Nonlinearity and Flight‐to‐Safety in the Risk‐Return Trade‐Off for Stocks and Bonds

• Published date: 01 August 2019
• DOI: http://doi.org/10.1111/jofi.12776
• Date: 01 August 2019
• Author: RICHARD K. CRUMP,TOBIAS ADRIAN,ERIK VOGT

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 4 •AUGUST 2019

Nonlinearity and Flight-to-Safety in the

Risk-Return Trade-Off for Stocks and Bonds

TOBIAS ADRIAN, RICHARD K. CRUMP, and ERIK VOGT∗

ABSTRACT

We document a highly significant, strongly nonlinear dependence of stock and bond

returns on past equity market volatility as measured by the VIX. We propose a

new estimator for the shape of the nonlinear forecasting relationship that exploits

variation in the cross-section of returns. The nonlinearities are mirror images for

stocks and bonds, revealing flight-to-safety: expected returns increase for stocks when

volatility increases from moderate to high levels while they decline for Treasuries.

These findings provide support for dynamic asset pricing theories in which the price

of risk is a nonlinear function of market volatility.

INVESTOR FLIGHT-TO-SAFETY IS pervasive in times of elevated risk (Longstaff

(2004), Beber, Brandt, and Kavajecz (2009), Baele et al. (2013)). Economic

theories of investor flight-to-safety predict highly nonlinear asset pricing rela-

tionships (Vayanos (2004), Weill (2007), Caballero and Krishnamurthy (2008),

Brunnermeier and Pedersen (2009)). Such nonlinear pricing relationships are

difficult to document empirically as the particular shape of the nonlinearity

is model-specific, and inference of nonlinear relationships presents economet-

ric challenges.

In this paper, we document an economically and statistically strong nonlin-

ear risk-return trade-off by estimating the relationship between stock market

volatility as measured by the VIX and future returns. The nonlinear risk-return

trade-off features evidence of flight-to-safety from stocks to bonds in times of

elevated stock market volatility consistent with the above cited theories. The

VIX strongly forecasts stock and bond returns up to 24 months into the future

∗Tobias Adrian is with the International Monetary Fund. Richard Crump is with the Federal

Reserve Bank of New York. Erik Vogt is with Citadel. The views expressed in this paper are those

of the authors and do not necessarily represent those of the International Monetary Fund, the Fed-

eral Reserve Bank of New York, the Federal Reserve System, or Citadel. The authors thank Ken

Singleton (the Editor), three anonymous referees, Michael Bauer, Tim Bollerslev, Andrea Buffa,

John Campbell, Itamar Drechsler, Rob Engle, Eric Ghysels, Arvind Krishnamurthy, Ivan Shalias-

tovich, Allan Timmermann, Peter VanTassel, Ken West, Jonathan Wright, as well as seminar and

conference participants at Boston University, the Federal Reserve Bank of New York, the Federal

Reserve Bank of San Francisco, the NYU Stern VolatilityInstitute, the NBER 2015 Summer Insti-

tute, and the 2016 AFA Annual Meeting for helpful comments and suggestions. Daniel Stackman,

Rui Yu, and Oliver Kim provided outstanding research assistance. The authors declare that they

have no relevant or material financial interests related to the research in this paper.

DOI: 10.1111/jofi.12776

1931

1932 The Journal of Finance R

Figure 1. Nonlinear expected returns. This figure shows the relationship between the six-

month cumulative equity market return (CRSP value-weighted U.S. equity market portfolio) and

the six-month lag of the VIX in red, as well as the relationship between the six-month cumulative

one-year Treasury return (CRSP one-year constant maturity Treasuryportfolio) and the six-month

lag of the VIX in blue. Both nonlinear relationships are estimated using sieve reduced-rank re-

gressions on a cross-section of stocks and bonds. The y-axisisexpressedasaratioofreturns

to the full-sample standard deviation. The x-axis shows the VIX. (Color figure can be viewed at

wileyonlinelibrary.com)

when the nonlinearity is accounted for, in sharp contrast to the insignificant

linear relationship.

