Risk Parity or Budgeting with Constraints

Define Problem

Start by defining the problem data.

% Assets variance
vol = [0.1;0.15;0.2;0.3];
% Assets correlation
rho = [1.00  0.50  0.50  0.50;
       0.50  1.00  0.50  0.50;
       0.50  0.50  1.00  0.75;
       0.50  0.50  0.75  1.00];
% Assets covariance matrix
Sigma = corr2cov(vol,rho);
% Risk budget
budget = [0.3;0.3;0.195;0.205];

Obtain the long-only, fully invested risk budgeting portfolio to use as benchmark.

% Long-only fully invested risk budgeting portfolio
w_simple = riskBudgetingPortfolio(Sigma,budget);
% Compute the risk contribution of the weight allocation
RC_simple = portfolioRiskContribution(w_simple,Sigma);
% Table with weights and risk contributions
T_simple = table(w_simple,RC_simple,budget, ...
    VariableNames={'RB Portfolio','RB Contribution','Budget'})

T_simple=4×3 table
    RB Portfolio    RB Contribution    Budget
    ____________    _______________    ______

      0.45053              0.3           0.3 
      0.30035              0.3           0.3 
      0.14668            0.195         0.195 
      0.10244            0.205         0.205 

As desired, the risk contribution of the risk budgeting portfolio matches the budget.

Add Constraints to Risk Parity or Budgeting Portfolio

When you add constraints, other than the long-only, fully invested constraints, to the risk parity formulation, you must find the portfolio allocation that satisfies all the constraints and minimizes the deviation from the target risk budget.

The percent risk contribution of asset $\mathit{i}$ is defined as

${\mathrm{RC}}_{\mathit{i}}\left(\mathit{w}\right)=\frac{{\mathit{w}}_{\mathit{i}}{\left(\Sigma \mathit{w}\right)}_{\mathit{i}}}{{\mathit{w}}^{\mathit{T}}\Sigma \mathit{w}}$,

and the goal is to find the portfolio that minimizes the deviation of ${\mathrm{RC}}_{\mathit{i}}\left(\mathit{w}\right)$ to ${\mathit{b}}_{\mathit{i}}$, the target risk budget of asset $\mathit{i}$. For risk parity problems, ${\mathit{b}}_{\mathit{i}}=\frac{1}{\mathit{n}}$, where $\mathit{n}$ is the number of assets.

A common way to measure deviation is to use a norm. In this example, use the 2-norm. The risk parity or budgeting problem with constraints results in

where $\Omega$ represents the feasible set that is defined by the desired constraints. The objective function is the sum of squares of the deviation of the risk contributions from the target. Notice that the objective function is nonconvex.

Next, introduce the constraint ${\mathit{x}}_{\mathit{i}}\le 0.3$, which means that no more than 30% of the capital can be invested in each asset.

% Define the Portfolio object
p = Portfolio(AssetCovar=Sigma);
% Set bound constraints
p = setBounds(p,0,0.3);
% Set budget constraint
p = setBudget(p,1,1);

Create a function handle for the nonconvex objective

${\sum _{\mathit{i}}\left(\frac{w_i{\left(\Sigma w\right)}_i}{w^T\Sigma w}-{\mathit{b}}_{\mathit{i}}\right)}^2$.

% Define objective function
objFun = @(w) sum(((w.*(Sigma*w))/(w'*Sigma*w) - budget).^2);

Solve the problem using estimateCustomObjectivePortfolio and riskBudgetingPortfolio.

% Risk budgeting portfolio with extra constraints
w_extra = estimateCustomObjectivePortfolio(p,objFun);
% Risk contribution
RC_extra = portfolioRiskContribution(w_extra,Sigma);
% Table with weights and risk contributions
T_extra = table(w_extra,RC_extra,budget, ...
    VariableNames={'RB Portfolio','RB Contribution','Budget'})

T_extra=4×3 table
    RB Portfolio    RB Contribution    Budget
    ____________    _______________    ______

          0.3           0.15505          0.3 
          0.3           0.24998          0.3 
       0.2465           0.30862        0.195 
       0.1535           0.28635        0.205 

Now that you have added additional constraints, the risk contribution of the resulting portfolio does not match the budget. Yet, the solution to the problem is minimizing the deviation of the risk contribution to the target risk budget. Because the risk budgeting portfolio without the extra constraints assigns weights larger than 30% to the first and second assets, the solution to the problem that minimizes the deviation assigns as much as possible to the first and second assets and distributes the rest to the other assets.

As explained in Role of Convexity in Portfolio Problems, because the objective function is nonconvex, this formulation cannot be solved by the Portfolio object solvers if you add cardinality constraints or conditional bounds to the problem.

Compare Risk Parity and Mean-Variance Portfolios

You can examine the difference in the allocation between the traditional minimum variance portfolio and the risk parity portfolio.

First compute the minimum variance portfolio. The traditional mean-variance framework requires the assets mean return to be defined before you compute any of the portfolios on the efficient frontier.

% Define assets mean
p.AssetMean = [0.1; 0.1; 0.15; 0.2];
% Minimum variance portfolio
wMinVar = estimateFrontierLimits(p,'min');

To compute the risk parity portfolio, make sure all assets contribute equally to the risk. Set ${\mathit{b}}_{\mathit{i}}=\frac{1}{\mathit{n}}$ and update the objective function.

% Update budget
budget = 1/p.NumAssets*ones(p.NumAssets,1);
% Update objective
objFun = @(w) sum(((w.*(Sigma*w))/(w'*Sigma*w) - budget).^2);

Solve the risk parity problem using estimateCustomObjectivePortfolio.

% Risk parity portfolio with extra constraints
wRP = estimateCustomObjectivePortfolio(p,objFun);

Compare the weight allocation concentrations.

% Plot pie charts
tiledlayout(1,2);
% Minimum variance portfolio
nexttile
pie(wMinVar)
title('Minimum Variance Portfolio',Position=[0,1.5]);
% Risk parity portfolio
nexttile
pie(wRP)
title('Risk Parity Portfolio',Position=[0,1.5]);

% Add legend
lgd = legend({'1','2','3','4'});
lgd.Layout.Tile = 'east';

The portfolio allocation of the minimum variance portfolio sets all assets with the smallest risk to their maximum (30%). On the other hand, the risk parity allocation sets only the first two assets to 30% and the last two are more balanced. This result is a simple example of how the risk parity allocation helps to diversify a portfolio.