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FEDERAL RESERVE BANK o f ATLANTA WORKING PAPER SERIES

Asset Co-movements: Features and Challenges
Nikolay Gospodinov
Working Paper 2017-11
November 2017
Abstract: This paper documents and characterizes the time-varying structure of U.S. and international
asset co-movements. Although some of the time variation could be genuine, the sampling uncertainty
and time series properties of the series can distort significantly the underlying signal dynamics. We
discuss examples that illustrate the pitfalls from drawing conclusions from local trends of asset prices.
On a more constructive side, we find that the U.S. main asset classes and major international stock
indices share a factor that is closely related to the business cycle. At even lower frequency, the common
asset co-movement appears to be driven by demographic trends.
JEL classification: G13, G14, G17
Key words: cross-asset, within-asset and international asset co-movements, rolling correlation, timevariability, persistence, higher moments, risk factors, sampling frequency

This paper was prepared for the 22nd Annual Financial Markets Conference, “Managing Global Financial Risks: Shifting
Sands and Shock Waves” (May 7–9, 2017). The author thanks Richard Crump, Paula Tkac, and Larry Wall for helpful
comments and suggestions. The views expressed here are the author’s and not necessarily those of the Federal Reserve Bank
of Atlanta or the Federal Reserve System. Any remaining errors are the author’s responsibility.
Please address questions regarding content to Nikolay Gospodinov, Research Department, Federal Reserve Bank of Atlanta,
1000 Peachtree Street NE, Atlanta, GA 30309-4470, nikolay.gospodinov@atl.frb.org.
Federal Reserve Bank of Atlanta working papers, including revised versions, are available on the Atlanta Fed’s website at
www.frbatlanta.org. Click “Publications” and then “Working Papers.” To receive e-mail notifications about new papers, use
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Identifying and quantifying the underlying co-movements in financial data have important
implications for asset allocation, portfolio valuation, regulatory enforcement, and policy analysis.
These co-movements can be defined broadly as common dynamics (i) within and across U.S. asset
classes (e.g., across sectors in equity space or bond-stock rotations), (ii) across international markets,
or (iii) across different parts of the joint distribution of asset returns. Since the 2008 financial crisis,
there have been several episodes when the correlations between different asset classes were elevated
but dropped off quickly afterwards and exhibited substantial variability. For example, the prevailing
view in the last few years is of an increased correlation across asset classes: the “risk-on/risk-off” view
of markets. Is this due to fundamental or transient (technical) factors? Can we reliably isolate, in real
time, the underlying source of risk from data and estimation noise? Is the observed time-variability of
these co-movements a true property of the data-generating process or a symptom of misspecification
that arises from omitted common factors and higher-moment distributional features?
To answer these questions, it is desirable first to document and characterize the main regularities in
the asset co-movements across asset classes over time. While the scope of the empirical analysis in
this paper is limited to some frequently studied co-movements within and across asset classes, we still
attempt to provide more general arguments about possible sources of common variation in asset
returns, emphasize some pitfalls that arise in commonly used measures of these co-movements, and
discuss the challenges of measuring statistically the shifts of the distribution at various frequencies.
The underlying sources of common asset variations can be crudely classified into long-run, transient,
and spurious factors. The long-run (fundamental) sources of risk include structural macroeconomic
and demographic factors at business-cycle or even lower frequency. The transient factors are roughly
interpreted as fluctuations prompted by macro, political, and market-structure (technical) events that
can force medium-term asset reallocations and rotations and induce movements in the asset risk
premia. Finally, the short-term co-movements can arise from unanticipated (oil supply, for example)
shocks or data noise. The data noise and finite samples can generate possible spurious co-movements
and observational equivalence between the different factors, which renders a sharp distinction
between the underlying sources of risk almost infeasible. Despite these limitations, it is often tempting
to ascribe “fundamental” structure to some of the observed short-term movements as illustrated by
some examples discussed in this paper.
The importance of reliable extraction and classification of common factors in asset dynamics for
investment management can be summarized as follows. First, inferring information about the risk
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factor structure that underlies the asset co-movements and then mimicking its implied dynamics using
tradeable proxies plays a key role in mimicking portfolio construction and factor investing. At the
same time, it is common practice to rely on rolling estimation of second-moment statistics (covariance,
correlation) for portfolio allocation, evaluation of asset-pricing models, and factor investing. In this
paper, we discuss several examples which provide strong evidence that if the multiple sources of
uncertainty, as well as data and model variation, are not properly accounted for, this may result in suboptimal investment strategies and mischaracterization of risk events.
The discussion in the paper will be focused around several general observations. First, if there is robust
evidence that the asset co-movements are genuinely time-varying, then it would be useful to identify
the source of this common variation. Market participants have suggested several sources of common
variation: regulatory changes and reduced market liquidity in some markets, correlated arbitrage,
growth of passive investment funds, particular investment strategies, and algorithmic trading, among
others. The heightened interdependence within and between asset classes can have important
implications for policymakers and can make the diversification and hedging implementation
increasingly challenging. On the one hand, the strengthened asset co-movements open up the
possibility of systematic market risk (such as concerns about the destabilizing effects of the “taper
tantrum” in 2013) and the proper calibration of monetary policy to economic conditions. On the other
hand, if monetary policy is a driver of the correlation change, then it is important to understand how
monetary policy decisions (and especially surprises) will affect markets; for example, how quantitative
easing could influence long-term bond and other asset prices. Identifying the sources of this increased
dependence in the joint distribution of asset returns may shed light on how to adapt to this shifting
and potentially unstable landscape.
Second, it should be acknowledged that another possibility for the increased time variation, and
potential instability, in the correlations across asset classes could be purely statistical due to the
limitations of the modeling framework and the use of second-moment measures.1 We argue in the
paper that particular attention should be paid to the sampling frequency and estimation uncertainty
associated with computing moments of the distribution, the effect of persistence on correlation-type
measures of co-movement, and differences in the correlation measures across different parts of the
There is a large and well-developed literature in financial econometrics on estimating dynamic conditional correlation
models (see Engle, 2002, for example). To avoid technical details, we do not discuss this literature here. However, some
of the remarks that we provide for time-varying correlations apply to this literature as well. For some caveats about
dynamics conditional correlation models, see Caporin and McAleer (2013).
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distribution. While most drawbacks of the standard measures of co-movements are thoroughly
discussed and documented in the literature, they are often downplayed, due to their computational
and interpretational convenience, by practitioners. In this respect, it is surprising how
underappreciated the sampling and estimation uncertainty could be in forming investment and policy
decisions based on nonrobust measures of second-moment variation. As a result, there is a tendency
to over-analyze some high-frequency movements and attribute them to a particular underlying signal
without taking fully into account the data noise and estimation uncertainty. Unlike high-frequency comovements that are largely elusive and fairly transitory, low-frequency analysis provides a more robust
and reliable way to evaluate these dependencies and suggests that the asset common variation appears
to exhibit more stable relationships with macroeconomic and slowly moving demographic factors at
business-cycle or longer duration. Furthermore, if sources of risk exhibit differential impacts on the
shape of the asset distribution over time (either by fattening the tail risk or shifting the tail risk from
one side of the distribution to the other), the information reflected in the second moments will be
incomplete and potentially misleading as it may exhibit substantial time variability even when the
underlying population correlations are constant. More broadly, if we seek to understand how investors
assess risk and reallocate their portfolios in response to the changing distribution of asset returns and
how their actions may themselves help to shape this distribution, more general dependence
measures—that reflect and summarize the information in the whole distribution—are therefore
necessary for identifying the underlying risk factors.2
In light of these remarks, it is prudent to approach the analysis of asset co-movements by explicitly
acknowledging the model and estimation uncertainty surrounding all investment and asset-allocation
decisions. Since all financial models are constructed to approximate a complex reality, they are
inherently misspecified. This is often done intentionally, as parsimonious models draw only partial or
incomplete maps of the latent objects of interest either to emphasize particular aspects or because the
underlying structure is completely unknown. How large is the effect of this model misspecification on
the quantity of interest (for example, portfolio weights, stochastic discount factor) is an empirical
question. While there is suggestive evidence of an increased asset-class interdependence and a shifting
market landscape, the exact source of this structural change (correlated arbitrage, interdependencies,

