Swiss Finance Institute

Research Paper Series

N°20-110
Predicting Corporate Bond Returns:
Merton Meets Machine Learning

Turan G. Bali
Georgetown University

Amit Goyal
University of Lausanne and Swiss Finance Institute

Dashan Huang
Singapore Management University

Fuwei Jiang
Central University of Finance and Economics (CUFE)

Quan Wen
Georgetown University
 Predicting Corporate Bond Returns: Merton Meets
Machine Learning*

Turan G. Bali„ Amit Goyal… Dashan Huang§ Fuwei Jiang¶ Quan Wen†

Abstract

We investigate the return predictability of corporate bonds using big data and machine
learning. We find that machine learning models substantially improve the out-of-
sample performance of stock and bond characteristics in predicting future bond returns.
We also find a significant improvement in the performance of machine learning models
when imposing a theoretically motivated economic structure from the Merton model,
compared to the reduced-form approach without restrictions. Overall, our work
highlights the importance of explicitly imposing the dependence between expected
bond and stock returns via machine learning and Merton model when investigating
expected bond returns.

This Version: May 2022

Keywords: Machine learning, big data, corporate bonds, hedge ratio, cross-sectional return
predictability
JEL Classification: G10, G11, C13.

* We thank John Y. Campbell, Allan Eberhart, Tom Knox, Jonathan Kluberg, Alejandro Lopez-Lira
(our discussant), Christopher Malloy, Markus Pelger, Alberto Rossi, Elvira Sojli, and Derek Vance for their
insightful and constructive comments. We also benefited from discussions with seminar participants at the
University of Bath School of Management, Arrowstreet Capital, the Center for Financial Markets and Policy
and Georgetown University Asset Management Conference, the 2020 Bank of America Global Quant and
Innovation Conference, the 2021 Microstructure Exchange seminars, the 2021 BI-SHoF Conference on Asset
Pricing and Financial Econometrics, and the 2022 EQD Barcelona conference.
„ Robert S. Parker Chair Professor of Finance, McDonough School of Business, Georgetown University,
Washington, D.C. 20057. Phone: (202) 687-5388, Fax: (202) 687-4031, Email: Turan.Bali@georgetown.edu
… Professor of Finance, Faculty of Business and Economics, University of Lausanne and Swiss Finance
Institute. Email: amit.goyal@unil.ch
§ Associate Professor of Finance, Lee Kong Chian School of Business, Singapore Management University.
Email: dashanhuang@smu.edu.sg
¶ Professor of Finance, School of Finance, Central University of Finance and Economics. Email:
jfuwei@gmail.com
† Associate Professor of Finance, McDonough School of Business, Georgetown University, Washington,
D.C. 20057. Email: Quan.Wen@georgetown.edu
1 Introduction
A substantial number of stock characteristics have been presented as statistically significant
predictors of the cross-section of stock returns since 1970 (Cochrane, 2011). Since then, a few
studies show that the majority of the predictive power associated with these characteristics are
most likely an artifact of data mining, data snooping, correlated multiple testing, or p-hacking,
especially when examined out-of-sample (Harvey, Liu, and Zhu, 2016; Green, Hand, and Zhang,
2017; Linnainmaa and Roberts, 2018; Hou, Xue, and Zhang, 2020). Despite the out-of-sample and
post-publication decline of a vast majority of stock characteristics (McLean and Pontiff, 2016),
recent studies have shown that machine learning methods are able to generate robust forecasting
power to predict stock returns, address the data-snooping concerns, and identify the marginal
contribution of new factors relative to the large set of existing ones (Feng, Giglio, and Xiu, 2020;
Gu, Kelly, and Xiu, 2020; Kozak, Nagel, and Santosh, 2020; Giglio, Liao, and Xiu, 2021).

Despite the proliferation of stock characteristics or factors to explain the cross-section of stock
returns, however, far fewer studies are devoted to predict future returns on corporate bonds. Recent
studies examine a few corporate bond characteristics related to default and term betas (Fama and
French, 1993; Gebhardt, Hvidkjaer, and Swaminathan, 2005), liquidity risk (Lin, Wang, and Wu,
2011), bond momentum (Jostova, Nikolova, Philipov, and Stahel, 2013), downside risk (Bai, Bali,
and Wen, 2019), and long-term reversal (Bali, Subrahmanyam, and Wen, 2021a), which exhibit
significant explanatory power for future bond returns. Kelly, Palhares, and Pruitt (2022) propose
a conditional factor model for corporate bond returns and find that the model with five factors
and time-varying factor loadings produces strong out-of-sample return predictions. Using standard
asset pricing tests such as the OLS cross-sectional regressions, other papers investigate whether
well-known equity market anomalies impact the cross-section of corporate bond returns and find
mixed evidence on the predictability (Chordia, Goyal, Nozawa, Subrahmanyam, and Tong, 2017;
Choi and Kim, 2018).

One common element in most of these studies is that they use standard linear methods to
analyze return predictability. However, bondholders are more sensitive to downside risk compared
to stockholders (Hong and Sraer, 2013; Bai, Bali, and Wen, 2019). Because of the nonlinear payoffs
of corporate bonds and the high correlation between many of the stock and bond characteristics,
machine learning is well suited for such challenging prediction problems by reducing the degrees of
freedom and condensing redundant variation among a large set of predictors, with an emphasis on
variable selection and dimension reduction techniques (Gu, Kelly, and Xiu, 2020).1
1
Recent studies use machine learning techniques to extract information from both the cross-section and
time-series of stock returns in identifying the most relevant stock characteristics or factors. For example,
Feng, Giglio, and Xiu (2020) propose a model selection method to systematically evaluate the contribution
to asset pricing of any new factor, above and beyond what a high-dimensional set of existing factors explains.
Lettau and Pelger (2020) develop a risk premium PCA estimator that adds to the traditional PCA objective
function a no-arbitrage penalty term that helps price the cross-section of equity returns. Freyberger, Neuhierl,

1
 In this paper, we provide a comprehensive study on the cross-sectional predictability of
corporate bond returns using a large set of stock and bond characteristics. Previous studies, in
general, rely on the reduced-form approach that examines cross-sectional bond return predictability,
without explicitly linking the functional forms of bond and stock expected returns. In this
article, we highlight the importance of imposing a theoretically motivated economic structure when
investigating expected bond return predictability, which tends to be largely understudied in the
aforementioned research. There are a few reasons to impose an economic structure and investigate
the dependence between expected bond and stock returns in a unified framework. First, stocks
and bonds issued by the same firm represent claims on the same underlying assets of the firm.
Hence, relevant information about the firm should have an impact on both the firm’s outstanding
stocks and its outstanding bonds, leading to co-movement between individual stock and bond
prices. Hence, it is not surprising that their returns should be correlated.2 Second, the typical
workhorse model to analyze the stock-bond connection is Merton (1974) structural credit risk
model, which explains how bonds and stocks should be jointly priced. Based on the model of Merton
(1974), if a variable/characteristic explains stock returns, then the model places restrictions on the
predictability of bond returns from this variable. As a result, motivated by the Merton (1974)
model, we impose the dependence between expected returns of bonds and stocks, and compare the
forecasted bond returns with such restrictions to the ones obtained from the reduced-form approach
that neglects any form of economic structure.

In light of the machine learning methods, we seek to answer the following questions: First,
without imposing any economic structure from the Merton (1974) model, do corporate bond
characteristics and stock characteristics, individually or combined, predict future bond returns? Do
stock characteristics improve the performance of bond-level characteristics in predicting future bond
returns? Second, is there any significant improvement in the performance of the machine learning
models when imposing the economic structure from the Merton (1974) model, compared to the ones
without restrictions? Overall, our results highlight that it is important to explicitly impose the
dependence between expected bond and stock return via the Merton (1974) model as such economic
structure significantly improves future bond return forecasts. Our results also show that once we
impose the Merton (1974) model structure, equity characteristics provide significant improvement
above and beyond bond characteristics for future bond returns, whereas the incremental power of
equity characteristics for predicting bond returns are quite limited in the reduced-form approach
when such economic structure is not imposed.
and Weber (2020) introduce a nonparametric method (i.e., the adaptive group LASSO) to study which
characteristics provide incremental information for the cross-section of stock returns. Nagel (2021) provides
a comprehensive overview of machine learning models and discusses the application of these techniques in
empirical research in asset pricing.
2
Kwan (1996) indeed finds that stock returns and bond yield changes are positively correlated. Kelly,
Palhares, and Pruitt (2022) find that the systematic components of bond and equity returns are roughly
twice as integrated as their total returns, whereas idiosyncratic bond and stock returns are substantially less
integrated than their systematic counterparts.

2
 We first build a comprehensive data library of 43 corporate bond-level characteristics that are
motivated by the existing literature on the cross-section of corporate bonds. This list of a broad set
of corporate bond return predictors is designed to be representative of (i) bond-level characteristics
such as issuance size, credit rating, time-to-maturity, and duration, (ii) proxies of risk such as bond
systematic risk, downside risk, and credit risk, (iii) proxies of bond-level illiquidity constructed using
daily and intraday transaction data and liquidity risk, (iv) past bond return characteristics such
as bond momentum, short-term and long-term reversals, and (v) the distributional characteristics
such as return volatility, skewness, and kurtosis.

We then combine them with the 94 stock characteristics used by Green, Hand, and Zhang (2017)
and Gu, Kelly, and Xiu (2020). Our final sample of the 137 stock- and bond-level characteristics
cover both the equity and debt markets, thus provide a wide range of predictors for corporate bond
returns. Focusing on a variety of machine learning methods proposed by Gu, Kelly, and Xiu (2020),
we compare and evaluate the out-of-sample performance of alternative machine learning models in
predicting the cross-sectional dispersion in future bond returns. The machine learning methods
include the dimension reduction models (PCA and PLS), penalized methods (Lasso, Ridge, and
Elastic Net), regression trees (Random Forests), and neural networks including the feed forward
neural networks (FFN). In addition to these methods, we use the long short-term memory neural
network (LSTM) proposed by Hochreiter and Schmidhuber (1997) to capture a long memory effect
(Lo, 1991). Moreover, we rely on the forecast combination method (Combination) which averages
individual expected return forecasts from the aforementioned sophisticated machine learning models
(Rapach, Strauss, and Zhou, 2010; Chen, Pelger, and Zhu, 2019).

We first show that the traditional unconstrained linear regression models such as the OLS fail to
deliver statistically significant out-of-sample forecasting power for future corporate bond returns.
The standard OLS regression methodology with all 43 bond characteristics produces a negative
2 ), whereas the machine learning models substantially improve the
out-of-sample R-squared (ROS
2 ranging from 1.85% to 2.37%. Using the Diebold and Mariano (1995) test
predictive power with ROS
for differences in out-of-sample predictive accuracy between two models, we find that all machine
learning models perform equally well and they significantly outperform the unconstrained OLS
model.

To further investigate the economic significance of machine learning approaches, we form

corporate bond portfolios based on machine learning forecasts using the 43 bond characteristics.
The machine learning driven bond portfolios are constructed based on the one-month-ahead out-
of-sample forecasts of bond returns, where the arbitrage (high-minus-low) portfolio corresponds to
the long-short portfolio that buys bonds with the highest one-month-ahead expected returns (decile
10) and sells bonds with the lowest one-month-ahead expected returns (decile 1). We find that all
machine learning forecasts generate economically and statistically significant return spreads on the
arbitrage portfolios, ranging from 0.33% to 0.79% per month, compared to the unconstrained OLS

3
model which delivers the smallest monthly return spread of 0.16%.

We proceed to identify corporate bond characteristics that are important determinants of the
cross-section of bond returns, while simultaneously controlling for the many other predictors.
Following the ranking and variable importance approach of Kelly, Pruitt, and Su (2019) and
Gu, Kelly, and Xiu (2020), we discover influential covariates by measuring the reduction in panel
2 , while holding the remaining model estimates fixed. This approach allows
predictive regression ROS
us to investigate the relative importance of individual bond characteristics for the out-of-sample
forecasting performance of each machine learning model. Our results demonstrate that all machine
learning models are in close agreement on the most influential bond-level characteristics, which can
be classified into four broad categories (i) bond characteristics related to interest rate risk such as
duration and time-to-maturity, (ii) risk measures such as downside risk proxied by Value-at-Risk
(VaR) and expected shortfall (ES), total return volatility (VOL), and systematic risk proxied by
the bond market beta, default beta, and term beta, (iii) bond-level illiquidity measures such as
the average bid and ask price (AvgBidAsk), and Amihud and Roll’s measures of illiquidity, and
(iv) past return characteristics related to bond momentum, short-term reversal, and long-term
reversal. To find out which one of the four groups of bond return predictors is the most important
determinant of the expected bond returns, we compute the sum of the importance measure of each
return predictor for each method, within each characteristic group. We find that the top two most
important groups are the characteristics related to bond-level illiquidity and illiquidity risk (i.e.,
Group III) and risk measures such as downside risk and systematic risk proxies (i.e., Group II).

Then, we examine whether a large number of stock characteristics improve the cross-sectional
return predictability of corporate bonds, using the reduced-form approach without explicitly linking
the functional forms of bond and stock expected returns via the Merton (1974) model. Recent
studies often draw from the well of cross-sectional predictors on a few stock characteristics and find
mixed evidence of predictability for corporate bonds (Chordia et al., 2017; Choi and Kim, 2018).
Compared to these studies, we extend the candidates to a much larger set of stock characteristics
and more importantly, we rely on machine learning methods to reduce redundant variation among
predictors that address overfitting bias. We show that all machine learning models substantially
improve the forecasting power of stock characteristics for future bond returns compared to the
standard OLS, for all sample of bonds.3 However, the marginal improvement of the forecasting
power of stock characteristics relative to bond characteristics is economically small and insignificant,
as most machine learning forecasts fail to deliver statistically significant positive return spreads on
the long-short bond portfolios.

It is important to note that so far we have only used different machine learning approaches
to model bond expected returns, which is a reduced-form approach that does not explicitly link
3 2
The machine learning models using stock characteristics deliver an ROS in the range of 1.61% and
2
2.02%, which is similar to the ROS obtained from using bond characteristics, which ranges from 1.85% to
2.37%.

4
the functional forms of bond and stock expected return. Motivated by Merton (1974) model, we
next impose the dependence between expected bond and stock return using hedge ratios. When
we use regression-based hedge ratios,4 we find that the machine learning model with such economic
structure generates economically and statistically significant return spreads on the long-short bond
portfolios, in the range of 0.55% and 0.92% per month, compared to the unconstrained OLS
model which delivers the smallest return spread of 0.18%. More importantly, there is significant
improvement in the performance of the machine learning models with imposing restrictions,
compared to the bond return forecasts obtained without restrictions using bond characteristics
alone or the combined stock and bond characteristics.

Finally, we further investigate the predictability of bond returns using Merton (1974) model
with hedge ratios estimated with machine learning models. Specifically, we model hedge ratio as a
function of bond characteristics and investigate the performance of expected bond return forecasts
with Merton (1974) restrictions and machine learning estimated hedge ratio. Our results show
a positive and statistically significant Diebold-Mariano test statistics for all the machine learning
models, compared to the bond return forecasts using only bond characteristics, the combined stock
and bond characteristics, or those generated using the Merton model restriction with an exogenously
specified hedge ratio. However, the results show that the economic significance of using machine
learning estimated hedge ratio is similar to that using exogenously specified hedge ratio, as the
return spreads generated from both approaches are similar in economic magnitude and they are
not statistically different from each other. Overall, we conclude that it is important to impose
Merton model restrictions along the lines of Schaefer and Strebulaev (2008) when estimating bond
expected returns, which significantly improves bond return predictability compared to the reduced-
form approach that does not explicitly model the dependence between bond and stock expected
returns.

