Applies to: @RISK for Excel, all releases

Error:

Matrix is invalid

Cause:

If you encounter a correlation matrix that appears invalid, this is due to the underlying data, not an error in StatTools or @RISK. Historical datasets often contain missing or incomplete data points, which can result in a correlation matrix that is not self-consistent.

A self-consistent correlation matrix has no negative eigenvalues. For more information, see [How @RISK Tests a Correlation Matrix for Validity].

Why Invalid Correlation Matrices Occur

According to Peter Jäckel in Monte Carlo Methods in Finance (Wiley, 2002, p. 60):

“Ad hoc correlation matrices, those calculated from incomplete data and those taken from news services, sometimes don't comply with the requirement of symmetry and positive semi-definiteness. … In practice, the problem of an invalid correlation matrix, i.e., one that has negative eigenvalues, can also very easily arise in the context of risk analysis for equity portfolios. This is because there are frequently asynchronous gaps in the historical stock exchange time series. The chance that slight inconsistencies in the data from which historical correlation coefficients are calculated can lead to negative eigenvalues grows rapidly as the size of the correlation matrix increases. … Since equity index or portfolio analysis typically involves many underlying assets, the risk of negative eigenvalues of the correlation matrix calculated from historical data is particularly large.”

In short, missing or asynchronous data can produce small inconsistencies that result in negative eigenvalues, especially for large matrices.

Solutions

1. Correct the Correlation Matrix Using @RISK

@RISK provides the function RiskCorrectCorrmat() to transform an invalid correlation matrix into a valid one. You have two options:

• Let @RISK automatically adjust the matrix without specifying preferences.

• Provide adjustment weights to influence how @RISK modifies the matrix.

For detailed guidance, see [How @RISK Adjusts an Invalid Correlation Matrix].

2. Restrict the Data Set

Redefine your StatTools dataset to include only rows where all variables have data. This ensures that the calculated correlation matrix is based on complete cases, reducing the risk of negative eigenvalues.

Apply Judgment

Even after correction with RiskCorrectCorrmat() or filtering your data:

• Use your own knowledge and judgment to assess correlations.

• Historical correlations may not fully reflect the relationships relevant to your specific model or scenario.

• In some cases, it may be desirable to specify correlations that differ from the historical values.

 

If you want, I can also add a small diagram showing how missing data leads to negative eigenvalues—that helps non-technical readers visualize the issue. Would you like me to do that?

According to Peter Jäckel, Monte Carlo Methods in Finance (Wiley, 2002), page 60:

Ad hoc correlation matrices, those calculated from incomplete data and those taken from news services, sometimes don't comply with the requirement of symmetry and positive semi-definiteness. ...

In practice, the problem of an invalid correlation matrix, i.e. one that has negative eigenvalues, can also very easily arise in the context of risk analysis for equity portfolios. This is because there are frequently asynchronous gaps in the historical stock exchange time series. The chance that slight inconsistencies in the data from which historical correlation coefficients are calculated can lead to negative eigenvalues grows rapidly as the size of the correlation matrix increases. ... Since equity index or portfolio analysis typically involves many underlying assets, the risk of negative eigenvalues of the correlation matrix calculated from historical data is particularly large.

Related Articles:

How @RISK Tests a Correlation Matrix for Validity.

How @RISK Adjusts an Invalid Correlation Matrix