Embrechts et al. (2002) provide a comprehensive discussion on the properties and pitfalls of correlation for measuring
general dependence.
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trading strategies, passive investment, algorithmic trading) is difficult to pin down due to short
samples, noisy data, and estimation uncertainty.
We approach this problem from an empirical perspective since economic theory provides only limited
guidance on the fundamental sources of time-varying co-movements across asset classes. There is a
well-established literature that studies the large directional swings in the covariance between U.S. bond
and stock returns in the post-World War II period. For instance, it is a stylized fact that this correlation
was largely positive between 1953 and the late 1990s but then turned negative, especially during the
2001 and 2007–09 recessions, with bonds providing a hedge against equity and macroeconomic risks.
Baele at al. (2010), Campbell et al. (2015), David and Veronesi (2016), and Song (2017), among others,
propose models that better fit the observed dynamics of the correlation between bond and stock
returns. However, developing a unifying framework for modeling jointly several major asset classes
(stocks, government bonds, corporate bonds, commodities, currencies) proves to be prohibitively
difficult. Cochrane (2015) identified the question “Once we find the factor structure in bonds, what
is the factor structure of expected returns across asset classes?” as one of the “bigger” questions facing
financial economics. On that note, dimension reduction techniques such as factor analysis prove to
be useful tools for summarizing the common variation across asset markets. Identifying the main
sources of risk via the estimated factors is of fundamental importance not only for guiding investment
decisions but also for monitoring financial stability, stress testing, and more.
While distinguishing among fundamental, transitory, and spurious sources of risk proves empirically
difficult, we find clear evidence of long-run co-movements at business-cycle or even lower frequency.
There is a reasonable basis for tying these co-movements to macroeconomic fundamentals and
demographic factors. While there is also some evidence of co-movement at the shorter end, one
needs to be careful in analyzing this evidence as there are several sources of possible error that make
the competing hypotheses observationally equivalent. These short-term co-movements, which force
asset reallocations and induce temporary co-movements in the asset-risk premia, tend to be more
unstable. Nevertheless, there is a tendency to ascribe more fundamental structure to these variations,
which warns against overreliance on various nonrobust measures of co-movement that are routinely
used for portfolio allocation, evaluation of asset–pricing models, and factor investing.

