For a market with $m$ assets and $T$ discrete trading sessions, Cover and Ordentlich (1998) found that the "Cost of Achieving the Best Rebalancing Rule in Hindsight" is $p(T,m)=\sum\limits_{n_1+\cdot\cdot\cdot+n_m=T}\binom{T}{n_1,...,n_m}(n_1/T)^{n_1}\cdot\cdot\cdot(n_m/T)^{n_m}$. Their super-replicating strategy is impossible to compute in practice. This paper gives a workable generalization: the cost (read: super-replicating price) of achieving the best $s-$stock rebalancing rule in hindsight is $\binom{m}{s}p(T,s)$. In particular, the cost of achieving the best pairs rebalancing rule in hindsight is $\binom{m}{2}\sum\limits_{n=0}^T\binom{T}{n}(n/T)^n((T-n)/T)^{T-n}=\mathcal{O}(\sqrt{T})$. To put this in perspective, for the Dow Jones ($30$) stocks, the Cover and Ordentlich (1998) strategy needs a 10,000-year horizon in order to guarantee to get within $1\%$ of the compound-annual growth rate of the best (30-stock) rebalancing rule in hindsight. By contrast, it takes 1,000 years (in the worst case) to enforce a growth rate that is within $1\%$ of the best pairs rebalancing rule in hindsight. For any preselected pair $(i,j)$ of stocks it takes 320 years. Thus, the more modest goal of growth at the same asymptotic rate as the best pairs rebalancing rule in hindsight leads to a practical trading strategy that still beats the market asymptotically, albeit with a lower asymptotic growth rate than the full-support universal portfolio.
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