# lpm

Compute sample lower partial moments of data

## Syntax

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

## Description

example

````lpm(Data)` computes lower partial moments for asset returns `Data` relative to a default value for `MAR` for each asset in a ```NUMORDERS x NUMSERIES``` matrix and a default value for `Order`.```

example

````lpm(Data,MAR,Order)` computes lower partial moments for asset returns `Data` relative to `MAR` for each asset in a ```NUMORDERS x NUMSERIES``` matrix.```

example

````Moment = lpm(Data,MAR,Order)` computes lower partial moments for asset returns `Data` relative to `MAR` for each asset in a ```NUMORDERS x NUMSERIES``` matrix `Moment`.```

## Examples

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This example shows how to compute the zero-order, first-order, and second-order lower partial moments for the three time series, where the mean of the third time series is used to compute `MAR` (minimum acceptable return) with the so-called risk-free rate.

```load FundMarketCash Returns = tick2ret(TestData); Assets```
```Assets = 1x3 cell {'Fund'} {'Market'} {'Cash'} ```
`MAR = mean(Returns(:,3))`
```MAR = 0.0017 ```
`LPM = lpm(Returns, MAR, [0 1 2])`
```LPM = 3×3 0.4333 0.4167 0.6167 0.0075 0.0140 0.0004 0.0003 0.0008 0.0000 ```

The first row of `LPM` contains zero-order lower partial moments of the three series. The fund and market index fall below `MAR` about 40% of the time and cash returns fall below its own mean about 60% of the time.

The second row contains first-order lower partial moments of the three series. The fund and market have large average shortfall returns relative to `MAR` by 75 and 140 basis points per month. On the other hand, cash underperforms `MAR` by about only four basis points per month on the downside.

The third row contains second-order lower partial moments of the three series. The square root of these quantities provides an idea of the dispersion of returns that fall below the `MAR`. The market index has a much larger variation on the downside when compared to the fund.

## Input Arguments

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Asset returns, specified as a `NUMSAMPLES`-by-`NUMSERIES` matrix with `NUMSAMPLES` observations of `NUMSERIES` asset returns.

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. Although the `lpm` function works for noninteger orders and, in some cases, for negative orders, this falls outside customary usage.

Data Types: `double`

## Output Arguments

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Lower partial moments, returned as 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). The `lpm` 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))```

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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.

 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.

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

 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.

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