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Cross-country equity risk allocation with statistical learning

Jupyter Notebook

Macroeconomic factors plausibly cause divergences in equity market returns across countries. Factors related to monetary policy, financial conditions, and competitiveness should all systematically help detect such divergences. In the presence of rational inattention, point-in-time indicators should also have predictive power.
We apply statistical learning to investigate the signalling value of nine candidate macro factors for cross-country excess returns within major equity sectors for a set of 12 countries. Most factors turn out to be relevant predictors. Cross-country trading signals based on point-in-time macro factors and models would have added material uncorrelated PnL value to an equity portfolio.

Please quote as “Costa, Michele and Sueppel, Ralph, ‘Cross-country equity risk allocation with statistical learning,’ Macrosynergy research post, February 2025.”

A Jupyter notebook for audit and replication of the research results can be downloaded here. The notebook operation requires access to J.P. Morgan DataQuery to download data from JPMaQS. Everyone with DataQuery access can download data, except for the latest months. Moreover, J.P. Morgan offers free trials on the complete dataset for institutional clients. An academic support program sponsors data sets for research projects.

Cross-country equity allocation: the challenges

Macro conditions of local economies influence the prospects for equity returns in local currency. Favourable conditions include those that support or indicate accommodative monetary policy, easy financial conditions, and improving competitiveness. These conditions vary across countries and can be approximately tracked by a range of macro indicators. If the market is not fully information efficient such indicators should have predictive power and, hence, are candidates for factors in a cross-country equity allocation strategy.

A previous post (“Cross-country equity futures strategies”) shows that relevant point-in-time macro indicators can be combined into thematic trading factors for “pure” equity positions, i.e., positions funded in local currency across countries. Note that these differ from dollar-funded positions across countries whose PnL include contributions from currency exposure. Signals based on simple averages of macro factors have produced respectable economic value through allocation to equity index futures across 16 countries.

However, cross-country equity strategies face one challenge that sets them apart from fixed-income and currency strategies: they trade structurally heterogeneous assets across countries. The equity indexes of, say, the U.S. and Switzerland differ in the number of stocks, types of businesses, and structures of companies. Moreover, all companies have specific idiosyncratic structures and sensitivities. As a consequence, there are many important factors aside from macroeconomic influences that influence relative returns. Hence, the predictive power of macro indicators is naturally quite subtle.

To mitigate these challenges, this post focuses on cross-country allocation within equity sectors and for “all sectors” equally weighted baskets rather than trading the leading indices against each other. Also, we use simple sequential machine-learning processes, one for each sector, to combine plausible factors into signals.

Cross-country positioning within sectors

We investigate whether cross-country differences in macro factors predict cross-country equity returns within the 11 sectors by the “Global Industry Classification Standards” (GICS). For an overview of the sectors, see Annex 1 below. The post looks at generic sectoral returns for 12 countries or currency areas, which are (alphabetically by currency symbol): Australia (AUD), Canada (CAD), Switzerland (CHF), the euro area (EUR), the UK (GBP), Israel (ILS), Japan (JPY), Norway (NOK), New Zealand (NZD), Sweden (SEK), Singapore (SGD), and the U.S. (USD). The underlying equity return data comes from the J.P. Morgan SIFT database via the J.P. Morgan Macrosynergy Quantamental System or “JPMaQS” (view documentation).

The focus is on volatility-targeted excess returns of one sector in one country versus an average of sectoral returns for all countries. Excess returns mean returns net of local funding costs. Country-sector positions are scaled to a 10% volatility target based on the historical standard deviation for an exponential moving average with a half-life of 11 days. They are rebalanced at the end of each month.

Long-term country differences in the performance of local risk-adjusted returns for an average basket of all (available) sectors have been sizable. For example, since 2000, the Norwegian basket has outperformed by around 75%, while the UK basket fell short by roughly 70%. Relative performance can show persistent trends. For example, the U.S. basket consistently underperformed in the 2000s but outperformed most years from 2010 to 2024.

