Correlation is a familiar concept used to describe the strength of the relationship between variables. Serial correlation (also known as autocorrelation) is the term used to describe the relationship between observations on the same variable over independent periods of time. If the serial correlation of observations is zero, observations are said to be independent. However, if observations are serially correlated, it means they don’t evolve in a random process, but rather they are related to their prior values.

Serial correlation is calculated as a function of the mean and variance.

If the observations are independent, the serial correlation is zero. However, if serial correlation has a statistically significant difference from zero, the observations are not likely to be independent. In this case, the observations may exhibit positive or negative serial correlation, and the proportional relationship between standard deviation and the square root of time does not hold.

- If observations are positively serially correlated, they are said to exhibit mean aversion. This means observations are prone to trends and returns measured over longer periods will have higher standard deviation than if the subperiod returns were independent.

- If observations are negatively serially correlated they are said to be mean reverting. Mean reversion indicates that observations tend towards the average value over time and returns measured over longer periods will have a lower standard deviation than if the subperiod returns were independent.

This can be used to relate the variability in returns in a single asset over different time period. For example, we can answer the following question:

How is the standard deviation of annual returns of a single asset related to the standard deviation of monthly returns for the same asset?

- Standard deviation of annual returns is Sqrt(12) times larger than standard deviations of monthly returns
- Standard deviation of annual returns is Sqrt(12) times larger than standard deviations of daily returns (assuming 250 business days in a year.)

As we can see, the standard deviation of annual returns is only Sqrt(12) times larger than monthly returns (and not 12 times larger). This is similar to diversification effect but within the asset, that is, losses in some months are offset by gains in other months.

This “square root of time” formula is also used to convert value-at-risk from one time horizon to another.

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