Validating and Adjusting Correlation Matrices in @RISK

Applies to: @RISK 6.x and Up

Question:
How does @RISK determine if a correlation matrix is valid, and how can I control adjustments for invalid matrices?

When Is a Correlation Matrix Valid?

A correlation matrix is valid if it is self-consistent, meaning all specified coefficients are mutually compatible.

• If a matrix is invalid (not self-consistent), it typically has one or more negative eigenvalues.

Automatic Validation

1. When you click Start Simulation, @RISK checks all correlation matrices.

2. If a matrix is invalid:

   • With an adjustment weight matrix: @RISK adjusts the matrix using the weights and continues.

   • Without an adjustment weight matrix: @RISK displays a warning:

Warning
The correlation matrix at ... is not self-consistent. @RISK can generate the closest self-consistent matrix. OK = corrected matrix, Cancel = stop simulation.

• Clicking Cancel lets you adjust the matrix manually or create a weight matrix before running the simulation.

Adjustment Weight Matrix

• Guides how much each correlation coefficient can be adjusted.

• Can be triangular or symmetric square matrix.

• Enter weights 0–100:

  • 100 = coefficient must not change

  • 0 = coefficient can change freely

  • Values between 0–100: higher = more importance on original coefficient

Naming convention:

• Name the weight matrix to match the correlation matrix with _Weights suffix (e.g., Matrix1_Weights).

Usage:

• During simulation: @RISK applies weights automatically if the name matches.

• Before simulation: Use RiskCorrectCorrmat() to place the adjusted matrix in your worksheet and reference it in your correlated inputs.

Technical details:

• Adjustment is a non-linear optimization: minimize the weighted sum of squared differences between the original and candidate consistent matrices.

• @RISK uses the limited-memory BFGS algorithm to perform the optimization.

• Weights are treated exponentially; they indicate importance rather than exact scaling.

Adjustments Without a Weight Matrix

• If no weight matrix is provided, @RISK adjusts an invalid matrix as follows:

1. Find the smallest eigenvalue (E₀).

2. Shift the matrix:

[
C' = C - E₀I
]

3. Normalize the diagonal to 1:

[
C'' = \frac{1}{1 - E₀} C'
]

• Resulting matrix is positive semi-definite and valid.

• This method may significantly alter some coefficients.

• Adjusted matrices are stored in CORRMAT.MTX in your temporary folder.

Recommendations

• For control over adjustments, create an adjustment weight matrix.

• For large matrices, consider pre-correcting with RiskCorrectCorrmat() to speed up simulations.

• If no weight matrix is used, review CORRMAT.MTX to verify adjustments or modify your matrix manually.

Attachments / Examples:

• KB75_AdjustDuringEverySimulation.xlsx – Adjustment during simulation

• KB75_RiskCorrectCorrmat.xlsx – Pre-correction using RiskCorrectCorrmat()

Related Articles:

limited-memory BFGS algorithm

How @RISK Tests a Correlation Matrix for Validity

Attachments:
KB75_AdjustDuringEverySimulation.xlsx
KB75_RiskCorrectCorrmat.xlsx