Correlation

In addition to choosing the right distributions for uncertain inputs, you also need to be concerned about properly modeling the correlation between uncertain inputs. Two inputs are correlated if they tend to move together. By default, all nodes that are defined as random are completely uncorrelated. That is, they move completely independently.

Complete or 100% correlation is easy to accomplish. In the first example of this chapter, Sales is modeled as an uncertain input. Variable costs are also uncertain, but they are 100% correlated with Sales. That is, if sales go up, variable costs go up proportionately. This correlation is accomplished by defining variable costs to be a function of sales.

Variable:=Sales*60%

Partial correlation is more difficult to model. One example of where you would need to model partial correlation is in simulating a portfolio of stocks. The future price of each stock is uncertain and can be modeled as separate random nodes. However, stock prices are partly correlated, even between completely unrelated stocks. For example, a change in interest rates affects the value of all stocks.

The following model simulates a portfolio of stocks without taking correlation into account.

This model will underestimate the volatility on the Portfolio node.

To cause the inputs (S1, S2, and S3) to be correlated, point to Monte Carlo in the Tools menu and click Correlate Inputs. This brings up the following dialog box:

Next, enter the names of the nodes you want to be correlated (S1,S2,S3) and click OK. When you do this, DecisionPro alters the definition of each input as follows:

DecisionPro also creates a data table named Cormat that looks like this:

You will need to enter correlation coefficients in the data table to reflect the level of correlation you want to use. The coefficients must lie in the range -1.0 to 1.0. For example, the following table assigns a correlation of 0.3 to the nodes S1 and S2. Similarly, S1 and S3 have a correlation of 0.4 and S2 and S3 have a correlation of 0.5.

Once the Cormat table is complete, you can simulate your model and DecisionPro will ensure that values for the inputs are chosen in a way that causes them to have the correlation you specify.

When you enter correlation coefficients, you are actually entering rank correlation coefficients. For example, if you simulate the model above over 100 iterations you will need to generate 100 values for each input. If you sort the values for each input and replace the actual value with its rank (1=smallest, 100=largest) these ranks will be correlated by the amount you chose. If the distribution shape for each input is the same, the difference between rank correlation and actual correlation is very small. However, if the distributions have vastly different shapes, the actual correlation can be somewhat different from the rank correlation.

It is always a good idea to test your model to verify that you are getting the correlation you expect. Select each input as an output when you run a simulation and then generate a correlation matrix (Tools, Monte Carlo, Show Correlation Matrix). You will probably find that the actual correlation is slightly lower than the value you entered in Cormat. You can adjust the Cormat values accordingly.