The rest of the paper proceeds as follows. Section 2 provides our theoretical motivation, presents
the corresponding prediction framework, and describes the performance metrics used to assess the
predictive power of stock and bond characteristics. Section 3 describes the data and variables used
in our empirical analyses. Section 4 relies on a reduced-form approach that does not explicitly
link the functional forms of bond and stock expected returns and investigates the performance of
machine learning models in predicting future bond returns without hedge ratios. Section 5 imposes
the dependence between expected bond and stock returns via Merton (1974) model and examines
the performance of machine learning models in predicting future bond returns using regression-
based hedge ratios. Section 6 presents results from predicting future bond returns with machine
learning based dynamic hedge ratios. We conclude in Section 7.
4
Schaefer and Strebulaev (2008) is the first paper to provide a comprehensive investigation of the
magnitude and statistical significance of the hedge ratio. Choi and Kim (2018) follow Schaefer and Strebulaev
(2008) in terms of the estimation methodology but rely on a different method to estimate the hedge ratio for
each firm and for each month based on a rolling regression using monthly returns over the past 36 months.

5
2 Methodology

2.1 Theoretical Motivation

We present a simple structural model to guide our empirical work. While the typical workhorse
model to analyze the stock-bond connection is Merton (1974) structural credit risk model, we follow
its extension in Du, Elkamhi, and Ericsson (2019). We assume that the value of the assets of the
firm, Vt , is governed by the following stochastic processes:

dVt = rVt dt + σt Vt dWt

dσt2 = κ(θ − σt2 )dt + γσt dZt , (1)

where the initial value of the assets V0 > 0 and r is the risk-free rate. The processes {Wt } and
{Zt } are two standard Brownian motions under the risk-neutral martingale measure Q and their
instantaneous correlation is ρ. κ is the speed of mean reversion, θ is the long-run mean variance,
and γ is the volatility parameter for asset variance. The firm issues a single class of debt, a zero-
coupon bond, with a face value B payable at time T . Default may happen only at time T , and
if default happens, creditors take over the firm without incurring any distress costs and realize an
amount VT . Otherwise, they receive B. Equation (1) differs from Merton (1974) by generalizing the
variance of the assets to follow a stochastic process (instead of assuming constant asset variance).
Du, Elkamhi, and Ericsson (2019) show that this relaxation can better describe the average credit
spreads levels.

When the asset variance is constant, Merton (1974) shows that the creditors take a short
position in a put option written on the assets of the borrowing firm with a strike B, the face value
of the debt, while the equity holders, who own the firm, borrow the amount B at time 0, and own
a put option on the assets of the firm with strike B, equivalently hold a call option on the assets of
the firm with strike B. As such, the equity and bond prices at any time t can be explicitly solved
by the Black and Scholes (1973) formula.

When the asset variance is stochastic, equity and bond prices cannot be expressed in closed
form. In this case, Hull and White (1987) propose an approximation method, which delivers a
closed form solution. We follow these authors and approximate the equity and bond prices as
√
−r(T −t) θγ
Et = Vt N (d1 ) − Be N (d2 ) − · ηt , (2)
8κ

where N (d1 ) and ϕ(d1 ) are the standard normal distribution and density functions, respectively.
The closed form ηt is provided in equation (A.8) of Appendix A. The debt value is then given by
Dt = Vt − Et . We note that the equity price in equation (2) differs from Hull and White (1987),
in that our variance follows a Cox, Ingersoll, and Ross (1985) process, while it follows a geometric

6
Brownian motion in Hull and White (1987).

With equation (2), we can analytically calculate the hedge ratio, following the definition of
Schaefer and Strebulaev (2008), as

def ∂Dt /∂Vt Et

ht = ×
∂Et /∂Vt Dt
1 − N (d1 ) + γ 2 ζt Et
= . (3)
N (d1 ) − γ 2 ζt Dt

At the same time, the equity and bond returns have the following relationship:

dDt dEt
− ht = αt dt. (4)
Dt Et

The expressions for ζt and αt are given in equation (A.15) of Appendix A.

Clearly, when the variance of the firm value is constant, i.e., γ = 0, Et and ht reduce to the case
in Schaefer and Strebulaev (2008), and αt = 0. Because the bond and equity prices are driven by
the firm value Vt only, the two markets are fully integrated or the systemic risk of the bond can be
perfectly hedged by the equity. In contrast, when the variance is stochastic, the bond and equity
prices are jointly driven by Vt and σt2 . The two markets are not fully integrated any more, which
is supported by the empirical fact that αt ̸= 0.

Equation (4) shows that any prediction of bond returns involves three components: (i)
predicting the hedge ratio, (ii) predicting the stock return, and (iii) predicting the ‘residual’ bond
return. This equation forms the basis of our empirical work.

2.2 Prediction Framework

We index assets (either a corporate bond or a stock) by i = 1, . . . , N and months by t = 1, . . . , T .

By definition, the excess return of asset i at time t + 1, Rit+1 , is equal to the sum of expected
return plus the error term:

Rit+1 = Et (Rit+1 ) + eit+1 , (5)

where Et (Rit+1 ) is the time-t expected return. Specifically, let RB and RS denote the realized
bond and stock return, respectively. We have:

RBit+1 = Et [RBit+1 ] + eBit+1 (6)

RSit+1 = Et [RSit+1 ] + eSit+1 , (7)

7
where eB and eS are the unexpected bond return and stock return, respectively. Using equation (4),
we have

Et (RBit+1 ) = hit × Et (RSit+1 ) + αit . (8)

Substituting equation (8) into equation (6), we get:

RBit+1 ≡ Et (RBit+1 ) + eBit+1

= hit × Et (RSit+1 ) + αit + eBit+1
= hit × RSit+1 + αit + (eBit+1 − hit × eSit+1 ). (9)

Define RBmRSit+1 as the difference between realized bond return (RB) and the product of the
hedge ratio and realized stock return (h × RS):

def
RBmRSit+1 = RBit+1 − hit × RSit+1
= αit + (eBit+1 − hit × eSit+1 ). (10)

Taking expectation, we see that Et (RBmRSit+1 ) = αit . We can, thus, express expected bond
returns as:

Et (RBit+1 ) = hit × Et (RSit+1 ) + Et (RBmRSit+1 ). (11)

The expectations in equation (11) are specified to be flexible functions of characteristics. For
instance, a generic time-t expected return, Et (Rit+1 ), is specified to be Et (Rit+1 ) = ϕ(Xit ), where
ϕ(·) is a flexible function of asset i’s P -dimensional characteristics, i.e., Xit = (Xi1t , . . . , XiP t )′ . We
discuss specific functional forms in the next Section 2.3.

We consider three variations of predicting bond returns:

1. Without the hedge ratios: The benchmark prediction model does not rely on the theoretical
framework as outlined in Section 2.1 and implicitly sets the hedge ratio, hit , to zero. As is
evident from equations (10) and (11), in this case RBmRS ≡ RB. Thus, the prediction task
simplifies to predicting just the bond returns with no cross-asset restrictions of the form (8).
We specify:
Et (RBit+1 ) = f1 (Xit ), (12)

where the characteristics X include combinations of bond characteristics, XB, and stock
characteristics, XS. Note that even though we do not formally use the hedge ratios in
this approach, the stock and the bond market are not assumed to be disconnected. For
instance, when X includes both bond and stock characteristics, we allow stock characteristics
(predictors of stocks returns) to predict bond returns too. Therefore, this approach can be

8
 considered as a reduced-form Merton (1974) approach. We discuss results from this approach
in Section 4.

2. With regression-based hedge ratios: In this prediction method, we estimate hedge ratios via
regressions of bond returns on stock returns. We then separately estimate Et (RSit+1 ) =
ψ1 (Xit ) and Et (RBmRSit+1 ) = ψ2 (Xit ) and then combine these predictions to obtain the
expected bond return as:

Et (RBit+1 ) = f2 (Xit ) = ĥit × ψ1 (Xit ) + ψ2 (Xit ). (13)

We investigate bond return prediction following this approach in Section 5.

3. With machine learning-based hedge ratios: In this prediction method, we let the hedge ratio
itself be a function of characteristics. Thus, we separately estimate three different machine
learning models Et (RSit+1 ) = ϕ1 (Xit ), Et (RBmRSit+1 ) = ϕ2 (Xit ), and hit = ϕ3 (Xit ), and
then combine these predictions to obtain the expected bond return as:

Et (RBit+1 ) = f3 (Xit ) = ϕ3 (Xit ) × ϕ1 (Xit ) + ϕ2 (Xit ). (14)

Note that the prediction of Et (RBmRSit+1 ) in this third variant is different from the
corresponding prediction in the second variant (ϕ2 (Xit ) ̸= ψ2 (Xit )) even if the same set
of characteristics is used in both predictions. The reason is that RBmRS in equation (10) is
defined using hedge ratio, hit , which is calculated differently in the two approaches. Section 6
provides further details on computations and the associated results.

2.3 Machine Learning and Performance Evaluation

Following Gu, Kelly, and Xiu (2020), we compare and evaluate a variety of machine learning
methods, including the ordinary least squares (OLS) with all covariates; penalized linear regression
methods such as LASSO, ridge regression (Ridge), and elastic net (ENet); dimension reduction
techniques such as principal component analysis (PCA) and partial least square (PLS); random
forests (RF); and feed-forward neural network (FFN). In addition to these methods, we use a long
short-term memory neural network (LSTM) to capture a long memory effect (Lo, 1991; Hochreiter
and Schmidhuber, 1997). Moreover, we rely on the forecast combination method (Combination)
which averages individual expected return forecasts from the aforementioned eight machine learning
models (Rapach, Strauss, and Zhou, 2010; Chen, Pelger, and Zhu, 2019). We provide a detailed
description of these methods in Section OA1 of the Online Appendix.

Following Gu, Kelly, and Xiu (2020), we use the out-of-sample R-squared as the performance

9
metric to assess the predictive power of individual bond return predictors,

(rit+1 − r̂it+1 )2
P
2 (i,t)∈T3
ROS =1− P 2 . (15)
(i,t)∈T3 rit+1

2
The ROS statistic pools prediction errors across bonds and over time into a grand panel-level
assessment of each model, and it measures the proportional reduction in mean squared forecast
error (MSFE) for each model relative to a naive forecast of zero benchmark, which assumes that
the one-month-ahead expected return on corporate bonds equals the time t + 1 risk-free rate. To
2 , we follow the most commonly used approach in the literature
estimate the out-of-sample ROS
and divide our full sample (July 2002 to December 2017) into three disjoint time periods; (i) the
first three years of “training” or “estimation” period, T1 , (ii) the second two years of “validation”
for tuning the hyperparameters, T2 , and (iii) the rest of the sample as the “test” period, T3 , to
evaluate a model’s predictive power, which represents the truly out-of-sample evaluation of the
model’s performance.

We use the mean squared forecast error (MSFE)-adjusted statistic of Clark and West (2007)
2 .
to test the statistical significance of ROS Considering the potentially strong cross-sectional
dependence among individual excess bond returns, we employ the modified MSFE-adjusted statistic
based on the cross-sectional average of prediction errors from each model instead of prediction errors
among individual returns. The p-value from the MSFE-adjusted statistic tests the null hypothesis
that the MSFE of a naive forecast of zero is less than or equal to the MSFE of a machine learning
model against the one-sided (upper-tail) alternative hypothesis that the MSFE of a naive forecast
2 ≤ 0 against H : R2 > 0).
of zero is greater than the MSFE of a machine learning model (H0 : ROS A OS

To compare the out-of-sample predictive power of two methods, we use the modified Diebold and
Mariano (1995) test, which accounts for the potentially strong cross-sectional dependence among
individual returns. Specifically, to compare the predictive powers of methods (1) and (2), we define
the modified Diebold-Mariano statistic as

DM12 = d¯12 /σ̂d¯, (16)

where d¯12 and σ̂d¯ are, respectively, the time-series mean and Newey-West standard error of d12,t+1
over the testing sample. d12,t+1 is the forecast error differential between the two methods, calculated
as the cross-sectional average of forecast error differentials from each model over each period t + 1,
n3  
1 X (1)
2 (2)
2
d12,t+1 = êit+1 − êit+1 , (17)
n3,t+1
i=1

(1) (2)
where êit+1 and êit+1 are the return forecast errors for individual asset i at time t + 1 generated by
two methods, and n3,t+1 is the number of assets in the testing sample.

10
3 Data and Variable Definitions
This section first describes the data and key variables used in our empirical analyses and then
provides summary statistics for the large set of corporate bond characteristics we construct.
Following Bessembinder, Maxwell, and Venkataraman (2006), who highlight the importance of
using TRACE transaction data, we rely on the transaction records reported in the enhanced version
of TRACE for the sample period from July 2002 to December 2017. The TRACE dataset offers
the best-quality corporate bond transactions, with intraday observations on price, trading volume,
and buy and sell indicators.5

For TRACE data, we adopt the filtering criteria proposed by Bai, Bali, and Wen (2019).
Specifically, we remove bonds that (i) are not listed or traded in the US public market; (ii) are
structured notes, mortgage backed/asset backed/agency backed/equity-linked; (iii) are convertible;
(iv) trade under $5; (v) have floating coupon rates; and (vi) have less than one year to maturity. For
intraday data, we also eliminate bond transactions that (vii) are labeled as when-issued or locked-
in or have special sales conditions, (viii) are canceled, (ix) have more than a two-day settlement,
and (x) have a trading volume smaller than $10,000. We then merge corporate bond pricing data
with the Mergent FISD to obtain bond characteristics such as the offering amount, offering date,
maturity date, coupon rate, coupon type, interest payment frequency, bond type, bond rating,
bond option features, and issuer information.

3.1 Corporate Bond Return

The monthly corporate bond return at time t is computed as

Pit + AIit + Cit

rit = − 1, (18)
Pit−1 + AIit−1

where Pit is the transaction price, AIit is accrued interest, and Cit is the coupon payment, if any,
of bond i in month t. We denote Rit as bond i’s excess return, Rit = rit − rf t , where rf t is the
risk-free rate proxied by the one-month Treasury bill rate.

With the TRACE intraday data, we first calculate the daily clean price as the trading volume-
weighted average of intraday prices to minimize the effect of bid-ask spreads in prices, following
Bessembinder, Kahle, Maxwell, and Xu (2009). We then convert the bond prices from daily to
monthly frequency following Bai, Bali, and Wen (2019), who discuss the conversion methods in
5
We use enhanced TRACE instead of the standard TRACE since it contains uncapped transaction
volumes and information on whether the trade is a buy, a sell, or an interdealer transaction, in addition
to the information contained in standard TRACE. The improvement of enhanced TRACE over standard
TRACE thus allows us to construct a variety measures of bond liquidity using daily and intraday transaction
data.

11
detail. Specifically, our method identifies two scenarios for a return to be realized at the end of
month t: (i) from the end of month t − 1 to the end of month t, and (ii) from the beginning of
month t to the end of month t. We calculate monthly returns for both scenarios, where the end
(beginning) of the month refers to the last (first) five trading days within each month. If there are
multiple trading records in the five-day window, the one closest to the last trading day of the month
is selected. If a monthly return can be realized in more than one scenario, the realized return in
the first scenario (from month-end t − 1 to month-end t) is selected.

Corporate bonds occasionally default prior to reaching maturity. If default returns are simply
treated as missing observations, return estimates can be overstated, particularly for high-yield bonds
and long-term losers. To address this potential return bias, we follow Cici, Gibson, and Moussawi
(2017) and Bali, Subrahmanyam, and Wen (2021a) and compute a composite default return for
all defaulted bonds. Specifically, we search for any price information on defaulted issues after the
default event. We then compute median returns on these defaulted issues in the (−1, +1) month
window around the default date and use the median return of −40.17% for defaulting investment-
grade (IG) issues and −17.67% for defaulting non-investment-grade (NIG) issues, which reflect
higher expected default probability for high yield ex-ante.6 For IG and NIG issues that default
without post-default prices, we use the corresponding IG and NIG default return averages as proxies
for default-month returns. Using the in-sample composite default-month returns for defaulting
bonds of similar credit quality, but without valid post-default pricing information, enables us to
avoid the delisting bias shown in previous research on equity returns (Shumway, 1997).

