Formulating Sharpe Return-Based Style Analysis as a Quadratic Program

Summary

The document frames return-based style analysis as finding nonnegative weights on index returns that sum to one and minimize the variance of the difference between a portfolio’s returns and the weighted index returns. This estimates how a portfolio’s return behavior can be represented by a mix of market indexes. The answer rewrites the residual as a portfolio of the index returns plus a fixed negative exposure to the investor’s portfolio, then expresses its variance through a covariance matrix and a quadratic objective.

This formulation points toward quadratic optimization methods and extends naturally to more indexes. However, the response does not provide Python code or recommend a specific library, despite the original question asking for both. Its written objective has a sign inconsistency: minimizing negative covariance quadratic form would generally maximize variance, whereas the stated goal is to minimize it. The expression also refers to returns as expected returns, though the objective requires return observations or their covariance estimates. These details should be checked before implementation.

Key ideas

• Style analysis fits portfolio returns with a weighted combination of index returns.
• The index weights are constrained to be nonnegative and sum to one.
• The variance of the residual can be represented using a return covariance matrix.
• A quadratic programming formulation can accommodate more indexes.
• The answer's displayed negative sign should be checked against the goal of minimizing variance.

Tags

# Implementing the Sharpe's return-based style analysis on Python


# Implementing the Sharpe's return-based style analysis on Python












I am trying to implement the Sharpe's return-based style analysis on Python. The problem is formulated as follows:

```
min Var(M-(c1a1 + c2a2 + c3a3 + c4a4))
subject to c1 + c2 + c3 + c4 = 1
           c1 >=0, c2 >= 0, c3 >= 0, c4 >= 0
where M = monthly or daily return of an investor's portfolio
      a1, a2, a3, a4 = monthly or daily return of an index
      and c1, c2, c4, c4 are the optimization decision variables.
```

Of course, the objective function (Variance) makes the problem nonlinear. I am trying to use Scipy to implement this, but I cannot find a good example of quadratic/nonlinear optimization similar to this problem.

What Python library should I use to do this? The example above has only 4 indices, but I want to make it more general and flexible as to handle many indices.

I would also very appreciate it if someone could show a Python code for the optimization problem above.

Thank you!!!

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/40413

You could restate your problem as:

$$ \min_x \quad Var \left (-\sum_{i=0}^n x_iR_i \right) $$ $$ \text{s.t.} \quad x_0 = -1, \quad \sum_{i=1}^n x_i = 1, \quad \text{non-negativity of }x_1:x_n $$ where $R_i$ are the expected returns of asset $i$ and $x_i$ are your solution variables.

The objective function can be expressed as: $$ \min_x \quad - \sum_{i,j} x_i x_j Cov(R_i, R_j)= - \mathbf{x^TQx} $$ where $\mathbf{x}=[x_0, ..., x_n]^T$ and $\mathbf{Q}$ is the covariance matrix of returns.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.