## The Electric Day Trader and Ruin

### How probability oppposes short-term investors.

Day trading is extraordinarily risky.

Mark Twain’s cat, having sat upon a hot stove lid, will never again sit upon a hot stove lid. Nor will he sit on a cold stove lid. Today’s electronic day trader and Mark Twain’s cat have a lot in common– both get more out of the experience than is in it.

### What We Know

Beginning with Louis Bachelier’s classic 1901 doctoral dissertation *Theorie de la Speculation,* which developed the mathematical theory of random processes, the financial community has steadily developed a body of evidence supporting the hypothesis that no useful information is contained in the sequence of past changes in stock prices. Current prices reflect all relevant known information, and therefore prices change only when new information arrives. New information, by definition, cannot be predicted ahead of time. Given that, technical analysis, which is based on premise that it is possible to identify and project market and individual stock trends, is suspect to say the least.

The mountain of evidence amassed to support the independence of changes in stock prices — and therefore the understanding that efficient financial markets have no memory — does not faze today’s electronic day trader, nor does it affect the addiction to buying and selling stocks several times a day. The apparent missing memory is the memory of past market blood baths where day traders have been reminded that they live in a risk-return world, and that the job of the markets is to price risk.

### Why We Behave the Way We Do

A rash of interest in behavioral finance has resulted from the great bull market of the 1990’s. The rise of risk taking in America has induced psychologists to come up with reasons why classical risk aversion preferences have been scuttled. One of the explanations is that the Internet trading world is the beneficiary of the tremendous wealth building of the past few years. This gives birth to the “house money” effect, which states that when a potential trader starts with zero in an account, the trader is likely to take a sure gain over a risky bet, even if the risky bet has a higher expected value. But when that same trader started with $30 instead of zero, far more people take the risky bet. The long bull market has increased the number of investors who believe they are starting with money in the account.

Another strong influence on day traders is the publicity about some wildly successful Internet initial public offerings (IPO’s). Everyone can envy the big winners, and, like Pavlov’s dog, it takes only an occasional winner to reinforce the belief – erroneous though it may be — that food will be in the bowl every time the bell rings.

When traders are reminded daily that current earnings seem not to be a prerequisite for wealth building (the “Amazon.com effect”), and mutual funds constantly advertise the reduction of risk acquired by taking the long view, it follows that the return required by shareholders for taking a unit of risk has steadily drifted downward. It is easy for the day trader to forget in this environment that not only are there behavioral reasons to question the long run wisdom of the national addiction to day trading, but also sound financial reasons exist as well.

### The Concept and Practice of Ruin

The financial community defines risk as the variability (standard deviation) of the returns received. Indeed, expected returns are determined by the risk taken. A well-diversified investor is paid proportionate to the level of risk taken. The works of several Nobel Laureates support the innate wisdom of this concept.

However, the day trader sees risk and return in a different light. The horizon is twenty minutes, not twenty years, away; and trading is a game dependent upon personal discipline and money management. For this reason, the day trader has to consider the probability of ruin, a different definition of risk which refers to the loss of all of the invested capital.

Assume you are a day trader with beginning capital of $10,000. You have brilliantly isolated and quantified a trading scheme that results in a 55 percent probability of a profit on any trade after commissions and taxes. Your chance for loss is therefore 45 percent. Your mathematical advantage on each trade is 10 per cent. Assume that you plan to trade indefinitely rather than aim toward some specific objective, that no change is expected in these presumed probabilities, and that your attitude toward your gains and losses will remain unchanged as your fortunes ebb and flow. You feel ready and willing to embark on a journey destined to bestow wealth beyond the dreams of Midas.

Before placing the first trade, consideration of ruin demands an early decision – perhaps the most critical decision the day trader will ever make. How much of the $10,000 will the trader risk on each trade? Assume the trader decides to use $2,500, or one-fourth, of the $10,000 capital for each trade, and plans to continue making $2,500 commitments indefinitely.

For simplicity’s sake, assume that 55% of the time the trader will double the amount invested on each trade, and 45% of the time will lose the entire amount of that bet. Any sequence of trades where the number of wins exceeds the number of losses results in a net profit. Conversely, any sequence of trades where the number of losses is greater results in a net loss.

There are two ways to consider the impact of ruin. The first is to consider the worst thing that can happen to the trader, which would be four consecutive losses. The chance of losing on the first trade is 45 percent. The chance of losing on all of the first four trades is (.45)^{4} or only 4.1 percent. So far, thinks the trader, “no problem.”