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An Illustrative Example: Oil Price Co-movements
From the beginning of 2014 to the middle of 2016, the price of crude oil dropped by more than 50
percent, a drop that was accompanied by a substantially elevated correlation between oil price and
several other asset classes. This unusually high correlation attracted the attention of market analysts
for at least two main reasons. First, this co-movement significantly restricted the set of diversification
and hedging opportunities in the market. Second, oil prices as the driving factor behind the asset comovements on a more sustained basis may pose risks to the stability of the financial system given the
highly volatile nature of commodity prices, with their larger exposure to geopolitical risk, supply
disruptions, and more.3

The median values of the option-implied volatility indices OVX and VIX (for oil prices and S&P 500, respectively) since
the beginning of 2014 (2015) are 37.78 (42.28) and 14.12 (14.48).
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Figure 1. Oil price dynamics against (1) 5-year, 5-year forward breakeven inflation; (2) U.S. dollar index
(DXY); (3) percent changes in S&P 500 index from a year ago; and (4) Barclays high yield index. The
solid lines denote the dynamics of each series from January 2014 to May 2016, and the dashed lines
represent the period June 2016 to February 2017.
Figure 1 plots the dynamics of several asset prices—5-year, 5-year forward breakeven inflation from
Treasury Inflation-Protected Securities (TIPS) and Treasury prices, U.S. dollar index (DAX), and oneyear changes in the S&P 500 index, and Barclays high yields index—versus the log oil price from the
beginning of 2014 to the end of February 2017. Ending in May 2016, the series (or some
transformations of them) are plotted as solid lines to illustrate the seemingly tight relationship between
the oil price and other asset prices that was widely documented and discussed by pundits and the
media at that time. In contrast, the dashed lines after May 2016 tend to suggest that this relationship
has substantially weakened.
Was this co-movement due to fundamental common factor/information driving all these variables
(global demand shock, for example), or was it caused by transitory (market-specific) factors reflected
only in the asset risk premia? While seeking answers to these questions, we want to emphasize some
statistical features—such as the possibility of spurious time variability due to persistence (trending
behavior) of the variables over this particular period or sensitivity to movements in other parts
(skewness, tails) of the joint or marginal distributions—that are often overlooked by practitioners.
To gain a better understanding of these issues, we focus on the relationship between breakeven
inflation (BEI), embedded in nominal and inflation-protected securities, and oil prices from the
beginning of 2014 to the middle of 2016. During this particular period, both oil prices and BEI
exhibited a downward trend, and this strong correlation was largely attributed to the effect of oil prices
on inflation expectations. This interpretation was challenged on several grounds (see, for example,
Gospodinov, Tkac, and Wei, 2016). First, BEI is not a clean measure of inflation expectations as it
also contains other, unobservable components such as liquidity and risk premium. Second, it is
somewhat counterintuitive for the day-to-day (transitory) variations in oil prices to affect inflation
expectations 5 to 10 years out. Finally, both variables are highly persistent and the sample correlation
may provide a spurious signal of a relationship between the two variables.
A proper decomposition of the observable BEI into its latent components (see Gospodinov and Wei,
2015) reveals the following. Despite the wide variations in BEI, long-term inflation expectations
appear to be stable and uncorrelated with oil prices. A large portion of the low-frequency variation in
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BEI can be attributed to the inflation risk premium. But most of the high-frequency variation in the
recent dynamics of BEI can be attributed to a “liquidity” premium. This “liquidity” factor captures a
wide range of market-related factors such as seasonal carry, deflation floor, limits to arbitrage, tenorspecific liquidity, redemptions, reallocations, and hedging in the TIPS market following an oil price
drop or global financial turbulence. For example, an event that drives flight-to-quality into nominal
Treasuries and/or forced sales of TIPS leads to a lower BEI without any change in inflation
expectations.
Figure 2 shows that almost all of the recent correlation between BEI and oil prices is being picked up
by the “liquidity” premium. As shown above, other asset classes (stock prices, high-yield bonds,
municipal bonds, exchange rate) have also exhibited an elevated correlation with oil prices during this
period. If genuine, it is useful to determine if this is due to a global demand driver, correlated risk, or
market specificities (such as forced de-risking and liquidations, reallocations, flight-to-safety, or
covering hedges).

