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Compute expected lower partial moments for normal asset returns


Moment = elpm(Mean,Sigma,MAR,Order)



NUMSERIES vector with mean returns for a collection of NUMSERIES assets.


NUMSERIES vector with standard deviation of returns for a collection of NUMSERIES assets.


(Optional) Scalar minimum acceptable return (default MAR = 0). This is a cutoff level of return such that all returns above MAR contribute nothing to the lower partial moment.


(Optional) Either a scalar or a NUMORDERS vector of nonnegative integer moment orders. If no order specified, default Order = 0, which is the shortfall probability. This function will not work for negative or noninteger orders.


Given NUMSERIES asset returns with a vector of mean returns in a NUMSERIES vector Mean, a vector of standard deviations of returns in a NUMSERIES vector Sigma, a scalar minimum acceptable return MAR, and one or more nonnegative integer moment orders in a NUMORDERS vector Order, compute expected lower partial moments (elpm) relative to MAR for each asset in a NUMORDERS-by-NUMSERIES matrix Moment.

The output, Moment, is a NUMORDERS-by-NUMSERIES matrix of expected lower partial moments with NUMORDERS Orders and NUMSERIES series, that is, each row contains expected lower partial moments for a given order.


To compute upper partial moments, reverse the signs of both the input Mean and MAR (do not reverse the signs of either Sigma or the output). This function computes expected lower partial moments with the mean and standard deviation of normally distributed asset returns. To compute sample lower partial moments from asset returns which have no distributional assumptions, use lpm.


See Expected Lower Partial Moments.


Vijay S. Bawa. "Safety-First, Stochastic Dominance, and Optimal Portfolio Choice." Journal of Financial and Quantitative Analysis. Vol. 13, No. 2, June 1978, pp. 255–271.

W. V. Harlow. "Asset Allocation in a Downside-Risk Framework." Financial Analysts Journal. Vol. 47, No. 5, September/October 1991, pp. 28–40.

W. V. Harlow and K. S. Rao. "Asset Pricing in a Generalized Mean-Lower Partial Moment Framework: Theory and Evidence." Journal of Financial and Quantitative Analysis. Vol. 24, No. 3, September 1989, pp. 285–311.

Frank A. Sortino and Robert van der Meer. "Downside Risk." Journal of Portfolio Management. Vol. 17, No. 5, Spring 1991, pp. 27–31.

Introduced in R2006b