The nature of the nonlinearity in the risk-return trade-offs for stocks and

bonds are virtually mirror images, as can be seen in Figure 1, estimated from

a cross-section of stocks and bonds. Both stock and bond returns have been

normalized by their unconditional standard deviation to plot them on the same

scale. Three notable regions characterize the nature of the nonlinear risk-

return trade-off. When the VIX is below its median of 18, both stocks and

bonds exhibit a risk-return trade-off that is relatively insensitive to changes

in the VIX. When the VIX is between 18 and its 99th percentile of 50, the

nonlinearity is highly pronounced: as the VIX increases above its unconditional

median, expected Treasury returns tend to fall while expected stock returns

rise. This finding is consistent with a flight-to-safety from stocks to bonds,

which increases expected returns to stocks and compresses expected returns to

bonds. For levels of the VIX above 50, which has only occurred in the aftermath

of the Lehman failure, this pattern reverses, with a further increase in the VIX

associated with lower stock and higher bond returns. The latter finding for

very high values of the VIX likely reflects the fact that severe financial crises

are followed by abysmal stock returns and aggressive interest rate cuts, due

to a collapse in real activity, and thus changes in cash flow expectations (see

Campbell, Giglio, and Polk (2013)).

What is most notable is that a linear regression using the VIX does not fore-

cast stock or bond returns significantly at any horizon. Nonlinear regressions,

in contrast, forecast stock and bond returns with very high statistical signif-

icance and reveal the striking mirror image property of Figure 1. We study

Nonlinearity and Flight-to-Safety in the Risk-Return Trade-Off 1933

the nature of the nonlinearity and the mirror image property using kernel re-

gressions, polynomial regressions, as well as nonparametric sieve regressions.

In all cases, and on subsamples considered, we find pronounced nonlinearities

within risky assets and reversed nonlinearities for safe assets in terms of both

statistical and economic magnitudes.

To estimate the shape of the nonlinearity in a robust way, we nonparamet-

rically estimate the shape using a sieve reduced-rank (SRR) regression on a

cross-section of stock and bond returns. We specify a nonlinear forecasting

function φh(v) such that

Rxi

t+h=ai

h+bi

h·φh(vixt)+εi

t+h,i=1,...,n,(1)

where hdenotes the forecasting horizon, idenotes the individual stock and bond

portfolios, and Rx denotes excess returns. The nonlinearity of the function

φh(v) is highly significant, and its forecasting power is strong. Importantly,

when we estimate φh(v) separately for stocks and bonds, we obtain statistically

indistinguishable functions (up to an affine transformation).

A major advantage of estimating φh(v) from a panel of stock and bond re-

turns is that it exploits additional cross-sectional variation unavailable in the

univariate regressions that are typical in the return forecasting literature. The

algebra for the estimator can be described intuitively in two stages. In the

first stage, returns to each asset are regressed in the time series on lagged

sieve expansions of the VIX. In the second stage, the rank of the matrix of

forecasting coefficients is reduced using an eigenvalue decomposition, and only

a rank-one approximation is retained (see Adrian, Crump, and Moench (2015)

for a related derivation). This dimensionality reduction is optimal when errors

are conditionally Gaussian and the number of regressors is fixed. The result-

ing factor φh(v) is a nonlinear function of volatility and is the best common

predictor for the whole cross-section of stock and bond returns.

In addition to the new estimator, we introduce asymptotically valid inference

procedures for four hypotheses of interest, which may be implemented using

standard critical values. The first is a joint test of significance for whether the

whole cross-section of test assets loads on φh. The second tests the null that φh

does not predict excess returns for a specific asset, while allowing it to predict

excess returns for other assets. The third test is a comparison of whether the

function φh(¯v) is different from zero at any fixed value ¯v, generating pointwise

confidence intervals for the unknown function. The fourth is a specification test

for the rank-one restriction that the same nonlinear function of volatility φh

drives expected stock and bond returns, which we use to test for flight-to-safety.

To conduct inference when estimating multihorizon returns with overlapping

data, we extend the “reverse regression” approach of Hodrick (1992)toour

reduced-rank, nonparametric setting.

Our finding that the VIX forecasts stock and bond returns in a nonlinear

fashion is robust to the inclusion of standard predictor variables such as the div-

idend yield, the BAA/10-year Treasury default spread, the 10-year/three-month

Treasury term spread, and the variance risk premium (VRP). Furthermore,