To create a fully specified CVaR portfolio optimization problem,
instantiate the PortfolioCVaR object using the PortfolioCVaR
function. For information
on the workflow when using PortfolioCVaR objects, see PortfolioCVaR Object Workflow.
Use the PortfolioCVaR
function
to create an instance of an object of the PortfolioCVaR
class.
You can use the PortfolioCVaR
function
in several ways. To set up a portfolio optimization problem in a PortfolioCVaR
object, the simplest syntax is:
p = PortfolioCVaR;
p
, such
that all object properties are empty. The PortfolioCVaR
function
also accepts collections of argument namevalue pair arguments for
properties and their values. The PortfolioCVaR
function
accepts inputs for public properties with the general syntax:
p = PortfolioCVaR('property1', value1, 'property2', value2, ... );
If a PortfolioCVaR object already exists, the syntax permits
the first (and only the first argument) of the PortfolioCVaR
function to be an existing
object with subsequent argument namevalue pair arguments for properties
to be added or modified. For example, given an existing PortfolioCVaR
object in p
, the general syntax is:
p = PortfolioCVaR(p, 'property1', value1, 'property2', value2, ... );
Input argument names are not casesensitive, but must be completely
specified. In addition, several properties can be specified with alternative
argument names (see Shortcuts for Property Names). The PortfolioCVaR
function
tries to detect problem dimensions from the inputs and, once set,
subsequent inputs can undergo various scalar or matrix expansion operations
that simplify the overall process to formulate a problem. In addition,
a PortfolioCVaR object is a value object so that, given portfolio p
,
the following code creates two objects, p
and q
,
that are distinct:
q = PortfolioCVaR(p, ...)
A CVaR portfolio optimization problem is completely specified with the PortfolioCVaR object if the following three conditions are met:
You must specify a collection of asset returns or
prices known as scenarios such that all scenarios are finite asset
returns or prices. These scenarios are meant to be samples from the
underlying probability distribution of asset returns. This condition can
be satisfied by the setScenarios
function
or with several canned scenario simulation functions.
The set of feasible portfolios must be a nonempty
compact set, where a compact set is closed and bounded. You can satisfy
this condition using an extensive collection of properties that define
different types of constraints to form a set of feasible portfolios.
Since such sets must be bounded, either explicit or implicit constraints
can be imposed and several tools, such as the estimateBounds
function,
provide ways to ensure that your problem is properly formulated.
You must specify a probability level to locate the
level of tail loss above which the conditional valueatrisk is to
be minimized. This condition can be satisfied by the setProbabilityLevel
function.
Although the general sufficient conditions for CVaR portfolio optimization go beyond the first three conditions, the PortfolioCVaR object handles all these additional conditions.
If you create a PortfolioCVaR object, p
,
with no input arguments, you can display it using disp
:
p = PortfolioCVaR; disp(p);
PortfolioCVaR with properties: BuyCost: [] SellCost: [] RiskFreeRate: [] ProbabilityLevel: [] Turnover: [] BuyTurnover: [] SellTurnover: [] NumScenarios: [] Name: [] NumAssets: [] AssetList: [] InitPort: [] AInequality: [] bInequality: [] AEquality: [] bEquality: [] LowerBound: [] UpperBound: [] LowerBudget: [] UpperBudget: [] GroupMatrix: [] LowerGroup: [] UpperGroup: [] GroupA: [] GroupB: [] LowerRatio: [] UpperRatio: []
The approaches listed provide a way to set up a portfolio optimization
problem with the PortfolioCVaR
function.
The custom set functions offer additional ways to set and modify collections
of properties in the PortfolioCVaR object.
You can use thePortfolioCVaR
function to
directly set up a "standard" portfolio optimization
problem. Given scenarios of asset returns in the variable AssetScenarios
,
this problem is completely specified as follows:
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; AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioCVaR('Scenarios', AssetScenarios, ... 'LowerBound', 0, 'LowerBudget', 1, 'UpperBudget', 1, ... 'ProbabilityLevel', 0.95);
LowerBound
property value
undergoes scalar expansion since AssetScenarios
provides
the dimensions of the problem.You can use dot notation with the function plotFrontier
.
p.plotFrontier;
An alternative way to accomplish the same task of setting up
a "standard" CVaR portfolio optimization problem, given AssetScenarios
variable
is:
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; AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioCVaR; p = setScenarios(p, AssetScenarios); p = PortfolioCVaR(p, 'LowerBound', 0); p = PortfolioCVaR(p, 'LowerBudget', 1, 'UpperBudget', 1); p = setProbabilityLevel(p, 0.95); plotFrontier(p);
This way works because
the calls to the are in this particular order. In this case, the
call to initialize AssetScenarios
provides the
dimensions for the problem. If you were to do this step last, you
would have to explicitly dimension the LowerBound
property
as follows:
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; AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioCVaR; p = PortfolioCVaR(p, 'LowerBound', zeros(size(m))); p = PortfolioCVaR(p, 'LowerBudget', 1, 'UpperBudget', 1); p = setProbabilityLevel(p, 0.95); p = setScenarios(p, AssetScenarios);
Note:
If you did not specify the size of 
The PortfolioCVaR
function
has shorter argument names that replace longer argument names associated
with specific properties of the PortfolioCVaR object. For example,
rather than enter 'ProbabilityLevel'
, the PortfolioCVaR
function accepts the caseinsensitive
name 'plevel'
to set the ProbabilityLevel
property
in a PortfolioCVaR object. Every shorter argument name corresponds
with a single property in the PortfolioCVaR
function.
The one exception is the alternative argument name 'budget'
,
which signifies both the LowerBudget
and UpperBudget
properties.
When 'budget'
is used, then the LowerBudget
and UpperBudget
properties
are set to the same value to form an equality budget constraint.
Shortcuts for Property Names
Shortcut Argument Name  Equivalent Argument / Property Name 



























For example, this call to the PortfolioCVaR
function
uses these shortcuts for properties:
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; AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioCVaR('scenario', AssetScenarios, 'lb', 0, 'budget', 1, 'plevel', 0.95); plotFrontier(p);
Although not recommended, you can set properties directly using dot notation, however no errorchecking is done on your inputs:
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; AssetScenarios = mvnrnd(m, C, 20000); p = PortfolioCVaR; p = setScenarios(p, AssetScenarios); p.ProbabilityLevel = 0.95; p.LowerBudget = 1; p.UpperBudget = 1; p.LowerBound = zeros(size(m)); plotFrontier(p);
Note:
Scenarios cannot be assigned directly to a PortfolioCVaR object.
Scenarios must always be set through either the 
estimateBounds
 PortfolioCVaR