# setConditionalBudget

## Description

sets up conditional budget constraints for `obj`

= setConditionalBudget(`obj`

,`ConditionalBudgetThreshold`

,`ConditionalUpperBudget`

)`Portfolio`

,
`PortfolioCVaR`

, or `PortfolioMAD`

objects. If the weight of an asset exceeds the
`ConditionalBudgetThreshold`

value, the weight of that
asset is added to the aggregate sum that is bound by the
`ConditionalUpperBudget`

value. For more information, see
Conditional Budget Constraints .

This constraint supports the Undertakings for Collective Investment in Transferable Securities (UCITS) Directive. The UCITS asset regulation states that any investments in excess of 5% must not exceed 40% of the total portfolio. This constraint is a conditional budget constraint. For more information, see Undertakings for Collective Investment in Transferable Securities and Adding Constraints to Satisfy UCITS Directive.

For details on the respective workflows when using these different objects, see Portfolio Object Workflow, PortfolioCVaR Object Workflow, and PortfolioMAD Object Workflow.

## Examples

### Add Conditional Budget Constraints

This example shows the workflow to add a conditional budget constraint for a portfolio optimization problem. The conditional budget constraint enforces a limit on the total proportion that you can invest in assets that pass a prescribed threshold.

In mathematical terms, the conditional budget constraint has the following form:

$$\sum _{\mathit{i}\in \mathit{I}}{\mathit{w}}_{\mathit{i}}\le \beta $$

where $\mathit{I}=\left\{\mathit{i}|{\mathit{w}}_{\mathit{i}}\ge {\alpha}_{\mathit{i}}\right\}$ and $\mathit{w}$ represents the asset weights.

**Create Portfolio Object**

To solve a portfolio problem with this constraint, start by creating a `Portfolio`

object with default constraints. In addition to using a `Portfolio`

object, you can also perform this workflow using a `PortfolioCVaR`

or `PortfolioMAD`

object.

% Load data load CAPMuniverse.mat tol = 1e-8; p = Portfolio('AssetList',Assets(1:12)); p = estimateAssetMoments(p,Data(:,1:12));

Using `setConditionalBudget`

, set a conditional budget constraint that limits the aggregate weight that can be invested in assets that exceed 10% to 50%.

```
% Assets with weights above 10% must not exceed 50% of the total portfolio
p = setConditionalBudget(p,0.1,0.5);
```

At this point, the portfolio problem is unbounded and computing any optimal portfolio results in an error. Use `setBounds`

to add bounds to the portfolio weights.

```
% Weights must be between -1 and 1
p = setBounds(p,-1,1);
```

**Compute Efficient Frontier**

The most common workflow is to estimate a set of portfolios on the efficient frontier. The efficient frontier is the curve that shows the tradeoff between the return and risk achieved by Pareto-optimal portfolios. For a given return level, the portfolio on the efficient frontier is the one that minimizes the risk while maintaining the desired return. Conversely, for a given risk level, the portfolio on the efficient frontier is the one that maximizes return while maintaining the desired risk level.

```
% Compute efficient frontier
w = estimateFrontier(p)
```

`w = `*12×10*
0.0000 0.0368 0.0737 0.1105 0.1473 0.1979 0.2408 0.2686 0.5000 0.5000
0.0000 0.0008 0.0016 0.0023 0.0031 0.0082 0.0108 0.0762 0.0317 0.1000
0.0000 -0.0222 -0.0444 -0.0665 -0.0887 -0.1084 -0.1294 -0.2283 -0.2164 -1.0000
-0.0000 -0.0218 -0.0436 -0.0654 -0.0871 -0.1144 -0.1386 -0.4499 -0.4361 -1.0000
0.0000 -0.0087 -0.0175 -0.0262 -0.0350 -0.0448 -0.0539 -0.0260 -0.0300 0.1000
-0.0000 0.0398 0.0796 0.1194 0.1592 0.2122 0.2578 0.2314 0.1000 0.1000
0.0000 0.0366 0.0732 0.1098 0.1464 0.1000 0.1000 0.1000 0.1000 0.1000
-0.0000 -0.0282 -0.0565 -0.0847 -0.1129 -0.1424 -0.1712 -0.1291 -0.2717 0.1000
0.0000 0.0060 0.0120 0.0180 0.0240 0.0506 0.0658 0.1000 0.1000 0.1000
-0.0000 0.0111 0.0222 0.0333 0.0444 0.0746 0.0941 0.1000 0.1000 0.1000
⋮

plotFrontier(p,w)

