Portfolio returns and consumption growth covariation in the frequency domain, real economic activity, and expected returns

• Published date: 01 September 2022
• Author: Louis R. Piccotti
• Date: 01 September 2022
• DOI: http://doi.org/10.1111/jfir.12282

Received: 12 August 2020

|

Accepted: 15 February 2022

DOI: 10.1111/jfir.12282

ORIGINAL ARTICLE

Portfolio returns and consumption growth

covariation in the frequency domain, real

economic activity, and expected returns

Louis R. Piccotti

Department of Finance, Spears School of

Business, Oklahoma State University,

Stillwater, Oklahoma, USA

Correspondence

Louis R. Piccotti, Department of Finance,

Spears School of Business, Oklahoma State

University, Stillwater, OK 74078, USA.

Email: louis.r.piccotti@okstate.edu

Abstract

The slope of the portfolio return and consumption growth

cospectrum contains predictive information about future

real economic activity, future recession probabilities, the

risk aversion coefficient, and future expected returns.

Commonly used economic variables do not subsume the

predictive power of the cospectrum slope and although the

interest rate term spread largely fails to predict the financial

crisis, the set of cospectrum slopes predicts the crisis

with a 75% probability. The cospectrum slope significantly

improves the fit of long‐horizon expected return models

and contains more significant predictive information than

the current dividend yield.

JEL CLASSIFICATION

E44, G01

1|INTRODUCTION

In the frequency domain, the expected return on a portfolio can be viewed as the expected return on a portfolio of

basis assets,

1

which are exposed to consumption risks of various arrival frequencies (Dew‐Becker & Giglio, 2016).

This frequency domain decomposition of the pricing kernel is expressed as a curve, analogous to the term structure

of interest rates. Neuhierl and Varneskov (2021) show that low‐frequency factor loadings using the Fourier

J Financ Res. 2022;45:513–549. wileyonlinelibrary.com/journal/JFIR

|

513

© 2022 The Southern Finance Association and the Southwestern Finance Association.

1

These basis assets can be viewed heuristically, similar to Arrow–Debreu securities, except that rather than having a payoff of 1 unit in a certain state and

0 in all others, these basis assets have some payoff if a shock arrives with a certain frequency and a 0 payoff for all other shocks arriving at different

frequencies.

transform fit expected portfolio returns better than market betas do. Bandi et al. (2021) form spectral (frequency

exposure) betas using the extended Wold decomposition methodology of Ortu et al. (2020) and show how different

factor portfolios load differently across frequencies and provide evidence that a spectral factor model prices

anomaly portfolios reasonably well.

2

I contribute to the literature on asset pricing in the frequency domain by

examining the information content that the cospectrum (the frequency domain decomposition of the unconditional

covariance) between portfolio returns and consumption growth has for future real economic activity.

The advantage of looking at the frequency domain, rather than the time domain, is that cyclic behavior is more

visible in the frequency domain. As a result, the slope of the cospectrum between portfolio returns and

consumption growth shows how susceptible portfolio returns are to low‐frequency consumption shocks, relative to

high‐frequency consumption shocks. Such a relation would be more difficult to identify in the time domain (e.g.,

with a vector autoregression [VAR] model). This information is valuable, as it is an indicator of investors' abilities to

weather short‐term shocks, such as recessionary periods. For example, van Binsbergen et al. (2013) show that high‐

frequency risks are priced in the market more heavily in recessionary periods. I extend the evidence across this

dimension by additionally showing that the slopes of the cospectra between small‐minus‐big (SMB), high‐minus‐low

(HML), and dividend‐sorted (DIV) portfolio returns and consumption growth are also leading indicators of

recessionary periods, where the SMB and HML cospectra slopes contain the most valuable information for

predicting future recessions.

First, I examine the frequency decomposition in the cross‐section of 25 size and book‐to‐market (BM)

portfolios and find that the cospectra for consumption growth with the market portfolio and the SMB portfolio are

downward sloping, with a greater portion of the risk premium due to low‐frequency risks than to high‐frequency

risks. The higher price for low‐frequency risks is consistent with the long‐run risk models of Bansal and Yaron

(2004) and Bansal et al. (2009). This pattern also indicates that investors in small stocks have a relative implied

preference to smooth their future (higher) consumption versus their present consumption (they display Epstein &

Zin, 1989; preferences). Conversely, the HML and DIV cospectra are upward sloping from low‐frequency risks to

high‐frequency risks. It appears that near‐term risks associated with dividend obligations and the default boundary

are more important to the investors in these portfolios, which suggests that the marginal investors in these stocks

are relatively more concerned with smoothing their near‐term consumption versus their long‐run consumption

(they display internal habit preferences; Abel, 1990; Constantinides, 1990). Although these results suggest a sort of

equities market segmentation, evidence of segmentation in preferences in equities is provided by Dorn and

Huberman (2010), Lease et al. (1976), Menzly and Ozbas (2010), and Wood and Zaichkowsky (2004), among others.

