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Risk Solver Engine provides powerful ways to
induce correlations among uncertain variables, even when they
are generated by dissimilar probability distributions. In
many modeling situations, both A1 and B1 are uncertain
variables, you have no way to directly compute one from the
other, yet A1 may statistically depend on B1. For example,
mortgage interest rates depend on bond market interest rates
(since mortgages are pooled and sold as securities in the bond
market), and both rates depend on inflation expectations, but it
is quite difficult to specify a formula that relates these
variables.
In cases like these, you’ll need to specify the
statistical dependence between uncertain variables using PSI
Property functions, supplied as arguments to PSI Distribution
functions. This will induce a correlation between uncertain
variables, that would otherwise be considered independent. For
example, you can write =PsiNormal (100, 10, PsiCorrIndep("MyCorr"))
in cell A1 and write =PsiUniform( 0, 100, PsiCorrDepen(("MyCorr",
0.9)) in cell B1 to specify that B1 depends heavily on A1. "MyCorr"
is an arbitrary string name.
The number 0.9 in the example above is a
Spearman rank order correlation coefficient. This is a
nonparametric measure of correlation that is computed from a rank
ordering of the trial values drawn for both variables. It can be
used to induce correlations between any two uncertain variables,
whether they have the same or different analytic distributions, or
even custom distributions.
Meaning of Rank Correlation Coefficients
 | A correlation coefficient of +1 forces
the sampled values of the uncertain variables to be exactly
positively correlated, meaning that the pth percentile value
from one distribution will be sampled on the same trial as the
pth percentile value from the other distribution. Coefficients
from 0 to +1 will produce varying degrees of positive
correlation.
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| A correlation coefficient of -1 forces
the sampled values of the uncertain variables to be exactly
negatively correlated, meaning that the pth percentile value
from one distribution will be sampled on the same trial as the
(100-pth) percentile value from the other distribution.
Coefficients from 0 to -1 will produce varying degrees of
negative correlation.
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| A correlation coefficient of 0 means
there is no induced relationship between the variables. In
practice, one usually uses coefficients less than +1 or -1, and
uses 0 only in a correlation matrix that defines relationships
among several variables. |
Using a Rank Correlation Matrix for Several Variables
What if you have three, four, or more uncertain
variables that should all be correlated with each other? You can
create a small table or matrix of rank correlation coefficients in a
cell range on the worksheet, and use this cell range in the PSI
Property function PsiCorrMatrix.
You pass PsiCorrMatrix (matrix cell range,
position) as an argument to the PSI Distribution function,
for example =PsiNormal (10,5,PsiCorrMatrix(A1:C3,1))
for the first uncertain variable covered by the correlation matrix.
You’d pass PsiCorrMatrix(A1:C3,2) to the PSI Distribution function
for the second variable, and PsiCorrMatrix(A1:C3,3) for the third.
Back to Risk Solver Engine Product
Overview
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