Encoding Sector Allocation Bands in Portfolio Optimization
Summary
The document shows how sector allocation limits can be represented as linear inequalities in a portfolio optimization problem. For each sector, a row of the constraint matrix selects that sector’s asset weights; paired rows can impose upper and lower bounds. These inequalities are added alongside other portfolio constraints. The example frames the objective as maximizing a Sharpe ratio, then uses a quadratic optimization formulation with a return normalization and covariance-based risk objective.
The post includes a CVXPY example and a corrected formulation that adjusts the constraint matrix by the sector bounds before solving and normalizes the resulting auxiliary weights. Its example uses generated returns and four assets, so it demonstrates construction rather than investment performance. The write-up does not discuss estimation error, transaction costs, or whether the resulting portfolio is robust out of sample; users must also ensure their matrix signs and bounds match their intended minimum and maximum allocations.
Key ideas
- Represent each sector’s total allocation as a sum of the weights of its constituent assets.
- Use paired linear inequalities to enforce upper and lower sector allocation limits.
- Combine sector bounds with portfolio constraints in the optimization problem.
- The example uses a quadratic risk objective and normalizes the auxiliary solution to obtain portfolio weights.
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Full text
# How to calculate optimal portfolio using sector constraints in python
# How to calculate optimal portfolio using sector constraints in python
I'm looking into CVXPY at the moment.
Main goal would be to be able to calculate the optimal portfolio, which in my opinion would mean that we need to maximise
(expected return - risk free) / volatility
To make it simpler I would like to drop risk free out of the equation (it's anyway near zero at the moment).
My problem is: I would like to constrain the weights in the portfolio based on maximum allocation that can be allocated to a certain sector.
```
Sector Stocks
A 1
2
3
B 4
5
6
```
how can i achieve, that the minimum and maximum of an allocation to every sector lies in between some bands?
I've already seen this post: industry level constraints
But i really would like to implement it with cvxpy as the answer suggested. Unfortunately I have no idea how to implement this?
I already got the covarmatrix, return estimates and bands on hand but simply can't find a way how to implement those constraints. Any help would be highly appreciated!
EDIT: I tried my very best and came up with this script: thanks to the comment from @Attack68 in combination with this paper: Sharpe Quadratic Optimization
```
import cvxpy as cp
import numpy as np
np.random.seed(101)
## NUMBER OF ASSETS
n_assets = 4
## NUMBER OF OBSERVATIONS
n_obs = 1000
## GENERATE RANDOM RETURNS
return_vec = np.random.randn(n_assets, n_obs)
## SET UP PROBLEM
C = np.asmatrix(np.cov(return_vec)) # Covar Matrix
mu = np.asmatrix(np.mean(return_vec,axis=1)) # return estimat
mu0 = -0.0075 # risk free rate
y = cp.Variable(n_assets) # "weights"
A = np.asmatrix([[0.6,0.6,0,0],[-1.2,-1.2,0,0],[0,0,0.2,0.2],[0,0,-1.2,-1.2]])
bounds = np.asmatrix([0.4,0.2,0.8,0.2])
# HOW CAN I SUBSTRACT THE BOUNDS FROM ONLY THE NON-NULL VALUES? (AS FAR AS THIS WILL BE NEEDED?)
A_mod = A #- bounds.T
## CREATE CONSTRAINTS
constraints = [(mu-mu0)@y==1,
y >= 0,
y@A_mod >= 0]
## FORM OBJECTIVE
obj = cp.Minimize(cp.quad_form(y,C))
## FORM AND SOLVE PROBLEM
prob = cp.Problem(obj, constraints)
prob.solve()
w = y.value/sum(y.value)
w
array([0.35386785, 0.17693393, 0.4034641 , 0.06573412])
```
Unfortunately this solution doesn't meet the constraints. What am I doing wrong?
## Answer by Attack68 (score 1, accepted)
https://quant.stackexchange.com/a/45955
This is very simply done. It involves ensuring the constraints are presented as part of the matrix standard form.
You will typically have the constraint that all assets sum to one, i.e. the matrix-vector equation:
$$ \delta^T x = 1 $$
If you want to create an inequality constraint for assets in a sector just isolate them:
$$ \begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\ -1 & -1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & -1 & -1 & -1 \\ \end{bmatrix} x \leq \begin{bmatrix} a_1 \\ a_2 \\ b_1 \\ b_2 \end{bmatrix}$$
Then add(stack) these matrix-vector rows to any existing inequality constraints you have.
## Answer by R. Steigmeier (score 1)
https://quant.stackexchange.com/a/45979
For those experiencing a similar problem, here is the solution that worked for me:
```
## OPTIMIZE PORTFOLIO WEIGHTS UNDER THE OBJECTIVE OF MAXIMIZING THE SHARPE RATIO
## WHILE CONSTRAINING THE WEIGHTS TO SECTOR BOUNDS
## PAPER: # ACCORDING TO: http://people.stat.sc.edu/sshen/events/backtesting/reference/maximizing%20the%20sharpe%20ratio.pdf
np.random.seed(101)
## NUMBER OF ASSETS
n_assets = 4
## NUMBER OF OBSERVATIONS
n_obs = 1000
## GENERATE RANDOM RETURNS
return_vec = np.random.randn(n_assets, n_obs)
## SET UP PROBLEM
C = np.asmatrix(np.cov(return_vec))
mu = np.asmatrix(np.mean(return_vec,axis=1))
mu0 = np.asmatrix(-0.0075)
## INITIATE WEIGHT VARIABLE
y = cp.Variable(n_assets)
# DEFINE CONSTRAINTS AND MODIFY FOR QUADRATIC PROBLEM
A = np.asmatrix([[1,1,0,0],[-1,-1,0,0],[0,0,1,1],[0,0,-1,-1]])
bounds = np.asmatrix([0.2,-0.5,0.2,-0.8])
A_mod = A - bounds.T
## CREATE CONSTRAINTS
constraints = [(mu-mu0)@y==1,
y >= 0,
A_mod@y.T >= 0]
## FORM OBJECTIVE
obj = cp.Minimize(cp.quad_form(y,C))
## FORM AND SOLVE PROBLEM
prob = cp.Problem(obj, constraints)
prob.solve()
## TRANSFORM FINAL WEIGHTS
w = y.value/sum(y.value)
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