# Correlation Coefficients

The 'asset class correlations', identify the extent to which (movement in) each individual asset class is correlated to (movement in) each other asset class.

The correlation coefficient indicates the strength and direction of a linear relationship between two random variables. The correlation coefficient ranges from +1, indicating a perfect positive linear relationship, to -1, indicating a perfectly negative linear relationship. One major caveat applies to the use of a correlation coefficient matrix in Voyant. The correlation coefficient must be positive definite. For this reason, the correlation coefficient matrix cannot be random, as random numbers usually result in matrices that are not positive-definite.

#### How this specifically relates to the Monte Carlo Insight:

A given set of market assumptions will contain the names of each asset class being used, as well as the ‘mean’, and standard deviation values for each asset class. The asset classes are more-or-less correlated, as defined in the 'correlation matrix'. In running a single iteration, the simulation will randomly select a percentile value for each asset class. This randomized percentile selection will then be repeated for each year of the plan.

For example: if a given client's timeline runs for 50 years, and your choice of 'market assumptions' contains 10 individual asset classes, then – for a single iteration (i.e. across the lifetime of the plan) – the simulation will generate 500 (i.e. 50 x 10) random percentile values.

The simulation will take the randomly-generated percentile values, and – given the assumed 'correlations' between the asset classes – will apply an algorithm to ensure (so far as possible) that investment outcomes (for each asset class) are consistent with the aforementioned correlations. This operation is called a 'Cholesky decomposition'. Some correlation matrices make it effectively impossible for the software to perform this operation, in which case the simulation will remain completely random. This is not necessarily a problem inasmuch as the correlations between asset classes represent only a 'tendency', and not an 'iron law'. In other words, there is nothing that prohibits even negatively-correlated asset classes from moving in the same direction over a given period of time.

Provided that it is possible to 'correlate' asset classes, the Monte Carlo simulation will adjust its randomly generated percentiles so as to take account of the relationship between them. For example, suppose that two asset classes are highly-correlated (e.g. a value of 0.8, as defined within one’s market assumptions), then the Monte Carlo simulation generates a random 90th percentile return for asset class 1 and a 15th percentile return for asset class 2. The Cholesky decomposition will then lead the software to adjust these two randomly-assigned percentiles, to values that are somewhat more consistent (or more probable) given the high correlation value. The percentile for asset class 1 might be adjusted downwards, say to the 85th percentile, while the percentile for asset class 2 could be adjusted upward, e.g. to the 60th percentile, thereby narrowing the difference between the two, based on the correlation value.

The Monte Carlo simulation does not guarantee that 'highly-correlated' asset classes will always move in the same direction (or that negatively-correlated asset classes will move in opposite directions) – but it puts weight behind the assumed tendency. The stronger the tendency, the greater the weight.

One recommended way to compute the correlation coefficients for a data set is to use the Excel CORREL function on the historical prices for each combination of asset classes.

1. Take the historical prices (dating back as far as possible) for each asset class and add them to an Excel spreadsheet.

2. Use Excel functions to compute an average and standard deviation for each asset class.

3. Then using the historical prices for each combination of 2 asset classes, use the Excel CORREL function to compute the correlation coefficient for the asset pair.