On the systemic nature of global inflation, its association with equity markets and financial portfolio implications

Inflation has been dormant for the best part of 20 years in most developed countries globally, providing an environment for increasing business prosperity and consistent valuation multiple expansion across a number of asset classes including equities. In the wake of the COVID-19 pandemic and the associated economic and financial market crises, we are left with a new question of great pertinence - is the current elevation in global inflation levels transitory, or symptomatic of something more pathological? Rather than seeking to predict what is likely to happen next, we analyze previous periods of inflationary pressure and the associated impacts on economies, equity indices, equity sectors and investor portfolios. Our analysis takes a data-driven and mathematical approach, seeking to learn from the results of bespoke experiments, rather than approaching this problem with (often rigid) econometric theory.

Inflation is a well researched problem in the quantitative finance and economics literature [1, 2, 3, 4]. A significant amount of prior work on the topic has applied well-established econometric theory [5] to gain insights into the time-varying nature of inflationary behaviours. In the academic literature, however, there is limited work exploring the explicit impact of inflation on equity markets. Anecdotally, low-to-moderate inflation is typically associated with buoyant periods for equity markets. Inflationary extremes, where inflation may be particularly high or low, is generally considered to be harmful to equity market returns. In this paper, we test this hypothesis in a variety of contexts. Importantly, we study periods of time consistent with significant changes in inflation, rather than simply studying periods of high or low inflation. This way, we may ensure that our findings are relevant and topical in a period that is characterised by such pronounced economic uncertainty.

More generally, however, the study of financial market dynamics has been a topic of great interest to the applied mathematics, econometrics and econophysics communities. In more recent years the study of time-varying dynamics and correlation structures [6, 7, 8] has attracted the interest of researchers. A wide variety of techniques have been used to study these evolutionary dynamics including principal components analysis (PCA) [7, 9, 10, 11], clustering [12, 13], change point detection [14, 15] and various statistical modelling frameworks [Tilfani2022, 16]. Such analysis has been applied to a wide variety of underlying security types including equities [11], foreign exchange [17], and fixed income instruments [18].

There has been a recent surge in interest among researchers studying various aspects of cryptocurrencies, the blockchain and other aspects of decentralised finance. Recent work that has attracted research interest include analyses of Bitcoin’s behaviour [19, 20, 21, 22, 23], fractal-type behaviour and fractal patterns, [24, 25, 26, 16], cross-correlations and various scaling effects [27, 28, 29, 30, 31], and trading strategies [32]. Quite separately, the topic of portfolio optimization has been of significant interest to the quantitative finance community for many years [33, 34], and the core problem has been extended upon considerably since its inception [35, 36, 37, 38, 39, 40]. Although there is literature examining the effectiveness of various asset classes’ role as a portfolio ‘safe-haven’, we are unaware of any work explicitly studying the benefit of cryptocurrencies in the context of extreme inflationary periods. This work studies the effect of cryptocurrencies in multi-asset portfolios, exploring their time-varying optimal portfolio allocation.

Many of these studies are concerned with the time-varying nature of such dynamics, or the behaviour of cryptocurrencies during various market regimes. Quite naturally, the impact of COVID-19 on cryptocurrency behaviours has been widely studied [41, 42, 43, 44]. For recent work on time-varying similarity and its implications for the cryptocurrency market, and the significant impact of societal physics on the economy, readers should see [45, 46, 47].

In Section 3, we examine the similarity in country CPI trajectories and the associated collective similarity in global inflation. We then introduce a new optimization framework to identify countries that are most "central" in global inflation, and which countries’ equity indices are most robust during periods of significant inflationary change. Then in Section 4 we turn to inflationary impacts on equity market performance, where we study the evolutionary correlation structure of various equity sectors. Finally in Section 5, we implement a time-varying portfolio optimization over the past 5.5 years to determine which asset class would serve as the optimal complement to a core equity exposure, with the purpose of maximizing an investor’s Sharpe ratio. Our methods demonstrate that, despite the associated volatility, cryptocurrency was the most essential additional asset class.

In Section 3 we study country CPI data between 01/01/1955-01/09/2021 and equity index data 01/01/1990 - 01/09/2021. This data comes from a variety of sources including The International Monetary Fund (https://www.imf.org/en/Home), Bank of Canada (https://www.bankofcanada.ca/), Organisation for Economic Cooperation and Development (https://www.oecd.org/australia/), Bank of Japan (https://www.boj.or.jp/en/index.htm/) and Bloomberg. In Section 4 we study equity sector behaviours based on underlying equity data between 01/01/2005 - 31/12/2009, this data is sourced from Bloomberg. In Section 5 we study various asset class performance (in the context of portfolio optimization) between 01/01/2016-30/06/2021. This data is sourced from Bloomberg and Coinmarketcap.com (https://coinmarketcap.com/).

