2 – M4 L4 02 Setting Up The Problem Alphas V5

Our goal is to set up a portfolio optimization problem using our alpha factors and risk model. In practice, it’s possible that we are doing this in order to design a portfolio from scratch, but it’s also possible that we are trying to guide the evolution of an existing portfolio. So, there may already be a portfolio in production with capital invested in a universe of assets and portfolio weights on the assets from a previous optimization, which may have evolved as the asset values appreciated or depreciated. So, how do we set up the optimization with our new or updated alpha factors and updated data? Thinking back to what we learned before about portfolio optimization, we know we want to do something like maximize return and limit risk as measured by variance. This is already great intuition for how to set up the problem. Can you guess where our alpha factors and risk model would go in the problem formulation following this intuition? Let’s make this explicit starting with the alpha factors. Let’s say we have just one alpha factor, we know that on a given day, our alpha factor is a vector of values, one value per stock that is hopefully predictive of the future mean return of each stock. But now we want a quantity in the objective function that represents the predicted portfolio return. Somehow, we need to sum the alpha values over the portfolio. Can you see what we want here? To calculate the portfolio alpha, we just take the dot product of the alpha with a vector of portfolio weights. Let’s plot this into the objective function. We make it negative because we want to maximize alpha.