By Hoi, Steven C. H.; Li, Bin

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**Example text**

M1 is named as Uniform Constant Rebalanced Portfolios (UCRP). Another special CRP is the optimal offline† CRP strategy, whose portfolio can be calculated as n b = arg max Sn (CRP(b)) = arg max bn ∈ b∈ m m b xt , t=1 which is convex and can be efficiently solved. The CRP with b is denoted as Best Constant Rebalanced Portfolios (BCRPs), which achieve a final cumulative wealth as Sn (BCRP) = max Sn (CRP(b)) = Sn (CRP(b )) b∈ m Note that BCRP is a hindsight strategy, which can only be calculated with complete market sequences.

Helmbold et al. (1998) proposed the EG strategy, which is based on the same algorithm for mixture estimation (Helmbold et al. 1997). 2, EG adopts relative entropy as its regularization term, that is, m R(b, bt ) = bi log i=1 bi . bt,i EG’s formulation is convex in b; however, it is hard to solve since the log function is nonlinear. Thus, the authors adopted log’s first-order Taylor expansion at bt , that is, log b · xt ≈ log(bt · xt ) + xt (b − bt ). bt · xt Then the nonlinear log term becomes linear and the optimization is easy to solve.

Finally, we introduce the idea of how to evaluate a strategy’s performance. Specifically, this chapter is organized as follows. 1 formally formulates the OLPS task as a sequential decision problem. 2 relaxes the transaction costs and margin buying constraints. 3 introduces several evaluation metrics for the task. 4 summarizes this chapter. † At the t-th period (t = 1, . . , n), the asset (close) prices are represented by a vector pt ∈ Rm + , and each element pt,i , i = 1, . . , m, represents the close price of asset i.