Check that none of the assets that exceed 10% go above 50%.

```
% Keep weights above 10% threshold
auxW = w;
auxW(w <= 0.1 + tol) = 0;
conditionalBudget = sum(auxW)
```

`conditionalBudget = `*1×10*
0 0 0 0.3397 0.4529 0.4101 0.4986 0.5000 0.5000 0.5000

### Set Different Conditional Budget Thresholds for Different Assets

This example shows how to set a different conditional budget threshold for one of the assets in the portfolio. `setConditionalBudget`

supports a vector input for the `CondtionalBudgetThreshold`

argument, where each entry represents the conditional threshold for each asset. In this example, assume that the conditional budget threshold for all assets is 10% except for IBM. The conditional budget threshold for IBM is 15%.

To solve a portfolio problem with this constraint, start by creating a `PortfolioCVaR`

object with default constraints. In addition to using a `PortfolioCVaR`

object, you can also perform this workflow using a `Portfolio`

or `PortfolioMAD`

object.

% Load data load CAPMuniverse.mat tol = 1e-8; startRow = find(~isnan(Data(:,6)),1); p = PortfolioCVaR(AssetList=Assets(1:12), ... AssetScenarios=Data(startRow:end,1:12),ProbabilityLevel=0.95); p = setDefaultConstraints(p);

The objective of the minimum variance problem is to find the weights (properties of the total investment) allocated to each asset in the portfolio that result in the lowest possible risk.

threshold = 0.1*ones(p.NumAssets,1); threshold(strcmpi('IBM',p.AssetList)) = 0.15; p = setConditionalBudget(p,threshold,0.4); % Solve the minimum risk problem wMin = estimateFrontierLimits(p,'min'); figure bar(Assets(1:12),wMin); hold on yline(0.1,'k--',LineWidth=2) yline(0.15,'r--',LineWidth=2) title('Minimum Risk Portfolio') ylabel('Weight')

All assets except IBM and MSFT do not exceed the 10% threshold, IBM does not exceed the 15% threshold, and the sum of all the assets above their conditional threshold (in this case only MSFT) is less than 40%.

### Excluding One Asset from Conditional Budget Constraint

This example shows how to exclude individual assets from the conditional budget constraint.

Create a `PortfolioMAD`

object. In addition to using a `PortfolioMAD`

object, you can also perform this workflow using a `Portfolio`

or `PortfolioCVaR`

object.

% Load data load CAPMuniverse.mat tol = 1e-8; % Create a PortfolioMAD object with default constraints startRow = find(~isnan(Data(:,6)),1); q = PortfolioMAD(AssetList=Assets(1:12), ... AssetScenarios=Data(startRow:end,1:12)); % Nonnegative and fully invested portfolios q = setDefaultConstraints(q);

To exclude any asset from the conditional budget constraint, set the `ConditionalBudgetThreshold`

of that asset to `Inf`

. In this example, you can exclude IBM from the conditional budget constraint.

% Conditional budget threshold = 0.1*ones(q.NumAssets,1); threshold(strcmpi('IBM',q.AssetList)) = Inf; q2 = setConditionalBudget(q,threshold,0.4); % Solve minimum risk problem wMin2 = estimateFrontierLimits(q2,'min'); figure bar(Assets(1:12),wMin2); hold on yline(0.1,'k--',LineWidth=2) title('Minimum Risk Portfolio') ylabel('Weight')

### Working with Conditional Budget and Other Constraints

This example shows how to combine the conditional budget constraint with other types of constraints. Specifically, the example shows how to add semicontinuous bounds on the assets together with a conditional budget constraint, while ensuring that the portfolio is fully invested.

Create a mean-variance `Portfolio`

object with default constraints.

% Load data load CAPMuniverse.mat tol = 1e-8; p = Portfolio('AssetList',Assets(1:12)); p = estimateAssetMoments(p,Data(:,1:12));

Use `setBounds`

to ensure that any asset that is chosen in the portfolio has a weight of at least 8% and at most 15%. Then, use `setBudget`

to set the fully invested constraint.

% Assets must be either 0 or between 8% and 15% p = setBounds(p,0.08,0.15,BoundType="conditional"); % Assets must sum to 1 p = setBudget(p,1,1); % Conditional budget p = setConditionalBudget(p,0.1,0.2);

To find the minimum tracking error portfolio that satisfies all the previously specified constraints, use the risk parity portfolio as the benchmark portfolio.