Cross-country performance has not been homogeneous across sectors. For example, while Sweden’s cross-sector average equity basket has been a long-term outperformer, its communication sector has underperformed since 2000. In the U.S., financials and industrials failed to match the outperformance of the overall cross-sector basket.

Conceptual factor candidates

There is a range of conceptual factors that could have an influence on cross-currency relative equity returns. To evaluate predictive power and trading value, however, factors must be built with quantamental indicators, which are time series of macroeconomic information states that represent for each date the value of the indicator that would have been known to the market on that date. To this end, we use macro-quantamental indicators from the J.P. Morgan Macrosynergy Quantamental System (JPMaQS), which are specifically designed for the development and backtesting of financial markets trading strategies.

We calculate nine-factor panels using only point-in-time information. Each factor represents the average of several quantamental categories that might represent or contribute to it. The choice and combination of the underlying categories are not optimized. Five conceptual factors have, in similar forms, been used in previous research (view post here). The other four have been added based on simple theory and plausibility. All factors have been formulated such that their theoretical impact on equity markets is positive. Predictive power is assumed to arise from rational inattention of markets (view post here), which explains the gradual dissemination of information due to costs and cognitive limitations.

In general, the conceptual factors are calculated in three steps.

  1. We take an average of the factor constituents to get a directional conceptual factor.
  2. We calculate the relative values of each factor as the difference between one country’s value and the average of all countries’ values. If a country value is missing the average is calculated based on the available ones.
  3. The relative values are normalized around their neutral zero level, whereby standard deviations at each point in time are estimated based on the full panel, i.e., all countries’ values for the factor up to that point in time. We call these series relative factor scores.

We consider three factors related to relative cost pressure and the monetary policy outlook:

  • Relative inflation shortfall: The factor uses four point-in-time inflation indicators: headline CPI and core CPI as % over a year ago and seasonally—and jump-adjusted headline and core CPI as % of the latest 6 months over the previous 6 months at an annualized rate (view documentation). From these rates, we subtract the concurrent effective inflation target of the currency area (view documentation) and take negative values. Below-target inflation calls for accommodative monetary policy and typically heralds easing cost pressure.
  • Relative labour market slack: The factor averages two categories. The first is the negative of the difference between employment growth as % over a year ago in 3-month moving averages (view documentation), and the 5-year trend in work force growth (view documentation). The second is simply the increase in the unemployment rate over a year ago in 3-month moving averages (view documentation). Labour market underperformance also calls for easing monetary policy and production costs.
  • Relative consumption shortfall: This is calculated as the negative average of real private consumption growth (view documentation) and real retail sales growth (view documentation), both as % over a year ago and a 3-month average or quarterly relative to medium-term GDP growth, i.e., its rolling 5-year median (view documentation). Weak past consumption growth bodes for accommodative monetary policy.

We consider three factors that are indicative of relative financial conditions:

  • Relative effective currency depreciation: The factor averages two constituents. The first is the negative of the difference between employment growth as % over a year ago in 3-month moving averages (view documentation), and the 5-year trend in work force growth (view documentation). The second is simply the increase in the unemployment rate over a year ago in 3-month moving averages (view documentation). Real or nominal depreciation bodes for improvements in the relation of sales prices and costs.
  • Relative real interest rate conditions: This factor averages the negatives of two measures. The first is the real short-term interest rate (view documentation) relative to productivity, i.e., the difference between medium-term real GDP growth (view documentation) and workforce growth (view documentation). The second is the change in this excess real interest rate over the past year. Low or declining real interest rates point to easier access to finance and greater demand for fixed capital.
  • Relative money growth: This is the average of two point-in-time measures: annual growth in the local narrow money aggregate (view documentation) and annual growth in the local broad money aggregate (view documentation). High money growth often indicates an unsustainable build-up in liquidity.