3.2 Corporate Bond and Equity Characteristics

We build a comprehensive data library of 43 corporate bond characteristics that are either
theoretically motivated or empirically identified by earlier studies on the cross-section of corporate
bond returns. This broad set of bond return predictors can be largely classified into (i) bond-
level characteristics such as issuance size, age, credit rating, time-to-maturity, and duration, (ii)
proxies of corporate bond downside risk, (iii) proxies of bond-level illiquidity and liquidity risk,
(iv) proxies of systematic risk such as default and term betas and volatility betas, (v) past bond
return characteristics such as bond momentum, short-term reversal, and long-term reversal, and
(vi) distributional characteristics including return volatility, skewness, and kurtosis. Appendix B
provides a detailed description of these 43 bond characteristics as well as the studies that we follow
closely to construct these measures. This list of corporate bond characteristics is not an exhaustive
analysis of all possible predictors of corporate bond returns. Nonetheless, our list is designed to be
representative of a broad set of corporate bond characteristics motivated in the literature for their
explanatory power for bond returns. For equity characteristics, we rely on a large set of 94 stock-
6
Consistent with Bali, Subrahmanyam, and Wen (2021a) who use a common dataset of bond returns
after July 2002, the frequency of default events is rare in our sample.

12
level predictors used by Green, Hand, and Zhang (2017).7 We restrain our equity characteristics
sample to begin from July 2002 and end in December 2017 because we focus on the common sample
period when our bond returns and characteristics become available in TRACE which starts in July
2002.

Our final sample includes 22,980 bonds issued by 1,841 unique firms, yielding a total of 146,085
firm-level bond-month return observations during the sample period from July 2002 to December
2017. Panel A of Table 1 reports the time-series average of the cross-sectional bond returns’
distribution and bond characteristics. The numbers are presented at the firm-level using value-
weighted average of firm-level bond returns and bond characteristic measures. The sample contains
bonds with an average rating of 10.08 (i.e., BBB-), an average issue size of $500 million, and an
average time-to-maturity of 8.05 years. Among the full sample of bonds, about 75% are investment-
grade and the remaining 25% are high-yield bonds. Panel B of Table 1 presents the correlation
matrix for some of the firm-level bond characteristics and risk measures. As shown in Panel B,
downside risk (i.e., proxied by the 5% Value-at-Risk) is positively associated with bond market
beta (β Bond ), illiquidity, and rating, with respective correlations of 0.61, 0.19, and 0.25. The
bond market beta, β Bond , is also positively associated with rating and illiquidity, with respective
correlations of 0.01 and 0.04. Bond maturity and duration are positively correlated with most risk
measures, implying that bonds with longer maturity or duration (i.e., higher interest rate risk)
have higher β Bond and higher ILLIQ. Bond size is negatively correlated with ILLIQ, indicating
that bonds with smaller size have higher ILLIQ.

4 Predicting Bond Returns without Hedge Ratios

4.1 Using only Bond Characteristics

We start our analysis with the baseline scenario of predicting bond returns without imposing
cross-asset restrictions and using bond characteristics. Using the notation from equation (12) of
Section 2.2, our goal in this subsection is to predict corporate bond returns as Et (RBit+1 ) =
f1 (XBit ).

4.1.1 Out-of-Sample Predictive Power

2 (in percentage) for the entire pooled sample of corporate bonds

Table 2 presents the monthly ROS
using all the 43 bond characteristics as covariates. The results are presented at the firm-level in
7
Details on each of the 94 equity characteristics can be found in the Appendix in Green, Hand, and
Zhang (2017) and Table A.6 in Gu, Kelly, and Xiu (2020). For missing equity (bond) characteristics, we
follow Gu, Kelly, and Xiu (2020) and replace them with the cross-sectional median characteristic of each
stock (bond) for each month.

13
 2
Table 2 using value-weighted average of firm-level bond returns. Panel A of Table 2 reports ROS
for the entire sample of corporate bonds. The first column shows that the OLS model with all
2
43 bond characteristics produces an ROS of −3.36%, indicating that the model fails to deliver
significant out-of-sample forecasting power for the expected corporate bond returns. However, the
2 .8
other columns of Table 2 show that the machine learning models substantially improve the ROS
For example, by forming a few linear combinations of predictors via dimension reduction, columns
2 to 2.07% and 2.03%, respectively.
(2) and (3) of Table 2 show that PCA and PLS improve the ROS
By introducing the penalized methods into the loss function, columns (4) to (6) show that LASSO,
2
Ridge, and ENet approach improve the ROS to 1.85%, 1.89%, and 1.87%, respectively.

Unlike the linear models in column (1), regression trees are fully nonparametric and can reduce
overfitting in individual bootstrap samples, and make the predictive performance more stable.
2
Consistent with this prediction, column (7) of Table 2 shows a significant increase in ROS to
2.19% using random forests (RF). In addition to nonparametric regressions, we investigate the
performance of different neural network models including the feed forward neural networks (FFN)
and the long short-term memory neural network (LSTM). As a typical neural network, feed forward
neural networks (FFN) produces more flexible prediction approach by adding hidden layers between
the inputs and output layer that aggregates hidden layers into the outcome prediction. The long
short-term memory neural network (LSTM) captures long-term dependencies as a flexible hidden
state space model for a large dimensional system. Columns (8) and (9) show that the FFN and
2
LSTM models produce significant ROS values of 2.37% and 2.28%, respectively. Finally, the last
column of Table 2 shows that the forecast combination model (Combination) significantly improves
2
the ROS to 2.09%.9

To make pairwise comparisons of the estimation methods, we use the Diebold and Mariano
(1995) test for differences in out-of-sample predictive accuracy between two models. Panel B
of Table 2 reports the Diebold-Mariano test statistics for pairwise comparisons of a column model
versus a row model. A positive statistic indicates that the column model outperforms the row model.
The first row of Panel B shows a positive and statistically significant test statistic for all the machine
learning models with Diebold-Mariano test statistics ranging from 2.89 to 3.85, compared to the
unconstrained OLS model. Thus, all machine learning methods produce statistically significant
improvements over the unconstrained OLS model. Comparisons between machine learning methods
8 2
In Table 2, all of the ROS statistics for the machine learning models are statistically significant with
p-values less than 1%.
9 2
Despite significant improvements in the forecasting performance, the ROS of 2.09% based on the
Combination model is slightly lower than those of RF (2.19%), FFN (2.37%), and LSTM (2.28%) models.
This is plausible because the mean squared forecast error (MSFE) can be decomposed into forecast variance
and the squared forecast bias (Rapach, Strauss, and Zhou, 2010) so that a model’s forecasting performance
depends on the tradeoff between the reduction in variance and bias. Combination model may significantly
reduce the forecast variance but increases the bias of estimation, whereas the individual machine learning
models such as RF and FFN may deliver better performance due to their ability to further reduce the
forecasting biases which outweigh the costs of increasing variance.

14
themselves show that there is little difference in the performance of dimension reduction methods
(PCA and PLS), penalized linear methods (LASSO, Ridge, ENet, and RF), and neural networks
(FFN and LSTM), as the test statistics are not significant. Finally, the last column of Panel B shows
that the forecast combination model (Combination) produces large and statistically significant
improvements over most individual machine learning models.

4.1.2 Which Bond Characteristics Matter?

Next, we identify the corporate bond characteristics that are important determinants of the
expected bond returns while simultaneously controlling for the many other predictors. We take the
value-weighted average of bond-level characteristics to generate the firm-level bond characteristic
measures. Following the ranking approach in Kelly, Pruitt, and Su (2019) and Gu, Kelly, and Xiu
(2020), we discover influential covariates from setting all values of predictor j to zero, while holding
the remaining model estimates fixed. The variable importance of the j th input variable is measured
2 , which allows us to investigate the relative importance
by the reduction in panel prediction ROS
of individual bond characteristics for the performance of each machine learning model. To begin,
2
for each of the nine machine learning methods, we calculate the reduction in ROS from setting all
values of a given predictor to zero within each training sample, and then average these into a single
importance measure for each predictor. Figure 1 reports the resulting forecasting performance of
the top 10 bond-level characteristics for each method, whereas Figure 2 reports overall rankings of
characteristics for all models.10

Figures 1 and 2 demonstrate that all machine learning models are generally in close agreement
regarding the most influential bond-level characteristics, which can be classified into four categories
(i) bond characteristics related to interest rate risk such as duration (DUR) and time-to-maturity
(MAT), (ii) risk measures such as downside risk proxied by Value-at-Risk (VaR) and expected
shortfall (ES), total return volatility (VOL), and systematic risk related to bond market beta,
default beta, term beta, and economic uncertainty beta (β Bond , β DEF , β T ERM , and β U N C ),
(iii) bond-level illiquidity measures such as the average bid and ask price (AvgBidAsk), Amihud
and Roll’s measures of illiquidity, and (iv) past return characteristics related to bond momentum
(MOM), short-term reversal (STR), and long-term reversal (LTR). Figure 1 shows that the risk
measures play an important role in the dimension reduction methods (PCA and PLS), whereas
bond-level characteristics related to interest rate risk are more prominent in the penalized methods
(Lasso, Ridge, and Enet). Regression trees such as the random forest model rely more heavily on
bond-level illiquidity measures such as the average bid and ask price and the Amihud measure.
Neural networks such as FFN and LSTM draw predictive information mainly from bond return
characteristics such as bond momentum and short-term reversal. Finally, the forecast combination
10
The color gradient within each column in Figure 2 shows the model-specific ranking of characteristics,
where the lightest (darkest) color indicates the least (most) important bond characteristics for each model.

15
model shows that bond momentum (MOM), return volatility (VOL), coskewness (COSKEW), and
illiquidity (ILLIQ) are the top important covariates for the predictive performance.

In addition to comparing the covariate importance across all 43 firm-level bond characteristics,
we further investigate their importance within each of the four characteristic groups. Panel A of
Figure 3 shows that time-to-maturity is the most important characteristic for the expected bond
returns within the Group I characteristics for all models, followed by duration. Panel B shows that
within the Group II characteristics, coskewness and downside risk measures including VaR and ES,
and systematic risk such as the macroeconomic uncertainty beta and default beta are the most
important covariates. Panel C shows that the illiquidity measures such as the average bid and ask
price play an important role across all machine learning models, whereas Panel D shows that higher
return moments such as VOL as well as past return characteristics related to bond momentum are
the top important covariates.

To find out which one of the four characteristic groups is the most important determinant
of the expected bond returns, we present the relative strength of the four characteristic groups,
respectively, in Figure 4, which shows a 10×4 bar chart representing the importance of each
characteristic group for all methods. The columns of Figure 4 correspond to individual models,
and color gradients within each column present a ranking from the most influential (dark blue) to
the least influential (white) characteristic group. Figure 4 shows that the top two most important
determinants are the characteristics related to bond-level illiquidity and liquidity risk (i.e., Group
III) and the risk measures such as downside risk and systematic risk proxies (i.e., Group II).

4.1.3 Machine Learning based Long-Short Portfolios

To further investigate the economic significance of the machine learning models, we form portfolios
based on the machine learning forecasts using the 43 bond characteristics. At the end of each
month, we calculate the one-month-ahead out-of-sample firm-level bond return predictions for each
of the ten methods (including the OLS). We then sort firm-level bond returns into deciles based
on each model’s forecasts of the one-month-ahead returns and then construct the value-weighted
long-short portfolios of corporate bonds.11 Table 3 reports the monthly performance results. “Low”
is the decile portfolio with the lowest one-month-ahead expected return forecast (decile 1), “High”
is the decile portfolio with the highest one-month-ahead expected return forecast (decile 10), and
“High−Low” denotes the long-short portfolio that buys the highest expected return bonds in decile
10 and sells the lowest expected return bonds in decile 1. The returns are in percent per month
and Newey-West t-statistics are reported in parentheses in the last column.

Table 3 presents the firm-level bond return results from long-short portfolios. Consistent with
11
Following Bai, Bali, and Wen (2019), we use the bond’s outstanding dollar values as weights. Since our
statistical objective functions minimize equally weighted forecast errors, we also repeat the analysis using
the equal-weighted portfolios and obtain qualitatively similar results.

16
 2
our earlier findings using ROS as the performance metric, Table 3 shows that all machine learning
forecasts generate economically and statistically significant return spreads on the long-short bond
portfolios, in the range of 0.33% and 0.79% per month, compared to the unconstrained OLS model
which delivers the smallest return spread of 0.16%. The top three best hedge portfolios are generated
by the RF, FFN, and LSTM, with the monthly return spread of 0.79% (t-statistic = 2.78), 0.75%
(t-statistic = 2.61), and 0.79% (t-statistic = 3.33), respectively. The forecast combination model
(Combination) also generates economically and statistically significant return spread of 0.67%
(t-statistic = 3.41). Overall, Table 3 shows that the machine learning approaches significantly
improve the forecasting performance for bond portfolios using firm-level bond characteristics as the
covariates.

In unreported results, we also calculate the alphas and their t-statistics for the four-factor model
of Bai, Bali, and Wen (2019) with the aggregate corporate bond market, the downside risk, the
credit risk, and the liquidity risk factors of corporate bonds. Consistent with the strong explanatory
power of these factors in explaining the cross-sectional variation in bond returns, we find that none
of the alpha spreads is statistically significant. This is not surprising given that downside risk,
credit risk, and liquidity risk as a whole are known to be pervasive and strong determinants of the
expected bond returns.

4.2 Using Stock Characteristics

Equity and corporate bonds are contingent claims on firm fundamentals but also differ in several
key features such as the payoff structure and the markedly different institutional and informational
frictions across equities and bonds. Motivated by these observations, a few studies investigate
whether a variety of stock characteristics predict corporate bond returns using cross-sectional
Fama-MacBeth regressions (Chordia et al., 2017; Choi and Kim, 2018). These studies find mixed
evidence on the role of stock characteristics for predicting future bond returns.12 Compared to
these studies which draw from the well of a limited number of predictors, we extend the list to a
much larger set of stock characteristics and more importantly, we rely on machine learning methods
to reduce redundant variation among predictors that address overfitting bias. Using the notation
from equation (12) of Section 2.2, our goal in this subsection is to predict corporate bond returns
as Et (RBit+1 ) = f1 (XSit ). In other words, while we do not impose the Merton (1974) model
restrictions explicitly, we do allow for linkages between stock and bond returns in allowing stock
characteristics (predictors of stock returns) to predict bond returns.
12
For example, Chordia et al. (2017) find that many equity characteristics, such as accruals, standardized
unexpected earnings, and idiosyncratic volatility, do not impact bond returns, whereas profitability and asset
growth are negatively related to corporate bond returns. In contrast, Choi and Kim (2018) find that some
variables (e.g., profitability and net issuance) fail to explain bond returns, and for others (e.g., investment
and momentum) bond return premia are too large compared with their loadings, or hedge ratios, on equity
returns of the same firms.

17
 2
Table 4 presents the ROS for the entire pooled sample of corporate bonds using all 94 stock
characteristics from Green, Hand, and Zhang (2017) and Gu, Kelly, and Xiu (2020) as the covariates.
The results in Table 4 are presented at the firm-level by constructing value-weighted average of firm-
level bond returns, as well as the firm-level value-weighted average of bond characteristics, using
amount outstanding as weights. Panel A of Table 4 shows that the OLS model with all 94 stock
2
characteristics produces an ROS of −3.09%, indicating that the model fails to deliver statistically
significant out-of-sample forecasting power for the expected corporate bond returns. However, the
2 .
other columns of Panel A show that the machine learning models substantially improve the ROS
2 of 1.61%, 1.57%, and
The penalized methods approach (LASSO, Ridge, and ENet) generate an ROS
1.62%, respectively, all of which are similar to those delivered by the dimension reduction approach
(PCA an PLS). Neural networks such as FFN and LSTM deliver significantly positive performance
2
and improve the ROS 2
to 1.88% and 2.00%, respectively. Figure 5 plots the ROS associated with
2
stock characteristics and shows that the ROS is in the range of 1.85% (Lasso) to 2.37% (FFN),
which is in similar magnitude to those generated by using corporate bond characteristics, also
presented in Figure 5.

