# elpm

Compute expected lower partial moments for normal asset returns

## Description

example

elpm(Mean,Sigma) compute expected lower partial moments (elpm) relative to a default value of MAR for each asset in a NUMORDERS-by-NUMSERIES matrix.

example

elpm(Mean,Sigma,MAR) computes expected lower partial moments (elpm) relative to a MAR for each asset in a NUMORDERS-by-NUMSERIES matrix.

example

elpm(Mean,Sigma,MAR,Order) computes expected lower partial moments (elpm) relative to a MAR and Order for each asset in a NUMORDERS-by-NUMSERIES matrix.

example

Moment = elpm(MeanSigmaMAROrder) computes expected lower partial moments (elpm) relative to a default value of MAR for each asset in a NUMORDERS-by-NUMSERIES matrix Moment.

## Examples

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This example shows how to compute expected lower partial moments based on the mean and standard deviations of normally distributed asset returns. The elpm function works with the mean and standard deviations for multiple assets and multiple orders.

Returns = tick2ret(TestData);
MAR = mean(Returns(:,3))
MAR = 0.0017
Mean = mean(Returns)
Mean = 1×3

0.0038    0.0030    0.0017

Sigma = std(Returns, 1)
Sigma = 1×3

0.0229    0.0389    0.0009

Assets
Assets = 1x3 cell
{'Fund'}    {'Market'}    {'Cash'}

ELPM = elpm(Mean, Sigma, MAR, [0 1 2])
ELPM = 3×3

0.4647    0.4874    0.5000
0.0082    0.0149    0.0004
0.0002    0.0007    0.0000

Based on the moments of each asset, the expected values for lower partial moments imply better than expected performance for the fund and market and worse than expected performance for cash. The elpm function works with either degenerate or nondegenerate normal random variables. For example, if cash were truly riskless, its standard deviation would be 0. You can examine the difference in average shortfall.

RisklessCash = elpm(Mean(3), 0, MAR, 1)
RisklessCash = 0

## Input Arguments

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Mean returns, specified as a NUMSERIES vector with mean returns for a collection of NUMSERIES assets.

Data Types: double

Standard deviation of returns, specified as a NUMSERIES vector with standard deviation of returns for a collection of NUMSERIES assets.

Data Types: double

(Optional) Minimum acceptable return, specified as a scalar numeric. MAR is a cutoff level of return such that all returns above MAR contribute nothing to the lower partial moment.

Data Types: double

(Optional) Moment orders, specified as a either a scalar or a NUMORDERS vector of nonnegative integer moment orders. If no order specified, the default Order = 0, which is the shortfall probability. The elpm function does not work for negative or a noninteger Order.

Data Types: double

## Output Arguments

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Expected Lower partial moments, returned as 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. The output Moment for the lower partial moment represents the moments of asset returns that fall below a minimum acceptable level of return.

Note

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.

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### Lower Partial Moments

Use lower partial moments to examine what is colloquially known as “downside risk.”

The main idea of the lower partial moment framework is to model moments of asset returns that fall below a minimum acceptable level of return. To compute lower partial moments from data, use lpm to calculate lower partial moments for multiple asset return series and for multiple moment orders. To compute expected values for lower partial moments under several assumptions about the distribution of asset returns, use elpm to calculate lower partial moments for multiple assets and for multiple orders.

## References

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

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

[3] Harlow, W.V. 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.

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

## Version History

Introduced in R2006b