Using Sample and Expected Lower Partial Moments
Introduction
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.
Sample Lower Partial Moments
This example shows how to use 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
(minimum acceptable return) with the so-called risk-free rate.
load FundMarketCash
Returns = tick2ret(TestData);
Assets
Assets = 1×3 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.
Expected Lower Partial Moments
This example shows how to 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))
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 = 1×3 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. 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 average shortfall.
RisklessCash = elpm(Mean(3), 0, MAR, 1)
RisklessCash = 0
See Also
sharpe
| inforatio
| portalpha
| lpm
| elpm
| maxdrawdown
| emaxdrawdown
| ret2tick
| tick2ret