careful, methodical system selection and validation are just as important as in developing ones own trading systems.

According to the Sturgeon principle, 90 percent of everything ever developed is totally worthless. After years of evaluating, developing and refining trading systems, I can safely state that in the trading system design arena, Sturgeon was too optimistic. Much of my time is spent testing and trying to improve commercial trading systems, or systems and ideas in books, magazines, or newsletters. I can honestly say that less than 1 percent of everything I have examined has proven to be worthwhile.

Efficient portfolios

Once several promising trading methods have been identified and developed, the question then arises of how best to incorporate and balance these approaches into a single overall system portfolio. Many traders, even professional ones, simply trade all systems and markets the same way. Others try to adjust for volatility by attempting to risk the same dollar amount on each trade. They then adjust each position size accordingly. Traders following these two approaches are hoping that diversification benefits will come into play automatically because of the different markets that are being traded. To some extent this is true, but there is a more scientific way to properly construct and balance system portfolios.

Treating all markets and systems alike is the same as admitting that we know nothing about the relative value of our systems and methods other than that they all have some merit. Adjusting position sizes to conform to market volatility goes a step further by assuming that past volatility will remain constant into the future. But beyond that, it still assumes that we know nothing more about the expected future performance of the markets and of the systems we use to trade them.

Modern Portfolio Theory (MPT), first developed by Nobel laureate Harry Markowitz in the 1950s, takes a different approach.1 Under MPT, we first gather what we know about the past returns, volatilities, and cross-correlations of the systems that we trade. We then apply a computer-based optimization model to this information that examines all possible ways of allocating trading capital. This computer model maps out all portfolios offering the highest expected return at every level of expected risk, or, conversely, those portfolios offering the lowest expected risk at each specified level of return. These are known as efficient portfolios, since they offer the best possible reward/risk characteristics based on past performance data. The set of all such efficient portfolios is known as the efficient frontier. It is then a simple matter to choose whatever portfolio we want along the efficient frontier. Often this is the portfolio offering the best overall reward-to-risk or Sharpe ratio.*

*The Sharpe ratio measures risk-adjusted rate of returns. The formula is (AAR-NRJ/SAR, where AR is the systems average annualized percentage rate of return, NR is the annualized rate of return of a no-risk investment (e.g., 5% yield on 90-day T-bills), and SAR is the standard deviation of the systems annual rates of returns. (Editor)

The investor may, alternatively, select a portfolio that offers the highest expected geometric growth rate of capital, or some other appropriate objective function.

Modern portfolio Theory in practice

When I first started applying MPT models to futures trading in the early 1980s, there were no commercially available software programs of this kind. I therefore had to develop my own. Now, however, at least several low-cost commercial programs are available that will do MPT calculations. A recent issue of the Journal of Finance had advertisements for two such programs. There is also an add-on module for commonly used Omega, Computrac, and MetaStock programs that will easily export the appropriate data from these programs to an MPT optimizer.2 MPT software can also be obtained as an inexpensive add-on disk to a portfolio theory textbook.3

Before looking on the application of MPT principles as a trading panacea, however, traders should be aware of its limitations. Like any form of modeling, MPT optimization relies heavily on the assumption that future conditions will be similar to those of the past. This is always a dangerous assumption, especially when it involves optimized system performance with its attendant estimation risk and selection bias problems. It may also be problematic relying on model assumptions such as constant price variance, stationary (fixed) underlying return distributions, and serial independence of trading results.

One way to partially overcome some of these problems is to use as much data as possible for portfolio optimization. Many markets go through long active and quiet periods. Therefore, the more data traders have to work with, the greater is the likelihood of capturing an accurate representation of expected long-term performance. It may also make sense to take all system cross-correlations that are calculated as less than zero and set them back to zero. A cross-correlation of zero means that there is no relationship at all between the performance of one system and the performance of another. This is a reasonable assumption given that we are now dealing with trading system results rather than prices themselves. Cross-correlations that are less than zero imply that gains from one system are associated with losses from the other, and vice versa. I can see no reason for this to be true, unless one system is designed as a hedge to the other. Negative cross-correlations are therefore most likely spurious results due to randomness in the data, and setting them to zero seems appropriate.

Correlations that are too high may also create problems for MPT analysis because they may lead to model instability and "confusion." This problem can be avoided by dropping the least desirable of the highly correlated systems or markets. Little should be lost by doing this, since the remaining system or market will capture most of what is worthwhile from both, given its high correlation with the other.

Finally, the true state of affairs probably exists somewhere between MPT modeling, with its attendant estimation risks, and a naive strategy that calls for equal allocations of capital among all investment opportunities. A methodology known as

Endnotes

1. Markowitz, Harry, Portfolio Selection: Efficient Diversification of Investments, New York: John Wiley & Sons, 1959.

2. Portfolio Manager. Ralph Vince, 46 Chagrin Plaza #110, Chagrin Falls, Ohio 44022

3. Haugen, Robert, Modem Investment Theory, Englewood Cliffs, NJ: Prentice Hall, 1996.

4. Bawa, Vijay, Brown, Stephen and Klein, Roger, Estimation Risk and Optimal Portfolio Choice, New York: North Holland, 1979.

Bayesian statistics can be useful for this purpose, but it goes well beyond the scope of this chapter.4

Conclusion

Diversification in unrelated markets and the application of different trading approaches can add considerable value. Portfolio theory concepts can help us capture much of this value. But modeling has its limitations and can only approximate reality. In this uncertain world, where past performance is indeed not necessarily indicative of future results, portfolio optimization combined with common sense is the best combination for investment success.