Second, I examine how the slope of the cospectra of portfolio returns and consumption growth changes over

time and how it is related to real economic activity in the United States. This test reveals how investor preferences

for consumption change over time and how these preference shifts are related to future economic growth. The

cospectra using the excess market (EMRKT) and SMB portfolios are generally downward sloping from low‐to high‐

frequency cycles, which suggests Epstein and Zin (1989) preferences (Dew‐Becker & Giglio, 2016); however,

leading up to recessionary periods, the cospectra become more upward sloping from low‐to high‐frequency cycles.

This indicates that leading up to recessionary periods, investors become increasingly concerned with near‐term

consumption needs as would be the case with internal habit preferences (Constantinides, 1990; Dew‐Becker &

Giglio, 2016). Conversely, the portfolio return and consumption growth cospectrum for the HML and DIV portfolios

is evenly split between upward and downward sloping, each approximately equal over time. However, the

cospectra for these portfolios also display sharp shifts toward being upward sloping from low‐to high‐frequency

cycles leading up to recessionary periods. These results are consistent with those found in van Binsbergen et al.

(2013), which show that high‐frequency risks are priced in the market portfolio more heavily surrounding

recessionary periods by examining the time series of dividend strips with varying maturities.

2

In particular, the small Fama–French portfolios load more heavily on risks with cycle periods of 4–8 months, and the high book‐to‐market Fama–French

portfolios load more heavily on risks with cycle periods of 32–64 months.

514

|

JOURNAL OF FINANCIAL RESEARCH

Third, because investors' implied preferences toward smoothing consumption change over time, which is

revealed by the time series of the slope of the portfolio return and consumption growth cospectrum, I examine the

predictive relation between the current cospectrum slope and future real economic growth and recession

probabilities. This complements the literature, which has focused on examining how risks of various frequencies

affect the stochastic discount factor. I find that the slope of the cospectrum is a significant predictor after

controlling for the growth rates of a commonly used set of economic variables. In many cases, the cospectrum slope

is a more significant predictor than the interest rate term spread, which has been shown to have substantial

predictive ability for growth rates in the real economy as well as for recession probabilities (Estrella & Hardouvelis,

1991; Harvey, 1988; Laurent, 1988). Whereas the term spread provides information about future consumption

demands (primarily through the inflation premium), the shape of the portfolio return and consumption growth

cospectrum provides information about investors' timing preferences for consumption. That is, the cospectrum

slope indicates whether investors need to shift future consumption to the present (e.g., to meet liquidity needs) or

whether their current consumption is durable and they have the capacity to shift consumption from the present to

the future to attain a higher level of consumption from investing.

The portfolio return and consumption growth cospectrum slope alone is a good predictor (using a probit model)

of 12‐month forward recession probabilities (a pseudo R

2

of 0.41), and the information contained in the cospectrum

slope is not simply a manifestation of the information in the interest rate term spread. The most compelling case for

this is that the term spread generates a 30% recession probability for a financial crisis, whereas the set of

cospectrum slopes generates a 75% recession probability. When the recession probability probit model includes

both the interest rate term spread and the set of cospectrum slopes, the model estimates 12‐month forward

recession probabilities very well. The full probit recession model probability model attains a pseudo R

2

of 0.658

versus a pseudo R

2

of 0.523 attained from the univariate interest rate term spread model and predicts the financial

crisis with a recession probability in excess of 80%.

Finally, I examine the relation between the portfolio return and consumption growth cospectrum slope and

expected returns. Long‐horizon expected return regressions show that the slope significantly predicts future excess

market returns, with the slope more significantly related to future expected returns than the current dividend yield

is (the second most significant predictor). Therefore, the cospectrum slope contains valuable information for

investors as well, which is further evidenced by the utility gains attained by using a trading strategy, which

conditions on the slope of the cospectrum.

2|LITERATURE REVIEW

This article fits into the literature on the term structure of the risk premium. van Binsbergen et al. (2012) and van

Binsbergen et al. (2013) examine the equity risk premium by looking at the pricing of dividend strips with various

maturities and find that the term structure is generally upward sloping in maturity. They attribute the shape and

time‐varying behavior of the equity term structure to dividend growth rates and risk premia. van Binsbergen and

Koijen (2016) further attempt to reconcile the shape of the term structure of equity with common asset pricing

models. They conclude that the pattern in dividend strip prices is not consistent with the Lucas (1978) consumption

capital asset pricing model (CAPM), external habit preferences (Abel, 1990; Campbell & Cochrane, 1999), long‐run

consumption risks (Bansal & Yaron, 2004), or by variable rare disasters (Gabaix, 2012). Otrok et al. (2002) cast doubt

on the ability of internal habit preference models (Constantinides, 1990) to explain the risk premium. In contrast to

these studies, Lettau and Wachter (2007,2011) show that the risk premium should be downward sloping with long‐

horizon equity (growth stocks) having lower expected returns than short‐horizon equity (value stocks). Croce et al.

(2015) further show that the term structure of equity can be downward sloping with bounded rationality, but is

upward sloping with full information. The differing shapes of the term structure of the risk premium can be

reconciled, however, when firms have dividend policies that lead to stationary leverage (Belo et al., 2015).

PORTFOLIO RETURNS AND CONSUMPTION GROWTH COVARIATION

|

515