Figure 2. Time plot of weekly TIPS liquidity premium with oil prices and oil option-implied (OVX)
volatility
For example, numerous studies have documented the existence of persistent mispricing in various
asset markets. It is conjectured that common factors could drive this mispricing and the resulting
arbitrage. Brunnermeier and Pedersen (2009) argue that availability of funding may have liquidity
effects on asset prices. Also, if capital returns slowly to the fixed-income funds following a period of
flat performance, then the arbitrage in various fixed-income markets (corporate, CDS, Treasury, TIPS)
can exhibit commonalities (Fleckenstein, Longstaff, and Lustig, 2014). Finally, the macroeconomic

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environment after the financial crisis may also have contributed to the strengthened asset comovements. For instance, Datta et al. (2017) provide evidence that oil and equity returns have become
more responsive to macroeconomic news during the zero lower bound period.4
While this and other (more direct or anecdotal) evidence on the 2014–16 episode suggests that the comovement of various asset prices and oil price can be attributed to market-structure factors, the next
section attempts to raise the awareness that a part of this co-movement can be purely coincidental or
unreliable. In particular, we explore the role of persistence of the series under considerations,
estimation, and sampling uncertainty, as well as the potential limitations of second-moment based
measures of co-movement to represent accurately changes in the joint distribution of interest.

Statistical Challenges for Measuring Asset Co-movements
Persistence and Measures of Co-movement
As illustrated above, the co-movement across asset classes is often analyzed using (commodity, for
example) prices and yields. While commodity prices and bond yields are usually modeled as meanreverting processes, they are strongly persistent with a very slow mean reversion. Because the high
persistence generates local trends, these local trends can induce spurious correlation even when the
underlying processes are completely unrelated to each other. Furthermore, even when the processes
are genuinely related, the serial correlation present in prices or yields can obfuscate the underlying
relationship if the serial correlation is not properly taken into account.
To illustrate the latter point, we examine the leverage effect (the negative relationship between stock
prices and volatility) that occupies a prominent place in financial economics and econometrics. Figure
3 plots the time series dynamics of the S&P 500 (SPX) and VIX indices (implied by options written
on SPX) from January 2000 to February 2017.

There is also evidence that the transmission mechanism of propagating the oil shocks through the US economic system
has changed due to the increased role of domestic oil production (Baumeister and Kilian, 2016).
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Figure 3. Time series dynamics of SPX and VIX indices

While the negative relationship is evident during several episodes of sharp stock declines, there are
periods (the beginning of the sample as well as the 2012–16 period) when any possible co-movement
may be masked by the trending behavior of the SPX index.
Figure 4 below suggests that the 120-day sample correlation between the SPX and VIX (blue line) is
highly volatile with even positive values over some sub-samples. It is tempting to conjecture that
removing the local trends in the SPX index by transforming it to SPX log returns may stabilize and
uncover better the leverage effect. The red line in figure 4 reveals that this transformation reduces the
variability of the sample correlation coefficient, but it leads to a severely biased measure that
underestimates significantly the negative co-movement between the SPX and VIX. This is due to the
fact that the SPX returns and the VIX index are highly unbalanced in terms of their statistical
properties, since the former is an uncorrelated (or only weakly correlated) process and the latter is a
highly persistent and bounded variable.

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Figure 4. 120-day correlation between SPX and VIX

To render the statistical properties of both series similar, we compute the log changes of the VIX
index. Figure 4 plots the 120-day rolling correlation5 of SPX returns and VIX changes in yellow. In
fact, this is akin to the way the leverage effect is modeled in financial econometrics. The correlation
now is much more stable and averages around -0.8, which is the magnitude of the leverage effect that
is typically estimated in the empirical literature. Note that some of the variability in the rolling
correlation coefficient can be attributed to overlapping estimation uncertainty as discussed below.
To illustrate the point that the observed correlation may be spurious, we generate data from two
uncorrelated, near-unit root processes (that match the persistence in bond yields and oil prices) and
plot in figure 5 one realized sample path of the two processes. The unconditional correlation between
the two series is 0.36 with an even stronger correlation over sub-samples that arises from “common”

In this paper, we use Spearman’s (rank) correlation instead of the standard Pearson’s correlation coefficient. This choice
is dictated by the robustness properties of the rank correlation. Also, when appropriate, we computed the correlation
coefficient using standardized returns using a GARCH(1,1) model for estimating the conditional variance of returns. The
results were very similar and are not reported, unless specified otherwise.
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local trending behavior. Any observed commonality is completely spurious, as the two series are
generated as independent of each other.

Figure 5. Time plot of two independent highly persistent processes
This should serve as a warning sign in analyzing and justifying observed co-movements of highly
persistent processes. Persistence in variables can mask or exacerbate time variation and can induce
spurious correlation. It also suggests that one should avoid computing correlations in levels and
analyze instead the transformed series after the persistence is removed. This is the approach that we
follow in the rest of the paper.

