# lpm

Compute sample lower partial moments of data

## 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.

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 Orders 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.

## 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