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## Tuesday, December 12, 2006

### CONSISTENCY AND SHARPE RATIO (1)

For me consistency has always been a very important and interesting factor in trading. By consistency I mean how long, in a given timeframe, let say 12 months or 12 weeks, my trades are giving a positive result. Can I expect 6 months out of 12 being profitable or more or less? This is one of the questions which I always try to give as best as possible an answer.

One of the most used criteria for evaluating fund managers is the Sharpe Ratio. The Sharpe Ratio is used for this because of its easy use: the bigger the Sharpe the better the fund or the fund manager. This almost has become fait accompli.

How is the Sharpe Ratio calculated? It is simple obtained by dividing the average net profit (mean or expectation E) by the standard deviation of the trades. The standard deviation is, as you may know, a measure for the spread of winners of losers around the mean. Because the deviation is a measure for the trade spread it is also considered as a measure for the risk of the fund or the fund manager, but this standard deviation is also defined as the volatility of the fund.

The Sharpe ratio is in this way defined as the mean profit of the fund (-manager) per point risk or volatility. The bigger the mean, the bigger the Sharpe the better the fund (-manager) is performing. That's the idea.

An example. Suppose trader A has a gain of 5 ES points per trade, a spread of 10 points, done over 100 trades in one year. This means that, when trader's A profits are normally distributed over the losses and the winners, 99,73% of his trade outcomes is lying between 5 minus 30 (=-20) and 5+ 30 (=35). The Sharpe ratio is now 5/10 = 0.5 .

Do you see how it works? When the trade outcomes would have a bigger spread but with the same mean the Sharpe ratio would decrease, the mean per point risk will decrease.

When you look closely at the Sharpe Ratio you will see and understand that is in fact a measure for consistency. To see this take the next example. Trader B takes 100 trades a year but with every trade being a winner (if this would be possible, but for clarity reasons taken here) for 5 points. His net profitability and mean is the same as for trader A but his Sharpe ratio becomes 5/∞ = ∞. His consistency is 100%.

We defined the Sharpe ratio as the mean defined by the spread. This is not precisely correct. The exact definition is the excess profit over a zero risk profit divided by the spread. A zero risk is generally the return of a 30 year T-bond but also a European long bond could be taken. Off course the risk is not exactly zero but is considered to be and also T- bond returns are changing over time (inflation).

Our definition of the Sharpe is a modified Sharpe, but it doesn't really matter, because the exact definition doesn't give more information. You only have bear in mind how the Sharpe is defined when it is used in comparisons.

There is a clear relation with so called z score of a normal distribution and a lookup in a table with a z 0f ∞ gives a 100% consistency. In this way consistency can be defined and I use it this way.

But before closing this post three remarks have to be made.

1. The Sharpe and the consistency is defined for normal distributed trade returns but in practice returns seldom are.

2. The spread as a measure is not a very good one because in this way it also takes positive returns as a risk and this is not what traders and investors generally feel about positive returns.

3. The longer a trader stays in business the bigger his draw downs will be. Or said otherwise: the more trades a fund manager takes the bigger his draw downs will be, or said it in another way: the more trades a fund manager takes, how bigger his drawdown. The consequence is that the longer a fund manager is in business (the longer a fund exists) he will be punished with a lowering of his Sharpe ratio. I will come back to this in my next post.

So the conclusion is that Sharpe is a useful concept but not more and so other concepts for consistency are needed. These are the Sortino Ratio, Callmar ratio and the Martin Ratio. I developed another concept of consistency which I called Excess Loss Wealth Function (ELWF) which I draw directly from the theory of random walks.