Time-Variability, Estimation Uncertainty, and Tail Behavior
The last decade has witnessed a proliferation of passive investment funds. Since 2005, the share of
S&P 500 ownership of passive mutual funds and exchange-traded funds (ETF) increased from 4.6
percent to 11.6 percent (Wall Street Journal, October 2016) while the share of active mutual funds and
ETFs remained flat around 17 percent. This same period has been characterized by an increased
correlation among U.S. equities due to convergence of equity betas of individual stocks. Passive
investing is believed to have benefited and possibly contributed to this empirical regularity, although
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this relationship is confounded by other factors that include proliferation of electronic and algorithmic
trading, globalization, unconventional monetary policy, and reduced market liquidity arising from
regulatory changes in the developed countries. This type of observation is typically made based on
computing some time-varying correlation coefficient and following its dynamics over time.
Figure 6 presents the 60-day and 120-day rolling correlations between large-cap (S&P 100) and smallcap (S&P 600) returns from January 2000 to February 2017. The figure reveals substantial variability
in these correlation coefficients with occasional sharp upward and downward turns (for example, the
one that occurred in the beginning of this year) that could affect directly the inflow or outflow of
funds and performance of passive and active investment strategies. What is less discussed, however,
is the substantial estimation uncertainty surrounding these estimates. There are two main sources of
this uncertainty. First, it is the length of the sample and the sampling frequency, as short samples and
high frequency data may be preferred to ensure that the recent momentum in the series is followed
nearly in real time. Second, rolling correlations involve overlapping observations and a large estimation
error can be accumulated and amplified and, combined with the short sample size, can persist.6

Figure 6. Rolling correlations of S&P 100 and S&P 600 (small cap) returns

Rolling estimators can also be interpreted as nonparametric estimators (Ang and Kristensen, 2012; Adrian et al., 2015)
with a slow rate of convergence and larger variability.
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It is instructive to investigate further how much of the time variation of the rolling correlation
coefficient can be attributed to estimation and sampling uncertainty. For this purpose, we generate
artificial data that is calibrated to the actual data by matching the sample size, estimated means,
unconditional covariance, and conditional variances of the S&P 100 and S&P 600 returns. The
standardized returns are drawn from a bivariate t-distribution with varying tail thickness (a degree of
freedom parameter that is drawn from a uniform distribution on the interval [4, 13]) but a constant
correlation parameter. Figure 7 plots the 60-day and 120-day rolling correlations for the simulated
data. Interestingly, these correlations exhibit similar time variation as the correlations in figure 6 even
though the true correlation is constant. Part of this variation is due to changes in the tails of the
distribution, arising from the time-specific degrees of freedom parameter of the t-distribution, but
most of the variation is a result of overlapping estimation uncertainty. While this is only suggestive, it
warns against overreliance (without properly taking into account the estimation uncertainty) on rolling
estimation that is routinely used for portfolio allocation, evaluation of asset-pricing models, and factor
investing (for a recent example, see Asness et al., 2017).
One natural way to address the question of whether the documented time-variability of these comovements is a genuine feature or a statistical fiction is to evaluate its statistical significance using a
reliable statistical procedure. This is precisely the point that we want to make in this paper. Since
overlapping estimation error, data noise, tail movements etc. can be confused for time variation,
monitoring only estimated rolling correlations can impact adversely portfolio allocations. While
bootstrap and other resampling techniques are useful tools for constructing confidence bands around
these estimates, accounting for all sources of uncertainty (model uncertainty, estimation uncertainty,
sampling uncertainty, distributional uncertainty etc.) is a daunting task without any knowledge about
the true data generating process. For the particular example in figure 7, where the data generating
process is known, the confidence bands for the rolling correlation – obtained by repeatedly simulating
data – are wide enough to cover most of the area in this figure.

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Figure 7. Rolling correlations for simulated returns data

DeMiguel et al. (2009) have demonstrated that the estimation imprecision may adversely affect asset
allocation decisions. We mimic their argument by computing out-of-sample Sharpe ratios from an
optimal mean-variance portfolio with estimated weights7 and the naïve portfolio with equal (1/N,
where N is the number of assets) weights. The asset returns, which are also used later in this paper,
are for four major U.S. asset classes—– S&P 500 index (SPX) returns, Bloomberg Barclays Treasury
total returns, Goldman Sachs commodity index (GSCI) returns, and USD index (DXY) returns—as
well as five international equity indices (converted in USD)—S&P 500 (SPX), FTSE 100 (UKX),
Nikkei 225 (NKY), DAX, and MSCI emergent markets (MXEF) index. The data are daily from
January 2000 to February 2017. We use a rolling window sample of 500 daily observations for
estimation of the weights for the mean-variance problem. These estimated weights, as well as the fixed

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The recent literature has suggested various ways to sharpen the estimation of the portfolio weights under the meanvariance risk measure. Another way is to replace the expectation bounded risk (such as the mean-variance risk measure) with
a more robust, which takes explicitly into account the tails of the distribution, coherent risk measure such as the conditional
value-at-risk (Assa and Gospodinov, 2014). Coherent risk measures are tightly linked to the Choquet expected utility,
which can distort the probability of different events by assigning, for example, larger weights to less favorable events and
smaller weights to more favorable ones (see Bassett et al., 2004).

