# Universal Portfolio Optimization & Trading Frequency

Readers with background in portfolio selection and optimization may be interested in a recent working paper by Hazan and Kale, presented at NIPS 2009: On Stochastic and Worst-case Models for Investing.

This paper builds upon Agarwal *et al.*, Algorithms for Portfolio Management Based on the Newton Method (2006); and Cover, Universal Portfolios (1991). Notable is this research theme is beginning to actually intersect with reality, suggesting the potential for practical trading applicability.

Specifically, Lemma 11 makes an interesting claim relating variance of the minimum variance CRP with trading frequency (p. 8); prose summary of this lemma is:

Thus, increasing the trading frequency decreases the variance of the minimum variance CRP, which implies that it gets less risky to trade more frequently; in other words, the more frequently we trade, the more likely the payoff will be close to the expected value. On the other hand, as we show in Theorem 10, the regret does not change even if we trade more often; thus, one expects to see improving performance of our algorithm as the trading frequency increases.

Interesting that portfolio optimization papers are now beginning to comment formally on trading frequency optimality, albeit in characteristically stylistic ways.

As is typical when computer science guys write papers about portfolio managament, implementation is brushed aside with comments like “it can be implemented very efficiently”. Regarding the paper by Agarwal et al., there isn’t much detail about how to actually implement the algo (though the algo is there). So far I’ve been messing around with the algo by Helmbold et al (1998) which is extremely easy to implement, lightning-fast and gives good results. But I’m tempted to test Agarwal et al.’s algo as well.

You can search for “Online Newton Step” presented in ICML 2006 for implementation.

The Nips 2009 paper is quite theoretical and I think its biggest contribution is the improved loss bound, which takes the advantage of GBM model.