Finally, we consider three factors relative to competitiveness:

  • Relative terms-of-trade improvement: This is calculated based on various rates of change of point-in-time commodity terms-of-trade (view documentation). These are % of the latest month over a year ago, % of the last month over the previous 1-year average, and % of the latest week over the previous for weeks. Rising export prices relative to import prices, augurs for an improving relation between revenues and costs in the local economy.
  • Relative industry confidence change: This is based on two versions of annualized seasonally adjusted changes in the local manufacturing business confidence score (view documentation): the difference of the last 3 months over the previous 3 months and the difference of the last 6 months over the previous 6 months. Improving confidence scores should indicate improving business conditions in a timely fashion.
  • Relative trade balance ratio change: The factor is the average of two (annualized) changes of seasonally adjusted merchandise trade balance ratios to GDP (view documentation). Those are the 3-month moving average over the previous 3 months and the 6-month moving average over the previous 6 months. Improving export-to-import ratios often indicates improving competitiveness.

Correlations between the nine conceptual factor candidates have been modest and in different directions, which reflects that they represent different influences on relative equity market performance. The largest negative correlation is between relative FX depreciation and terms-of-trade improvements, which reflects that higher terms-of-trades typically drive a stronger currency. From a local equity perspective, it is sensible to consider both effects in conjunction. The largest positive correlation is between relative FX depreciation and labor market slack and seems to reflect

Benefits and design of simple statistical learning

A key challenge is to select and combine conceptual factors for a single signal for cross-country vol-targeted positions within each GICS sector. As emphasized in previous research (FX trading signals: Common sense and machine learning) there are two valid methods to do so.

  • Conceptual parity simply takes the average of all (available) conceptual factor scores for each country, forsaking any optimization or a-priori statistical evaluation. In the present case, this would also imply that the same set of signals is applied for all sectors.
  • Sequential machine learning applies point-in-time optimization of models, hyperparameters and parameters, starting with a minimum data sample and progressing over its expansion. This means that signal calculation models change over time, and, in the present case, different cross-country signals can be calculated for different equity sectors.

The empirical analysis uses data from 1997-2000 onwards, depending on the country. To have a reasonable minimum initial history, we start using learning results for evaluation from 2003. The conceptual parity signals are used as [1] an indication of the quality of the underlying data and their choice and [2] a benchmark for the success of statistical learning. Statistical learning does not generally beat conceptual parity in historical correlation analysis and backtests. However, it delivers a more objective basis for backtests, as selection and weights are never influenced by hindsight.

We apply one statistical learning process per GICS sector and for an “all sectors” equally weighted basket of equity. The learning process considers several types of panel regression and derives inference from relative factors and returns for all 12 markets together. Learning is implemented with the scikit-learn package and various wrapper functions of the Macrosynergy package. The basic structure has been explained in a previous post (Optimizing macro trading signals – A practical introduction). Here, it is simply extended to the three-dimensional nature of the data set, which features time, countries and sectors.

  1. We set a model and hyperparameter grid. It defines all model versions that are considered for predicting cross-country returns based on the nine relative factor scores.
    There is one common grid for all sectors. The models considered are OLS regression, non-negative least square regression (forcing all coefficients to be positive), time-weighted least squares regression (imposing exponentially decaying weights on observations), and non-negative time-weighted least squares. Also, for time-weighted least squares, the learning process considers half-lives between 1 and 5 years. Signals are calculated based on regression coefficients that are adjusted for statistical precision (see post How to adjust regression-based trading signals for reliability). Other regression models and hyperparameters could be optimized (see Regression-based macro trading signals). However, more options come at the expense of greater model instability and a longer run time of the example code.
  2. We also apply a single set of cross-validation rules for training and testing the different model versions using the expanding development data sets. As a cross-validation criterion, we use the mean squared error of the test set predictions. Cross-validation splitting is done using the expanding k-fold panel splitter, as implemented by the ExpandingKFoldPanelSplit class of the Macrosynergy package, with a minimum of three splits.
  3. Sequential model selection and optimized signal calculation are executed separately for each GICS sector and a sector average basket by using the Macrosynergy package’s  SignalOptimizer. It operates scikit-learn model selection and cross-validation classes for expanding samples. In particular, the calculate_predictions method produces signals on each rebalancing day, respecting the panel structure of the underlying data. Using this class allows looking “under the hood” of the learning process, particularly at the choice of optimal model and the effective weight of factors over time.