If the trader has a net profit after the first four trades, i.e., has won at least three times, the larger level of capital means that four successive losses can no longer cause ruin. However, now the trader must confront the second source of ruin, four net losses at any point in time.

The probability R of eventual ruin is given as follows: R=[(1-A)/(1+A)]^{C}

where A is our day trader’s advantage on each trade expressed in decimal form (10 percent in the above example), and C is the number of beginning trading units (4 in our example). As shown in Table I, even with the 10% advantage, our hypothetical trader’s resulting probability of eventual ruin is an appalling 45 percent!

### Table I: Probability of Ruin and Survival

10 percent Advantage per Bet

Number of beginning trade units | Probability of ruin (10 percent advantage per bet) | Probability of survival (10 percent advantage per bet) |
---|---|---|

1 | 81.82% | 18.18% |

2 | 66.94% | 33.06% |

3 | 54.77% | 45.23% |

4 | 44.81% | 55.19% |

5 | 36.66% | 63.34% |

6 | 30.00% | 70.00% |

7 | 24.54% | 75.46% |

8 | 20.08% | 79.92% |

9 | 16.43% | 83.57% |

10 | 13.44% | 86.56% |

11 | 11.00% | 89.00% |

12 | 9.00% | 91.00% |

13 | 7.36% | 92.64% |

14 | 6.02% | 93.98% |

15 | 4.93% | 95.07% |

16 | 4.03% | 95.97% |

17 | 3.30% | 96.70% |

18 | 2.70% | 97.30% |

19 | 2.21% | 97.79% |

20 | 1.81% | 98.19% |

Let’s restate that finding. The day trader has completed proprietary research which shows a ten percentage point probability advantage in each trade, more than twice the advantage that the gambling casino enjoys in the game of roulette. Yet, if the trader plays the game indefinitely, betting 25% of initial capital on each trade, the probability of ruin is perilously close to the results of a coin flip!

Examination of this depressing outcome leads to a crucial insight. Assuming the research is correct, the only variable that can change is C, the number of trading units with which the trader begins. Feverishly, the trader substitutes $1,000 of risk and return for the results of each trade, and enters a C of 10 in the formula. The chance of losing all of the $10,000 capital in the first ten trades drops to (.45)^{10} or less than one chance in 3,000. However, the probability of ultimate ruin is higher than that. Table I shows that our trader’s probability of ruin is approximately 13 percent, while the expected gain from each trade has dropped from $250 = [.55($2,500) + .45(-$2,500)] to $100 = [.55($1,000) + .45(-$1,000)]. That difference can easily be made up over time, and with a low probability of ruin, time is exactly what you get.

The next demon our day trader will have to confront is the decision of what to do if the trading account grows on schedule, thereby affirming the validity of the research. If the original capital of $10,000 has grown to $20,000, the chance of being ruined by trading in $1,000 chunks has become much smaller than the original 13 percent because now it would take twenty net losses rather than ten to cause ruin. The probability of this happening is only 1.81 percent. The chance of continuing to add to capital over time becomes more and more certain if our trader does not increase the scale of the trading commitment.

Unfortunately, it is at this point that compulsion, ego, and environmental mania are likely to take over. Success, no matter how short-lived, is extrapolated to infinity. When the initial capital is doubled, that leaves only six doublings necessary to top $1 million. The trader decides to begin trying for $40,000 by doubling the size of his trading units.

Unfortunately, while it took ten net profits to take the trader from $10,000 to $20,000 at $1,000 per trade, at $2,000 per trade it takes only five net losses to return the trader to his original capital base of $10,000. This realization is sobering because it raises the probability of ruin significantly.

If each doubling of capital is accompanied by a doubling of the “bet unit,” a frightening scenario ensues. Recall that the probability of ruin facing the trader who plays the original game indefinitely was only 13 percent. That infers that the long-run probability of successful survival was 87 percent. The probability of reaching the trader’s goal of $1 million or more by doubling capital and then doubling the bet size seven successive times is therefore (.87)^{7}, or about 38 percent. The chance of being ruined en route is 62 percent.

It is important to note that the typical day trader probably exaggerates the real chance of success for at least two reasons:

- The presumed ten percent advantage undoubtedly has some high hopes mixed in with the research.
- Independence of results, while desired, often does not exist, which makes runs of so-called “bad luck” more likely than the computed probabilities would indicate.

