Quadratic Programming

Will any one be able to explain the use of “quadratic programming” in the context of curve construction via bootstrapping.

Heard it for the first time today!!

Some context might make any answer better. I have no idea what you mean by “curve construction”. What curve and how is it being constructed? Without knowing that, I couldn’t tell you how quadratic programming and bootstrapping would relate to it.

In case you’re not familiar with the other terms:

Quadratic programming is basically a way to minimize a quadratic function subject to constraints. For instance, minimizing the variance of a portfolio such that the weights sum to 1 is a quadratic programming problem. Also, it is possible to estimate least squares subject to constraints using a quadratic optimizer.

Bootstrapping is statistical technique used to generate distributions. I think wikipedia’s explanation is better than any I could do: “we take our original data set of N heights, and, using a computer, make a new sample (called a bootstrap sample) that is also of size N. This new sample is taken from the original using sampling with replacement so it is not identical with the original “real” sample. We repeat this a lot (maybe 1000 or 10,000 times), and for each of these bootstrap samples we compute its mean (each of these are called bootstrap estimates). We now have a histogram of bootstrap means. This provides an estimate of the shape of the distribution of the mean from which we can answer questions about how much the mean varies.”

Sounds like you are wanting to use quadratic programming (which is mostly used as an optimization technique in finance) to construct an efficient frontier, given means and covariances.

You’re likely to use bootstrapping to create a set of return sequences based on your initial data sample, but whose time-ordering of returns is randomized (this is also a useful purpose of bootstrapping - to see what your equity curve might look like if you mix up the order of your returns stream). Then repeat this over and over.

The idea is that the means and correlations might vary a little bit from realization to realization (one realization being one simulated returns stream). Since the constituents of the optimal portfolio can sometimes change dramatically based on small changes in inputs, this provides a one method to try to increase robustness of a portfolio allocation to small changes in the distributional characteristics.