In Panel B of Table 4, we form the long-short bond portfolios based on the machine learning
forecasts using stock characteristics only (XS). Consistent with our earlier findings using the out-
of-sample R-squared as the performance metric, Panel B shows that all machine learning forecasts
generate economically and statistically significant return spreads on the long-short bond portfolios,
in the range of 0.24% and 0.52% per month, compared to the unconstrained OLS model which
delivers the smallest return spread of 0.02% (t-statistic = 0.12). Overall, Table 4 shows that
the machine learning approaches significantly improve the return prediction performance for bond
portfolios using the stock characteristics as the covariates.

4.3 Do Stock Characteristics Improve the Predictive Power of

Bond Characteristics for Future Bond Returns?

The results so far suggest that all machine learning models produce significantly positive predictive
power using either set of characteristics, and the predictive performance with using the bond
characteristics is similar to that using the stock characteristics. In this section, we test whether
the stock characteristics provide incremental power in predicting future bond returns relative
to the bond characteristics. We start by predicting corporate bond returns as Et (RBit+1 ) =
f1 (XBit , XSit ).
2
Panel A of Table 5 reports ROS from alternative estimation methods implemented with
combining the 43 bond characteristics and 94 stock characteristics. Consistent with our previous
2
findings, the traditional OLS model produces an ROS of −5.38%, indicating that the model fails
to deliver statistically significant out-of-sample forecasting power for the expected corporate bond

18
returns. The other columns in Table 5, Panel A, show that the machine learning models using
2
the combined 137 characteristics deliver significantly positive ROS ranging from 1.60% (Ridge) to
2.11% (LSTM).

In Panel B, Table 5, we examine the improvement in the predictive power by comparing the
machine learning bond portfolios formed based on the 137 characteristics, f1 (XB, XS), to those
formed using the 43 bond characteristics, f1 (XB) from Section 4.1, or the 94 stock characteristics,
f1 (XS) from Section 4.2. Specifically, we calculate the difference in the High−Low long-short
portfolio that takes a long position in the highest expected return bonds and a short position in
the lowest expected return bonds based on different kinds of forecasts.

As shown in the last two rows of Panel B, Table 5, the economic significance of using both
bond and stock characteristics is small compared to using bond characteristics alone. We find
that most machine learning forecasts fail to deliver significantly positive return spread, indicating
that there is no difference in the performance of the machine learning models when adding stock
characteristics to the bond characteristics in forecasting future bond returns. In contrast, the
last row of Panel B shows that most of the models deliver significantly positive return spreads,
indicating the improvement in the models’ performance when adding bond characteristics to the
stock characteristics in predicting future bond returns. Overall, we conclude that although stock
characteristics produce significant explanatory power for bond returns when used alone, their
incremental predictive power relative to bond characteristics is economically insignificant, whereas
bond characteristics play a major role and improve the performance of stock characteristics in
predicting future bond returns.

4.4 Robustness Checks

4.4.1 Transaction Cost

Table OA1 of the Online Appendix provides robust checks of the main results in Table 3 and reports
the monthly performance of value-weighted decile portfolios sorted on out-of-sample machine
learning return forecasts using the 43 bond characteristics after accounting for transaction costs.
Following Bao, Pan, and Wang (2011), we use the Roll (1984) measure of effective spreads calculated
from autocovariances of bond returns and calculate transaction costs as the product of the portfolio
turnover and the time-series mean of the cross-sectional average effective spread. Consistent with
the findings in Table 3, Table OA1 shows that the machine learning approaches provide significantly
positive long-short portfolio returns net of transactions costs.13
13
A relatively low transaction cost is mainly driven by a low portfolio turnover, due to the persistence of
predicted bond returns.

19
4.4.2 Time-varying Performance

We investigate the time-varying performance of the machine learning bond portfolio returns
generated in Table 3. Table OA2 of the Online Appendix provides robustness checks and reports
the conditional portfolio performance across different economic states based on the Chicago Fed
National Activity Index (CFNAI).14 The results in Table OA2 show that the machine learning
bond portfolios exhibit significantly positive returns in both states of the economy, whereas the
unconstrained OLS model delivers insignificant return spread of 0.14% (t-statistic = 1.38) and
0.11% (t-statistic = 1.32) in good and bad economic state, respectively.

4.4.3 Maturity-matched Bond Excess Returns

Throughout the paper we measure bond excess return as the difference between bond return and
the risk-free rate proxied by the one-month Treasury bill rate. Table OA3 of the Online Appendix
provides robust checks of the main results in Table 3 using maturity-matched Treasury returns
to calculate bond excess returns. Consistent with the findings in Table 3, Table OA3 shows that
the machine learning approaches provide significantly positive long-short portfolio returns after
accounting for maturity-matched Treasury returns, with return spreads in the range of 0.31% and
0.73% per month.

4.4.4 Removing Financial Firms

We investigate whether our results are sensitive to the exclusion of financial firms. Following Fama
and French (1992), we exclude financial firms with SIC codes between 6000 and 6999 because the
high leverage that is normal for these firms probably does not have the same implication for non-
financial firms, where high leverage is more likely to indicate financial distress. Consistent with our
earlier findings, Table OA4 of the Online Appendix replicates the main findings in Table 2 (Panel
A), Table 4 (Panel B), Table 5 (Panel C), Table 6 (Panel D), Table 7 (Panel E), Table 8 (Panel F),
and Table 9 (Panel G) and shows similar results.
14
The CFNAI is a monthly index designed to assess overall economic activity and related inflationary
pressure. The CFNAI is a weighted average of 85 existing monthly indicators of national economic activity.
It is constructed to have an average value of zero and a standard deviation of one. An index value below
(above) zero corresponds to a good (bad) economic state.

20
5 Predicting Bond Returns with Regression-Based
Hedge Ratios
We have so far shown that the marginal improvement of the forecasting power of stock
characteristics relative to bond characteristics is economically small and statistically insignificant
in predicting future bond returns. The results of the previous section thus seem to provide prima
facie evidence of segmentation in the two markets. However, as noted in Section 2.2, the approach
in the previous section is a reduced-form approach that does not explicitly link the functional
forms of bond and stock expected return. In this section, we impose the dependence between
expected bond and stock return via Merton (1974) model and investigate the incremental power of
stock characteristics for future bond returns. The steps involved in estimating equation (13) are as
follows:

1. Estimate hedge ratios via regressions. Following Choi and Kim (2018), our baseline estimate
of the hedge ratio (ĥit ) is based on the 36-month rolling window regression,

RBis = αi + hit RSis + eBis , s = t − 35, . . . , t, (19)

where RBis is the firm-level excess bond returns in month s and RSis is the excess equity
return of the same firm i in month s. The output is ĥit .

2. Calculate RBmRSit+1 = RBit+1 − ĥit × RSit+1 , following equation (10).

3. Run separate machine learning models to predict the expected stock return Et (RSit+1 ) and
RBmRSit+1 .

Et (RBmRSit+1 ) = ψ2 (XBit )
Et (RSit+1 ) = ψ1 (XSit ). (20)

Taking Et (RBmRSit+1 ) = ψ2 (XBit ) as an example, the machine learning model is given

bond characteristics (XBit ) inputs to forecast the “dependent variable” (RBmRSit+1 ). The
output is a number ψ2 (XBit ) for each firm i and month t.

4. The prediction for expected bond return, a function of stock and bond characteristics and
the hedge ratio, is then given by plugging in the estimated quantities in equation (13) to
obtain:
Et (RBit+1 ) = f2 (XBit , XSit , ĥit ) = ĥit × ψ1 (XSit ) + ψ2 (XBit ). (21)

We then compare the forecasted bond returns in equation (21) to the ones from the previous
Section 4 without hedge ratios. We consider predictions using only bond characteristics, f1 (XB)

21
from Section 4.1, and using both bond and stock characteristics, f1 (XB, XS) from Section 4.3,
and evaluate whether or not f2 (XB, XS, ĥ) significantly outperforms f1 (·).15

Table 6 presents the forecasted bond returns based on equation (21). Consistent with our earlier
2
findings using ROS as the performance metric, Panel A shows that all machine learning forecasts
2 , in the range of 1.93% (LASSO) to 4.95%
generate economically and statistically significant ROS
(Combination). Panel B of Table 6 reports the Diebold-Mariano test statistics for comparisons of
f2 (XB, XS, ĥ) versus f1 (XB) and f1 (XB, XS). We find a positive and statistically significant test
statistic for six of the nine machine learning models with Diebold-Mariano test statistics ranging
from 0.21 (PCA) to 2.86 (Combination), compared to the bond return forecasts using only bond
characteristics, f1 (XB). Finally, the last row of Panel B shows a positive and statistically significant
test statistic for all machine learning models, indicating superior performance of f2 (XB, XS, ĥ)
compared to bond return forecasts generated using the combined stock and bond characteristics,
f1 (XB, XS).

To further investigate the economic significance of our findings, we form the long-short bond
portfolios based on the machine learning forecasts based on equation (21). Consistent with our
2
earlier findings using the ROS as the performance metric, Table 7 shows that f2 (XB, XS, ĥ)
generates economically and statistically significant return spreads on the long-short bond portfolios,
in the range of 0.55% and 0.92% per month, compared to the unconstrained OLS model which
delivers the smallest return spread of 0.18% (t-statistic = 1.07). Finally, the last two rows of
Table 7 examine the improvement in the predictive power by comparing the machine learning bond
portfolios formed based on the restrictions to those without restrictions. Specifically, we calculate
the average return (double) differences of the machine learning High−Low bond portfolios, (i)
formed from sorting on forecasts f2 (XB, XS, ĥ) and those on f1 (XB), and (ii) formed from sorting
on forecasts f2 (XB, XS, ĥ) and those on f1 (XB, XS). As shown in Table 7, the average return
differences of the machine learning bond portfolios are all economically large and statistically
significant, indicating that there is improvement in the performance of the machine learning
models when we impose restrictions from the Merton (1974) model. Overall, we conclude that
it is important to impose such restrictions when estimating bond expected returns, where equity
characteristics provide significant improvement above and beyond bond characteristics for future
bond returns.
15
Step 3 above involves predicting stock returns using stock characteristics. Gu, Kelly, and Xiu (2020)
find that machine learning offers an improved description of expected return relative to traditional methods
in forecasting future stock returns. Consistent with their findings, Table OA5 of the Online Appendix shows
that the machine learning methods provide strong forecasting power using the stock characteristics.

22
6 Predicting Bond Returns with Machine Learning-
Based Hedge Ratios
The previous section uses firm-specific hedge ratios estimated via regressions. Since we estimate
rolling window regressions, the hedge ratios are allowed to vary over time. An alternative approach
is to use a full-scale structural model and use the estimated parameters to calculate hedge ratios. For
example, Schaefer and Strebulaev (2008) is the first article to provide a comprehensive investigation
of the magnitude and statistical significance of the hedge ratio. In this section, we follow the spirit
of their approach by estimating hedge ratio using different machine learning approaches. The hedge
ratio estimated in this section is time-varying and also a function of bond characteristics, that is,
hit = ϕ3 (XBit ). The steps involved in estimating equation (14) are as follows:

1. Estimate the hedge ratio, ĥ(XBit ), based on the following functional form using a 36-month
rolling window:
RBis = h(XBis−1 )RSis + uBis , s = t − 35, . . . , t, (22)

where RBi,s is the firm-level excess bond returns in month s, calculated as the value-weighted
average excess returns of individual bonds issued by firm i, and RSi,s is the excess equity
return of the same firm i in month s. The machine is given inputs including the bond
characteristics (XBis−1 ), realized bond returns (RBis ), and realized stock returns (RSis ).
The machine outputs a “fitted value” Ê(RBis |XBis−1 , RSis ) = ĥ(XBis−1 ) × RSis , which
could be linear or non-linear, depending on the specific machine learning model used. Using
the outputs of the machine we can calculate both the out-of-sample fitted value ĥ(XBit ) ×
RSit+1 and the hedge ratio ĥ(XBit ) = ϕ3 (XBit ).16

2. Calculate RBmRSit following equation (10) as

RBmRSit+1 = RBit+1 − ĥ(XBit ) × RSit+1 .

3. Run separate machine learning models to predict RSit+1 and RBmRSit+1 .

Et (RBmRSit+1 ) = ϕ2 (XBit )
Et (RSit+1 ) = ϕ1 (XSit ). (23)
16
As an illustrative example, consider prediction using neural
 network. Given XBis−1 , the machine
(k) (k)
generates K units of neurons for each layer l as XBl = g θl XBis−1 , where g(·) is the nonlinear
(k)
activation function.
 Then, in the last layer, we multiply each XBL by RSis . The output is the fitted
(k)
value RB is = g XB RSis = ĥ(XBis−1 ) × RSis . We can calculate the out-of-sample fitted value as
d
L

Ê(RBit+1 |XBit , RSit+1 ) = ĥ(XBit ) × RSit+1 . When needed, the hedge ratio itself can be recovered by
‘setting’ the stock return to be one to obtain ĥ(XBit ) = Ê(RBit+1 |XBit , 1).

23
 Taking Et (RBmRSit+1 ) = ϕ2 (XBit ) as an example, the machine learning model is given
inputs including the bond characteristics (XBit ) to forecast the “dependent variable”
(RBmRSit+1 ). The output is a number ϕ2 (XBit ) for each firm i and month t. Note that
the prediction of the stock return in equation (23) is the same as that in equation (20),
ϕ1 (XSit ) = ψ1 (XSit ).

4. The prediction for expected bond return, a function of stock and bond characteristics and
the hedge ratio, is then given by plugging in the estimated quantities in equation (14) to
obtain:

Et (RBit+1 ) = f3 (XBit , XSit , ĥ(XBit )) = ϕ3 (XBit ) × ϕ1 (XSit ) + ϕ2 (XBit ). (24)

We then compare the forecasted bond returns from equation (24), f3 (XB, XS, ĥ(XB)), to
f2 (XB, XS, ĥ) from Section 5, and evaluate whether or not forecasts from machine learning based
hedge ratios significantly outperform bond returns forecasted using the regression-based hedge
ratios.

Table 8 presents the forecasted bond returns based on equation (24). Consistent with our
2
earlier findings using ROS as the performance metric, Panel A shows that all machine learning
2 , in the range of 2.04% to 5.70%.
forecasts generate economically and statistically significant ROS
Panel B of Table 8 compares the forecasted bond returns with Merton model restriction and
machine learning estimated hedge ratio (i.e., f3 (XB, XS, ĥ(XB)) to the bond return forecasts from
Section 4 obtained using bond characteristics, f1 (XB), the combined stock and bond characteristics,
f1 (XB, XS), and bond return forecasts from Section 5 using f2 (XB, XS, ĥ). Panel B shows a
positive and statistically significant test statistic for all the machine learning models with Diebold-
Mariano test statistics, compared to the bond return forecasts using only bond characteristics,
f1 (XB), or the combined stock and bond characteristics, f1 (XB, XS). Finally, the last row
of Panel B shows a positive and statistically significant test statistic for all machine learning
models, indicating superior performance of f3 (XB, XS, ĥ(XB)) compared to bond return forecasts
generated using regression-based hedge ratios, f2 (XB, XS, ĥ).

