estimateAssetMoments
Estimate mean and covariance of asset returns from data
Description
estimates mean and covariance of asset returns from data for a
obj
= estimateAssetMoments(obj
,AssetReturns
)Portfolio
object. For details on the workflow, see Portfolio Object Workflow.
estimates mean and covariance of asset returns from data for a Portfolio object
with additional options for one or more obj
= estimateAssetMoments(___,Name,Value
)Name,Value
pair
arguments.
Examples
Estimate Mean and Covariance of Asset Returns from Data for a Portfolio Object
To illustrate using the estimateAssetMoments
function, generate random samples of 120 observations of asset returns for four assets from the mean and covariance of asset returns in the variables m
and C
with the portsim
function. The default behavior portsim
creates simulated data with estimated mean and covariance identical to the input moments m
and C
. In addition to a return series created by the portsim
function in the variable X
, a price series is created in the variable Y
:
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; m = m/12; C = C/12; X = portsim(m', C, 120); Y = ret2tick(X);
Given asset returns and prices in the variables X
and Y
from above, the following examples demonstrate equivalent ways to estimate asset moments for the Portfolio object. A Portfolio object is created in p
with the moments of asset returns set directly in the Portfolio
object and a second Portfolio object is created in q
to obtain the mean and covariance of asset returns from asset return data in X
using the estimateAssetMoments
function.
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; m = m/12; C = C/12; X = portsim(m', C, 120); p = Portfolio('mean',m,'covar',C); q = Portfolio; q = estimateAssetMoments(q, X); [passetmean, passetcovar] = getAssetMoments(p)
passetmean = 4×1
0.0042
0.0083
0.0100
0.0150
passetcovar = 4×4
0.0005 0.0003 0.0002 0
0.0003 0.0024 0.0017 0.0010
0.0002 0.0017 0.0048 0.0028
0 0.0010 0.0028 0.0102
[qassetmean, qassetcovar] = getAssetMoments(q)
qassetmean = 4×1
0.0042
0.0083
0.0100
0.0150
qassetcovar = 4×4
0.0005 0.0003 0.0002 0.0000
0.0003 0.0024 0.0017 0.0010
0.0002 0.0017 0.0048 0.0028
0.0000 0.0010 0.0028 0.0102
Notice how either approach yields the same moments. The default behavior of the estimateAssetMoments
function is to work with asset returns. If, instead, you have asset prices, such as in the variable Y
, the estimateAssetMoments
function accepts a parameter name 'DataFormat'
with a corresponding value set to 'prices'
to indicate that the input to the method is in the form of asset prices and not returns (the default parameter value for 'DataFormat'
is 'returns'
). The following example compares direct assignment of moments in the Portfolio object p
with estimated moments from asset price data in Y
in the Portfolio object q
:
m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; m = m/12; C = C/12; X = portsim(m', C, 120); Y = ret2tick(X); p = Portfolio('mean',m,'covar',C); q = Portfolio; q = estimateAssetMoments(q, Y, 'dataformat', 'prices'); [passetmean, passetcovar] = getAssetMoments(p)
passetmean = 4×1
0.0042
0.0083
0.0100
0.0150
passetcovar = 4×4
0.0005 0.0003 0.0002 0
0.0003 0.0024 0.0017 0.0010
0.0002 0.0017 0.0048 0.0028
0 0.0010 0.0028 0.0102
[qassetmean, qassetcovar] = getAssetMoments(q)
qassetmean = 4×1
0.0042
0.0083
0.0100
0.0150
qassetcovar = 4×4
0.0005 0.0003 0.0002 0.0000
0.0003 0.0024 0.0017 0.0010
0.0002 0.0017 0.0048 0.0028
0.0000 0.0010 0.0028 0.0102
Estimate Mean and Covariance of Asset Returns from Timetable Data for a Portfolio Object
To illustrate using the estimateAssetMoments
function with AssetReturns
data continued in a timetable
object, use the CAPMuniverse.mat
which contains a timetable
object (AssetTimeTable
) for returns data.
