Maximal wealth portfolios.
详细信息
文摘
We analyze wealth maximization problem for constant rebalanced portfolios. We show how to compute a combination of assets producing an appropriate index of past performance. The desired index is equal to SmaxT = maxb&barbelow; ST( b&barbelow;) which is the maximum of T-period investment return ST(b&barbelow;) = t=1T b&barbelow; · x&barbelow; t, where x&barbelow;i is the vector of returns for the tth investment period, and b&barbelow; is the portfolio vector specifying the fraction of wealth allocated to each asset. We provide an iterative algorithm to approximate this index, where at step k the algorithm produces a portfolio with at most k assets selected among M available assets. We show that the multi-period wealth factor ST( bk) converges to the maximum SmaxT as k increases. Furthermore, in the exponent the wealth factor is within c2/k of the maximum, where c is determined by the empirical volatility of the stock returns, and we compare this computation to what is achieved by general procedures for convex optimization. This SmaxT provides an index of historical asset performance which corresponds to the best constant rebalanced portfolio with hindsight. Surprisingly, we find empirically that a small handful of stocks among hundreds of candidate stocks are sufficient to have come close to SmaxT .;Universal portfolios are strategies for updating portfolios each period to achieve actual wealth with exponent provably close to what is provided by SmaxT . A new mixture strategy for universal portfolios based on subsets of stocks. Under a volatility condition, this mixture strategy universal portfolio achieves a wealth exponent that drops from the maximum not more than order logM T . We also solve an integer problem for the size of stock subsets.;Although our main results do not depend on any stochastic assumptions, we do discuss in stochastic settings about the hellinger risk bound on wealth exponent function. We show a bound for the wealth exponent by using historical observed density of the return sequences on future unknown return sequences. We also discuss on the characterization of the wealth of constant rebalanced portfolios of stocks an options. We show a compounded wealth can be decompose into a production of three easily interpretable factors.