Why Minimize Negative Log Likelihood?
One of the wonders of machine learning is the diversity of divergent traditions from which it originates, from classical statistics (both frequentist and Bayesian) to information and control theories, plus a significant dose of pragmatism from computer science. For those interested in the historical relationship between statistics and machine learning, see Breiman’s Two Cultures.
This diversity is reflected in the surprising complexity in answering simple-sounding questions, which often speaks to the heart of trading using computational machine learning models—ranging from estimating HMM models via MLE (e.g. vol / correlation regime models) to non-convex optimization via non-standard likelihood or loss functions (e.g. portfolio optimization via omega):
Why is minimizing the negative log likelihood equivalent to maximum likelihood estimation (MLE)?
Manifold Learning
One of the perennial quant guessing games is speculating on RenTech (e.g. see amusing 5-year NP thread), particularly given the fascinating background of Jim Simons (see his arXiv for recent work on differential cohomology). Ignoring public commentary, whose veracity is obviously questionable, careful consideration of historical hiring trends and corresponding employee backgrounds are suggestive. While such speculation is amusing, potential relevance arises in assisting in filtering the exploration of research.
Empirical Distributions: Minimum Variance
Trading experience reminds Quantivity the distributions of security returns are rarely normal (or log-normal), despite ubiquitous mathematical presumptions to the contrary. Yet, this begs an obvious question (to beginners and experts alike): if returns are not normal, then what distribution are they?
This question is particularly interesting in the context of minimum variance portfolios (MVPs), as they refute the risk premium and demonstrate significant performance differences compared with standard equity benchmarks. Further, attempts to build intuition about the higher and mixed moments of MVPs depend upon understanding the corresponding returns distribution.
This post analyzes empirical distributions to posit an answer, rather than theorize.
Higher Moments and Minimum Variance
Several subsequent posts will analyze the relevance and comparative performance of higher and mixed moments to minimum variance portfolios (along with CVaR / ES and lower partial moments), extending previous US sector and global rotation models. To help motivate this, consider the following quote from Amaya and Vasquez [2010], which conjectures that compensation for volatility depends on skewness level (p. 16):
Global Rotation with Minimum Variance
Analyzing foreign equity tilt within asset allocation has always been a macroecon crapshoot, given geopolitical turbulence, currency risk, and country-sector concentration. Towards eliminating discretionary prediction, this post applies minimum variance rotation to prominent equity markets within EAFE. The results are quite interesting in comparison with the previous 1999 – 2010 analysis of US sectors.
The following international ETFs are included in this analysis: Germany (EWG), China (FXI), Brazil (EWZ), Japan (EWJ), Australia (EWA), South Korea (EWY), Italy (EWI), France (EWQ), Canada (EWC), Sweden (EWD), Mexico (EWW), and Taiwan (EWT). While many additional international ETFs exist today, their inception dates are too recent to accommodate a longitudinal panel of sufficient length. Global rotation will be benchmarked against EAFE (via EFA), as S&P 500 does for the US market via SPY.
Minimum Variance Sectors: Part 2
Continuing analysis of sector rotation with minimum variance portfolios, John Hall inquired about performance of minimum variance sector rotation outside the period illustrated in the previous post. This post expands analysis to consider the period extending back to inception of US sector ETFs (circa late 1998), unveiling several unexpected delights.
The period 1999 – 2010 includes a diversity of market price regimes: two bubbles, mixed up/down trending, and ample mean reversion. Begin minimum variance sector rotation analysis with MVP sector weights, which are unexpectedly quite interesting:
Minimum Variance Sector Rotation
Numerous readers inquired how to rethink asset allocation in a world where Portfolio Theory is Dead. One approach is embracing dynamic asset allocation while assuming zero risk premium, recognizing that estimating portfolio return moments via standard longitudinal time series analysis turns out to be flawed in practice (irregardless of robust statistical estimators). In this world, the notion of strategic asset allocation is nonsense and thus buy-and-hold investors are unknowingly gambling.
Acknowledging this trend, the historical distinction between short-term “trading” and long-term “investing” is gradually blurring. This post rethinks sector rotation by applying minimum variance portfolios. Although Quantivity dislikes classic sector rotation as it’s both discretionary and predictive, applying minimum variance is interesting: systematic, prediction-free way to model rotation. In doing so, providing a quantitative lens to analyze rotation both ex post and ex ante.
Minimum Variance Portfolios
As recounted in Portfolio Theory is Dead, Now What?, the minimum variance portfolio is the optimal portfolio in a world with zero risk premium. This post expands on the topic via practical implementation in R, preceded by walk through of the corresponding mathematics.
Portfolio Theory is Dead, Now What?
Readers often ask what Quantivity thinks of long-term quantitative strategies, and thus corresponding relevance of modern portfolio theory and asset allocation (strategic and tactical). In short, Quantivity is not a fan.
That said, holistic understanding of how and why these theories are wrong is insightful and relevant for both short- and long-term quantitative strategies. This perspective is informed by standard institutional and retail portfolio management as exemplified by Grinold and Kahn and Faber, along with academic background in both economics and finance.
Review: Machine Learning
The following is a review of Machine Learning: An Algorithmic Perspective by Marsland.
Machine learning (ML) is one of those topics that elicits widely varying responses. Some folks think it’s rubbish for trading, recalling memories of failed early biologically-inspired AI techniques like neural nets. Other folks think it’s the holy grail (it’s not). And yet other folks are uncertain of its applicability, confounded by its multidisciplinary nature and corresponding lack of historic pedagogical coherence.
Quantivity 