For cross-country equity allocation, the model choice has mostly converged over time for a non-negative least squares regression, i.e., the simplest and most restrictive model version. Preference for restrictions by theoretical priors, i.e., the sign of the factor impact, reflects the scarcity of historical experience and the small amount of equity performance that can be explained by simple macroeconomic factors. The timelines below show the evolution of the optimal model choice for the “all sectors” signal generation. In this case, non-negative least squares (which is the OLS setting “positive == True” in scikit learn) have prevailed since 2003. For most sectors that convergence took longer, however.

Convergence on a model that uses expanding data samples without time weighting also means that factor weights converge. For the “all sectors” models, six of the nine candidate relative factor scores ultimately pass the test of statistical relevance. The three most powerful ones have been relative terms-of-trade changes, relative FX depreciation, and relative money growth. Relative inflation and consumption shortfalls, as well as relative trade balance changes, have not prevailed as factors in signal generation.

However, different sectoral learning processes have selected different sets of factors and different weights, as shown by examples of financials and consumer staples below. Sectoral differences can be both a strength and a weakness of the current learning design. The strength is that coefficients can adapt to differences in the importance of relative macro factors across sectors. The weakness is that in the present setting of limited historical experience, the influence of sector and even stock-specific dynamics unrelated to macro influence is magnified vis-à-vis a learning process that consolidates cross-country experiences for all sectors.

An assessment of aggregate value generation

The influence of cross-country pure equity allocation signals for positions that exclude exchange rate effects is subtle, particularly if the number of countries to which it is applied is limited. However, empirical evidence suggests that long-term performance contributions are nevertheless material and uncorrelated with overall equity exposure.

Here, we estimate the economic benefit by generating standard “naïve PnLs”. Each sector and the “all sectors” basket rebalance monthly volatility-targeted positions in accordance with the optimized signals and conceptual parity signals under the assumption of a 1-day slippage for trading. The naïve PnL does not consider transaction costs, risk management, or compounding, as all of these depend on context. Sectoral PnLs have been scaled to an annualized volatility of 10% to represent them jointly in one graph.

If one had traded only the “all sectors” basket, the learning-based sequentially optimized signal would have produced a Sharpe ratio of 0.4 and a Sortino ratio of 0.5-0.6. Correlation with the S&P500 would have been near zero. The conceptual parity signal would have delivered a slightly higher Sharpe but a slightly lower Sortino. Also, the seasonality of the conceptual parity signal would have been greater.

The various sectoral PnLs produced individual Sharpe ratios between 0.1 and 0.6, all with S&500 correlations near zero. An unweighted average of sectoral cross-country PnLs would have delivered a long-term Sharpe ratio of 0.5 and a Sortino ratio of 0.7 with pronounced seasonality. As a stand-alone leveraged strategy, that would not be very high, particularly in comparison with cross-country fixed-income and FX strategies. However, as a cross-country overlay, a risk-adjusted return of this magnitude of risk-adjusted value generation is material. Value generation is not only incremental to overall equity exposure but also to macro-based cross-sector strategies (see Statistical learning for sectoral equity allocation) and related FX positions.

 

Historic success of cross-country allocation by sectors

According to naïve PnLs, value generation has been very different across sectors. The cross-country strategy worked best for consumer staples, materials, industrials and consumer discretionary, all with Sharpe ratios of 0.4-0.6 and Sortino ratios of 0.6-0.9.

