# Input Variable Correlation in Monte Carlo Simulation

When conducting a Monte Carlo simulation, correlation among input variables is an important factor to consider.  If input random variables are treated as independent, when they are actually correlated, risk can be under or over estimated.

Let's think about how this occurs, when two input variables have positive correlation, the value for each should be relatively high in a given simulation iteration and both relatively low in another iteration.  For negatively correlated inputs, one should be at the high end of possible values while the other should be at the low end for a given iteration.

We will consider three simple examples to illustrate how input variable correlation affects simulation output.

## Revenue Model

Consider a very simple model of revenue that has demand and price as inputs.  Demand and price are negatively correlated.  When price increases, demand decreases and vice versa.  The model is below with formulas shown.  Of course, knowing if correlation is present may be a difficult question to answer, but for our example we will assume we know the correlation.

We will run two simulations to compare independent inputs versus correlated inputs.  For demand, we will assume a triangular distribution with 10 as worst case, 20 as most likely, and 35 as best case.  For price we will assume a triangular distribution with 125 as worst case, 150 as most likely, and 190 as best case.

In the first simulation we will assume that demand and price are independent.

Revenue Model Without Correlation
Revenue Results Without Correlation

In the second simulation we will assume a Spearman rank correlation coefficient of -.5 between demand and price.  We will use rank order correlation to simulate the input variable correlation.

Revenue Model With Correlation Matrix
Revenue Results With Correlation

After running the simulations we see that independent inputs resulted in a wider spread of outcomes.

Independent input simulation revenue range: 1375 to 6443

Correlated input simulation revenue range: 1588 to 5750

As expected, the revenue variance using independent inputs is greater as well.  In this example, if we assume independent inputs we would be over estimating risk.

## Generic Model 1

In the second example, we have a generic model with two random variables.  The output is the product of the two random variables.  One random variable follows the logistic distribution and the second is normally distributed.

First, we simulate the model with the random variables being independent.

Generic 1 Results Without Correlation

Next, we'll simulate assuming the random variables are correlated with a 0.5 correlation coefficient.

Generic 1 Results With Correlation

After running the simulations we have an interesting difference.

Output range with independent inputs: -11.35 to 105.48

Output range with correlated inputs: -4.37 to 107.96

Note that the mean of each simulation is nearly identical.  In this example we have a risk shift to the right.  Without correlation, the range of outcomes is still larger.

## Generic Model 2

In the third example, we have a generic model with two random variables.  The output is the product of the two random variables.  Both random variables are normally distributed.

First, we simulate the model with the random variables being independent.

Generic 2 Results Without Correlation

Next, we'll simulate assuming the random variables are correlated with a 0.5 correlation coefficient.

Generic 2 results with correlation

After running the simulations we see that there is more risk in the correlated simulation.

Output range with independent inputs: 29.65 to 73.32

Output range with correlated inputs: 26.87 to 78.96

Note that the mean of each simulation is nearly identical.  In this example if we assumed independent inputs, we would have underestimated risk.

## Final Thoughts on Input Variable Correlation

We've looked at three examples where input variable correlation affects the outcome differently.  By assuming independence of inputs, we could be under or over estimating risk.  Also, there could also be a shift in the outcomes to the left or right.

Are you looking for a Monte Carlo simulation add-in packed with features at a reasonable price?  If so, visit the Simulation Master product page.

To learn about input variable correlation methods in Simulation Master, check out this article.

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