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The final element for a complete specification of a portfolio optimization problem is the set of feasible portfolios, which is called a portfolio set. A portfolio set $$X\subset {R}^{n}$$ is specified by construction as the intersection of sets formed by a collection of constraints on portfolio weights. A portfolio set necessarily and sufficiently must be a nonempty, closed, and bounded set.

When setting up your portfolio set, ensure that the portfolio set satisfies these
conditions. The most basic or “default” portfolio set requires portfolio
weights to be nonnegative (using the lower-bound constraint) and to sum to
`1`

(using the budget constraint). The most general portfolio set
handled by the portfolio optimization tools can have any of these constraints:

Linear inequality constraints

Linear equality constraints

`'Simple'`

Bound constraints`'Conditional'`

Bond constraintsBudget constraints

Group constraints

Group ratio constraints

Average turnover constraints

One-way turnover constraints

Cardinality constraints

*Linear inequality constraints* are general linear
constraints that model relationships among portfolio weights that satisfy a system
of inequalities. Linear inequality constraints take the form

$${A}_{I}x\le {b}_{I}$$

where:

*x* is the portfolio (*n* vector).

*A _{I}* is the linear inequality constraint
matrix (

*b _{I}* is the linear inequality constraint
vector (

*n* is the number of assets in the universe and
*n _{I}* is the number of constraints.

`PortfolioCVaR`

object properties to specify linear inequality
constraints are:

`AInequality`

for*A*_{I}`bInequality`

for*b*_{I}`NumAssets`

for*n*

The default is to ignore these constraints.

*Linear equality constraints* are general linear constraints
that model relationships among portfolio weights that satisfy a system of
equalities. Linear equality constraints take the form

$${A}_{E}x={b}_{E}$$

where:

*x* is the portfolio (*n* vector).

*A _{E}* is the linear equality constraint
matrix (

*b _{E}* is the linear equality constraint
vector (

*n* is the number of assets in the universe and
*n _{E}* is the number of constraints.

`PortfolioCVaR`

object properties to specify linear equality
constraints are:

`AEquality`

for*A*_{E}`bEquality`

for*b*_{E}`NumAssets`

for*n*

The default is to ignore these constraints.

`'Simple'`

*Bound constraints* are specialized linear constraints that
confine portfolio weights to fall either above or below specific bounds. Since every
portfolio set must be bounded, it is often a good practice, albeit not necessary, to
set explicit bounds for the portfolio problem. To obtain explicit bounds for a given
portfolio set, use the `estimateBounds`

function. Bound
constraints take the form

$${l}_{B}\le x\le {u}_{B}$$

where:

*x* is the portfolio (*n* vector).

*l _{B}* is the lower-bound constraint
(

*u _{B}* is the upper-bound constraint
(

*n* is the number of assets in the universe.

`PortfolioCVaR`

object properties to specify bound constraints are:

`LowerBound`

for*l*_{B}`UpperBound`

for*u*_{B}`NumAssets`

for*n*

The default is to ignore these constraints.

The default portfolio optimization problem (see Default Portfolio Problem) has
*l _{B}* =

`0`

with
`'Conditional'`

Bound Constraints`'Conditional'`

*Bound constraints*, also called semicontinuous constraints,
are mixed-integer linear constraints that confine portfolio weights to fall either
above or below specific bounds *if* the asset is selected;
otherwise, the value of the asset is zero. Use `setBounds`

to
specify bound constraints with a `'Conditional'`

`BoundType`

. To mathematically formulate this type of constraints,
a binary variable *v*_{i} is
needed. *v*_{i} = 0
indicates that asset *i* is not selected and
*v*_{i} indicates
that the asset was selected. Thus

$${l}_{i}{v}_{i}\le {x}_{i}\le {u}_{i}{v}_{i}$$

where

*x* is the portfolio (*n* vector).

*l* is the conditional lower-bound constraint
(*n* vector).

*u* is the conditional upper-bound constraint
(*n* vector).

*n* is the number of assets in the universe.

`PortfolioCVaR`

object properties to specify the bound constraint are:

`LowerBound`

for*l*_{B}`UpperBound`

for*u*_{B}`NumAssets`

for*n*

The default is to ignore this constraint.

*Budget constraints* are specialized linear constraints that
confine the sum of portfolio weights to fall either above or below specific bounds.
The constraints take the form

$${l}_{S}\le {1}^{T}x\le {u}_{S}$$

where:

*x* is the portfolio (*n* vector).

`1`

is the vector of ones (*n* vector).

*l _{S}* is the lower-bound budget constraint
(scalar).

*u _{S}* is the upper-bound budget constraint
(scalar).

*n* is the number of assets in the universe.

`PortfolioCVaR`

object properties to specify budget constraints are:

`LowerBudget`

for*l*_{S}`UpperBudget`

for*u*_{S}`NumAssets`

for*n*

The default is to ignore this constraint.