load CAPMuniverse
AssetsTimeTable.Properties;
head(AssetsTimeTable,5)
Time AAPL AMZN CSCO DELL EBAY GOOG HPQ IBM INTC MSFT ORCL YHOO MARKET CASH ___________ _________ _________ _________ _________ _________ ____ _________ _________ _________ _________ _________ _________ _________ __________ 03-Jan-2000 0.088805 0.1742 0.008775 -0.002353 0.12829 NaN 0.03244 0.075368 0.05698 -0.001627 0.054078 0.097784 -0.012143 0.00020522 04-Jan-2000 -0.084331 -0.08324 -0.05608 -0.08353 -0.093805 NaN -0.075613 -0.033966 -0.046667 -0.033802 -0.0883 -0.067368 -0.03166 0.00020339 05-Jan-2000 0.014634 -0.14877 -0.003039 0.070984 0.066875 NaN -0.006356 0.03516 0.008199 0.010567 -0.052837 -0.073363 0.011443 0.00020376 06-Jan-2000 -0.086538 -0.060072 -0.016619 -0.038847 -0.012302 NaN -0.063688 -0.017241 -0.05824 -0.033477 -0.058824 -0.10307 0.011743 0.00020266 07-Jan-2000 0.047368 0.061013 0.0587 -0.037708 -0.000964 NaN 0.028416 -0.004386 0.04127 0.013091 0.076771 0.10609 0.02393 0.00020157
Notice that GOOG
has missing data (NaN
), because it was not listed before Aug 2004. The estimateAssetMoments
function has a name-value pair argument 'MissingData'
that indicates with a Boolean value whether to use the missing data capabilities of Financial Toolbox™ software. The default value for 'MissingData'
is false which removes all samples with NaN
values. If, however, 'MissingData'
is set to true, estimateAssetMoments
uses the ECM algorithm to estimate asset moments.
r = Portfolio; r = estimateAssetMoments(r,AssetsTimeTable,'dataformat','returns','missingdata',true);
In addition, the estimateAssetMoments
function also extracts asset names or identifiers from a timetable object when the name-value argument 'GetAssetList'
set to true
(its default value is false
). If the 'GetAssetList'
value is true
, the timetable column identifiers are used to set the AssetList
property of the Portfolio object. To show this, the formation of the Portfolio object r
is repeated with the 'GetAssetList'
flag set to true
.
r = estimateAssetMoments(r,AssetsTimeTable,'GetAssetList',true);
disp(r.AssetList')
{'AAPL' } {'AMZN' } {'CSCO' } {'DELL' } {'EBAY' } {'GOOG' } {'HPQ' } {'IBM' } {'INTC' } {'MSFT' } {'ORCL' } {'YHOO' } {'MARKET'} {'CASH' }
Estimate Mean and Covariance of Asset Returns from Data for a Portfolio
Object with Integrality Constraints
Create a Portfolio
object for three assets.
AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; AssetMean = AssetMean/12
AssetMean = 3×1
0.0008
0.0004
0.0011
AssetCovar = AssetCovar/12
AssetCovar = 3×3
10-3 ×
0.2705 0.0192 0.3503
0.0192 0.0416 0.0160
0.3503 0.0160 0.6367
X = portsim(AssetMean', AssetCovar, 120); p = Portfolio('AssetMean',AssetMean, 'AssetCovar', AssetCovar); p = setDefaultConstraints(p);
Use setBounds
with semi-continuous constraints to set xi=0
or 0.02
<=xi
<=0.5
for all i=1
,...NumAssets.
p = setBounds(p, 0.02, 0.5,'BoundType', 'Conditional', 'NumAssets', 3);
When working with a Portfolio
object, the setMinMaxNumAssets
function enables you to set up cardinality constraints for a long-only portfolio. This sets the cardinality constraints for the Portfolio
object, where the total number of allocated assets satisfying the nonzero semi-continuous constraints are between MinNumAssets
and MaxNumAssets
. By setting MinNumAssets
=MaxNumAssets
=2, only two of the three assets are invested in the portfolio.
p = setMinMaxNumAssets(p, 2, 2);
Use estimateAssetMoments
to estimate mean and covariance of asset returns from data for a Portfolio
object.
p = estimateAssetMoments(p, X); [passetmean, passetcovar] = getAssetMoments(p)
passetmean = 3×1
0.0008
0.0004
0.0011
passetcovar = 3×3
10-3 ×
0.2705 0.0192 0.3503
0.0192 0.0416 0.0160
0.3503 0.0160 0.6367
The estimateAssetMoments
function uses the MINLP solver to solve this problem. Use the setSolverMINLP
function to configure the SolverType
and options.