Having explored equity index robustness, we now focus on the performance of underlying equities, grouped by sector, during a more recent period of time. Here, we study correlation behaviours of S&P 500 equities to determine differences in sector behaviours during periods of significant inflationary pressure. We partition these equities into $S=11$ sectors, and study the evolutionary correlation dynamics of each sector. Each underlying equity is grouped into one sector from Communication Services, Consumer Discretionary, Consumer Staples, Energy, Financials, Healthcare, Industrials, Information Technology, Materials, Real Estate and Utilities. We study the evolution of intra-sector correlation dynamics over a relatively recent period that exhibited extreme inflationary behaviours, 1/1/2005-31/12/2009. Let $t_{2}=1,...,T_{2}$ be an index in time reflecting this date range. Let each sector’s collection of underlying equity log returns be the vector $\mathbf{R}^{S}(t_{2})$, where $S$ indexes one of 11 sectors, and $t_{2}$ denotes a specific point in time for the vector of log return values. We use a rolling window of 120 days. For each sector, we implement a time-varying correlation matrix as follows:

and study the evolution of each sector’s correlation behaviours. For each evolutionary correlation matrix $\Omega_{ij}^{S}(t)$, we wish to compute an average sector correlation $\tilde{\mu}^{S}$ during our window of interest. To determine the average correlation of each sector during the period 1/1/2005-31/12/2009, we compute

where $S_{n}$ is the number of equities in each sector, $t_{2}=1307$ and $t_{2}=2610$ correspond to 1/1/2005 and 31/12/2009 respectively.

There are several reasons why we elect to study this period of time. First, rather than simply studying periods of high inflation, we wish to study equity performance during periods of significant changes in inflation (we are indifferent to the direction of such changes). In the current economic climate, the presence of inflation may be transient - and we could see a sharp spike in CPI followed by a reversion to more stable levels. Alternatively, we could see a structural increase in global inflation, and these levels may remain elevated for a prolonged period of time (such as that seen across the world in the 1970s and 1980s). Given the vast range of possibilities, we deliberately choose a country (USA) which experienced profound inflationary changes (both up and down) for a substantial period of time. Second, given the wealth of market commentary currently speculating on an imminent market crisis, we wish to study the behaviour of various equity sectors during a more prolonged crash such as the Global Financial Crisis (GFC). We avoid examining equity behaviours during the COVID-19 market crisis due to its brevity and the immediate nature of this market shock.

Figure 7 displays the rolling average correlation exhibited by each equity sector. Some general patterns are clear. First, the ranking of various sectors’ average correlation over time is remarkably consistent. That is, most sectors’ average correlation trajectory appear to deviate from any other by some translation $\tau$, which is approximately constant over time. The second most notable insight is the obvious spike in average correlation levels during the GFC market crisis. This is consistent with a significant body of prior research, demonstrating that equity behaviours become more homogenous during market crises, with average correlations increasing. That being said, the degree of this increase varies significantly based on the sector - and this warrants further investigation.

The findings of our analysis are shown in Table 4. There is a striking difference in the average correlation between difference sectors. During our period of analysis, the sector with the lowest average correlation is healthcare and the sector with the highest average correlation is energy. This is consistent with prior portfolio research, which has highlighted the typically low portfolio beta displayed by healthcare, and the high portfolio beta displayed by more volatile sectors such as energy. One interpretation of this analysis for those seeking to make stock selection decisions during periods of meaningful inflationary pressure, is the degree of homogeneity within candidate equity sectors. Sectors such as: Utilities, Real estate and Energy, all of which exhibit average correlation levels above $0.6$, display more homogenous behaviours and will provide less opportunities for relative alpha generation. By contrast, sectors such as Consumer Staples and Healthcare, both of which have average correlation below $0.35$, exhibit greater heterogeneity in equity behaviours. These sectors are likely to provide more opportunity for alpha generation during an environment of extreme inflationary pressure.

We now explore a similar theme within the context of managing a portfolio of different asset classes. One notable development over the past several years has been the proliferation of digital currencies such as Bitcoin. Although investments in many coins such as Bitcoin and Ethereum provided very strong returns over the past several years, this was accompanied by significant volatility. But just how beneficial was cryptocurrency exposure to an investor focused on maximizing their risk-adjusted returns? We now try to address this pertinent and topical question. In particular, we study (in the context of managing a multi-asset financial portfolio), which asset classes would have provided the greatest utility to a core equity portfolio over the past $\sim$ 5 years? We take a collection of $a=1,...,A$ daily asset class returns where we let $A=7$, and study their behaviour over a date range from 01/01/2016 - 30/06/2021. We index these date $t_{3}=1,...,T_{3}$. The 7 securities studied include S&P 500 equities, the Bloomberg LNG Index, London Metals Exchange, S&P 500 energy index, CRB Commodity Index, Bloomberg REIT index and Bitcoin (as a representative of the cryptocurrency asset class). Let each asset classes’ daily log returns be $R_{a}(t_{3})$, where $t_{3}=1,...,T_{3}$ references a point in time $t_{3}$ during our window of analysis. We implement a time-varying portfolio optimization, where we use a rolling window of $250$ days, and study the resulting functions of time-varying optimal portfolio weights. We carefully select a $250$ day rolling window to simulate a portfolio manager who examines data over the past year to inform trading decisions. In the optimization problem below, we seek to learn portfolio weights $w_{1},...,w_{A}$ such that we maximize the following