Cross-country signals for information technology, real estate, communication services, and utilities would have produced almost no PnL contribution. However, no sector displayed a negative PnL over the past 22 years.

Annex 1: Equity sectors

The analysis in this post refers to the following equity 11 sectors by the “Global Industry Classification Standards” (GICS) developed in 1999 jointly by MSCI and Standard & Poor’s. The purpose of the GICS is to help asset managers classify companies and benchmark individual company performances:

  • Energy: The sector comprises companies that support the production and transformation of energy. There are two types. The first type focuses on exploring, producing, refining, marketing, and storing oil, gas, and consumable fuels. The second type provides equipment and services for the oil and gas industries, including drilling, well services, and related equipment manufacturing.
  • Materials: The sector encompasses a wide range of companies engaged in discovering, developing, and processing raw materials. These include chemicals, construction materials (such as cement and bricks), container and packaging materials (such as plastic and glass), base and precious metals, industrial minerals, paper, and other forest products.
  • Industrials: The sector contains a broad range of companies involved in producing goods used in construction and manufacturing (capital goods) as well as providing commercial services and transportation. The area of capital goods includes aerospace and defence, building products, construction and engineering, electrical equipment, industrial conglomerates, and machinery. The commercial services sub-sectors include waste management, office supplies, security services, and professional services (consulting, staffing, and research). The transportation area includes air freight and logistics, airlines, marine transportation, road and rail transportation, and transportation infrastructure companies.
  • Consumer discretionary: This sector comprises companies producing consumer goods and services considered non-essential but desirable when disposable income is sufficient. The main areas are automobiles, consumer durables, apparel, consumer services (such as hotels and restaurants), and various retail businesses.
  • Consumer staples: This sector includes companies that produce and distribute presumed essential consumer products that households purchase regardless of economic conditions. These products mainly include food, beverages, household goods, and personal care items.
  • Health care: The sector includes companies that provide medical services, manufacture medical equipment, or produce drugs. It has two main areas. The first features health care equipment and services. It includes manufacturers of medical products and supplies, providers of health care services (such as hospitals and nursing homes), and companies that provide technology services (such as electronic health records). The second area features research, development, and production of pharmaceuticals, biotechnology, and life sciences tools.
  • Financials: This sector provides financial services, including banking, investment services, insurance, and financial technology (fintech). The four main subsectors are banks, diversified financials (such as asset management, credit cards, and financial exchanges), insurance, and investment trusts.
  • Information technology: This sector includes companies that produce software, hardware, and semiconductors, as well as those that provide IT services, internet services, and interactive media. Software companies produce application software and systems software. Hardware companies provide computers, networking equipment, and consumer electronics. The semiconductor sector manufactures semiconductors and the equipment used for producing the former. IT services include consulting, data processing and outsourced services. Internet services encompass cloud computing, web hosting and data centres. Interactive media include digital platforms, such as Google and Facebook.
  • Communication services: This sector features companies that broadly provide communication services and entertainment content. It contains two main areas. The first is telecommunication services, which provide the means for telecommunication, including traditional fixed-line telephone services, broadband internet services, and wireless telecommunication services. The second area is media and entertainment, which focuses on the creation and distribution of content for broadcasting, home entertainment, movies, music, video games, social media platforms, search engines, and so forth.
  • Utilities: This sector includes companies that provide essential utility services such as electricity and water. Their activities include generation, transmission, and distribution, and they are typically subject to tight regulations. Standard classification distinguishes five types of utilities: electric utilities, gas utilities, water utilities, multi-utilities, and independent power and renewable electricity producers.
  • Real estate: This sector focuses on real estate development and operation. It encompasses property ownership, development, management, and leasing. It also includes Real Estate Investment Trusts (REITs) that invest in various property types.

 

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