I have been reading a few papers lately on the subject of fund performance results and their statistical analysis. I must admit I was very surprised and at the same time very disappointed by the way this subject is approached, especially by academicians. I am not going to mention names or make reference to any of the papers. I think it will be too embarrassing to the authors after they read this post. Why? Because most academicians do not understand what is involved in trading, how markets and their participants operate and, needless to say, most have never traded a single share or contract in their life. Yet, they create a lot of confusion to the community of traders and investors with their studies, which supposedly try to investigate whether the stellar performance of some funds or even particular trading systems is due to luck, or skill, two terms involved in the null hypothesis they often construct for statistical analysis.
I will start by stating that the argument whether there is skill or luck involved when applied to the returns of professional traders who have outperformed their respective benchmark or index is a red herring, i.e. it is a distracting issue away from the real issues. Yet, there are scores of academic papers with pages and pages of statistical analysis trying to provide an answer to this logical fallacy. How could someone in the right state of mind question 20 years of positive performance, year-after-year, of a certain fund manager or trading system on the basis that it may be the result of persistent luck? Is it possible that a professional managing hundreds of millions or even billions is so persistently lucky for the last 20 years, for example, to have zero, or even just 1 or 2 marginally losing years, while having outperformed significantly a benchmark index? Academicians claim that if you have a large population of fund managers some of them will end up with extraordinary performance. So far, we all agree with that. This is basically the outcome of a law of nature known as the Power Law and the associated Pareto Principle. Twenty percent of fund managers and traders will outperform the other 80%. The problem is that academicians go one step further after employing some fallacious logic and apply all sorts of statistical analysis methods to try to come up with some measure of the probability or significance of performance results to control for luck. These are pointless exercises motivated by a lack of understanding of how the markets work. More details about this after I define what a trading edge is.
The trading edge, E, can be defined as the expected value of the random variable T, the P/L of trades. It can be shown that
E[T] = w × avgW -(1-w) × avgL (1)
where w is the win rate, avgW is the average winning trade and avgL is the average losing trade.
Some authors in the trading system literature have claimed that the value of the expected gain is the trading edge if E[T] > 0. However, this is mathematically equivalent to the trivial claim that the profit factor is greater than 1. The profit factor is the ratio of the sum of winning trades to the sum of losing trades. The equivalence is demonstrated as follows using equation (1)
E[T] > 0 => w × avgW – (1-w) × avgL > o => w × ∑W/Nw – (1-w) × ∑ L/NL > 0 => (Nw/N) × ∑W/Nw – (NL/N) × ∑ L/NL > 0 or
∑W/∑L > 1 (2)
where Nw and NL are the number of winning and losing trades, respectively, and ∑W and ∑L the sum of winning and losing trades respectively. Recall that w = Nw/N and 1-w = NL/N.
Thus, what some authors present as a fancy formula for the trading edge, equation (1), reduces to the trivial and intuitive statement that having an edge is equivalent to having more winnings than losses, equation (2). It is that simple. This shows that the mathematics of an edge are trivial and there is actually nothing complicated involved: if the sum of the winnings trades is greater than the sum of the losing trades, then you have an edge. So, what is the problem all those academicians try to study with fancy statistics even when the edge, as just defined, is present?
Again, the claim is that given a large pool of fund managers and traders, the results of those who outperform the mean can be due to sheer luck rather than due to skill. In other words, a fund manager, trader, or trading system with 10 or 20 years of positive results may have achieved that performance simply by chance. Often, the distribution of the mean returns is obtained using resampling methods and based on how far on the right tail of the distribution of returns scores a particular mean return, the null hypothesis that the results are due to luck rather than skill is rejected or not. Any performance that is close to the mean performance is called mediocre and possibly the outcome of luck.
Common sense and understanding of the markets and the way participants interact forces us to label such studies naive, to say the least. Those of us who actually trade and have done this for a long time know how hard it is to make money and finish the year with a positive cumulative P/L. It takes special skills and tremendous effort to trade real money, by competing in a game with all sorts of market participants, like for example commercials, market makers, high frequency robots, insiders, and at the end of the day taking a share of the losses of the losers in a zero-sum game (trading is a zero-sum game with a few exemptions that are outside of the scope of this post). The winner participants share the losses of the losers. It is a true achievement to be able to maintain a profit factor much greater than 1 for an extended period of time and the question whether there is luck involved is more a metaphysical one than pragmatic. It is the wrong question to ask and actually a distracting issue away from the real issue to be discussed below. Even if a fund manager has a mean monthly or yearly performance close to the mean performance of a large pool, this is in itself an achievement and should not be called a mediocre performer, as some authors who have never traded think. I claim that even if the performance of a manager is less than the mean performance of a large pool, as long as it is positive and the profit factor is much greater than 1, it reflects some type of an edge.
So what is the real issue here? The real issue is the quality of an edge in the context of probability of ruin. It has to do with the leverage to achieve the positive performance. A fund manager who uses a leverage of 2 to achieve an average 10% yearly return is better than a manager who uses a leverage of 4 to achieve the same return. However, the leverage information is not easily accessible for the most part. It is hard to know the actual level of risk of ruin from looking at monthly, quarterly or yearly results. That is difficult even when trade-by-trade P/L results are available. From the individual trader point of view, if someone trades one S&P 500 mini futures contract per $50K of equity and another two contracts per $20K and both achieve about the same return then the former trader has superior skill or edge, depending on whether he is a discretionary trader or uses mechanical trading systems. In both cases there is an edge but this is not the important issue. The difference is the quality of the edge and in particular, the probability of ruin involved. Thus, the real and important issue, given positive performance, is not whether there was luck or skill involved in achieving it, but what was the probability of ruin as a function of time. Traders and fund managers with high probability of ruin, even when profitable, have a lower quality edge but they do have an edge. If a trader or fund manager does not have an edge, in these competitive markets it will probably show in the first 6 or 12 months. The probability of two or three years in a row of positive returns above the mean market return is so low that it is for all practical purposes zero. This is dictated by experience and common sense. Thus, I do not know what all those studies investigating the persistence of luck in performance results are all about other than the fact that their authors lack trading experience and common sense.