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1/N weights, are then interacted with the next-day return and these out-of-sample returns are used to
compute the corresponding Sharpe ratio. The Sharpe ratios for the U.S. across-asset portfolio are
0.205 for the equal-weight and 0.160 for the mean-variance portfolio, with the difference being
statistically significant at the 5 percent significance level (p-value of 0.028). For the international
portfolio, the Sharpe ratios are 0.274 for the U.S. across-asset portfolio and 0.261 for the international
portfolio, with the difference between the Sharpe ratios being statistically insignificant. This is in line
with the results in DeMiguel et al. (2009) showing the effects of estimation error on optimal portfolio
allocation.8
Finally, part of the observed time variability in the second-moment statistics (covariance, correlation)
can be attributed to movements in the higher moments of the distribution if the latter are ignored and
not adequately modeled. It has been documented (Ang and Chen, 2002; Longin and Solnik, 2001;
Karolyi and Stulz, 1996; among others) that the correlations increase, often dramatically, during
extreme downward market movements. These increased correlations diminish the benefits of
portfolio diversification and hedging, especially in situations when they are needed the most.9 This
changing correlation structure calls for a more holistic approach to studying dependence across assets
that incorporates information in the whole distribution. While the description of this “maximum
entropy” approach is beyond the scope of this paper, it is worth emphasizing that many “anomalies”
and “puzzles”, documented in the empirical finance literature, are largely diminished or even
eliminated when general loss/risk functions and robust dependence measures are allowed for.

Risk Factor Extraction
It is widely believed that financial asset returns contain a small low-frequency, persistent component;
see, for example, Bansal and Yaron (2004) for the role of long-run risks in explaining the equity
premium puzzle. Despite its theoretical appeal, the empirical evidence on the existence of such longrun components is rather weak, as the observed asset returns do not appear to exhibit any persistence.
The potential explanation for this tension between theory and empirics is that the underlying slowIt may be useful to remind the readers that any asset return can be decomposed (by identity) as the product of its sign
and its absolute value (Anatolyev and Gospodinov, 2010, 2017). If the commonality is primarily in the sign (directional)
changes across assets but not in the volatility, then this could be another reason why the fixed-weight portfolio may
dominate the mean-variance optimization.
9
For some drawbacks and biases in evaluating and testing the change in the correlation during market turmoil, which is
accompanied with elevated asset volatility, see Boyer et al. (1999), Campbell et al. (2008), and Forbes and Rigobon (2002).
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moving component is overwhelmed by higher frequency noise and volatility, and quantifying its
impact and empirical detection prove challenging using only time series data on intrinsically more
volatile asset returns. However, provided that this trend component is common across assets, one
could use a cross-section of asset returns and estimate their common variation by the method of
principal components. Thus, cross-sectional information can be useful to extract more precise signals
about common variation and factors.
Unlike the correlation and dependence measures discussed in Section 3 that apply largely to pair-wise
relationships and can exhibit some idiosyncrasies, principal components and factor analysis are a
convenient tool to summarize the co-movements in a large cross-section of asset returns. Another
advantage of this approach is that the precision of the estimation increases with the dimension of the
asset returns included in the analysis. Below, we attempt to isolate common low-frequency movements
in asset returns and relate them to some macroeconomic and demographic factors.

Business Cycle Co-movements
Here, we follow Bai and Ng (2004) by estimating the common factor from returns and then integrate
the process to obtain the common stochastic trend component.10 This method guards against the
possibility of spurious trends and is particularly effective when the number of assets is large. An
alternative approach is to model the common variation in asset prices through cointegrating vectors.
Given the differences in the persistence of the individual series in levels and the lack of robustness of
the co-integration approach to deviations from the exact unit root for each process (Elliott, 1998), we
have adopted the principal component estimation of Bai and Ng (2004).
We first analyze the factor structure in the four U.S. asset classes considered before: S&P 500 index
(SPX), Bloomberg Barclays Treasury total return index, Goldman Sachs commodity index (GSCI),
and USD index (DXY). All series are daily returns for the period January 2000–February 2017. The
common factor is estimated from these returns by the method of principal components, and figure 8
plots the integrated and linearly detrended process.

10

The integrated process is linearly detrended and demeaned.

