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# lpm

Compute sample lower partial moments of data

## Syntax

```lpm(Data)
lpm(Data, MAR)
lpm(Data, MAR, Order)
Moment = lpm(Data, MAR, Order)
```

## Arguments

 `Data` `NUMSAMPLES`-by-`NUMSERIES` matrix with `NUMSAMPLES` observations of `NUMSERIES` asset returns. `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. Although this function works for noninteger orders and, in some cases, for negative orders, this falls outside customary usage.

## Description

Given `NUMSERIES` assets with `NUMSAMPLES` returns in a `NUMSAMPLES`-by-`NUMSERIES` matrix `Data`, a scalar minimum acceptable return `MAR`, and one or more nonnegative moment orders in a `NUMORDERS` vector `Order`, `lpm` computes lower partial moments relative to `MAR` for each asset in a `NUMORDERS x NUMSERIES` matrix `Moment`.

The output `Moment` is a ```NUMORDERS x NUMSERIES``` matrix of lower partial moments with `NUMORDERS` `Order`s and `NUMSERIES` series, that is, each row contains lower partial moments for a given order.

 Note:   To compute upper partial moments, reverse the signs of both `Data` and `MAR` (do not reverse the sign of the output). This function computes sample lower partial moments from data. To compute expected lower partial moments for multivariate normal asset returns with a specified mean and covariance, use `elpm`. With `lpm`, you can compute various investment ratios such as Omega ratio, Sortino ratio, and Upside Potential ratio, where:```Omega = lpm(-Data, -MAR, 1) / lpm(Data, MAR, 1)``````Sortino = (mean(Data) - MAR) / sqrt(lpm(Data, MAR, 2))``````Upside = lpm(-Data, -MAR, 1) / sqrt(lpm(Data, MAR, 2))```

## 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.

## See Also

#### Introduced in R2006b

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