I read the Ph.D thesis of Cengiz Y. Belentepe titled "A Statistical View of Universal Portfolios" because I'd been looking for a concrete paper on universal portfolios (UP).

The UP is a portfolio allocation scheme that is supposed to perform as well as the best constantly rebalanced portfolio constructed with the benefit of hindsight. A constantly rebalanced portfolio is one where you choose certain weights for each asset and then rebalance your allocations whenever one rises or falls to match the original weights. With hindsight means you can choose the best portfolio among all fixed-weight portfolios knowing what the future will be like. The UP is supposed to be able to match this because it is a dynamic, rather than static, allocation scheme.

However, it's important to know that the UP is only supposed to perform as well in the sense of a lower bound. If you know computer science then this is like saying that the Big-O efficiency (in terms of profits not time) is the same. However, a constant multiple can be enormously important in investments.

Anyway, the thesis was really interesting (although I imagine that for a thesis committee it wouldn't be obfuscated by enough complicated and excessive but impressive math).

Here are some of the most interesting bits I learned with page numbers so you can look them up too (p. book/pdf).

1. Universal Portfolios are, down to an approximation, equivalent to Markowitz's mean-variance optimization. (p.36/52)

2. A constantly rebalanced portfolio is much more powerful than it appears. For example it can "turn two nags (assets A and B) into a thoroughbred (equity curve)":

In this case we have two assets that just go up and down with -1 correlation and our fixed weights are half of our equity in each asset, rebalanced at the end of every period. He also shows a similar result on real stocks. Essentially the equal-weight constantly rebalanced portfolio is the simplest mean-reversion strategy. (p.22/38)

3. There is a nice quote by Cover summarizing the UP on page 33/49

4. There is an extremely concise and practical explanation of two methods of intelligently estimating a covariance matrix - by exponential smoothing to decrease the influence of points in the past and eigendecomposition to shrink/regularize/decrease overfitting. (p.62/74)

5. The UP can be beaten by the most naive portfolio allocation scheme - uniform allocation. He humorously calls it the capitulation portfolio. Results on real stocks show that this can hold in the real world too. (p.63/75)

6. He uses Cover's own data to show that the UP is not that effective. (p.60/76)

7. And finally lists some good heuristic learning rates to try with an exponentiated gradient (EG) online learner. I won't go into details here (probably later) but EG is a powerful, efficient, theoretically pretty algorithm good for adversarial learning settings. I always like seeing heuristics because most academic papers are missing them but in practice they are so helpful. (p.66/82)

Overall I enjoyed it a lot more than I expected and it was an easy read. Please leave a comment if you know of any good sources on modern UP work.