On‐Line Portfolio Selection Using Multiplicative Updates

Abstract

We present an on-line investment algorithm which a c hieves almost the same wealth as the best constant-rebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algorithm is very simple to implement and requires only constant storage and computing time per stock i n e a c h trading period. We tested the performance of our algorithm on real stock data from the New York Stock Exchange accumulated during a 22-year period. On this data, our algorithm clearly outperforms the best single stock a s w ell as Cover's universal portfolio selection algorithm. We also present results for the situation in which the investor has access to additional side information."

Optimal Search and One-Way Trading Online Algorithms

“…For example, Raghavan [27] and Chou et al [9], [10] study an online two-way trading problem against statistical adversaries, which must produce exchange rate sequences that conform to some statistical constraints. Cover and Ordentlich [13], [25], Helmbold et al [18], and Blum and Kalai [6] study the general portfolio selection and give algorithms which are competitive relative to a constrained optimal offline algorithm. In particular, they compare their online algorithm to the best (offline) constant rebalanced algorithm that must keep a fixed proportion invested in each of the N assets.…”

Internal Regret in On-Line Portfolio Selection

“…Thus, the minimization of the above upper bound on the worst-case logarithmic wealth ratio may be cast as a sequential prediction problem as described in Section 2. Observing that the eg investment algorithm is just the exponentially weighted average predictor for this prediction problem, and using the performance bound (1) we obtain the cited inequality of Helmbold, Schapire, Singer, and Warmuth (1998). Note that in (5), we could replace the fixed η by a time-adaptive η t = (m/M ) (8 ln N )/t.…”

“…In this section we describe the problem of sequential portfolio selection, recall some previous results, and take a new look at the eg strategy of Helmbold, Schapire, Singer, and Warmuth (1998).…”

“…Helmbold, Schapire, Singer, and Warmuth (1998) prove that if the market values x i,t all fall between the positive constants m and M , then the worst-case logarithmic wealth ratio of the eg investment strategy is bounded by ln N η + nη 8…”

“…The problem of sequential portfolio allocation is well-known to be closely related to the on-line prediction of individual sequences under expert advice, see, for example, Cover (1991), Cover and Ordentlich (1996), Helmbold, Schapire, Singer, and Warmuth (1998), Ordentlich and Cover (1998), Blum and Kalai (1999), Cesa-Bianchi and Lugosi (2000). In the on-line prediction problem the goal is to minimize the predictor's cumulative loss with respect to the best cumulative loss in a pool of "experts".…”