Table 9 investigates the long-short portfolios of corporate bonds constructed with the machine
learning forecasts based on f3 (XB, XS, ĥ(XB)). Consistent with our earlier findings using the out-
of-sample R-squared as the performance metric, Table 9 shows that f3 (XB, XS, ĥ(XB)) generates
economically and statistically significant return spreads on the long-short bond portfolios, in the
range of 0.54% and 1.00% per month, compared to the unconstrained OLS model which delivers the
smallest return spread of 0.16% (t-statistic = 0.53). Moreover, the average return spreads on the
machine learning bond portfolios are all economically large and statistically significant, indicating
that there is improvement in the performance of the machine learning models when we impose
restrictions from the Merton (1974) model with machine learning estimated hedge ratio. Finally,

24
the last row of Table 9 shows small and insignificant return spreads, indicating a relatively small
improvement in economic significance between f3 (XB, XS, ĥ(XB)) estimated in equation (24) and
f2 (XB, XS, ĥ) estimated from equation (21). Overall, we conclude that machine learning based
hedge ratios provide more accurate predictions than the regression-based hedge ratios in terms of
statistical significance. However, the economic significance of the predictions from both approaches
turns out to be similar. One possible reason is that regression-based hedge ratios, being calculated
over rolling windows, already account for time-variation in hedge ratios. We investigate these hedge
ratios next.

To what extent the regression-based and machine-learning-based hedge ratios differ from each
other? To answer this question, we choose the stochastic variance-based hedge ratio, equation (3) of
Section 2.1, as the benchmark, and compare the mean squared error (MSEs) of the regression-based
hedge ratio with the machine-learning-based ones. Specifically, let hit and ĥit be the benchmark
and alternative hedge ratios, respectively. The MSE can be defined as

MSE = E(ĥit − hit )2 . (25)

To calculate the benchmark hedge ratio, we estimate the asset variance following equation (8) of
Schaefer and Strebulaev (2008), with which we calculate the long-term mean (θ) and volatility of
volatility (γ). We assume the speed of mean reversion κ = 4 across all firms.

Table 10 reports the MSEs of different hedge ratios. We find that the MSE for the regression-
based hedge ratio is 0.051, similar to those delivered by the machine learning-based hedge ratios,
in the range of 0.053 (Ridge) and 0.057 (FFN). Both the regression-based and machine learning-
based hedge ratio MSEs are much smaller than the unconstrained OLS model, which delivers the
highest MSE of 0.097. The next two rows in this table report the MSE for the subsample based on
the firm-level credit rating of individual bonds, and show smaller MSEs for non-investment-grade
bonds than investment-grade bonds. The last two rows of the table show the smallest MSE for
short-maturity bonds compared to the medium- and long-maturity bonds. Overall, the results are
consistent with our earlier findings in Section 6 that the economic significance of using machine
learning-based hedge ratio is similar to that using regression-based hedge ratio.

7 Conclusion
Using a variety of machine learning methods, we provide a comprehensive study of the cross-
sectional pricing of corporate bonds using a large set of 94 stock characteristics and 43 bond
characteristics. Because of the nonlinear payoffs of corporate bonds and the high correlation
between many of the stock and bond characteristics, machine learning approaches are well suited
for such challenging prediction problems by mitigating overfitting biases and uncovering complex

25
patterns and hidden relationships.

Motivated by the Merton (1974) model that both equity and corporate bonds are contingent
claims on firms, we explicitly link the functional forms of bond and stock expected returns by
imposing economic structure when investigating bond expected returns. We find that the traditional
linear regression models such as the OLS perform poorly, whereas the machine learning methods
substantially improve the out-of-sample performance in predicting the cross-sectional differences in
future bond returns. We show that using the reduced-form approach, the incremental improvement
of stock characteristics relative to bond characteristics is economically and statistically small
in forecasting future bond returns. However, after imposing the dependence between expected
returns of bonds and stocks via the Merton (1974) model, we find economically and statistically
large improvement in all machine learning forecasting models compared to the ones without any
restrictions. Overall, our work highlights the importance of explicitly imposing the dependence
between expected bond and stock returns when investigating expected bond returns.

26
Appendices

A Derivation of ht and αt
This section provides the analytical solutions for ht and αt in Section 2.1.
When the variance of the firm value is constant, according to Merton (1974), the equity price
is equal to the European call price:

Et = Ct (σ 2 ) = Vt N (d1 ) − Be−r(T −t) N (d2 ), (A.1)

2 /2)(T −t) √
where d1 = ln(Vt /B)+(r+σ
√
σ T −t
, d2 = d1 − σ T − t, and N (·) is the standard normal cumulative
distribution function. When the variance is stochastic, Hull and White (1987) propose to
2 ] with Taylor expansion as
approximate Et at E[σ̄t,T

1 ∂ 2 Ct (σ 2 ) 

2 2
Et ≈ Ct (E[σ̄t,T ]) + · Var(σ̄t,T ), (A.2)
2 ∂σ 2 σ2 =E[σ̄2
t,T ]

RT
2
where σ̄t,T = T 1−t t σs2 ds is average variance of stochastic variance over time t to maturity T .
Given equation (A.1), together with Cox, Ingersoll, and Ross (1985), we have

1 T
e−κt − e−κT 2
Z
2
E[σ̄t,T ] = E[σs2 |σ02 ]ds = θ + (σ0 − θ). (A.3)
T −t t κ(T − t)
Z T
2 1
Var(σ̄t,T ) = Var(σs2 |σ02 )ds
T −t t
θγ 2 γ 2 (e−κt − e−κT ) 2 γ 2 (e−2κt − e−2κT )
= + (σ0 − θ) + (θ − 2σ02 ). (A.4)
2κ κ(T − t) 4κ2 (T − t)

In the literature of asset pricing with stochastic volatility, κ is sufficiently positive. For example,
Aı̈t-Sahalia and Kimmel (2007) suggest κ > 4 for pricing equity index options, which implies a
fairly fast speed of σt2 converting to its long-run mean θ. Thus, equations (A.3) and (A.4) can be
approximated as
2
E[σ̄t,T ] = θ, (A.5)
2 θγ 2
Var(σ̄t,T ) = . (A.6)
2κ

With some algebra, we can rewrite equation (A.2) as

√
−r(T −t) θγ 2
Et = Vt N (d1 ) − Be N (d2 ) − · ηt , (A.7)
8κ
where
" #
√ (T − t) − 2θ ln( VBt ) 1
ηt = Vt ϕ(d1 ) T − t d1 · p + , (A.8)
4 θ(T − t) 2θ

27
and ϕ(·) is the probability density function of standard normal distribution. Clearly, equation (A.7)
reduces to equation (A.1) when γ = 0 and σt2 = θ, the case with constant variance.
Now we derive the hedge ratio as follows. According to Schaefer and Strebulaev (2008), the
ratio is defined as:
" #
∂Et −1

Et
ht = −1 . (A.9)
∂Vt Dt

With constant variance, from equation (A.7) we get:

1 − N (d1 ) Et
 
ht = . (A.10)
N (d1 ) Dt

In this case, the systemic risk of bond returns can be perfectly hedged by equity returns (because
both bond and equity are driven by the dynamics of firm value alone):

dDt dEt
− ht = 0. (A.11)
Dt Et

In the case of stochastic variance, the hedge ratio of equation (A.9) is

1 − N (d1 ) + γ 2 ζt Et
ht = · , (A.12)
N (d1 ) − γ 2 ζt Dt

where
" #
ϕ(d1 ) (T − t) − 2θ ln( VBt )  2 p  pθ(T − t) − 2d
1
ζt = p d1 + d1 θ(T − t) − 1 + . (A.13)
8κ 4 θ(T − t) 2θ

Because bond and equity are driven by both the dynamics of firm value and variance, the systemic
risk of bond returns cannot be perfectly hedged by equity returns, and they have the following
relation:
dDt dEt
− ht = αt dt, (A.14)
Dt Et

where
p
θ(T − t) δ1 δ2 (1 − d21 ) − δ2 d1 /2θ + a1 + a2
 
γ 2 ϕ(d1 )Vt
αt = · hp
 i (A.15)
8κDt N (d1 ) − γ2
− 1 2 d1
8κ ϕ(d1 ) θ(T t) δ d
1 1 + 2θ − δ1 (1 − d1 ) + θ

in which
ln(Vt /B)
ln(Vt /B) + (r + θ/2)(T − t) (T − t) − 2/θ · ln(Vt /B) − (r + 2θ )
d1 = p , δ1 = p , δ2 = T −t p ,
θ(T − t) 4 θ(T − t) 2 θ(T − t)
2
h i
2
 2

σ 2 δ d (3 − d2 ) + 3+d1 − θd δ − 1/2
σ δ1 (1 − d1 ) − d1 /θ − d1 /4 t 1 1 1 2θ 1 1
a1 = t , a2 = .
2θ(T − t)
p
2 θ(T − t)

28
B Corporate Bond Characteristics
This section describes a broad set of the 43 corporate bond characteristics, designed to be
representative of (i) bond-level characteristics such as issuance size, credit rating, time-to-maturity,
and duration, (ii) proxies of risk such as bond systematic risk, downside risk, and credit risk, (iii)
proxies of bond-level illiquidity constructed using daily and intraday transaction data and liquidity
risk, (iv) past bond return characteristics such as bond momentum, short-term and long-term
reversals, and the distributional characteristics such as return volatility.

1. Credit rating (Rating ). We collect bond-level rating information from Mergent FISD
historical ratings. All ratings are assigned a number to facilitate the analysis, for example, 1
refers to a AAA rating, 2 refers to AA+, ..., and 21 refers to CCC. Investment-grade bonds
have ratings from 1 (AAA) to 10 (BBB−). Non-investment-grade bonds have ratings above
10. A larger number indicates higher credit risk, or lower credit quality. We determine a
bond’s rating as the average of ratings provided by S&P and Moody’s when both are available,
or as the rating provided by one of the two rating agencies when only one rating is available.

2. Time-to-maturity (MAT ). The number of years to maturity.

3. Issuance size (Size). The natural logarithm of bond amount outstanding.

4. Age (Age). Bond age since the first issuance, in the number of years.

5. Duration (DUR). A bond’s price sensitivity to interest rate changes, measured in years.

6. Downside risk proxied by the 5% VaR (VaR5 ). Following Bai, Bali, and Wen (2019),
we measure downside risk of corporate bonds using VaR, which determines how much the
value of an asset could decline over a given period of time with a given probability as a
result of changes in market rates or prices. Our proxy for downside risk, 5% Value-at-Risk
(V aR5), is based on the lower tail of the empirical return distribution, that is, the second
lowest monthly return observation over the past 36 months. We then multiply the original
measure by −1 for convenience of interpretation.17

7. Downside risk proxied by the 10% VaR (VaR10 ). This measure is defined as the
fourth lowest monthly return observation over the past 36 months. We then multiply the
original measure by −1 for convenience of interpretation.

8. Downside risk proxied by the 5% Expected Shortfall (ES5 ). An alternative measure

of downside risk, “expected shortfall,” is defined as the conditional expectation of loss given
that the loss is beyond the VaR level. In our empirical analyses, we use the 5% expected
shortfall (ES5) defined as the average of the two lowest monthly return observations over the
past 36 months (beyond the 5% VaR threshold).

9. Downside risk proxied by the 10% Expected Shortfall (ES10 ). An alternative

measure of downside risk, “expected shortfall,” is defined as the conditional expectation of
loss given that the loss is beyond the VaR level. In our empirical analyses, we use the
17
Note that the original maximum likely loss values are negative since they are obtained from the left tail
of the return distribution. After multiplying the original VaR measure by −1, a positive regression coefficient
and positive return/alpha spreads in portfolios are interpreted as the higher downside risk being related to
the higher cross-sectional bond returns.

29
 10% expected shortfall (ES10) defined as the average of the four lowest monthly return
observations over the past 36 months (beyond the 10% VaR threshold).

10. Illiquidity (ILLIQ). A bond-level illiquidity measure. We follow Bao, Pan, and Wang
(2011) to construct the measure, which aims to extract the transitory component from bond
price. Specifically, let ∆pitd = pitd − pitd−1 be the log price change for bond i on day d of
month t. Then, ILLIQ is defined as

ILLIQ = −Covt (∆pitd , ∆pitd+1 ).

11. Roll’s daily measure of illiquidity (Roll ). As an alternative measure of bond-level

illiquidity using daily bond returns, the Roll (1984) measure is defined as,
 p
2 −cov(rd , rd−1 ) if cov(rd , rd−1 ) < 0,
Roll =
0 otherwise,

where rd is the corporate bond return on day d. Given the fact that corporate bonds do not
trade frequently, this measure crucially depends on two conditions. First, a bond is traded
for two days in a row so that we can calculate its daily return. Second, a bond has at least
a number of daily returns calculated each month so that we can calculate its covariance. We
set the threshold equal to five. A bond’s monthly Roll measure will be missing if that bond
does not have five daily returns calculated that month.

12. Roll’s intraday measure of illiquidity (TC Roll). Following Dick-Nielsen, Feldhütter,
and Lando (2012), we employ an intraday version of the Roll (1984) estimator for effective
spreads,  p
2 −cov(ri , ri−1 ) if cov(ri , ri−1 ) < 0,
T C Roll =
0 otherwise,
Pi −Pi−1
where ri = Pi−1 is the return of the ith trade.

13. High-low spread estimator(P HighLow). Following Corwin and Schultz (2012), we use
the ratio between the daily high and low prices on consecutive days to approximate bid-ask
spreads. With such motivation, their effective spread proxy is defined as

2(eα − 1)
P HighLow = ,
1 + eα
√ √
2β − β γ
r
α = √ − √ ,
3−2 2 3−2 2
1   2
X Ht+j
β = ln ,
Lt+j
j=0
  2
Ht,t+1
γ = ln .
Lt,t+1

Ht (Lt ) is the highest (lowest) transaction price at day t, and Ht,t+1 (Lt,t+1 ) is the highest
(lowest) price on two consecutive days t and t + 1. Again, we take the mean of the daily
values in a month to get a monthly spread proxy for each bond.

14. Illiquidity measure based on zero returns (P Zeros). Following Lesmond, Ogden,

30
 and Trzcinka (1999), we use the proportion of zero return days as a measure of liquidity.
Lesmond, Ogden, and Trzcinka (1999) argue that zero volume days (hence zero return days)
are more likely to reflect lower liquidity. We compute their measure on a monthly basis with
T as the number of trading days in a month,
# of zero return days
P Zeros = ,
T
The number of zero return days comprises two parts, the sequential days with no price change
hence zero returns, and the days with zero trading volume.

15. Modified illiquidity measure based on zero returns (P FHT). Fong, Holden, and
Trzcinka (2017) propose a new bid-ask spread proxy based on the zeros measure in Lesmond,
Ogden, and Trzcinka (1999). In their framework, symmetric transaction costs of S/2 leads
to observed returns of 
∗ S ∗ S
R + 2 if R < − 2 ,

R= 0 if − S2 < R∗ < S2 ,

 ∗ S
R − 2 if S2 < R∗ ,
where R∗ is the unobserved true value return, which they assume to be normally distributed
with mean zero and variance σ 2 . Hence, they equate the theoretical probability of a zero
return with its empirical frequency, measured via P Zeros. Solving for the spread S, they
get  
−1 1 + P Zeros
P F HT = S = 2 · σ · Φ
2
where Φ−1 is the inverse of the cumulative standard normal distribution. We compute a
bond’s σ for each month and then calculate P F HT .

16. Amihud measure of illiquidity (Amihud ). Following Amihud (2002), the measure is
motivated to capture the price impact and is defined as,
N
1 X |rd |
Amihud = ,
N Qd
d=1

where N is the number of positive-volume days in a given month, rd the daily return, and
Qd the trading volume on day d, respectively.

17. An extended Roll’s measure (PI Roll ). Goyenko, Holden, and Trzcinka (2009) derive
an extended transaction cost proxy measure, which for every transaction cost proxy tcp and
average daily dollar volume Q in the period under observation is defined as
Roll
P I Roll = .
Q

18. An extended FHT measure based on zero returns (PI FHT ).

P F HT
P I F HT = .
Q

where P F HT is the modified illiquidity measure based on zero returns (Fong, Holden, and

31
 Trzcinka, 2017) and Q is the average daily dollar volume in the period under observation.
19. An extended High-low spread estimator (PI HighLow ).