The effects of such exaggeration are shown in Table II which demonstrates the trader’s probability of success with various strategies from a “sure thing” (100 percent winners) down to doing no better than having winners and losers equal. The table assumes the money is divided into four equal units and is bet in equal units over time.

### Table II: Probability of Ruin or Survival at Different Advantages per “Bet”

Advantage per bet | Percent winners | Percent losers | Probability of ruin (25 percent of initial stake per investment) | Probability of survival (25 percent of initial stake per investment) |
---|---|---|---|---|

100.00% | 100.00% | 0.00% | 0.00% | 100.00% |

90.00% | 95.00% | 5.00% | 0.00% | 100.00% |

80.00% | 90.00% | 10.00% | 0.02% | 99.98% |

70.00% | 85.00% | 15.00% | 0.10% | 99.90% |

60.00% | 80.00% | 20.00% | 0.39% | 99.61% |

50.00% | 75.00% | 25.00% | 1.23% | 98.77% |

40.00% | 70.00% | 30.00% | 3.37% | 96.63% |

30.00% | 65.00% | 35.00% | 8.41% | 91.59% |

20.00% | 60.00% | 40.00% | 19.75% | 80.25% |

10.00% | 55.00% | 45.00% | 44.81% | 55.19% |

9.00% | 54.50% | 45.50% | 48.58% | 51.42% |

8.00% | 54.00% | 46.00% | 52.66% | 47.34% |

7.00% | 53.50% | 46.50% | 57.07% | 42.93% |

6.00% | 53.00% | 47.00% | 61.84% | 38.16% |

5.00% | 52.50% | 47.50% | 67.01% | 32.99% |

4.00% | 52.00% | 48.00% | 72.60% | 27.40% |

3.00% | 51.50% | 48.50% | 78.66% | 21.34% |

2.00% | 51.00% | 49.00% | 85.21% | 14.79% |

1.00% | 50.50% | 49.50% | 92.31% | 7.69% |

0.00% | 50.00% | 50.00% | 100.00% | 0.00% |

Table II shows that even with a five percent advantage, day traders will face ruin over two-thirds of the time. The truth is that positive expected values do not lurk unnoticed in strategies based on the computation of moving averages or various trend-following strategies. Nor is it possible to discern consistently the trading ranges which will reward the strategy of selling strength and buying weakness.

Experience suggests that day traders violate the fundamental principle that games with positive expected values should be played slowly to avoid ruin. A distant second choice is to play games with positive-expected values with heavy commitments and hope to be lucky. No reasonable person should expect good luck to last forever.

If there are no consistent, positive expected values to be achieved, then the day trader will be in the position of paying commissions and execution costs to play a fair game–that is, a game with a zero expected value. A game thus defined has a negative expected value, the playing of which will lead inevitably to ruin.

### Conclusion

The probability that day trading on the Internet will have an outcome similar to the tulip bulb craze in Holland during the 1600’s is high. It seems that the capacity of the untrained human mind to resist the intrusion of knowledge continues to be virtually infinite.

The bottom line is that electronic day trading is extraordinarily risky. To avoid the inevitable confrontation with ruin, the day trader should ask, “Why not do something less risky, like throwing all my money off the roof of a tall building and keeping what the wind blows back?”

*For more information on this area of research, you may refer to Richard J. Teweles, Charles V. Harlow and Herbert L. Stone, The Commodity Futures Game, McGraw-Hill, 1974, Chapter 10, “Money Management.” This chapter is basically the same in the next edition of this book by Teweles and Jones (1987) in which the work of Dr. Harlow is acknowledged.*

### About the Author(s)

Michael D. Kinsman, PhD, CPA, is a member of the Graziadio Business Review Advisory Board. He is a professor of finance and accounting at the Graziadio School of Business and Management. A former systems analyst for General Electric Corporation, a financial analyst for Pacific Telephone, and a consultant for a variety of large and small firms, Dr. Kinsman operates a CPA and consulting firm in Laguna Beach. He has written and lectured on a variety of subjects and has been published in the Journal of Finance, the Journal of Accountancy, and other periodicals.

Issue: 1999 Volume 2 Issue 4

Topic: Finance / Investing / Accounting

Tags: behavioral finance, stock market