Markowitz Model is a famous method in the Portfolio Investment Theory. This model provides efficient portfolios, (i.e. portfolios with the highest profitability and lowest risk possible) through mathematical programming.

The set of portfolios composes the efficient frontier. The strategy is based on quadratic optimisation, therefore **minimizing the estimated risk** and negative return. We often calculate the efficient frontier by using a fixed temporal window based on historical data. The real market doesn’t use big windows of data to calculate the mean-variance portfolio, but the portfolio is updated more frecquently; every day, for example.

It’s interesting to analyse the evolution of the frontier based on the timeline. Here, we have included the results of a dynamic frontier in a static representation with two alternatives: expansive method and six months rolling method.

## Expansive Method

We consider a three-year window of value returns of the companies that compose the index IBEX-35 between 02/01/2012 and 02/01/2015. The first frontier was calculated using 120 observations. The next one has 121, and then we continue adding daily returns. The final frontier will have every historic dataset observation.

Each color represents the year of the last observation added. That provides 261 lines of each color, except 2012 and 2015, as we only have parts of these years in the dataset. The first thing we notice is the extreme risk values at **2014 ** (all green lines). Furthermore, incorporating new observations into the dataset results in decreasing return.

## Six Month Rolling Method

In finance, it’s very common to analyse a series’ evolution by choosing a window which is moved along the time. This method is knowed as *Rolling,* and it consists of performing a progressive **scanning of the serie**. As we can imagine, the window’s width will have an influence on the results. Below, we chose a six-month window (126 days, to be exact).

When we consider a fixed period, without accumulating old observations, risk and renturn of the portfolios move in different way. In 2012 the risk increases in comparison to other years. Specifically, it’s over 0.3%, whereas in 2013 and 2014 it falls bellow 0.1%. Return is higher in most of 2013, despite parts where risk shot up.

So, **dynamic efficient frontier** can be the answer to see at a glance the effect of adding new observations with a fixed initial date, or in a rolling period. However, as conclusions can be very different according to the method and parameters we choose, care must be taken with this approach.