P HighLow
P I HighLow = .
Q

where P I HighLow is the high-low spread estimator following Corwin and Schultz (2012)
and Q is the average daily dollar volume in the period under observation.
20. Std.dev of the Amihud measure (Std Amihud). The standard deviation of the daily
Amihud measure within a month.
21. Lambda (PI Lambda). Hasbrouck (2009) proposes Lambda as a high-frequency price
impact measure for equities. PI Lambda (λ) is estimated in the regression,
q
rτ = λ · sign(Qτ ) · |Qτ | + ϵτ,

where rτ is the stock’s return and Qτ is the signed traded dollar volume within the five
minute period τ . Following Hasbrouck (2009) and Schestag, Schuster, and Uhrig-Homburg
(2016), we take into account the effects of transaction costs on small trades versus large
trades (Edwards, Harris, and Piwowar, 2007) and run the adjusted regression,
p
ri = α · Di + λ · Di · Qi + ϵi ,

where λ is estimated in the equation above excluding all overnight returns and Di is an
indicator variable of trades defined as the following,

1
 if trade i is a buy,
Di = 0 if trade i is an interdealer trade,

−1 if trade i is a sell.


22. Difference of average bid and ask prices (AvgBidAsk). Following Hong and Warga
(2000) and Chakravarty and Sarkar (2003), we use the difference between the average
customer buy and the average customer sell price on each day to quantify transaction costs:

PtBuy − PtSell
AvgBidAsk =
0.5 · (PtBuy + PtSell )

Buy/Sell
where Pt is the average price of all customer buy/sell trades on day t. We calculate
AvgBidAsk for each day on which there is at least one buy and one sell trade and use the
monthly mean as a monthly transaction cost measure.
23. Interquartile range (TC IQR). Han and Zhou (2007) and Pu (2009) use the interquartile
range of trade prices as a bid-ask spread estimator. They divide the difference between the
75th percentile Pt75th and the 25th percentile Pt25th of intraday trade prices on day t by the
average trade price Pt of that day:

Pt75th − Pt25th
T C IQR = ,
Pt

32
 We calculate TC IQR for each day that has at least three observations and define the monthly
measure as the mean of the daily measures.

24. Round-trip transaction costs (RoundTrip). Following Feldhütter (2012), we aggregate

all trades per bond with the same volumes that occur within a 15-minute time window to a
round-trip transaction. We then compute the estimator for round-trip transaction costs as the
doubled difference between the lowest and highest trade price for each round-trip transaction.
To obtain a relative spread proxy, we divide the round-trip transaction cost estimator by the
mean of the maximum and the minimum price. A bond’s monthly round-trip measure is
then obtained by averaging over all round-trip trades in a month.

25. Pastor and Stambaugh’s liquidity measure (GammaPS, γP S ). Pástor and Stambaugh
(2003) develop a measure for price impact based on price reversals for the equity market. It
is given by the estimator for γ in the following regression:
e
rt+1 = θ + ψ · rt + γ · sign(rte ) · Qt + ϵt ,

where rte is the security’s excess return over a market index return, rt is the security’s return
and Qt is the trading volume at day t. For corporate bond market index, we use Merrill
Lynch aggregate corproate bond index. γ should be negative and a larger price impact leads
to a larger absolute value. As liquidity measures generally assign larger (positive) values to
more illiquid bonds, we define γP S = −γ expect it to be positively correlated with the other
liquidity measures.

26. Bond market beta (β Bond ). We estimate the bond market beta, β Bond , for each bond
from the time-series regressions of individual bond excess returns on the bond market excess
returns (MKTBond ) using a 36-month rolling window. We compute the bond market excess
return (MKTBond ) as the value-weighted average returns of all corporate bonds in our sample
minus the one-month Treasury-bill rate.18

27. Default beta (β DEF ). We estimate the default beta for each bond from the time-series
regressions of individual bond excess returns on the bond market excess returns (MKTBond )
and the default factor using a 36-month rolling window. Following Fama and French (1993),
the default factor (DEF) is defined as the difference between the return on a market portfolio
of long-term corporate bonds (the composite portfolio on the corporate bond module of
Ibbotson Associates) and the long-term government bond return.

28. Term beta (β T ERM ). We estimate the default beta for each bond from the time-series
regressions of individual bond excess returns on the bond market excess returns (MKTBond )
and the term factor using a 36-month rolling window. Following Fama and French (1993), the
term factor (TERM) is defined as the difference between the monthly long-term government
bond return (from Ibbotson Associates) and the one-month Treasury bill rate.

29. Illiquidity beta (β LW W ). Following Lin, Wang, and Wu (2011), it is estimated as the
exposure to the bond illiquidity factor, which is defined as the average return difference
between the high liquidity beta portfolio (decile 10) and the low liquidity beta portfolio
(decile 1).
18
We also consider alternative bond market proxies such as the Barclays Aggregate Bond Index and
Merrill Lynch Bond Index. The results from these alternative bond market factors turn out to be similar to
those reported in our tables.

33
30. Downside risk beta (β DRF ). Following Bai, Bali, and Wen (2019), for each bond and each
month in our sample, we estimate the factor beta from the monthly rolling regressions of
excess bond returns on the downside risk factor (DRF) over a 36-month fixed window after
controlling for the bond market factor (MKTBond ).

31. Credit risk beta (β CRF ). Similar to the construction of downside risk beta, for each
bond and each month in our sample, we estimate the factor beta from the monthly rolling
regressions of excess bond returns on the credit risk factor (CRF) over a 36-month fixed
window after controlling for the bond market factor (MKTBond ).

32. Illiquidity risk beta (β LRF ). Similar to the construction of downside risk and credit risk
beta, for each bond and each month in our sample, we estimate the factor beta from the
monthly rolling regressions of excess bond returns on the liquidity risk factor (LRF) over a
36-month fixed window after controlling for the bond market factor (MKTBond ).

33. Volatility beta (β V IX ). Following Chung, Wang, and Wu (2019), we estimate the following
bond-level regression

Ri,t = αi + β1,i M KTt + β2,i SM Bt + β3,i HM Lt + β4,i DEFt + β5,i T ERMt + β6,i ∆V IXt + ϵi,t ,

where Ri,t is the excess return of bond i in month t, and M KTt , SM Bt , HM Lt , DEFt ,
T ERMt , and ∆V IXt denote the aggregate corporate bond market, the size factor, the book-
to-market factor, the default factor, the term factor, and the market volatility risk factor,
respectively.

34. Macroeconomic Uncertainty Beta (β U N C ). Following Bali, Subrahmanyam, and Wen

(2021b), for each bond-month in our sample, we estimate the uncertainty beta from monthly
rolling regressions of excess bond returns on the change in the economic uncertainty index
(∆UNC) and the excess bond market returns (MKT), using the past 24 to 36 months of data
(as available):
UNC M KT
Ri,t = αi,t + βi,t · ∆U N Ct + βi,t · M KTt + ϵi,t ,
where Ri,t is the excess return of bond i in month t, ∆U N Ct is the change in the economic
uncertainty index in month t based on Jurado, Ludvigson, and Ng (2015), M KTt is the
aggregate corporate bond market, βi,tU N C is the uncertainty beta of bond i in month t.

35. Short-term reversal (REV ). The bond return in previous month.

36. Six-month momentum (MOM6 ). Following Jostova et al. (2013), it is defined as the
cumulative bond returns over months from t − 7 to t − 2 (formation period), skipping the
short-term reversal month.

37. Twelve-month momentum (MOM12 ). It is defined as the cumulative bond returns over
months from t − 12 to t − 2 (formation period), skipping the short-term reversal month.

38. Long-term reversal (LTR). Following Bali, Subrahmanyam, and Wen (2021a), it is defined
as the past 36-month cumulative returns from t − 48 to t − 13, skipping the 12-month
momentum and short-term reversal month.

39. Volatility (VOL). Following Bai, Bali, and Wen (2016), it is estimated using a 36-month

34
 rolling window for each bond in our sample
n
1 X
V OLi,t = (Ri,t − Ri )2 .
n−1
t=1

40. Skewness (SKEW ). Similar to the construction of volatility, skewness is estimated using
a 36-month rolling window for each bond in our sample
n 3
Ri,t − Ri

1X
SKEWi,t = .
n σi,t
t=1

41. Kurtosis (KURT ). Similar to the construction of volatility and skewness, kurtosis is
estimated using a 36-month rolling window for each bond in our sample
n 4
Ri,t − Ri

1X
KU RTi,t = − 3.
n σi,t
t=1

42. Co-skewness (COSKEW ). Harvey and Siddique (2000), Mitton and Vorkink (2007),
and Boyer, Mitton, and Vorkink (2010) provide empirical support for the three-moment
asset pricing models that stocks with high co-skewness, high idiosyncratic skewness, and
high expected skewness have low subsequent returns. Following the aforementioned studies,
we decompose total skewness into two components; systematic skewness and idiosyncratic
skewness, which are estimated based on the following time-series regression for each bond
using a 36-month rolling window:
2
Ri,t = αi + βi · Rm,t + γi · Rm,t + εi,t .

where Ri,t is the excess return on bond i, Rm,t is the excess return on the bond market
portfolio, γi is the systematic skewness (co-skewness) of bond i.
43. Idiosyncratic skewness (ISKEW ). The idiosyncratic skewness (ISKEW ) of bond i is
defined as the skewness of the residuals (εi,t ) in co-skewness regression equation.

C Four Groups of Corporate Bond Characteristics

We classify all 43 corporate bond characteristics into the following four broad categories,
ˆ Group I: Bond characteristics related to interest risk or maturity: Rating, M AT , Size, Age,
DU R.
ˆ Group II: Bond characteristics related to risk measures such as downside risk or systematic
risk: VaR5, VaR10, ES5, ES10, β Bond , β DEF , β T ERM , β U N C , β V IX , β LW W , β DRF , β CRF ,
β LRF , COSKEW .
ˆ Group III: Bond-level illiquidity and illiquidity risk: ILLIQ, Roll, T CRoll, P HighLow,
P Zeros, P F HT , Amihud, P I Roll, P I F HT , P I HighLow, Std Amihud, P I Lambda,
AvgBidAsk, T C IQR, Roundtrip, γP S .
ˆ Group IV: Past return characteristics: REV , MOM6, MOM12, LT R, V OL, SKEW , KU RT ,
ISKEW .

35
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39
 Table 1: Descriptive statistics
Panel A reports the total number of observations, the cross-sectional mean, median, standard deviation and monthly return percentiles of corporate
bonds, and bond characteristics including credit rating, time-to-maturity (Maturity, year), amount outstanding (Size, $ million), duration, downside
risk (5% Value-at-Risk, VaR), illiquidity (ILLIQ), and the CAPM beta based on the corporate bond market index, β Bond . The numbers are presented
at the firm-level using value-weighted average of firm-level bond returns and bond characteristic measures. Ratings are in conventional numerical
scores, where 1 refers to an AAA rating and 21 refers to a C rating. Higher numerical score means higher credit risk. Numerical ratings of 10 or
below (BBB- or better) are considered investment grade, and ratings of 11 or higher (BB+ or worse) are labeled high yield. Downside risk is the 5%
Value-at-Risk (VaR) of corporate bond return, defined as the second lowest monthly return observation over the past 36 months. The original VaR
measure is multiplied by −1 so that a higher VaR indicates higher downside risk. Bond illiquidity is computed as the autocovariance of the daily price
changes within each month, multiplied by −1. β Bond is the corporate bond exposure to the excess corporate bond market return, constructed using
the Merrill Lynch U.S. Aggregate Bond Index. The betas are estimated for each bond from the time-series regressions of bond excess returns on the
excess bond market return using a 36-month rolling window estimation. Panel B reports the time-series average of the cross-sectional correlations.
The sample period is from July 2002 to December 2017.

Panel A: Cross-sectional statistics over the sample period of July 2002 – December 2017
Percentiles
40

N Mean Median SD 5th 25th 75th 95th

Bond return (%) 146,085 0.59 0.52 3.18 −3.04 −0.33 1.47 4.33
Rating 146,085 10.08 9.50 3.94 4.33 7.33 13.16 16.50
Time to maturity (maturity, year) 146,085 8.05 6.75 5.24 2.46 4.86 9.60 17.98
Amount out (size, $million) 146,085 500.17 390.00 413.65 150.00 250.00 594.64 1271.74
Duration (DUR) 146,085 4.63 4.55 2.40 0.09 3.19 5.97 8.83
Downside risk (5% VaR) 146,085 2.35 1.38 4.02 0.00 0.00 2.79 8.37
Illiquidity (IILIQ) 146,085 1.07 0.21 3.74 -0.02 0.05 0.68 4.20
Bond market beta (β Bond ) 146,085 0.42 0.31 0.56 0.00 0.00 0.67 1.27

Panel B: Average cross-sectional correlations

Maturity Size DUR VaR5 ILLIQ β Bond
Rating −0.26 −0.18 −0.15 0.25 0.07 0.01
Maturity 1.00 0.06 0.56 −0.04 0.04 0.07
Size 1.00 0.08 0.00 −0.08 0.13
DUR 1.00 −0.09 −0.02 0.11
VaR5 1.00 0.19 0.61
ILLIQ 1.00 0.04
 Table 2: Predicting corporate bond returns with bond characteristics
2
Panel A of this table reports out-of-sample R-squared (ROS , in percentage) for the entire panel of corporate bonds using the 43 bond characteristics,
following equation (12) as f1 (XB). The results are presented at the firm-level by constructing value-weighted firm-level bond returns, as well as the
firm-level value-weighted bond characteristics, using amount outstanding as weights. The models include OLS with all bond characteristics (OLS),
principal component analysis (PCA), partial least square (PLS), LASSO, ridge regression (Ridge), elastic net (ENet), random forest (RF), feedforward
2
neural network (FFN), long short-term memory neural network (LSTM), and forecast combination (Combination). The ROS pools prediction errors
across firms and over time into a grand panel-level assessment of each model and is defined as,
P 2
2 (it)∈T3 (rit+1 − r̂it+1 )
ROS = 1 − P 2 .
(it)∈T3 rit+1

2
p-values associated with ROS are reported using one-sided test. The full sample covers the periods from July 2002 to December 2017 and is divided
into three disjoint time periods i) the training subsample (the first three years, T1 ) to estimate the model, ii) the validation subsample (the following
two years, T2 ) to tune the hyperparameters, and iii) the test subsample (the rest of the sample, T3 ) used to evaluate a model’s predictive performance.
2
All of the ROS associated with machine learning models from column (2) to column (10) are statistically significant with p-values less than 1%.
2
Panel B reports pairwise Diebold-Mariano test statistics comparing the out-of-sample firm-level bond return prediction performance (ROS ) among
the models used in Table 2. Positive numbers indicate the column model outperforms the row model. Numbers in bold denote statistical significance
at the 5% level or better.
41

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination
2
Panel A: Out-of-sample ROS
2
ROS −3.36 2.07 2.03 1.85 1.89 1.87 2.19 2.37 2.28 2.09
Panel B: Comparison of monthly out-of-sample prediction using Diebold-Mariano tests
OLS 3.07 2.89 3.45 3.53 3.59 3.82 3.85 3.28 3.38
PCA 1.14 −1.32 −1.26 −1.40 2.10 1.78 0.28 1.85
PLS −0.79 −0.57 −0.65 1.78 1.70 0.13 1.14
LASSO 0.44 0.40 1.60 1.18 0.86 2.05
Ridge 0.15 1.78 1.96 0.86 2.00
Enet 1.81 1.08 0.86 2.10
RF 1.10 1.74 1.91
FFN −0.80 1.20
LSTM 1.15
 Table 3: Performance of machine learning bond portfolios using corporate bond characteristics
This table reports the monthly performance of value-weighted decile portfolios sorted on out-of-sample machine learning return forecasts
using the 43 bond characteristics (i.e., r̂it+1 where (it) ∈ T3 , the test subsample). At the end of each month, we calculate one-month-ahead
out-of-sample firm-level bond return predictions for each method, where the firm-level bond returns are value-weighted using amount
outstanding as weights. We then sort firms into deciles based on each model’s forecasts and construct the value-weighted portfolio (e.g.,
using the sum of all bonds amount outstanding within the firm as weights) based on the out-of-sample forecasts. Low corresponds to the
portfolio with the lowest expected return (decile 1), High corresponds to the portfolio with the highest expected return (decile 10), and
High−Low corresponds to the long short portfolio that buys the highest expected return bonds (decile 10) and sells the lowest (decile
1). The returns are in monthly percentage and Newey-West t-statistics are reported in the last column.

