# Time varying weights in a portfolio

As I have seen in my portfolio theory class, we define the weights of some assets and quantify the risk and return of the whole portfolio. In this setup, the weights do not change in time. What if the weights vary in time, like every second... Is this out of the scope of portfolio theory? What should I study?

Thanks

• It is out of the scope of Markowitz Portfolio Theory, which is a one period model. But it is within the scope of Continuous Time portfolio theories such as the work of R.C. Merton. In these theories there can be changes over time in the investment opportunity set, which the investor has to deal with. – noob2 May 4 '20 at 20:32
• There's a chapter in Arbitrage Theory in Continuous Time by Björk called "The martingale approach to optimal investment", is that also similar to this? I've tried reading it but honestly even after doing a master's in finance (with one course being on the book (but not the chapter)) I'm still completely lost trying to understand it. The title certainly is elusive though – Oscar May 4 '20 at 21:39
• The math behind Continuous Time financial optimization (=> Stochastic Control Theory) is complex , that is why many people in practice use the (suboptimal) approach Tosh described in his answer below, of running a sequence of 1 period models in succession. – noob2 May 5 '20 at 19:51
• Does this answer your question? How can we quantify time varying portfolios? – develarist May 6 '20 at 7:33

Suppose you are given the historical data of 20 days. You calculate the asset returns and covariance matrix. Then you minimise the variance $$Min \ \sigma^2 = w\Sigma w^T \\ s.t. wI^T = 1 \\ and \ w\mu^T = r_p \\$$ Where $$\mu$$ is the asset returns and $$r_p$$ is the target return, to find the optimal weight allocation. This is a prediction of what your weight allocation should be for the 21st day. So if you are studying ticks by data, that is how stock prices change every second, you can experience change in weights. Hope this helps.