This paper describes multi-portfolio internal rebalancing processes used in the finance industry. Instead of trading with the market to externally rebalance, these internal processes detail how portfolio managers buy and sell between their portfolios to rebalance. We give an overview of currently used internal rebalancing processes, including one known as the banker process and another known as the linear process. We prove the banker process disadvantages the nominated banker portfolio in volatile markets, while the linear process may advantage or disadvantage portfolios.We describe an alternative process that uses the concept of market-invariance. We give analytic solutions for small cases, while in general show that the n-portfolio solution and its corresponding 'market-invariant' algorithm solve a system of nonlinear polynomial equations. It turns out this algorithm is a rediscovery of the RAS algorithm (also called the iterative proportional fitting procedure) for biproportional matrices. We show that this process is more equitable than the banker and linear processes, and demonstrate this with empirical results.The market-invariant process has already been implemented by industry due to the significance of these results.
This paper studies Stochastic Shortest Path (SSP) problems in known and unknown environments from the perspective of convex optimisation. It first recalls results in the known parameter case, and develops understanding through different proofs. It then focuses on the unknown parameter case, where it studies extended value iteration (EVI) operators. This includes the existing operators used in Rosenberg et al. [25] and Tarbouriech et al. [30] based on the 1 norm and supremum norm, as well as defining EVI operators corresponding to other norms and divergences, such as the KL-divergence. This paper shows in general how the EVI operators relate to convex programs, and the form of their dual, where strong duality is exhibited.This paper then focuses on whether the bounds from finite horizon research of Neu and Pike-Burke [20] can be applied to these extended value iteration operators in the SSP setting. It shows that similar bounds to [20] for these operators exist, however they lead to operators that are not in general monotone and have more complex convergence properties. In a special case we observe oscillating behaviour. This paper generates open questions on how research may progress, with several examples that require further examination.
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