Evaluating Portfolios Built with Connectedness Matrices
Summary
The document describes a portfolio-construction approach that substitutes a connectedness matrix for the covariance matrix and active returns for expected returns within a mean-variance optimization framework. The matrix is built from pairwise dependence measures between equities, with quantities relating to estimated returns, active returns, market sensitivity, and market returns entering the proposed construction.
The author asks what the optimization objective means and how to compare portfolios formed for different equity markets. The text mentions minimizing a cost function but does not fully explain its weighting term or provide a clear objective interpretation. It supplies no performance results or recommended evaluation metric, so it raises an important comparison question without resolving it; portfolio evaluation would need to be tied to the method’s actual objective and the investor’s goals.
Key ideas
- The described method replaces covariance inputs with a connectedness matrix in mean-variance optimization.
- Active returns take the place of expected returns in the stated portfolio construction.
- The author is uncertain about the cost function and the role of its weighting term.
- The document does not establish a single evaluation metric for comparing resulting portfolios.
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Full text
# Which performance evaluation measure to assess "Connectedness Matrix" based porfolios?
# Which performance evaluation measure to assess "Connectedness Matrix" based porfolios?
#### 1. Question
- Which performance evaluation measure would be best to assess the portfolios built on 'connectedness matrix'? The connectedness matrix is the concept introduced in the academic paper "Learning connections in financial time series".
- The problem is that I don't know what this 'connectedness matrix' optimization method is trying to minimize or maximize. For example, the classical MVO framework tries to find the portfolio with the maximized sharpe ratio, and the CVaR method is trying to minimize the CVaR while maximizing the expected return.
- After reading the paper or the brief walk-through, you will understand how to build a portfolio using 'connectedness matrix'; it's simple that it just replaces the covariance matrix and expected means with 'connectedness matrix' and 'active return'
- I can't figure out what this method is trying to minimize or maximize, so not sure of which performace evaluation measure is suitable to compare two portfolios. (Ex. " I have used the 'connectedness matrix' method to construct two portfolios with French stocks and British stocks. But not sure which country portfolio is better than the other!")
#### 2. How to construct a portfolio using the 'connectedness matrix'
- You can calculate the connectedness(dependence) between a pair of stocks with the following equation.
- $\hat{r}$t,k : the estimated return of the equity k on day t. ak : the active return of equity k in general through a certain trading period. bk : equity k's general sensitivity to market through a certain trading period. Wj,k : the 'connectedness' between 'equity j' and 'equity k'. j and k are not same. rt,Λ : the return of general market(e.g. S&P 500) on the day t. dt,k : the sum of ak and bk*
- With the Wj,k calcuated through the equation mentioned above, we will try to minimize the following function (the cost function):
- I don't know why they put the f(rt,k) as a weight for the cost.
- Now with the 'connectedness' of the equities, we construct a 'connectedness matrix' like below:
- As a final step, you just optimize the portfolio through the MVO framework with the 'connectedness matrix' and 'active return', instead of 'covariance matrix' and 'mean of returns'.
- In other words, you need to replace C (Covariance matrix) with G (connectedness matrix) and $\bar{r}$j with the active return' ajShown 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.