We generalize Kahler information manifolds of complex-valued signal processing filters by introducing weighted Hardy spaces and generic composite functions of transfer functions. We prove that the Riemannian geometry induced from weighted Hardy norms
for composite functions of its transfer function is the Kahler manifold. Additionally, the Kahler potential of the linear system geometry corresponds to the square of the weighted Hardy norms for composite functions of its transfer function. By using the properties of Kahler manifolds, it is possible to compute various geometric objects on the manifolds from arbitrary weight vectors in much simpler ways. Additionally, Kahler information manifolds of signal filters in weighted Hardy spaces can generate various information manifolds such as Kahlerian information geometries from the unweighted complex cepstrum or the unweighted power cepstrum, the geometry of the weighted stationarity filters, and mutual information geometry under the unified framework. We also cover several examples from time series models of which metric tensor, Levi-Civita connection, and Kahler potentials are represented with polylogarithm of poles and zeros from the transfer functions when the weight vectors are in terms of polynomials.
We introduce diversified risk parity embedded with various reward-risk measures and more generic allocation rules for portfolio construction. We empirically test advanced reward-risk parity strategies and compare their performance with an equally-wei
ghted risk portfolio in various asset universes. The reward-risk parity strategies we tested exhibit consistent outperformance evidenced by higher average returns, Sharpe ratios, and Calmar ratios. The alternative allocations also reflect less downside risks in Value-at-Risk, conditional Value-at-Risk, and maximum drawdown. In addition to the enhanced performance and reward-risk profile, transaction costs can be reduced by lowering turnover rates. The Carhart four-factor analysis also indicates that the diversified reward-risk parity allocations gain superior performance.
We review the information geometry of linear systems and its application to Bayesian inference, and the simplification available in the Kahler manifold case. We find conditions for the information geometry of linear systems to be Kahler, and the rela
tion of the Kahler potential to information geometric quantities such as $alpha $-divergence, information distance and the dual $alpha $-connection structure. The Kahler structure simplifies the calculation of the metric tensor, connection, Ricci tensor and scalar curvature, and the $alpha $-generalization of the geometric objects. The Laplace--Beltrami operator is also simplified in the Kahler geometry. One of the goals in information geometry is the construction of Bayesian priors outperforming the Jeffreys prior, which we use to demonstrate the utility of the Kahler structure.
We prove the correspondence between the information geometry of a signal filter and a Kahler manifold. The information geometry of a minimum-phase linear system with a finite complex cepstrum norm is a Kahler manifold. The square of the complex cepst
rum norm of the signal filter corresponds to the Kahler potential. The Hermitian structure of the Kahler manifold is explicitly emergent if and only if the impulse response function of the highest degree in $z$ is constant in model parameters. The Kahlerian information geometry takes advantage of more efficient calculation steps for the metric tensor and the Ricci tensor. Moreover, $alpha$-generalization on the geometric tensors is linear in $alpha$. It is also robust to find Bayesian predictive priors, such as superharmonic priors, because Laplace-Beltrami operators on Kahler manifolds are in much simpler forms than those of the non-Kahler manifolds. Several time series models are studied in the Kahlerian information geometry.
We empirically test predictability on asset price by using stock selection rules based on maximum drawdown and its consecutive recovery. In various equity markets, monthly momentum- and weekly contrarian-style portfolios constructed from these altern
ative selection criteria are superior not only in forecasting directions of asset prices but also in capturing cross-sectional return differentials. In monthly periods, the alternative portfolios ranked by maximum drawdown measures exhibit outperformance over other alternative momentum portfolios including traditional cumulative return-based momentum portfolios. In weekly time scales, recovery-related stock selection rules are the best ranking criteria for detecting mean-reversion. For the alternative portfolios and their ranking baskets, improved risk profiles in various reward-risk measures also imply more consistent prediction on the direction of assets in future. In the Carhart four-factor analysis, higher factor-neutral intercepts for the alternative strategies are another evidence for the robust prediction by the alternative stock selection rules.
We implement momentum strategies using reward-risk measures as ranking criteria based on classical tempered stable distribution. Performances and risk characteristics for the alternative portfolios are obtained in various asset classes and markets. T
he reward-risk momentum strategies with lower volatility levels outperform the traditional momentum strategy regardless of asset class and market. Additionally, the alternative portfolios are not only less riskier in risk measures such as VaR, CVaR and maximum drawdown but also characterized by thinner downside tails. Similar patterns in performance and risk profile are also found at the level of each ranking basket in the reward-risk portfolios. Higher factor-neutral returns achieved by the reward-risk momentum strategies are statistically significant and large portions of the performances are not explained by the Carhart four-factor model.
We test the price momentum effect in the Korean stock markets under the momentum universe shrinkage to subuniverses of the KOSPI 200. Performance of the momentum strategy is not homogeneous with respect to change of the momentum universe. It is found
that some submarkets generate the higher momentum returns than other universes do but large-size companies such as the KOSPI 50 components hinder the performance of the momentum strategy. The observation is also cross-checked with size portfolios and liquidity portfolios. Transactions by investor groups, in particular, the trading patterns by foreign investors can be a source of the momentum universe shrinkage effect in the momentum returns.