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Figure 8. Left: Common variation across asset classes with shaded areas representing NBER-dated
recessions. Right: R2 from projecting individual asset returns on the common factor
The estimated factor exhibits sharp decreases during recessions, with the drop during the 2007–09
recession being particularly pronounced. The factor also appears to exhibit some higher-frequency
cycles, but they are more difficult to identify. The right panel in figure 10 reports the R2 from projecting
the individual asset returns on the common factor. The largest loading on the factor is for the S&P
500 index, although bonds and commodity prices also contribute significantly to the common factor
variation.
It would be interesting to see if this common variation is specific to the United States or if it reflects
some global factor as well. For this purpose, we compute the common factor from international stock
returns on S&P 500, FTSE in UK, Nikkei in Japan (NKY), DAX in Germany, and MSCI emerging
markets index (MXEF) in both USD and local currency. Figure 9 plots the common factors.
Interestingly, these returns share very similar dynamics with the U.S. common factor. Moreover, the
largest contributors to these dynamics are FTSE, DAX, and MXEF and not S&P 500. Nikkei appears
to have a large idiosyncratic component. Consistent with figure 10, where DXY exhibits little
correlation with the common factor, the common variation in USD and local currency is fairly similar.
The results in figure 10 suggest a substantial integration of the international equity markets in the post2000 period that is accompanied with a decline in the benefits from global diversification (see also

17

Cotter et al., 2016).11

Figure 9. Left: Common variation in international stock returns with shaded areas representing
NBER-dated recessions. Right: R2 from projecting individual stock returns on the common factor
Despite the fairly short time span of the data, it would be instructive to relate more convincingly this
common variation with some underlying macroeconomic signal or risk factor. To provide some
suggestive evidence, figure 10 presents the Atlanta Fed/New York Fed turning point indicator of the
labor market, along with the Chicago Fed national activity index (CFNAI). The labor market indicator
is computed from vintage data and is available “almost” in real time. It is particularly useful for
monitoring since it is a one-sided filter with no distortionary effects on the cyclical dynamics. The
CFNAI series is relatively noisy and is constructed with data that are released with a delay. While the
smoothing in the Atlanta Fed/New York Fed indicator induces a phase shift in the series, it provides
a much improved estimate of the turning points and the local trends in the underlying signal.

Pukthuanthong and Roll (2009) provide an insightful analysis on the flaws of correlation as a measure of global market
integration. Instead, they recommend the use of global factor exposure for gauging financial market integration.
11

18

Figure 10. Chicago Fed national activity index (CFNAI) and Atlanta Fed and New York Fed
(AtlFed/NYFed) labor market indicator. Shaded areas are NBER-dated recessions
There are a couple of observations that warrant some remarks. First, the smoothed labor market
indicator identifies the turning points of the business cycle in advance, dating the recessions and the
expansions, which would be of great value to policymakers. The noisy monthly reports in the datadependent policy or the volatile asset dynamics in asset allocation can be supplemented and validated
with this lower-frequency, local-trend information. Moreover, it is striking how closely the smoothed
labor indicator underlies the dynamics of the common variation in asset returns, suggesting the
presence of a strong business cycle component in both domestic and global financial asset prices.
Pinning down these business cycle components in asset prices has some important implications for
medium-term investment and policy decisions. For example, this information can be used in preparing
supervisory scenarios for annual stress tests by the Federal Reserve. More specifically, a factoraugmented vector autoregressive model, with an asset-pricing factor estimated as above, can be used
to generate conditional forecast paths (see Waggoner and Zha, 1999), under various scenarios, for the
variables of interest. Incorporating this asset-pricing factor, using both domestic and international
data, should further improve the fit of the model and the accuracy of the conditional forecasts.
19

Long Cycles: Low Frequency Information in Demographic Variables
In this section, we investigate if there is any other common variation in asset prices—and stock prices
in particular—at even lower frequency than that of the business cycle. Recent literature has established
the usefulness of low-frequency demographic variables for long-horizon stock market returns. This is
because savings rates, and possibly risk preferences, vary substantially over the life cycle, with savings
rates peaking in middle age and then being drawn down in old age. These savings directly impact the
pool of funds available for investment in the stock market. In fact, they may explain and predict some
of the very persistent, low-frequency movements in stock market valuation ratios, such as the dividend
or earnings price ratios.
To try to address this issue, we use annual data for the period 1946–2015 for the S&P 500 returns,
returns on long-term and intermediate-term U.S. government bonds, changes in S&P 500 pricedividend ratio, changes in S&P 500 dividend yield, and changes in S&P 500 earnings-price ratio.12 The
data for the government bond returns are from Ibbotson Associates and the stock price data are from
Robert Shiller’s and Amit Goyal’s websites.
We also collect annual demographic data from the Census Bureau and construct the variable middleyoung (MY) ratio as the ratio of middle-aged (40–49) and young (20–29) cohorts. In addition to the
historical data, the Census Bureau provides projections until 2060. Figure 11 plots the smoothed
common factor estimated from the asset returns, as described in the previous section, along with the
middle-young ratio.