p.solverOptionsMINLP
ans = struct with fields:
MaxIterations: 1000
AbsoluteGapTolerance: 1.0000e-07
RelativeGapTolerance: 1.0000e-05
NonlinearScalingFactor: 1000
ObjectiveScalingFactor: 1000
Display: 'off'
CutGeneration: 'basic'
MaxIterationsInactiveCut: 30
ActiveCutTolerance: 1.0000e-07
IntMainSolverOptions: [1x1 optim.options.Intlinprog]
NumIterationsEarlyIntegerConvergence: 30
ExtendedFormulation: 0
NumInnerCuts: 10
NumInitialOuterCuts: 10
Input Arguments
obj
— Object for portfolio
object
Object for portfolio, specified using a Portfolio
object. For more information on creating a portfolio object, see
Data Types: object
AssetReturns
— Matrix, table, or timetable that contains asset price data that can be converted to asset returns
matrix | table | table
Matrix, table
, or timetable
that contains asset
price data that can be converted to asset returns, specified by a
NumSamples
-by-NumAssets
matrix.
AssetReturns
data can be:
NumSamples
-by-NumAssets
matrix.Table of
NumSamples
prices or returns at a given periodicity for a collection ofNumAssets
assetsTimetable object with
NumSamples
observations andNumAssets
time series
Use the optional DataFormat
argument to convert
AssetReturns
input data that is asset prices into
asset returns. Be careful when using asset price data because portfolio
optimization usually requires total returns and not simply price
returns.
Data Types: double
| table
| timetable
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: p =
estimateAssetMoments(p,Y,'dataformat','prices')
DataFormat
— Flag to convert input data as prices into returns
'Returns'
(default) | character vector with values 'Returns'
or
'Prices'
Flag to convert input data as prices into returns, specified as the
comma-separated pair consisting of 'DataFormat'
and a
character vector with the values:
'Returns'
— Data inAssetReturns
contains asset total returns.'Prices'
— Data inAssetReturns
contains asset total return prices.
Data Types: char
MissingData
— Flag indicating whether to use ECM algorithm or exclude samples with NaN
values
false
(default) | logical with value true
or
false
Flag indicating whether to use ECM algorithm or excludes samples with
NaN
values, specified as the comma-separated pair
consisting of 'MissingData'
and a logical with a
value of true
or false
.
To handle time series with missing data (indicated with
NaN
values), the MissingData
flag either uses the ECM algorithm to obtain maximum likelihood
estimates in the presences of NaN
values or excludes
samples with NaN
values. Since the default is
false
, it is necessary to specify
MissingData
as true
to use the
ECM algorithm.
Acceptable values for MissingData
are:
false
— Do not use ECM algorithm to handleNaN
values (excludeNaN
values).true
— Use ECM algorithm to handleNaN
values.
For more information on the ECM algorithm, see ecmnmle
and Multivariate Normal Regression.
Data Types: logical
GetAssetList
— Flag indicating which asset names to use for asset list
false
(default) | logical with value true
or
false
Flag indicating which asset names to use for the asset list, specified
as the comma-separated pair consisting of
'GetAssetList'
and a logical with a value of
true
or false
. Acceptable
values for GetAssetList
are:
false
— Do not extract or create asset names.true
— Extract or create asset names from a table or timetable object.
If a table
or timetable
is passed into
this function using the AssetReturns
argument and
the GetAssetList
flag is true
, the
column names from the table or timetable object are used as asset names
in obj.AssetList
.
If a matrix is passed and the GetAssetList
flag is
true
, default asset names are created based on
the AbstractPortfolio
property
defaultforAssetList
, which is
'Asset'
.
If the GetAssetList
flag is
false
, no action occurs, which is the default
behavior.
Data Types: logical
Output Arguments
obj
— Updated portfolio object
object for portfolio
Updated portfolio object, returned as a Portfolio
object. For more information on creating a portfolio object, see
Tips
You can also use dot notation to estimate the mean and covariance of asset returns from data.
obj = obj.estimateAssetMoments(AssetReturns);
Version History
Introduced in R2011aR2018a: Support for timetable input for AssetReturns
Using a fints
object for the AssetReturns
argument of estimateAssetMoments
is not recommended. Use
timetable
instead for financial time
series. For more information, see Convert Financial Time Series Objects (fints) to Timetables.
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