%
68,115
992
99.20%
99.30%
151,604
5
0.50%
99.80%
235,093
0
0.00%
99.80%
318,583
0
0.00%
99.80%
402,072
0
0.00%
99.80%
485,561
0
0.00%
99.80%
569,050
1
0.10%
99.90%
652,540
0
0.00%
99.90%
736,029
0
0.00%
99.90%
819,518
1
0.10%
100.00%

This says that 1 time out of 1000, we will go on a heater and win $736,029 - $820,000!  However, it is way more likely that we will lose money with this system.

Limitations on the Spreadsheet

1.  During all of the runs, we don’t simulate getting a blackjack and winning 3:2 or 2:1.  We always win what we bet.  I would imagine that this would skew the numbers up a bit.

2.  It allows us to bet an unlimited amount at anytime.  Again, most casinos wouldn’t let you bet $100,000 on one hand!  This happened more than once on our “heater.”

3.  We don’t take into account surrendering, insurance and all of the other fun stuff that goes into playing blackjack. 

4.  It assumes we play “perfectly” at all times. 

5.  We assume that you can lose more than your starting bankroll.  Can you say “marker”?

Changing the Multipliers and Winning Percentage

Again, just for fun, let’s change some of the parameters around so that we can see what happens.

What if we were expert card counters and could have an Edge of 1% and a Winning Multiplier of 1.1?

Profit/Loss
Minimum
(5683)
Maximum
8856
Average
1114
Median
1007
Standard Deviation
2536
Low
High
1-SD (68% Confidence)
(1423)
3650
2-SD (95% Confidence)
(3959)
6186

Profit/Loss
Bins
Occurrences
Ind %
Cum %
(5,683)
1
0.10%
0.10%
(4,230)
13
1.31%
1.41%
(2,776)
44
4.42%
5.82%
(1,322)
97
9.74%
15.56%
132
208
20.88%
36.45%
1,586
227
22.79%
59.24%
3,040
191
19.18%
78.41%
4,494
113
11.35%
89.76%
5,948
66
6.63%
96.39%
7,402
22
2.21%
98.59%
8,856
14
1.41%
100.00%

What if we were expert card counters and decided to increase our bets when we lost and when we won – a losing multiplier of 1.1 and a winning multiplier of 1.1?

Profit/Loss
Minimum
(7347)
Maximum
9026
Average
1276
Median
1101
Standard Deviation
2749
Low
High
1-SD (68% Confidence)
(1473)
4024
2-SD (95% Confidence)
(4222)
6773

Profit/Loss
Bins
Occurrences
Ind %
Cum %
(7,347)
1
0.10%
0.10%
(5,709)
6
0.60%
0.70%
(4,072)
23
2.31%
3.01%
(2,435)
54
5.42%
8.43%
(797)
132
13.25%
21.69%
840
234
23.49%
45.18%
2,477
212
21.29%
66.47%
4,114
189
18.98%
85.44%
5,752
89
8.94%
94.38%
7,389
43
4.32%
98.69%
9,026
13
1.31%
100.00%







No comments:

Post a Comment