Understanding Systematic Risk: A High‐Frequency Approach

• DOI: http://doi.org/10.1111/jofi.12898
• Author: MARKUS PELGER
• Published date: 01 August 2020
• Date: 01 August 2020

THE JOURNAL OF FINANCE •VOL. LXXV, NO. 4 •AUGUST 2020

Understanding Systematic Risk: A

High-Frequency Approach

MARKUS PELGER∗

ABSTRACT

Based on a novel high-frequency data set for a large number of firms, I estimate

the time-varying latent continuous and jump factors that explain individual stock

returns. The factors are estimated using principal component analysis applied to a

local volatility and jump covariance matrix. I find four stable continuous systematic

factors, which can be well approximated by a market, oil, finance, and electricity

portfolio, while there is only one stable jump market factor. The exposure of stocks to

these risk factors and their explained variation is time-varying. The four continuous

factors carry an intraday risk premium that reverses overnight.

ONE OF THE MOST POPULAR methods for modeling systematic risk is estimation

of factor models—finding the “right” systematic factors has become the central

question of asset pricing. I contribute to our understanding about systematic

risk by shedding light on the following three questions: (i) what are the factors

that explain the systematic comovement in individual stocks, (ii) how does the

systematic factor structure for stocks change over time, and (iii) what are the

asset pricing implications of the systematic factors?

Todo so, the approach that I follow has three key elements: First, rather than

using a prespecified (and potentially misspecified) set of factors, I estimate the

statistical factors, which can explain most of the common comovement in a

large cross section of stock returns. Second, I use high-frequency data which

allow me to study the time-variation in the factor structure under minimal

∗Markus Pelger is at the Department of Management Science & Engineering, Stanford Uni-

versity. I thank Jason Zhu for excellent research assistance. I thank Yacine A¨

ıt-Sahalia; Torben

Andersen; Robert M. Anderson; Svetlana Bryzgalova; Mikhail Chernov; John Cochrane; Frank

Diebold; Darrell Duffie; Noureddine El Karoui; Steve Evans; Jianqing Fan; Kay Giesecke; Lisa

Goldberg; Valentin Haddad; Michael Jansson; Martin Lettau; Ulrike Malmendier; Stefan Nagel

(Editor); Olivier Scaillet; Ken Singleton; George Tauchen; ViktorTodorov; Neil Shephard; Dacheng

Xiu; two anonymous referees; and audience participants at UC Berkeley, Stanford, University of

Pennsylvania, University of Bonn and SoFiE, INFORMS, FERM, Econometric society, and NBER

Time-Series meetings. This work was supported by the Center for Risk Management Research at

UC Berkeley. I have read The Journal of Finance disclosure policy and have no conflict of interest

to disclose.

Correspondence: Markus Pelger, Department of Management Science and Engineering, Stan-

ford University, 312 Huang Engineering Center, 475 Via Ortega, Stanford, CA 94305; e-mail:

mpelger@stanford.edu.

DOI: 10.1111/jofi.12898

C2020 the American Finance Association

2179

2180 The Journal of Finance R

assumptions as I can analyze very short time horizons independently.Allowing

for time-variation in the factor structure is crucial as individual stocks do not

have constant risk exposure in contrast to some characteristic-sorted portfolios

(see Lettau and Pelger (2020)). And third, I separate high-frequency returns

into continuous intraday returns and intraday and overnight jumps which

allows me to decompose the systematic risk structure into its smooth and

rough components, which have different asset pricing implications.

The statistical theory underlying my estimations, developed by Pelger (2019),

combines high-frequency econometrics and large-dimensional factor analysis

and is very general. Specifically,under the assumption of an approximate factor

model, it estimates an unknown factor structure for general continuous-time

processes based on high-frequency data. Using a truncation approach, I can

separate the continuous and jump components of the price processes, which I

use to construct a “jump covariance” matrix and a “continuous risk covariance”

matrix. The latent continuous and jump factors can be separately estimated by

principal component analysis (PCA).

My empirical investigation is based on a novel high-frequency data set of

five-minute returns of the stocks in the NYSE Trade and Quote Milliseconds

(TAQ millisecond) database for the period 2004 to 2016 . My main findings

are as follows. First, I find four high-frequency factors, with time-variation in

the exposure of stocks to these risk factors and the amount of variation that

they explain (e.g., one factor is only systematic during the financial crisis).

