## What Is Cross-Correlation?

Cross-correlation is a measurement that tracks the movements of two or more sets of time series data relative to one another. It is used to compare multiple time series and objectively determine how well they match up with each other and, in particular, at what point the best match occurs. Cross-correlation may also reveal any periodicities in the data.

### Key Takeaways

- Cross-correlation is used to track the similarities in the movement of two factors over time.
- Stock investors use it to determine the degree to which two stocks move in tandem.
- Portfolio diversification requires selecting stocks and other assets that move in opposite directions in order to hedge losses.

Cross-correlation is used in portfolio management to measure the degree of diversification among the assets contained in a portfolio. Investors increase the diversification of their assets in order to reduce the risk of big losses. That is, the prices of two technology stocks might move in the same direction most of the time, while a technology stock and an oil stock might move in opposite directions. Cross-correlation helps the investor pin down their patterns of movement more precisely.

Of course, cross-correlation can only measure patterns of historical data. It cannot predict the future.

## Understanding Cross-Correlation

Cross-correlation is generally used when measuring information between two different time series. The possible ranges for the correlation coefficient of the time series data is from -1.0 to +1.0. The closer the cross-correlation value is to 1, the more closely the sets are identical.

Investors and analysts use cross-correlation to understand how the prices of two or more stocks (or other assets) perform against one another. This is particularly important for correlation trades such as dispersion strategies and pairs trading.

## Formula for Cross-Correlation

In its simplest version, it can be described in terms of an independent variable, X, and two dependent variables, Y and Z. If independent variable X influences variable Y and the two are positively correlated, then as the value of X rises so will the value of Y. If the same is true of the relationship between X and Z, then as the value of X rises, so will the value of Z. Variables Y and Z can be said to be cross-correlated because their behavior is positively correlated as a result of each of their individual relationships to variable X.

### Cross-Correlation and Stock Markets

Cross-correlation can be used to gain perspective on the overall nature of the larger market. For example, back in 2011, various sectors within the S&P 500 exhibited a 95% degree of correlation.

That means that all of the sectors moved virtually in lockstep with each other. It was difficult to pick stocks that outperformed the broader market during that period. It was also hard to select stocks in different sectors to increase the diversification of a portfolio. Investors had to look at other types of assets to help manage their portfolio risk.

On the other hand, the high market correlation meant that investors could buy shares in index funds to gain exposure to the market, rather than attempting to pick individual stocks.

### How Cross-Correlation Is Used

Cross-correlation is used in portfolio management to measure the degree of diversification among the assets contained in a portfolio. Modern portfolio theory (MPT) uses a measure of the correlation of all the assets in a portfolio to help determine the most efficient frontier. This concept helps to optimize expected returns against a certain level of risk.

Including assets that have a low correlation to each other helps to reduce the overall risk in a portfolio. Still, cross-correlation can change over time. It can only be measured historically. Two assets that have had a high degree of correlation in the past can become uncorrelated and begin to move separately.

This is, in fact, one shortcoming of MPT. It assumes stable correlations among assets.

### How Cross-Correlation Works

A correlation coefficient is computed to see how well one series predicts the values in the other. Then the series are shifted using a time lag and the process is repeated.

The lag refers to how far the series are offset, and its sign determines which series is shifted. Note that as the lag increases, the number of possible matches decreases because the series “hang out” at the ends and do not overlap. The value of the lag with the highest correlation coefficient represents the best fit between the two series.

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