The default portfolio optimization problem (see Default Portfolio Problem) has
*l _{S}* =

`1`

, which
means that the portfolio weights sum to `1`

. If the portfolio
optimization problem includes possible movements in and out of cash, the budget
constraint specifies how far portfolios can go into cash. For example, if
`0`

and
`1`

, then
the portfolio can have 0–100% invested in cash. If cash is to be a portfolio choice,
set `RiskFreeRate`

(*Group constraints* are specialized linear constraints that
enforce “membership” among groups of assets. The constraints take the
form

$${l}_{G}\le Gx\le {u}_{G}$$

where:

*x* is the portfolio (*n* vector).

*l _{G}* is the lower-bound group constraint
(

*u _{G}* is the upper-bound group constraint
(

*G* is the matrix of group membership indexes
(*n _{G}*-by-

Each row of *G* identifies which assets belong to a group
associated with that row. Each row contains either `0`

s or
`1`

s with `1`

indicating that an asset is part
of the group or `0`

indicating that the asset is not part of the
group.

`PortfolioCVaR`

object properties to specify group constraints
are:

`GroupMatrix`

for*G*`LowerGroup`

for*l*_{G}`UpperGroup`

for*u*_{G}`NumAssets`

for*n*

The default is to ignore these constraints.

*Group ratio constraints* are specialized linear constraints
that enforce relationships among groups of assets. The constraints take the
form

$${l}_{Ri}{({G}_{B}x)}_{i}\le {({G}_{A}x)}_{i}\le {u}_{Ri}{({G}_{B}x)}_{i}$$

for *i* = 1,..., *n _{R}*
where:

*x* is the portfolio (*n* vector).

*l _{R}* is the vector of lower-bound group
ratio constraints (

*u _{R}* is the vector matrix of upper-bound
group ratio constraints (

*G _{A}* is the matrix of base group
membership indexes
(

*G _{B}* is the matrix of comparison group
membership indexes
(

*n* is the number of assets in the universe and
*n _{R}* is the number of constraints.

Each row of *G _{A}* and

Each row contains either `0`

s or `1`

s with
`1`

indicating that an asset is part of the group or
`0`

indicating that the asset is not part of the group.

`PortfolioCVaR`

object properties to specify group ratio
constraints are:

`GroupA`

for*G*_{A}`GroupB`

for*G*_{B}`LowerRatio`

for*l*_{R}`UpperRatio`

for*u*_{R}`NumAssets`

for*n*

The default is to ignore these constraints.

*Turnover constraint* is a linear absolute value constraint
that ensures estimated optimal portfolios differ from an initial portfolio by no
more than a specified amount. Although portfolio turnover is defined in many ways,
the turnover constraints implemented in Financial Toolbox™ computes portfolio turnover as the average of purchases and sales.
Average turnover constraints take the form

$$\frac{1}{2}{1}^{T}|x-{x}_{0}|\le \tau $$

where:

*x* is the portfolio (*n* vector).

`1`

is the vector of ones (*n* vector).

*x _{0}* is the initial portfolio
(

*τ* is the upper bound for turnover (scalar).

*n* is the number of assets in the universe.

`PortfolioCVaR`

object properties to specify the average turnover
constraint are:

`Turnover`

for*τ*`InitPort`

for*x*_{0}`NumAssets`

for*n*

The default is to ignore this constraint.

*One-way turnover constraints* ensure that estimated optimal
portfolios differ from an initial portfolio by no more than specified amounts
according to whether the differences are purchases or sales. The constraints take
the forms

$${1}^{T}\times \mathrm{max}\left\{0,x-{x}_{0}\right\}\le {\tau}_{B}$$

$${1}^{T}\times \mathrm{max}\left\{0,{x}_{0}-x\right\}\le {\tau}_{S}$$

where:

*x* is the portfolio (*n* vector)

`1`

is the vector of ones (*n* vector).

*x _{0}* is the Initial portfolio
(

τ_{B} is the upper bound for turnover
constraint on purchases (scalar).

τ_{S} is the upper bound for turnover
constraint on sales (scalar).

To specify one-way turnover constraints, use the following properties in the
`Portfolio`

, `PortfolioCVaR`

, or
`PortfolioMAD`

object:

`BuyTurnover`

for τ_{B}`SellTurnover`

for τ_{S}`InitPort`

for*x*_{0}

The default is to ignore this constraint.

**Note**

The average turnover constraint (see Average Turnover Constraints) with τ
is not a combination of the one-way turnover constraints with τ =
τ_{B} =
τ_{S}.

*Cardinality constraint* limits the number of assets in the
optimal allocation for an `PortfolioCVaR`

object. Use `setMinMaxNumAssets`

to specify the `'MinNumAssets'`

and
`'MaxNumAssets'`

constraints. To mathematically formulate this
type of constraints, a binary variable
*v*_{i} is needed.
*v*_{i} = 0 indicates
that asset *i* is not selected and
*v*_{i} = 1
indicates that the asset was selected. Thus

$$MinNumAssets\le {\displaystyle \sum _{i=1}^{NumAssets}{v}_{i}\le MaxNumAssets}$$

The default is to ignore this constraint.

- Creating the PortfolioCVaR Object
- Working with CVaR Portfolio Constraints Using Defaults
- Hedging Using CVaR Portfolio Optimization
- Compute Maximum Reward-to-Risk Ratio for CVaR Portfolio