Universal portfolios with side informationUniversal portfolios with side informationCover, Thomas M. and Ordentlich, Erik1996

Paper summarycdmurray80This paper is overly mathematical for what it accomplished (117 equations) but is somewhat interesting. Cover considers CRPs (constantly rebalanced portfolios), where a certain portion of wealth is "fixed" in each stock and the portfolio is rebalanced every period: CRPs can earn positive return even if none of the stock's do. Finding the best CRP represents an optimization over a simplex (since we optimize over the fraction to invest in each stock), while the best stock is just a corner of the simplex, so the best CRP does at least as well as the best stock. He considers the strategy of initially investing equal money in each CRP (i.e. over the continuous space of CRPs) and then letting the wealth in each CRP grow: doing this, he derives a bound that the terminal wealth of this strategy will only lag the terminal wealth of the best CRP chosen in hindsight by a factor $(T+1)^{(N-1)}$, when there are $T$ periods and $N$ assets. Thus the strategy earns the same asymptotic growth as the best CRP. In practice I found that this technique isn't useful and the bound is very very weak (even over 50 years of data). However, the idea of buy-and-hold having good asymptotic performance and comparing well to the best portfolio chosen in hindsight is a good one.

This paper is overly mathematical for what it accomplished (117 equations) but is somewhat interesting. Cover considers CRPs (constantly rebalanced portfolios), where a certain portion of wealth is "fixed" in each stock and the portfolio is rebalanced every period: CRPs can earn positive return even if none of the stock's do. Finding the best CRP represents an optimization over a simplex (since we optimize over the fraction to invest in each stock), while the best stock is just a corner of the simplex, so the best CRP does at least as well as the best stock. He considers the strategy of initially investing equal money in each CRP (i.e. over the continuous space of CRPs) and then letting the wealth in each CRP grow: doing this, he derives a bound that the terminal wealth of this strategy will only lag the terminal wealth of the best CRP chosen in hindsight by a factor $(T+1)^{(N-1)}$, when there are $T$ periods and $N$ assets. Thus the strategy earns the same asymptotic growth as the best CRP. In practice I found that this technique isn't useful and the bound is very very weak (even over 50 years of data). However, the idea of buy-and-hold having good asymptotic performance and comparing well to the best portfolio chosen in hindsight is a good one.