Surprisingly, the portfolio weights used to construct the statistical factors are

stable over time. Second, these four factors have an economically meaningful

interpretation and can be closely approximated by market, oil, finance, and

electricity factors, whereas the size, value, and momentum factors of the Fama-

French-Carhart model cannot span the statistical factors. Third, the factor

structure for smooth continuous movements is different from that for rough

jump movements. In particular, there seems to be only one intraday jump

market factor, while the continuous factors have the same composition as the

high-frequency factors including the jumps. Fourth, PCA factors estimated at

lower frequencies (e.g., weekly or monthly returns) are different from high-

frequency PCA factors. The lower frequencies result in a loss in information

and the resulting PCA factors have a less interpretable structure and explain

less of the time-series variation in stock returns than the high-frequency PCA

factors. Fifth, the high-frequency factors carry an intraday risk premium that

reverses overnight. Decomposing returns into their intraday and overnight

components, I document a strong reversal pattern in individual stock returns,

which is captured by the high-frequency factors. Finally, the high-frequency

factors explain the expected return structure of industry portfolios, while the

Fama-French-Carhart factors better explain size- and value-sorted portfolios.

This result suggests that time-varying factors that explain the comovement in

individual stocks are indeed informative about cross-sectional pricing, but are

not necessarily related to characteristics.

This paper addresses the central question in empirical and theoretical asset

pricing of what constitutes systematic risk. There are essentially three common

Understanding Systematic Risk 2181

approaches to identify the factors that describe the systematic risk. Under the

first approach, factor selection is based on theory and economic intuition. The

capital asset pricing model (CAPM) of Sharpe (1964), in which the market is

the only common factor, belongs to this category. Under the second approach,

factors are based on firm characteristics. The three-factor model of Fama and

French (1993) is the most famous example of this approach. Under the third

approach—the one to which this paper belongs—factor selection is statistical.

This approach is motivated by the arbitrage pricing theory (APT) of Ross (1976).

Factor analysis can be used to analyze the covariance structure of returns. This

approach yields estimates of factor exposures as well as returns to underlying

factors, which are linear combinations of returns on underlying assets. The

notion of an “approximate factor model” is introduced by Chamberlain and

Rothschild (1983), which allows for a nondiagonal covariance matrix of the

idiosyncratic component. Connor and Korajczyk (1988,1993) study the use of

PCA in the case of an unknown covariance matrix, that has to be estimated.1

One distinctive feature of the factor studies above is that they employ a

constant factor model that does not allow for time-variation. The objects of

interest are sorted portfolios based on previously established knowledge about

the empirical behavior of average returns. Lettau and Pelger (2020) show that

characteristic-sorted portfolios are well described by a constant factor model.

However,a shortcoming of the characteristic-sorted approach isthat the results

depend on the choice of conditioning variables to generate the portfolios (Nagel

(2013)).

In this paper, I work directly on a large cross section of individual stocks.

However, Lettau and Pelger (2020) show that a constant loading model is not

appropriate to model individual stock returns over longer time horizons. The

dominant approach is to model time-variation in risk exposure through time-

varying characteristics. Kelly, Pruitt, and Su (2017) and Fan, Liao, and Wang

(2016b) follow this approach.2In effect, these studies apply a version of PCA

to characteristic-managed portfolios. Their results therefore depend on the

choice of characteristics and the basis functions used to model the functional

relationship between characteristics and loadings. I propose an alternative

approach that accounts for time-variation and is completely general in that

it does not depend on the choice of characteristics or basis functions. As a

result, I identify the factors that explain most of the variation in individual

stocks without using any (potentially incorrect) prior about the characteristics

that drive the variation. Interestingly, I find that the factors that explain the

variation in individual stocks are not related to popular characteristic-based

factors such as the Fama-French-Carhart factors.3

1The general case of a static large-dimensional factor model is treated in Bai (2003) and Bai and

Ng (2002). Fan, Liao, and Mincheva (2013) study an approximate factor structure with sparsity.

2In a related approach, Fama and French (2020) extract factors by cross-sectional regressions on

prespecified characteristics and model time-varying loadings through time-varying characteristics.

3For a prespecified set of factors, prior studies show that time-varying systematic risk factors

capture the data better. Time-varying systematic risk factors contain the conditional version of

the CAPM as a special case, which appears to explain systematic risk significantly better than its