Documentation

# elpm

Compute expected lower partial moments for normal asset returns

## Syntax

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

## Arguments

 `Mean` `NUMSERIES` vector with mean returns for a collection of `NUMSERIES` assets. `Sigma` `NUMSERIES` vector with standard deviation of returns for a collection of `NUMSERIES` assets. `MAR` (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. `Order` (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.

## Description

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` `Order`s 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`.

## References

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.