MoD Moments of Distribution

Moments of Distributions accumulate the first Four Moments of your Portfolio's distribution:

• Mean - Average
• Variance
• Skew
• Kurtosis

This then allows us to Characterize de Distribution.

	=Sum Of returns
	=Sum Of Second Moments or Sum of return variance.
	=Sum Of Third Moment or Sum of return skew
	=Sum Of Fourth Moment or Sum of return kurtosis.

By the same token, we can accumulate other specific values, for example potential future exposure, peak exposures, threshold (min, max) as well as other vital statistics.

Characterize the Distribution of your Portfolio(s)

 

Once the whole computation is over, we can proceed as follows to

characterize the Portfolio's distribution:

 

	= number of paths
	Mean Value, which by assumption might be 0.

 

The mean / average return can be used to proxy the expected return or we can simply assume a mean zero distribution and thus substitute 0.A Budgeted mean can also be substituted in place in some implementations.

 

We can then proceed to compute densities via our moments, which are stored per aggregation and can be computed for the whole universe (portfolio) but also for each individual aggregation, from which joint densities can be extracted:

Mean  ( )
Standard Deviation
Skew
Kurtosis

Moments are very useful to:

• Gauge distribution properties: test mean-zero, normal distributions,  assumptions.
• Estimate Gamma Risk, Convexity.
• Probability / Quantile Distribution.
• Fat-tails, T-test.
• Combine with Histograms and Quantiles to narrow down the asset's distribution.