12

More specifically, dividends and earnings are 12-month moving sums. The dividend-price and earnings-price ratios are
constructed as the difference between the log of dividends (earnings) and log of prices. The dividend yield is the difference
between the log of dividends and log of lagged prices. Due to the near-unit root behavior of these series, we work with
their first differences.

20

Figure 11. Common factor in asset returns and middle-young ratio

During the 1946–2015 period, the long-cycle factor in asset returns seems to co-move closely with the
demographic trend, with a common trough in the early 1980s and a common peak in the early 2000s.
This is consistent with other results in the literature on the relationship of the demographic variables
with stock valuation ratios (Favero et al., 2011) and interest rates (Favero et al., 2016).
A unique feature of the demographic variables is that their low frequency trends can be projected with
reasonable accuracy decades into the future. The projections of the MY ratio by the Census Bureau
until 2040 (represented on the graph with a dashed line) thus facilitate the forecasting of both the
market valuations and the small low-frequency component of market returns. The MY ratio is
projected to fall until 2020, which implies a downward pressure on the stock valuation ratios and
interest rates due to demographic factors. After 2020, the MY ratio starts to increase again until 2040,
when it reaches a new peak. It is expected that during this period, the downward pressure on valuation
ratios and interest rates from demographics is diminished and even reversed.

21

It is instructive to discuss further the implications of the demographic-trend projections on longhorizon forecasting for stock returns and valuation ratios. The literature has explored many predictors
for asset returns, with varying and often debated success (for a comprehensive review, see Goyal and
Welch, 2008). However, the future trajectory of most predictors is itself highly uncertain, making them
less useful for longer-term forecasts. Also, the innovations from the predictive regression for stock
returns and the dynamic model for the predictor are often strongly correlated. For example, when the
lagged dividend-price ratio is used as a predictor, this correlation is -0.95 implying that a fall (rise) in
the dividend-price ratio is associated with positive (negative) returns. But this strong contemporaneous
relationship between stock returns and dividend-price ratio cannot be exploited for prediction without
knowledge of the future values of dividend-price ratio.
Demographic variables allow us to address both of these shortcomings. With the Census Bureau
projections for the MY ratio, we can construct recursive forecasts for dividend-price ratio that could
be included in the predictive regression for future stock returns. Some tentative findings from this
exercise can be summarized as follows. The model suggests that near-term returns are expected to be
low by historical standards until 2020 due to the projected fall of the MY ratio. After 2020, the MY
ratio rises again, with more middle-aged savers putting upward pressure on stock prices. Thus, after
2020, the long-term stock returns are projected to increase again but settle at a level that is lower than
the historical average over the last 70 years.
While these projections reflect pure demographic information13 and are only suggestive about the
future long-term path of U.S. stock returns, they conform to a broader set of arguments put forward
in the literature. A consensus is now emerging that the changing demographic structure in the
developed economies has contributed to the recent decline in the equilibrium real interest rates
(Gagnon et al., 2016). Historically, periods of low real interest rates14 are associated with lower asset
returns in the next five years (Dimson et al., 2013). These lower expected returns then pose a direct
risk to institutional investors with long-term commitments.

We acknowledge that the middle-young ratio could be just a convenient proxy for other slowly moving socioeconomic
and political factors such as safety-net development and political polarization. Also, the demographic projections are based
on assumptions about future immigration dynamics, which are influenced by policy decisions.
13

It should be noted that most of the previous episodes of low real interest rates were due to above-average inflation
instead of low nominal interest rates, as in the years after the 2007–09 recession.
14

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Concluding Remarks
This paper discussed and summarized some issues arising in the analysis of asset co-movements for
the purposes of asset allocation and portfolio diversification, as well as macroeconomic and macroprudential policy. One of the main findings that emerges from this review is that a judiciously
performed factor analysis appears to identify a common variation across domestic and international
asset classes at business and longer cycles. While the reported evidence is only tentative, a more
comprehensive empirical analysis with a larger cross-section of U.S. and international asset returns
would shed further light on the important sources of risks across asset classes, the integration of the
global markets, and their implications for diversification, re-allocation and policy analysis. Attaching a
risk factor interpretation to short- and medium-term co-movements appears to be more difficult due
to some statistical challenges in analyzing the data. Explicitly acknowledging the estimation and model
uncertainty as well as shifting the focus to more general measures of dependence would robustify the
decision-making process and reduce the risk of false positives/negatives in signal extraction and
performance evaluation.
One interesting topic that was omitted from the discussion is the increased importance of factor
(“smart beta”) investing. Studying more formally the dependence structure of these investment factors
and evaluating their performance using rigorous statistical criteria is an area of ongoing research. While
most of this research has focused on equities, constructing factors across divergent asset classes
enhances the ability of capturing multiple sources of systematic risk that are difficult to identify and
estimate statistically. Thus, a sufficient heterogeneity of spread factors across asset classes could
potentially span the underlying factor space (Roll, 2013) and contribute beneficially to risk
diversification and financial system stability.

23

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