Note: This page has been translated by MathWorks. Please click here

To view all translated materials including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materials including this page, select Japan from the country navigator on the bottom of this page.

Expected return and covariance from return time series

`[ExpReturn,ExpCovariance,NumEffObs] = ewstats(RetSeries)`

`[ExpReturn,ExpCovariance,NumEffObs] = ewstats(___DecayFactor,WindowLength)`

`[`

computes estimated expected returns (`ExpReturn`

,`ExpCovariance`

,`NumEffObs`

] = ewstats(`RetSeries`

)`ExpReturn`

), estimated
covariance matrix (`ExpCovariance`

), and the number of
effective observations (`NumEffObs`

). These outputs are
maximum likelihood estimates which are biased.

`[`

adds optional input arguments for `ExpReturn`

,`ExpCovariance`

,`NumEffObs`

] = ewstats(___`DecayFactor`

,`WindowLength`

)`DecayFactor`

and
`WindowLength`

.

For a return series
*r*(1),…,*r*(*n*), where
(*n*) is the most recent observation, and *w*
is the decay factor, the expected returns (`ExpReturn`

) are
calculated by

$$E(r)=\frac{(r(n)+wr(n-1)+{w}^{2}r(n-2)+\mathrm{...}+{w}^{n-1}r(1))}{NumEffObs}$$

where the number of effective observations `NumEffObs`

is
defined as

$$NumEffObs=1+w+{w}^{2}+\mathrm{...}+{w}^{n-1}=\frac{1-{w}^{n}}{1-w}$$

*E*(*r*) is the weighed average of
*r*(*n*),…,*r*(`1`

).
The unnormalized weights are *w*,
*w*^{2}, …,
*w*^{(n-1)}. The unnormalized weights
do not sum up to `1`

, so `NumEffObs`

rescales
the unnormalized weights. After rescaling, the normalized weights (which sum up to
`1`

) are used for averaging. When *w* =
`1`

, then `NumEffObs`

=
*n*, which is the number of observations. When *w*
< `1`

, `NumEffObs`

is still interpreted as
the sample size, but it is less than *n* due to the down-weight on
the observations of the remote past.

There is no relationship between `ewstats`

function and
the RiskMetrics® approach for determing the expected return and covariance
from a return time series.

Was this topic helpful?