Measuring the "Smoothness" of the Equity Curve

This section shows how to use linear regression analysis to measure the smoothness of an equity curve. We will use contrived data to perform actual calculations. You will understand how to use standard error and how to calculate the risk reward ratio. In later sections of this chapter we apply these ideas to market data and trading system calculations. The main advantage of using linear regression analysis is that it provides a consistent framework to analyze every equity curve.

The equity curve of your trading account or system is simply its daily equity. The daily equity is the sum of your starting account bal­ance, plus the profit or loss of all closed trades, plus the profit or loss of all open trades. Ideally, we want an equity curve that rises steadily in time, as shown for the hypothetical data in Figure 6.1. The slope of this equity line is $100 per day, all the points lie exactly on a straight line through zero, and the standard error is zero. This line shows an account whose equity increases exactly $100 each day.

Since we all have some trades that lose money, the equity curve is never a perfectly straight line. As you begin to compare the equity curves of different trading systems, you need a way to measure their

Measuring the "Smoothness" of the Equity Curve 181

-Perfectly Smooth" Equity Curve: Slope = $100, SE=0

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Figure 6.1 The perfectly smooth equity curve.

"smoothness." If you compare two systems with similar performance, the one with the smoother equity curve is preferred. We assume here that you are comparing system performance over the same time unit (days) and similar length (months or years). You. could compare systems over other time units and length, but you must recognize that some­times you may not be comparing these systems on a consistent basis.

We will use linear regression analysis to determine smoothness. One of the outputs of linear regression analysis is the residual sum squares (RSS). RSS is the sum of the squared vertical distance between the actual data and the fitted regression line at each point. The next step is to divide the BSS by the number of data points minus two, and then to take the square root, to calculate the standard error. The standard er­ror measures the smoothness. If all the points fall exactly on the best fit linear regression line, then RSS is automatically zero, and the standard error is also zero, for the ultimate smoothness in an equity curve.

182 Equity Curve Analysis

The curve in Figure 6.2 shows more hypothetical data. The slope of the best-fit linear regression line through zero is again $100. However, the points are scattered on either side of the best fit line. The standard error for these data is $82. If you measured the vertical dis­tance between the actual equity value and the best fit line every day, on average, this absolute, average vertical distance is $82. Thus, the standard error tells you typically how far a point is from the best-fit line.

In Figure 6.3, which uses even more hypothetical data, the slope of the best-fit equity curve is still $100, but there is a lot more scatter in the data on either side of the best-fit line. As expected, the SE is almost four times bigger, at $318.

You can get a better feel for what standard error means by looking at Figure 6.4, page 184, which contains the data in Figure 6.3 plus two lines one standard error away from the best-fit line. The data points are

Hypothetical Equity Curve: Slope $100, SE = $82

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Figure 6.2 These hypothetical data have a slope of $100, and the scatter about the regression line increases the standard error to $82.

Measuring the "Smoothness" of the Equity Curve 183

Hypothetical Equity Curve: Slope= $100, SE = $318

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Figure 6.3 These data have a $100 slope, but the large scatter about the linear regression line increases the standard error to $318.

inside, or close to, the standard error lines. Remember we find the standard error by squaring the vertical distance between the actual point and the best-fit line, summing this up, and dividing by the number of points less two. Hence, the standard error is the average "offset" on either side of the best-fit line, and the data clearly lie inside or close to the "offset" or standard error.

Thus, the standard error from linear regression analysis is a good measure of the smoothness of the equity curve. Note that the linear re­gression method can be applied to any number of time periods and to any equity curve. The standard error offers a general, consistent, and powerful method to measure smoothness.

The combined SE of two or more equity curves will be smaller than the SE of the individual curves only if the curves are negatively cor­related. Negative correlation means that when one increases, the other decreases. For a data set that is exactly negatively correlated to the data

184 Equity Curve Analysis