# Z SCORE

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The Z Score is a statistical measure used by financial traders to determine whether there is a dependency, or correlation, among their trades. A trader might suspect a dependency if he or she experiences a run of several consecutive profitable trades, or a run of several consecutive unprofitable trades. "Obviously, there was some kind of dependency or serial correlation among your trades [in this case], where winners were followed by winners and losers were followed by more losers," Thomas Stridsman wrote in the April 1998 issue of Futures. "If this happens again, you'll want to exploit the good times and perhaps avoid trading altogether in bad times."

Traders can verify the existence of a dependency among their trades by calculating the Z Score of their trading system or strategy. The Z Score indicates whether the trading system results in more or fewer streaks of consecutive wins or losses than would occur randomly. Ideally, traders can apply this information to future trades in order to increase profits and decrease losses by adjusting the amount of money invested in each trade, depending on the results of the previous trade. It is important to note, however, that the Z Score is useful only to traders who use a trading system, and only when that system is working on a particular market.

The formula for calculating the Z Score of a trading system is:

where N = the total number of trades (for the formula to work effectively, the Z Score must be calculated for a minimum of 30 trades),
R = the total number of runs (a new run begins each time a profitable trade is followed by an unprofitable one, or vice versa),
X = 2 × W × L ,
W = the total number of winning trades,
L = the total number of losing trades.

If this calculation results in a negative Z Score, it means that the trading strategy has fewer streaks or runs than would occur randomly. In other words, there is some dependency or correlation among trades because winners tend to follow winners and losers tend to follow losers. If the calculation results in a positive Z Score, it indicates that the trading strategy has more streaks than would occur randomly. There is a reverse correlation among trades because winners tend to follow losers and losers tend to follow winners.

The closer the Z Score to zero, the lower the likelihood that the trader will be able to rely on a dependency among trades to increase profits or decrease losses. On the other hand, traders are likely to be able to take advantage of a high or low Z Score (above + 2 or below - 2) to improve their results. For example, a trader with a Z Score of—2 should increase the size of his or her next trade after a winner, because the correlation between trades indicates that the next trade should also be a winner. Similarly, a trader with a Z Score of + 2 should increase the size of his or her next trade following a loser, because the correlation indicates that the next trade should be a winner.

The majority of trading systems yield a Z Score between I and—1, showing a limited dependency between trades. But this is not necessarily a bad thing. "What do you do if you don't know your system's Z Score or serial correlation, or if you find out they aren't high enough to be exploited profitably?" Stridsman noted. "Investigate whether you can further improve your system because—and it may sound strange—the truth is that a system or a trading strategy that shows signs of any kind of dependency or correlation among its trades is not optimized to its maximum potential."

## A DIFFERENT Z SCORE PREDICTS BANKRUPTCY

The most current definition of Z Score is the one used by financial traders to determine whether there is a dependency among their trades. It is worth noting, however, that another definition exists. Financial economist Edward 1. Altman developed a Z Score for predicting commercial bankruptcy in 1968. Altman's model determines the probability that a company will enter bankruptcy within any 12-month period, using five financial ratios that can be calculated from basic financial reports.

The formula for manufacturing companies is:

where A = working capital divided by total assets,
B = retained earnings divided by total assets,
C = earnings before interest and taxes (EBIT) divided by total assets,
D = market value of preferred stock and common stock divided by total liabilities,
E = sales divided by total assets (for nonmanufacturing companies, element E is omitted from the formula).

This Z Score provides an objective measure of a firm's financial health that can be used for credit evaluation, investment analysis, insurance underwriting, legal analysis, and turnaround management. It was 95 percent accurate in predicting bankruptcy in Altman's initial study, and between 82 and 85 percent accurate in independent follow-up studies.

[ Laurie Collier Hillstrom ]

## FURTHER READING:

Auchterlonie, David L. "A Paean to the Z Score and Its Commercial Bankruptcy Prediction." Journal of Lending and Credit Risk Management, September 1997, 50.

Stridsman, Thomas. "If It's Broke, Don't Fix It." Futures, May 1998, 44.

——. "Scoring High and Low." Futures, April 1998, 46.