By Hoi, Steven C. H.; Li, Bin
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Extra info for Online portfolio selection : principles and algorithms
Besides the absolute return metrics, it is also important to evaluate a strategy’s risk and risk-adjusted return (Sharpe 1963, 1994). One common criterion is the annualized standard deviation of portfolio period returns to measure volatility risk and ∗ That is, if one has $100 stock (down or collateral) one can borrow at most $100 cash (loan). † Such a measure is not perfect since it manually changes the margin component, although less than 5 per dataset. One may refer to Györfi et al. (2012, Chapter 4) for other solutions to the possibility of ruin.
3 summarizes the follow the loser principle. 1 Mean Reversion Besides the momentum-related idea that assumes that the stock price will continue its previous trend, there exists another different idea, or the mean reversion (contrarian) idea, which assumes that the assets’ prices will revert to their means. Thus, the follow the loser algorithms will transfer the wealth from outperforming assets to underperforming assets. This section illustrates a simple but convincing example to show the mean reversion idea.
At the beginning of period t, the portfolio manager rebalances his or her wealth to a new portfolio bt , from last close price adjusted portfolio bˆ t−1 , each component of b ×xt−1,i which is calculated as bˆt−1,i = t−1,i . Such rebalance incurs a transaction cost b x t−1 t−1 ˆ of γ2 × m i=1 bt,i − bt−1,i , where the initial portfolio is set to (0, . . , 0). Thus, the cumulative wealth after n periods can be expressed as γ n (bt · xt ) × 1 − S n = S0 t=1 γ × 2 m bt,i − bˆt−1,i . i=1 Another practical issue is margin buying, which allows the portfolio managers to buy securities with cash borrowed from securities brokers, using their own equity positions as collateral.