1-29. Advanced Portfolio Optimization

Remember how earlier we talked about the possibility of setting yourself a problem that doesn’t have a solution because there is no answer that satisfies all the constraints? Well, that can happen in practice. When you’re working with two or three dimensional problems where you can graph the functions and observe what’s going on, it … Read more

Let’s think for a moment why what we’ve come up with now is superior to the way we were doing optimization before. Remember how before we had this large covariance matrix of assets that we were using to estimate portfolio variance. Well, now we have a smaller number of factors. Common commercial risk models have … Read more

Another common type of constraint limits the extent to which you are exposed to any individual factor. This is where you might impose specific limitations on your exposure to common risk model factors. Individual sectors, momentum, value, or company size. Remember that the factor exposure matrix has dimensions of assets and factors. When you multiply … Read more

Another important constraint is the leverage constraints. With this constraint, we limit the leverage ratio which, as you recall, is the sum of the absolute dollar value of all positions, long and short, divided by whatever actual capital we devote to supporting those positions. In our portfolio optimization problem, we work with portfolio weights which … Read more

But we’re not quite done. As you know, we can impose lots of different constraints on the problem. These usually have to do with real-world constraints on the way we want our portfolio to behave. Sometimes they just make sure our results aren’t really wacky. Let’s discuss some common constraints. First off, we need to … Read more

As of right now, we have this as our optimization objective function. But there’s one other term that it’s a good idea to add to it at this point. This is what’s called a regularization term. Here we use the L2 norm of the portfolio weights as the regularization term. In other types of problems, … Read more

Okay, now where does the risk model come in? Well, we discussed previously that if we put the quantity representing the mean in the objective, one option is to place a constraint on the portfolio variance. We already know how to calculate the portfolio covariance matrix using a risk model. All we do to get … Read more

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 … Read more

Wow. Just wow. I am really proud of you. You made it through factor models and all the portfolio optimization content. This is tough stuff, no bones about it. This is a huge accomplishment on your part, and I congratulate you. Maybe, take a few minutes and think back to everything you have learned since … Read more

Let’s take a moment to have a final discussion about what is going on here. We started with our alpha vector, which is a vector of numbers that we think will be proportional to future returns. We want to maximize our alpha times our weights and minimize risk as modeled by our risk model, and … Read more

Why will two portfolios employing the same underlying strategy run on the same as at universe during the same period yield a different portfolio if started at different points in time? The reason is that transitions between different sets of portfolio weights can be more or less costly. Consider two portfolios trying to make the … Read more

As you already know, transaction costs are a hugely important thing to think about. Can you do anything to mitigate transaction costs and the optimization problem itself? Well, you know you want to minimize transaction costs. Can you just plop some quantity that measures transaction costs somehow into the objective function? That sounds like a … Read more

You have come so far. Just think of all the amazing things we’ve told you in the past few lessons, I bet your head is spinning in a good way. We’ve been talking about factors, how to think about them, how to find them, how to code them up, how to make them robust, and … Read more