% Solve minimum tracking error portfolio trackingPort = riskBudgetingPortfolio(p.AssetCovar); TE = @(w) (w-trackingPort)'*p.AssetCovar*(w-trackingPort); wTE = estimateCustomObjectivePortfolio(p,TE); figure bar(Assets(1:12),[trackingPort wTE]); hold on yline(0.1,'k--',LineWidth=2) yline(0.08,'b--',LineWidth=2) title('Minimum Tracking Error Portfolio') ylabel('Weight') legend('Tracking Portfolio','Constrained Min TE Port', ... Location='northwest')

All assets for the constrained portfolio are either zero or have at least a 0.08 weight, which means that the semicontinuous bound is satisfied. Furthermore, the sum of the assets above the 0.1 threshold is less than 0.2, and this result is also consistent with the conditional budget.

## Input Arguments

`obj`

— Object for portfolio

`Portfolio`

object | `PortfolioCVaR`

object | `PortfolioMAD`

object

Object for portfolio, specified using the `Portfolio`

,
`PortfolioCVaR`

, or `PortfolioMAD`

object. For more information on creating a portfolio object, see:

**Data Types: **`object`

`ConditionalBudgetThreshold`

— Weights threshold above which asset is accounted for in conditional budget

scalar numeric | numeric vector

Weights threshold above which the asset is accounted for in the
conditional budget, specified as a scalar or a
`NumAssets`

-by-`1`

vector for a
`Portfolio`

, `PortfolioCVaR`

, or
`PortfolioMAD`

input object (`obj`

).
The `ConditionalBudgetThreshold`

value can be a positive or
negative value. However, this value cannot be `NaN`

or
`-Inf`

.

The `ConditionalBudgetThreshold`

value indicates whether
an asset should be accounted for in the aggregate sum or not. If the weight
of an asset exceeds the `ConditionalBudgetThreshold`

value,
the weight of that asset is added to the aggregate sum that is bound by the
`ConditionalBudgetThreshold`

value.

**Note**

All assets for the `ConditionalBudgetThreshold`

value
are larger than or equal to the
`ConditionalUpperBudget`

value, then the
conditional budget constraint is always feasible, so Financial Toolbox™ issues the following message: ```
Warning: Redundant
conditional budget constraint
```

.

**Data Types: **`double`

`ConditionalUpperBudget`

— Maximum aggregate weight of portfolio that can be invested in assets that exceed `ConditionalBudgetThreshold`

scalar numeric

Maximum aggregate weight of the portfolio that can be invested in assets
that exceed the `ConditionalBudgetThreshold`

value,
specified as a scalar for a `Portfolio`

,
`PortfolioCVaR`

, or `PortfolioMAD`

input object (`obj`

). The
`ConditionalUpperBudget`

value can be positive or
negative. However, this value cannot be `NaN`

or
`-Inf`

.

**Data Types: **`double`

## Output Arguments

`obj`

— Updated portfolio object

Updated `Portfolio`

object | Updated `PortfolioCVaR`

object | Updated `PortfolioMAD`

object

Updated portfolio object, returned as a `Portfolio`

,
`PortfolioCVaR`

, or `PortfolioMAD`

object. For more information on creating a portfolio object, see:

## More About

### Undertakings for Collective Investment in Transferable Securities

*Undertakings for Collective Investment in Transferable
Securities* (UCITS) is a regulatory framework of the European Union
that creates a harmonized regime throughout Europe for the management and sale of
mutual funds.

UCITS funds are subject to strict rules regarding investment diversification and risk management to protect investors against concentration risk. One of the core principles of UCITS is investment diversification. This principle reduces the risk associated with concentrating investments in a single issuer or asset class. By spreading investments across various assets, UCITS funds aim to mitigate individual asset volatility and reduce the overall risk profile of the portfolio.

The diversification rules for a UCITS fund state that a fund cannot hold more than 10% of its assets in securities or money market instruments issued by a single body. Additionally, any investments in excess of 5% must not exceed 40% of the total portfolio. This rule is known as the 5/10/40 rule.

## Tips

You can also use dot notation to set up the conditional budget constraints.

obj = obj.setConditionalBudget(ConditionalBudgetThreshold,ConditionalUpperBudget);

## Version History

**Introduced in R2024b**

## See Also

### Topics

- Adding Constraints to Satisfy UCITS Directive
- Conditional Budget Constraints
- Conditional Budget Constraints
- Conditional Budget Constraints
- Supported Constraints for Portfolio Optimization Using Portfolio Objects
- Supported Constraints for Portfolio Optimization Using PortfolioCVaR Object
- Supported Constraints for Portfolio Optimization Using PortfolioMAD Object

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