The expertise of the principal establishes the foundation for an analytical discipline employing a macro, top-down, bottom-up perspective in a framework tailored to the asset allocation challenges facing investment professionals, professional & proprietary traders, brokers, accredited investors, hedge fund & asset managers whose main purpose is to maximize the return. We are uniquely positioned to provide high value-added in the quest for superior returns with an approach geared to the portfolio investors need to choose from among competing asset classes on low risk/high reward (Prudent) basis taking into account the difficulty of picking winners, commission, slippage costs and tax consequences.

Our client services are primarily delivered through http://collective2.com e-mails. Our Model Positions are representative positions that put our best economic forecasts to work. These are not recommendations to buy or sell specific instruments, nor are they personalized investment advice.

Model Positions are created at a size representing a fixed % of the portfolio. Larger versions of similar positions may involve market impact costs or other costs that we do not take into account.

The risk of a trade is defined as the dollar amount that the trade would lose per contract if it were a loss. Commonly, the trade risk is taken as the size of the money management stop applied, if any, to each trade. If your system doesnt use protective (money management) stops, the risk can be taken as the largest historical loss over a period of 30 recent trades. This is a modification of the approach Vince adopted in his book "Portfolio Management Formulas," John Wiley & Sons, New York, 1990. http://www.adaptrade.com/Articles/article-ffps.htm

Day-trading systems specialize in one market, do very well for a while & then suddenly fall to pieces. Many day-trading systems are taking extremely large positions which, in the event of any large intra-day moves or breakdown in exchange trading functions or server outages which happen from time to time, expose the account to wipe-out, even negative equity. Please also note that as with many systems that go for longer terms moves (although not all trades do this), the open equity plot favoured here at C2 sometimes provides a misleading, or we should say incomplete, picture. The most important negative points are those that involve actual (realized) account losses, which are not the same as an open equity (unrealized) drawdown. There are going to be open drawdowns. But in order to get the big moves, you have to be willing to give those profits a chance to run. Most times it works, sometimes it doesnt. What is not shown here on the C2 graph is the closed equity line. Usually, though not always, it shows a far smoother ride than looking at the open equity plot alone, which does tend to oscillate far more, and also with far larger moves than any day-trading system would permit.

Dr. David Druz says, "The more robust a system, the more volatile it tends to be! This is because robust systems are not optimized to particular markets or market conditions. The converse is also true. You can design systems with excellent returns & low volatility on historical testing, but which work only for given periods in given markets. These systems tend to be curve-fit or market-fit & are not robust." This quote comes from: http://www.tacticalnet.com/cgi-bin/t2.exe/VolatiltiyPaper.htm

The Formula for the ETF Long Term Return

The formula for the long term compound annual growth rate of a leveraged ETF cannot be written in terms of just the benchmark return and volatility. It also involves terms containing the skewness and kurtosis of the benchmark. It is derived using a Taylor series expansion. It does not assume that benchmark returns are Gaussian or that returns are continuous as do formulae derived using Itos lemma. But it turns out that for the worlds stock markets and for low levels of leverage (up to about 3) the formula can be approximated by this formula:

R = km - 0.5k^2s^2/(1 + km)

where R is the compound daily growth rate of the ETF, k is the ETF leverage (not necessarily an integer or positive), m is the mean daily return of the benchmark, and s is the daily

volatility (i.e. standard deviation) of the daily return of the benchmark. R is the quantity you use to calculate the long term buy-and-hold return of the ETF. You can see from the formula that if the volatility is zero then R = km so that the return of the ETF is k times the return of the benchmark. The 0.5k^2s^2=(1 + km) term is the volatility drag. Since k^2s^2 is always positive and (1+km) is always close to 1 then the volatility drag is always positive. R is a quadratic function of k with a negative coefficient for the square term. That means we will always get the parabola shape and we will always have a maximum for some value of k. Some algebra shows that the maximum is approximately (for small km) at

k = m/s^2

This clearly shows the return/volatility trade-off that determines the optimal leverage. This formula occurs in an appropriate form in the Kelly Criterion (Thorp 2006) and Mertons Portfolio Problem (Merton 1969). Its appearance here as the result of an optimisation is no surprise.