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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.

The following example demonstrates `lpm` 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` (with the so-called
risk-free rate).

```
load FundMarketCash
Returns = tick2ret(TestData);
Assets
MAR = mean(Returns(:,3))
LPM = lpm(Returns, MAR, [0 1 2])
```

which gives the following results:

Assets = 'Fund' 'Market' 'Cash' MAR = 0.0017 LPM = 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 expected shortfall
returns relative to `MAR` by 75 and 140 basis points
per month. On the other hand, cash underperforms `MAR` by
about only 4 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.

To compare realized values with expected values, use `elpm` 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))
Mean = mean(Returns)
Sigma = std(Returns, 1)
Assets
ELPM = elpm(Mean, Sigma, MAR, [0 1 2])
```

which gives the following results:

Assets = 'Fund' 'Market' 'Cash' ELPM = 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. Note that this 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 expected shortfall.

RisklessCash = elpm(Mean(3), 0, MAR, 1)

which gives the following result:

RisklessCash = 0

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