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# ewstats

Expected return and covariance from return time series

## Syntax

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

## Arguments

 RetSeries Return Series: number of observations (NUMOBS) by number of assets (NASSETS) matrix of equally spaced incremental return observations. The first row is the oldest observation, and the last row is the most recent. DecayFactor (Optional) Controls how much less each observation is weighted than its successor. The kth observation back in time has weight DecayFactor^k. DecayFactor must lie in the range: 0 < DecayFactor <= 1.Default = 1, the equally weighted linear moving average model (BIS). WindowLength (Optional) Number of recent observations in the computation. Default = NUMOBS.

## Description

[ExpReturn, ExpCovariance, NumEffObs] = ewstats(RetSeries, DecayFactor, WindowLength) computes estimated expected returns, estimated covariance matrix, and the number of effective observations. These are maximum likelihood estimates which are generally biased.

ExpReturn is a 1-by-NASSETS vector of estimated expected returns.

ExpCovariance is an NASSETS-by-NASSETS estimated covariance matrix. The standard deviations of the asset return processes are given by

```     STDVec = sqrt(diag(ExpCovariance))
```

The correlation matrix is

```     CorrMat = ExpCovariance./( STDVec*STDVec' )
```

NumEffObs is the number of effective observations = (1-DecayFactor^WindowLength)/(1-DecayFactor).

A smaller DecayFactor or WindowLength emphasizes recent data more strongly but uses less of the available data set.

## Examples

expand all

### Compute Estimated Expected Returns and Estimated Covariance Matrix

This example shows how to compute the estimated expected returns and the estimated covariance matrix.

```RetSeries = [ 0.24 0.08
0.15 0.13
0.27 0.06
0.14 0.13 ];

DecayFactor = 0.98;

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

0.1995    0.1002

ExpCovariance =

0.0032   -0.0017
-0.0017    0.0010

```