In the above expression, $\bm{w}(t_{3})$ is a vector of portfolio weights at each point in time, $\Sigma_{ij}(t_{3})=\text{cov}(R_{a,i}(t_{3}),R_{a,j}(t_{3}))$ is the time-varying covariance matrix between rolling log returns and $R_{f}$ is the current risk free rate which we set to 0.25%. The constraints in this optimisation are as follows. We hold the S&P 500 weight constant $a_{1}=0.4$, and restrict weights for the remaining portfolio assets $0.025\leq w_{a}\leq 0.3,a=2,...,A$ and assume that the portfolio is always fully invested in a long-only capacity $\sum^{A}_{a=1}w_{a}=1$. At each point in time, the optimization generates a vector of optimal portfolio weights which we denote $w_{a}^{*}(t_{3})$. For each asset class, we compute the mean and variance of the portfolio weights which we denote $\mu_{a}^{w^{*}}$ and $\sigma_{a}^{2,w^{*}}$. These results are shown in table 8 which displays the mean and variance of the time-varying optimal portfolio weights among our collection of $A=7$ representative securities/indices. In conjunction with Figure 8 which displays the optimal weight trajectories, there are several interesting findings.

First, Bitcoin has the highest average optimal weight of 14.9%. This suggests that for an investor with a core allocation to equities over the past 5.5 years, cryptocurrency was the most important asset in increasing the overall portfolio Sharpe Ratio. This was followed closely by the London Metals Exchange index with a total weight of 13%. Although having an average optimal weight of 10.8%, the REIT index exhibited the highest variance in its optimal weight over time. This suggests that REITs exposure may have to be monitored more closely than other asset classes, and may require more frequent rebalancing. Finally, although the LNG Index’s average optimal allocation was only 4.5%, it exhibited the lowest variance in its weight trajectory. This indicates that for a security exposure such as this, portfolio managers may require less trading and rebalancing.

This paper uses various data, mathematical techniques and frameworks to provide a holistic view of inflation, and its impact on equity markets and (predominantly) equity market investors. We believe this is the first work to explore this topic from a macroeconomic, financial market dynamics and portfolio optimization perspective.

In Section 3 we study the similarity of evolutionary CPI inflation patterns between a collection of 8 critically important economies. We demonstrate that Australia and Japan exhibited the most anomalous CPI trajectories, and this was primarily due to their hyper-inflationary onset preceding that of other major economies in the 1970s and 1980s. Our eigenvalue analysis demonstrated the strength of collective behaviours among country inflation patterns, reinforcing the systemic and largely ubiquitous nature of global inflation. We then introduce an inner-product based optimization framework to identify countries that are most central in global inflation. Japan is shown to be the least central (or most anomalous) country with respect to the timing of its inflationary patterns, with the majority of other countries forming one predominant cluster. Finally, we apply the Wasserstein distance between two conditional distributions of country equity index log returns to demonstrate that the USA and UK equity indices possess the most robustness during times of immense inflationary pressure.

In Section 4 we partition S&P 500 equities into 11 disjoint sectors, and study the evolutionary correlation behaviours during a period of extreme inflationary behaviours. Our analysis reveals significant differences between various equity sectors’ correlation during times of inflationary crisis. Consumer staples and Healthcare exhibited the lowest average correlation of $0.34$ and $0.32$ respectively, while energy and real estate exhibited the highest average correlation among constituents with $0.7$ and $0.66$ respectively. Sectors with lower average correlation levels may produce more opportunities for alpha generation, relative to the remaining collection of equities, given the heterogeneity in dynamics. By contrast, sectors with higher average levels of correlation may provide less opportunities for portfolio alpha, with more homogeneity among the collection.

Finally in Section 5 we turn our attention to core equity investors over the past several years who are seeking favourable portfolio diversification. Among our collection of candidate asset classes one could use to diversify portfolio returns, cryptocurrency (Bitcoin) was identified as the most effective complement to a core exposure of equities. The second most effective asset class was the London Metals Exchange. In studying the resulting time-varying optimal weight trajectories, we computed the variance of each function as an indication as to how frequently asset classes may have to be rebalanced. Our analysis indicates that REITS exhibit the highest variance, and may have to be monitored most closely by portfolio managers.

There are several avenues for further work. First, one could explore similar questions using different types of mathematical and statistical analysis. One possible methodology for the study of country inflation trajectories could be the application of multiplex network analysis, for example. Second, one could apply the techniques introduced in this manuscript to answer more general macroeconomic questions, not focused on inflation specifically. Specific modelling techniques, such as our time-varying linear regression, could be replaced with moving average or autoregressive moving average models. Finally, our equity sector and portfolio optimization analysis could be applied on a more varied collection of underlying securities. Equity securities could include equities outside the USA, and the optimization section could include more asset classes, such as fixed income, for example.