# 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(MeanSigmaMAROrder)``

## 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.

```load FundMarketCash 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` `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`.

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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