# 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' aj