Low 2 3 4 5 6 7 8 9 High High−Low

OLS 0.60 0.67 0.62 0.61 0.56 0.59 0.63 0.60 0.53 0.76 0.16 (1.38)
PCA 0.68 0.61 0.68 0.64 0.65 0.70 0.63 0.67 0.65 1.19 0.51 (2.51)
PLS 0.51 0.55 0.57 0.55 0.57 0.58 0.67 0.65 0.68 1.14 0.63 (2.86)
LASSO 0.57 0.50 0.47 0.40 0.42 0.42 0.46 0.59 0.58 0.96 0.39 (2.54)
Ridge 0.58 0.53 0.46 0.46 0.52 0.45 0.62 0.60 0.67 0.91 0.33 (2.15)
42

Enet 0.54 0.52 0.48 0.35 0.43 0.41 0.45 0.58 0.55 0.97 0.43 (2.67)
RF 0.57 0.69 0.54 0.51 0.52 0.50 0.59 0.55 0.49 1.37 0.79 (2.78)
FFN 0.61 0.63 0.48 0.55 0.49 0.59 0.50 0.59 0.56 1.36 0.75 (2.61)
LSTM 0.53 0.64 0.60 0.53 0.47 0.55 0.56 0.62 0.58 1.32 0.79 (3.33)
Combination 0.71 0.63 0.58 0.50 0.52 0.60 0.65 0.61 0.59 1.38 0.67 (3.41)
 Table 4: Predicting corporate bond returns with stock characteristics
Panel A of this table reports out-of-sample R-squared (ROS 2 , in percentage) for the entire panel of corporate bonds using the 94 stock

characteristics, following equation (12) as f1 (XS). The results are presented at the firm-level by constructing value-weighted firm-level
bond returns, as well as the firm-level value-weighted bond characteristics, using amount outstanding as weights. The models include
OLS with all variables (OLS), principal component analysis (PCA), partial least square (PLS), LASSO, Ridge regression (Ridge), Elastic
Net (ENet), Random Forest (RF), feed forward neural network (FFN), long short-term memory neural network (LSTM), and forecast
combination (Combination). The ROS 2 pools prediction errors across firms and over time into a grand panel-level assessment of each
model and is defined as, P 2
2 (it)∈T3 (rit+1 − r̂it+1 )
ROS = 1 − P 2 .
(it)∈T3 rit+1

The full sample covers the periods from July 2002 to December 2017 and is divided into three disjoint time periods i) the training
subsample (the first three years, T1 ) to estimate the model, ii) the validation subsample (the following two years, T2 ) to tune the
hyperparameters, and iii) the test subsample (the rest of the sample, T3 ) used to evaluate a model’s predictive performance. All of the
2
ROS associated with machine learning models from column (2) to column (10) are statistically significant with p-values less than 1%.
Panel B reports the monthly performance of value-weighted bond portfolios (i.e., High−Low return) sorted on out-of-sample machine
learning return forecasts.
43

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination
2
Panel A: ROS using stock characteristics
Using f1 (XS) −3.09 1.70 1.71 1.61 1.57 1.62 1.80 1.88 2.00 2.02
Panel B: Performance of machine learning High−Low bond portfolio using stock characteristics
Using f1 (XS) 0.02 0.36 0.43 0.24 0.26 0.24 0.43 0.48 0.52 0.52
(0.12) (2.35) (2.67) (2.12) (2.11) (2.03) (2.28) (2.25) (3.09) (3.13)
 Table 5: Predicting corporate bond returns with bond and stock characteristics
Panel A of this table reports out-of-sample R-squared (ROS2 , in percentage) for the entire panel of corporate bonds using the combined 137

stock and bond characteristics, following equation (12) as f1 (XB, XS). The results are presented at the firm-level by constructing value-
weighted firm-level bond returns, as well as the firm-level value-weighted bond characteristics, using amount outstanding as weights. The
models include OLS with all variables (OLS), principal component analysis (PCA), partial least square (PLS), LASSO, Ridge regression
(Ridge), Elastic Net (ENet), Random Forest (RF), feed forward neural network (FFN), long short-term memory neural network (LSTM),
and forecast combination (Combination). The ROS 2 pools prediction errors across firms and over time into a grand panel-level assessment

of each model and is defined as, P 2

2 (it)∈T3 (rit+1 − r̂it+1 )
ROS = 1 − P 2 .
(it)∈T3 rit+1

The full sample covers the periods from July 2002 to December 2017 and is divided into three disjoint time periods i) the training
subsample (the first three years, T1 ) to estimate the model, ii) the validation subsample (the following two years, T2 ) to tune the
hyperparameters, and iii) the test subsample (the rest of the sample, T3 ) used to evaluate a model’s predictive performance. All of
2
the ROS associated with machine learning models from column (2) to column (10) are statistically significant with p-values less than
1%. Panel B reports the monthly performance of value-weighted bond portfolios (i.e., High−Low return) formed using both stock and
characteristics (XS + XB) versus using only stock characteristics (XS) or bond characteristics (XB).
44

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination
2
Panel A: ROS using stock and bond characteristics
Using f1 (XB, XS) −5.38 1.74 1.70 1.62 1.60 1.66 1.89 1.97 2.11 2.09
Panel B: Comparing machine learning High−Low bond portfolio
Using f1 (XB, XS) 0.11 0.51 0.57 0.41 0.37 0.44 0.68 0.64 0.71 0.65
(1.18) (2.45) (2.35) (2.15) (2.13) (2.25) (3.13) (3.11) (3.19) (3.08)
Using f1 (XB, XS) − Using f1 (XB) −0.05 0.00 −0.06 0.02 0.04 0.01 −0.11 −0.11 −0.08 −0.02
(−0.97) (0.02) (−1.01) (0.22) (0.99) (0.68) (−1.26) (−1.35) (−1.45) (−0.92)
Using f1 (XB, XS) − Using f1 (XS) 0.09 0.15 0.14 0.18 0.11 0.21 0.25 0.16 0.19 0.13
(2.33) (1.81) (1.88) (1.78) (1.38) (1.77) (2.86) (2.22) (2.00) (2.15)
 Table 6: Predicting corporate bond returns with regression-based hedge ratios
2 , in percentage) for the entire panel of corporate bonds, based on equation (21).
Panel A of this table reports out-of-sample R-squared (ROS
Specifically, we generate bond return forecasts, f2 (XB, XS, ĥ), as a function of stock and bond characteristics, as well as the regression-
based hedge ratios (ĥ). Panel B of the table compares the forecasted bond returns with hedging ratio, f2 (XB, XS, ĥ), to the bond return
forecast obtained using bond characteristics, f1 (XB) (Table 2), or the combined stock and bond characteristics, f1 (XB, XS) (Table 5),
based on the Diebold-Mariano test statistics. The results are presented at the firm-level by constructing value-weighted firm-level bond
returns, using amount outstanding as weights. The ROS 2 pools prediction errors across firms and over time into a grand panel-level
assessment of each model and is defined as, P 2
2 (it)∈T3 (rit+1 − r̂it+1 )
ROS = 1 − P 2 .
(it)∈T3 rit+1

The full sample covers the periods from July 2002 to December 2017 and is divided into three disjoint time periods i) the training
subsample (the first three years, T1 ) to estimate the model, ii) the validation subsample (the following two years, T2 ) to tune the
hyperparameters, and iii) the test subsample (the rest of the sample, T3 ) used to evaluate a model’s predictive performance. All of the
2
ROS associated with machine learning models in Panel A from column (2) to column (10) are statistically significant with p-values less
than 1%. Numbers in bold in Panel B denote statistical significance at the 5% level or better.
45

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination
2
Panel A: ROS
Using f2 (XB, XS, ĥ) −4.37 2.28 2.88 1.93 1.95 1.95 3.05 3.11 4.89 4.95
Panel B: Comparison of monthly out-of-sample prediction using Diebold-Mariano tests
Using f2 (XB, XS, ĥ) − Using f1 (XB) −1.01 0.21 0.85 0.08 0.06 0.08 0.86 0.74 2.61 2.86
Using f2 (XB, XS, ĥ) − Using f1 (XB, XS) 1.01 0.54 1.18 0.31 0.35 0.29 1.16 1.14 2.78 2.86
 Table 7: Performance of machine learning bond portfolios using regression-based hedge ratios
This table reports the monthly performance of value-weighted bond portfolios (i.e., High−Low return) formed using regression-based
hedge ratios based on equation (21), f2 (XB, XS, ĥ), versus using bond characteristics only, f1 (XB), or combined stock and bond
characteristics, f1 (XB, XS). Numbers in bold denote statistical significance at the 5% level or better.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination
Using f2 (XB, XS, ĥ) 0.18 0.64 0.69 0.55 0.57 0.57 0.86 0.89 0.92 0.84
(1.07) (2.16) (2.33) (2.60) (2.75) (2.77) (3.01) (2.68) (2.69) (3.27)
Using f2 (XB, XS, ĥ) 0.02 0.13 0.06 0.16 0.24 0.14 0.07 0.14 0.13 0.17
− Using f1 (XB) (0.35) (2.35) (1.93) (2.27) (2.45) (2.36) (2.04) (2.54) (2.38) (2.81)
Using f2 (XB, XS, ĥ) 0.07 0.13 0.12 0.14 0.20 0.13 0.18 0.25 0.21 0.19
− Using f1 (XB, XS) (0.76) (2.44) (2.87) (2.15) (2.43) (2.22) (2.36) (2.77) (2.63) (2.60)
46
 Table 8: Predicting corporate bond returns with machine learning-based hedge ratios
Panel A of this table reports out-of-sample R-squared (ROS2 , in percentage) for the entire panel of corporate bonds, based on equation (24).

Specifically, we generate bond return forecasts, f3 (XB, XS, ĥ(XB)), as a function of stock and bond characteristics, as well as machine-
learning-based hedge ratios, ĥ(XB). Panel B of the table compares the forecasted bond returns with machine-learning-based hedge
ratios, f3 (XB, XS, ĥ(XB)), to the bond return forecast obtained using bond characteristics, f1 (XB) (Table 2), or the combined stock
and bond characteristics, f1 (XB, XS) (Table 5), or using regression-based hedge ratios, f2 (XB, XS, ĥ) (Table 6), based on the Diebold-
Mariano test statistics. The results are presented at the firm-level by constructing value-weighted firm-level bond returns, using amount
outstanding as weights. The ROS 2 pools prediction errors across firms and over time into a grand panel-level assessment of each model
and is defined as, P 2
2 (it)∈T3 (rit+1 − r̂it+1 )
ROS = 1 − P 2 .
(it)∈T3 rit+1

The full sample covers the periods from July 2002 to December 2017 and is divided into three disjoint time periods i) the training
subsample (the first three years, T1 ) to estimate the model, ii) the validation subsample (the following two years, T2 ) to tune the
hyperparameters, and iii) the test subsample (the rest of the sample, T3 ) used to evaluate a model’s predictive performance. All of the
2
ROS associated with machine learning models in Panel A from column (2) to column (10) are statistically significant with p-values less
than 1%. Numbers in bold denote statistical significance at the 5% level or better.
47

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination
2
Panel A: ROS
Using f3 (XB, XS, ĥ(XB)) −4.59 2.35 3.07 2.04 2.05 2.05 3.30 3.53 5.67 5.70
Panel B: Comparison of out-of-sample prediction using Diebold-Mariano tests
Using f3 (XB, XS, ĥ(XB)) − Using f1 (XB) −1.23 0.28 1.04 0.19 0.16 0.18 1.11 1.14 3.39 3.61
Using f3 (XB, XS, ĥ(XB)) − Using f1 (XB, XS) 0.79 0.61 1.37 0.42 0.45 0.39 1.41 1.56 3.56 3.61
Using f3 (XB, XS, ĥ(XB)) − Using f2 (XB, XS, ĥ) −0.22 0.07 0.19 0.11 0.10 0.10 0.25 0.42 0.78 0.75
 Table 9: Performance of machine learning bond portfolios using machine learning-based hedge ratios
This table reports the monthly performance of value-weighted bond portfolios (i.e., High−Low return) formed using machine-learning-
based hedge ratios based on equation (24), f3 (XB, XS, ĥ(XB)), versus using bond characteristics only, f1 (XB), or combined stock and
bond characteristics, f1 (XB, XS), or using regression-based hedge ratios, f2 (XB, XS, ĥ). Numbers in bold denote statistical significance
at the 5% level or better.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination
Using f3 (XB, XS, ĥ(XB)) 0.16 0.65 0.71 0.54 0.57 0.58 0.89 0.93 1.00 0.89
(0.53) (2.61) (2.49) (2.49) (2.52) (2.47) (2.71) (2.84) (3.22) (4.68)
Using f3 (XB, XS, ĥ(XB)) 0.00 0.14 0.08 0.15 0.24 0.15 0.10 0.18 0.21 0.22
− Using f1 (XB) (0.02) (2.14) (1.81) (2.43) (2.55) (2.61) (2.05) (2.07) (2.12) (2.41)
Using f3 (XB, XS, ĥ(XB)) 0.05 0.14 0.14 0.13 0.20 0.14 0.21 0.33 0.29 0.24
− Using f1 (XB, XS) (0.76) (2.25) (2.41) (2.42) (2.56) (2.41) (2.21) (2.44) (2.51) (2.75)
Using f3 (XB, XS, ĥ(XB)) −0.02 0.01 0.02 −0.01 0.00 0.01 0.03 0.04 0.08 0.05
48

− Using f2 (XB, XS, ĥ) (−0.22) (0.21) (0.44) (−0.10) (0.02) (0.10) (0.43) (0.54) (1.15) (0.79)
 Table 10: Comparison of hedge ratios
This table provides comparison of different hedge ratios based on the mean square errors (MSEs), defined as the the average sum of
squared differences between the regression-based or machine-learning-based hedge ratios and the benchmark hedge ratio. The hedge ratios
include (1) the regression-based hedge ratios in Section 5, and (2) the machine learning-based hedge ratios in Section 6. The benchmark
hedge ratio used to calculate MSEs is based on equation (3) in Section 2.1. Panel B reports the MSEs for the subsample based on the
firm-level credit rating of individual bonds and Panel C reports the MSEs for the subsample based on the firm-level time-to-maturity of
individual bonds.

Regression- Machine learning-based

based OLS PCA PLS LASSO Ridge Enet RF FFN LSTM Combination
All 0.051 0.097 0.053 0.054 0.053 0.053 0.054 0.055 0.057 0.056 0.054

Investment-grade (Rating ≤ 10) 0.033 0.064 0.032 0.031 0.033 0.032 0.032 0.031 0.031 0.031 0.031
Non-investment-grade (Rating > 10) 0.018 0.033 0.021 0.023 0.021 0.021 0.021 0.024 0.025 0.025 0.023
49

Short-maturity (1 ≤ Maturity < 3) 0.003 0.007 0.003 0.003 0.004 0.004 0.004 0.004 0.005 0.004 0.004
Medium-maturity (3 ≤ Maturity < 7) 0.024 0.049 0.026 0.027 0.026 0.026 0.026 0.026 0.027 0.026 0.026
Long-maturity (Maturity ≥ 7) 0.024 0.042 0.024 0.024 0.023 0.023 0.024 0.025 0.025 0.025 0.024
 Figure 1: Variable importance by model for corporate bond return prediction
This figure presents the variable importance for the top 10 most influential firm-level bond characteristics in each model for corporate
bond returns, using the 43 bond characteristics as the covariates. For each model, we calculate the reduction in ROS 2 from setting all
values of a given predictor to zero within each training sample, and average these into a single importance measure for each predictor.
Variable importance is an average over all training samples.

