correlation

The correlation tables show that there are many pairs of sectors that have a high correlation with each other. A high correlation means a linear relationship exists between the behaviour of the two sectors. If one sector's performance is positive or negative during a month then the other's will tend to be positive or negative by a proportional amount in the same month.

dependence

When analysing the linear relationship between sectors one sector is called the independent sector and the other the dependent sector. However there may or may not be an underlying, real-world dependence of one sector on the other. Most often the two sectors both depend to some extent on the same external factors. The 'dependent' and 'independent' terms are used mostly for convenience.

example

The best way to illustrate a linear relationship is to plot an x-y scatter graph of the monthly performance values of one sector against the other. The independent sector values are measured along the horizontal x axis and the corresponding dependent sector values are measured along the vertical y axis. The following shows an x-y scatter graph for the Japan and Japanese Small Companies sectors. The graph is 'based on' Japan meaning that we have chosen Japan as the independent sector.

      

regression

Linear regression is the process of fitting a straight line through a set of x-y scatter points. Regression finds a line through the points such that all the points are as close to the line as possible. (Mathematically, it is the sum of the squares of the y-component difference that is minimised). The higher the correlation, the smaller the average distance of the points from the line.

equation

A straight line is defined by two numbers - the line's 'slope' and its 'intercept' with the y axis. These numbers define the equation of the line:

            y =  slope * x + intercept

The equation for the line in the example appears on the chart:

            y =0.9524 * x - 0.198

prediction

From the line's equation we can predict the performance of the dependent sector (y) given any month's performance by the independent sector (x). There are two stages in the calculation: first multiply the x value by the slope and then add the intercept.

slope

The slope of the line measures the rate of change of y relative to x. In the example, if x (Japan sector) gains 1% in a month then y (Japanese Small Companies sector) can be expected to gain 0.9524% in the same month. If x makes a loss of 1% then y should lose 0.9524%.

Because the slope of the line is close to 1 in the example it means the dependent sector changes by almost the same amount as the independent sector in a month. This is unusual. Generally the slope will be greater than zero but either more or less than 1.0. (A negative slope corresponds to a negative correlation).

intercept

Whatever the result of the slope * x term we always add a constant value, the intercept. This means that every month whatever the performance of the independent sector a fixed amount is added (or subtracted) when predicting the dependent sector's performance.

In the example 0.198% is subtracted in the calculation, as the intercept is negative. This means the dependent sector (Japanese Small Companies) underperforms the independent sector (Japan) every month by 0.198% whatever the performance of the independent sector.

risk

The relationship between the two sectors indicates their relative risk and reward. If the slope is less than 1.0 then the dependent sector will not gain as much as the independent sector in a profitable month - but will not lose as much in a loss-making month. If the slope is greater than 1.0 the situation is reversed.

The intercept also affects the relative performance. A negative intercept means the independent sector is constantly underperforming the dependent sector. This may or may not be a major factor in the long run. To find how major, look at the Total% figure for the independent sector and divide by the number of months to find the sector's average monthly performance. Compare the size of the intercept correction to this figure.

m-x

Four of the points on the graph are coloured and labelled m-0 to m-3 in the legend. These are the most recent points on the graph and give some indication as to whether the line is changing over time.

If the four points are evenly scattered either side of the line then their relationship is similar to the long-term relationship for all the points and the line remains valid.

If the four points are all below the line the dependent sector is performing worse than the long-term trend would suggest. This means the relationship, and therefore the position of the best-fit line, is changing. If the four points are above the line the dependent sector's relative performance is improving. Always keep an eye out for economic, market or political factors that might cause a long-term linear relationship to change.