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 Figure 2: Characteristic importance for corporate bond return prediction
This figure presents the overall rankings of firm-level bond characteristics in each model for
corporate bond return prediction. For each model of the nine machine learning methods, we
calculate the reduction in ROS2 from setting all values of a given predictor to zero within each
training sample, and average these into a single importance measure for each predictor. The
importance of each characteristic for each method is ranked and then summed into a single rank.
Columns correspond to individual models, and color gradients within each column indicate the
most in influential (dark blue) to least in influential (white) variables.

MAT 1.0
PFHT
VOL
COSKEW
TCIQR
PIRoll
Rating
AvgBidAsk
MOM6 0.8
DUR
VaR10
PIHighLow
PZeros
BetaBond
VaR5
ISKEW
Amihud
PIFHT 0.6
GammaPS
REV
Size
KURT
TCRoll
ILLIQ
MOM12
SKEW 0.4
BetaTERM
Age
Roll
PILambda
PHighLow
ES5
BetaDEF
ES10
StdAmihud 0.2
Roundtrip
BetaVIX
BetaLWW
BetaDRF
BetaCRF
BetaLRF
BetaUNC
LTR 0.0
OLS PCA PLS SO Ridge ENet RF FFN LST
M ion
LAS 51 b inat
Com
 Figure 3: Characteristic importance for corporate bond return prediction
based on four characteristic groups
This figure presents the overall rankings of firm-level bond characteristics in each model, within
each of the four characteristic groups, for corporate bond return prediction. We classify the 43
corporate bond characteristics into four broad categories (i) bond characteristics related to interest
rate risk such as duration and time-to-maturity, (ii) risk measures such as downside risk proxied by
Value-at-Risk (VaR) or expected shortfall (ES), total return volatility (VOL), and systematic risk
proxied by the bond market beta, default beta, term beta, and macroeconomic uncertainty beta (iii)
bond-level illiquidity measures such as average bid and ask price (AvgBidAsk), Amihud and Roll’s
measures of illiquidity, and (iv) past return characteristics related to bond momentum, short-term
reversal, and long-term reversal. For each model of the nine machine learning methods, we calculate
the reduction in ROS2 from setting all values of a given predictor to zero within each training sample,

and average these into a single importance measure for each predictor. The importance of each
characteristic for each method is ranked and then summed into a single rank. Columns correspond
to individual models, and color gradients within each column indicate the most in influential (dark
blue) to least in influential (white) variables.

Panel A: Group I characteristics related to interest risk or maturity

MAT 1.00
Rating 0.75
DUR 0.50
Size 0.25
Age 0.00
OLS PCA PLS SO Ridge t RF FFN M ion
LAS ENe LST inat
m b
Co
Panel B: Group II characteristics related to risk measures
COSKEW 1.0
BetaBond
VaR5 0.8
VaR10
ES5
ES10 0.6
BetaTERM
BetaDEF
BetaVIX 0.4
BetaLWW
BetaDRF
BetaCRF 0.2
BetaLRF
BetaUNC 0.0
OLS PCA PLS SO Ridge t RF FFN M n
LAS ENe LST inatio
b
Com

52
 Panel C: Group III characteristics related to illiquidity and illiquidity risk
PFHT 1.0
TCIQR
PIRoll
PIHighLow 0.8
PZeros
Amihud
PIFHT 0.6
GammaPS
TCRoll
ILLIQ 0.4
Roll
PILambda
PHighLow 0.2
StdAmihud
Roundtrip
AvgBidAsk 0.0
OLS PCA PLS SO Ridge t RF FFN M n
LAS ENe LST in atio
b
Com
Panel D: Group IV characteristics related to past return characteristics
VOL 1.0
MOM6 0.8
ISKEW
REV 0.6
KURT 0.4
MOM12
SKEW 0.2
LTR 0.0
OLS PCA PLS SO Ridge t RF FFN M n
LAS ENe LST inatio
b
Com

53
 Figure 4: Which characteristic group matter for corporate bond return
prediction?
This figure presents the importance of the four characteristic group (Group I to IV), respectively,
for all machine learning models. We classify the 43 firm-level corporate bond characteristics into
four broad categories (i) bond characteristics related to interest rate risk such as duration and time-
to-maturity, (ii) risk measures such as downside risk proxied by Value-at-Risk (VaR) or expected
shortfall (ES), total return volatility (VOL), and systematic risk proxied by the bond market beta,
default beta, term beta, and macroeconomic uncertainty beta (iii) bond-level illiquidity measures
such as average bid and ask price (AvgBidAsk), Amihud and Roll’s measures of illiquidity, and (iv)
past return characteristics related to bond momentum, short-term reversal, and long-term reversal.
The figure reports the sum of the importance of each characteristic for each method, within each
characteristic group. Columns correspond to individual models, and color gradients within each
column indicate the most in influential (dark blue) to least in influential (white) characteristic
group.

Group 1 1.0
Group 2 0.5
Group 3
Group 4 0.0
OLS PCA PLS SO Ridge t RF FFN n
LAS ENe LSTM atio
bin
Com

54
 2
Figure 5: Out-of-Sample ROS of Corporate Bond Returns
2 , in percentage) of firm-level corporate bond
This figure presents the out-of-sample R-squared (ROS
returns using OLS, principal component analysis (PCA), partial least square (PLS), LASSO, ridge
regression (Ridge), elastic net (Enet), random forest (RF), feedforward neural network (FFN),
long short-term memory neural network (LSTM), and forecast combination (Combination). Figure
reports the ROS2 for using 43 bond characteristics only (f1 (XB)), 94 stock characteristics only
(f1 (XS)), and the 137 combined bond and stock characteristics (f1 (XB, XS)).

6.0

4.0

2.0

OLS
0.0
PC

PL

LA

Rid

EN

RF

FF

LS

Co
N
S

SS

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mb
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ina
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-8.0

-10.0 Using the 43 bond charactersitics

Using the 94 stock characteristics
-12.0 Using the combined 137 bond and stock characteristics

55
 Predicting Corporate Bond Returns: Merton Meets
Machine Learning
Online Appendix

Table OA1 provides robust checks of the main results in Table 3 and reports the monthly
performance of value-weighted decile portfolios sorted on out-of-sample machine learning return
forecasts using the 43 bond characteristics after taking into account of transaction costs.

Table OA2 provides robust checks of the main results in Table 3 and reports the conditional portfolio
performance, across different economic states based on the Chicago Fed National Activity Index
(CFNAI).

Table OA3 provides robust checks of the main results in Table 3 using maturity-matched Treasury
returns to calculate bond excess returns.

Table OA4 reports robust checks of the main results by excluding financial firms with with SIC
codes between 6000 and 6999.

Table OA5 shows that the machine learning methods provide strong forecasting power using the
stock characteristics, consisting with the findings of Gu, Kelly, and Xiu (2020).

Section OA1 provides details on various machine learning methods.

1
 Table OA1: Performance of machine learning bond portfolios using corporate bond characteristics after
transaction costs
This table reports the monthly performance of value-weighted decile portfolios sorted on out-of-sample machine learning return forecasts
using the 43 bond characteristics (i.e., r̂it+1 where (it) ∈ T3 , the test subsample), after taking into account transaction costs. At the
end of each month, we calculate one-month-ahead out-of-sample firm-level bond return predictions for each method, where the firm-level
bond returns are value-weighted using amount outstanding as weights. We then sort firms into deciles based on each model’s forecasts
and construct the value-weighted portfolio (e.g., using the sum of all bonds amount outstanding within the firm as weights) based on the
out-of-sample forecasts. Low corresponds to the portfolio with the lowest expected return (decile 1), High corresponds to the portfolio
with the highest expected return (decile 10), and High−Low corresponds to the long short portfolio that buys the highest expected return
bonds (decile 10) and sells the lowest (decile 1). The returns are in monthly percentage and Newey-West t-statistics are reported in the
last column.

Low 2 3 4 5 6 7 8 9 High High−Low

OLS 0.58 0.65 0.60 0.59 0.54 0.57 0.61 0.58 0.51 0.74 0.13 (1.25)
PCA 0.67 0.60 0.67 0.63 0.64 0.69 0.63 0.66 0.64 1.19 0.49 (2.44)
PLS 0.50 0.54 0.56 0.54 0.55 0.57 0.66 0.64 0.66 1.13 0.61 (2.78)
LASSO 0.55 0.48 0.45 0.38 0.40 0.40 0.44 0.57 0.56 0.94 0.35 (2.32)
2

Ridge 0.56 0.51 0.44 0.44 0.50 0.43 0.60 0.58 0.65 0.89 0.29 (2.04)
Enet 0.52 0.50 0.47 0.33 0.41 0.39 0.43 0.56 0.54 0.95 0.40 (2.57)
RF 0.56 0.67 0.52 0.49 0.50 0.48 0.57 0.54 0.47 1.35 0.75 (2.63)
FFN 0.60 0.61 0.46 0.53 0.47 0.58 0.48 0.57 0.54 1.34 0.71 (2.55)
LSTM 0.51 0.62 0.58 0.51 0.45 0.53 0.54 0.60 0.56 1.31 0.76 (3.12)
Combination 0.69 0.61 0.56 0.48 0.50 0.58 0.63 0.59 0.58 1.37 0.64 (3.25)
 Table OA2: Performance of machine learning bond portfolios in different
economic states
This table reports the monthly performance of High− Low value-weighted decile portfolios sorted on
out-of-sample machine learning return forecasts using the 43 bond characteristics (i.e., r̂it+1 where
(it) ∈ T3 , the test subsample), in different economic states based on the Chicago Fed National
Activity Index (CFNAI). At the end of each month, we calculate one-month-ahead out-of-sample
firm-level bond return predictions for each method, where the firm-level bond returns are value-
weighted using amount outstanding as weights. We then sort firms into deciles based on each
model’s forecasts and construct the value-weighted portfolio (e.g., using the sum of all bonds amount
outstanding within the firm as weights) based on the out-of-sample forecasts. Low corresponds to
the portfolio with the lowest expected return (decile 1), High corresponds to the portfolio with the
highest expected return (decile 10), and High−Low corresponds to the long short portfolio that
buys the highest expected return bonds (decile 10) and sells the lowest (decile 1). The returns are
in monthly percentage and Newey-West t-statistics are reported.

CFNAI > 0 (good economic state) CFNAI < 0 (bad economic state)
Average return t-stat Average return t-stat
OLS 0.14 (1.38) 0.11 (1.32)
PCA 0.58 (2.66) 0.40 (2.51)
PLS 0.70 (2.79) 0.51 (2.56)
LASSO 0.44 (2.34) 0.27 (2.24)
Ridge 0.38 (2.21) 0.20 (2.12)
Enet 0.49 (2.39) 0.30 (2.27)
RF 0.86 (3.52) 0.65 (2.84)
FFN 0.85 (3.64) 0.57 (3.21)
LSTM 0.91 (3.45) 0.60 (3.11)
Combination 0.79 (3.66) 0.48 (3.24)

3
 Table OA3: Performance of machine learning bond portfolios using corporate bond characteristics using
maturity-matched Treasury returns
This table reports the monthly performance of value-weighted decile portfolios sorted on out-of-sample machine learning return forecasts
using the 43 bond characteristics (i.e., r̂it+1 where (it) ∈ T3 , the test subsample), after adjusting for maturity-matched Treasury returns.
At the end of each month, we calculate one-month-ahead out-of-sample firm-level bond return predictions for each method, where the
firm-level bond returns are value-weighted using amount outstanding as weights. We then sort firms into deciles based on each model’s
forecasts and construct the value-weighted portfolio (e.g., using the sum of all bonds amount outstanding within the firm as weights)
based on the out-of-sample forecasts. Low corresponds to the portfolio with the lowest expected return (decile 1), High corresponds to
the portfolio with the highest expected return (decile 10), and High−Low corresponds to the long short portfolio that buys the highest
expected return bonds (decile 10) and sells the lowest (decile 1). The returns are in monthly percentage and Newey-West t-statistics are
reported in the last column.

Low 2 3 4 5 6 7 8 9 High High−Low

OLS 0.49 0.56 0.49 0.47 0.42 0.45 0.49 0.44 0.37 0.58 0.09 (0.95)
PCA 0.57 0.50 0.55 0.51 0.51 0.56 0.49 0.51 0.49 1.01 0.44 (2.21)
PLS 0.40 0.44 0.44 0.41 0.43 0.44 0.53 0.49 0.52 0.96 0.56 (2.33)
4

LASSO 0.46 0.39 0.34 0.27 0.28 0.28 0.32 0.43 0.42 0.78 0.31 (2.24)
Ridge 0.47 0.42 0.33 0.32 0.38 0.31 0.48 0.44 0.51 0.73 0.26 (2.00)
Enet 0.43 0.41 0.35 0.22 0.29 0.27 0.31 0.42 0.39 0.79 0.36 (2.27)
RF 0.46 0.58 0.41 0.37 0.38 0.36 0.45 0.39 0.33 1.19 0.73 (2.43)
FFN 0.50 0.52 0.35 0.42 0.35 0.45 0.36 0.43 0.40 1.17 0.67 (2.41)
LSTM 0.42 0.53 0.47 0.40 0.33 0.41 0.42 0.46 0.42 1.12 0.70 (2.91)
Combination 0.59 0.52 0.45 0.37 0.38 0.46 0.51 0.45 0.43 1.18 0.59 (3.04)
 Table OA4: Robustness checks excluding financial firms
This table reports robust checks of the main results by excluding financial firms with with SIC codes between 6000 and 6999. Panel A
replicates the main results in Table 2 on the out-of-sample R-squared (ROS2 , in percentage) for the entire panel of corporate bonds using

the 43 bond characteristics, following equation (12) as f1 (XB). Panel B replicates the main results in Table 4, following equation (12)
as f1 (XS). Panel C replicates the main results in Table 5, following equation (12) as f1 (XB, XS). Panel D replicates the main results
in Table 6, based on equation (21) as f2 (XB, XS, ĥ). Panel E replicates the main results in Table 7, using regression-based hedge ratios
based on equation (21), f2 (XB, XS, ĥ), versus using bond characteristics only, f1 (XB), or combined stock and bond characteristics,
f1 (XB, XS). Panel F replicates the main results in Table 8, based on equation (24). Panel F replicates the main results in Table 9,
based on equation (24), f3 (XB, XS, ĥ(XB)), versus using bond characteristics only, f1 (XB), or combined stock and bond characteristics,
f1 (XB, XS), or using regression-based hedge ratios, f2 (XB, XS, ĥ). Numbers in bold denote statistical significance at the 5% level or
better.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination
Panel A: Do corporate bond characteristics predict corporate bond returns?
2
ROS −3.12 2.07 2.14 2.02 2.01 2.04 2.25 2.30 2.27 2.55
Panel B: Do stock characteristics predict corporate bond returns?
2
ROS −3.04 1.73 1.76 1.66 1.60 1.66 1.75 1.99 2.13 2.20
5

Panel C: Comparing machine learning High−Low bond portfolio

Using f1 (XB, XS) − Using f1 (XS) 0.04 0.13 0.16 0.18 0.11 0.23 0.26 0.36 0.26 0.19
(1.81) (2.27) (1.99) (1.90) (1.97) (2.05) (3.12) (3.27) (2.33) (2.15)
Using f1 (XB, XS) − Using f1 (XB) −0.14 −0.04 −0.14 −0.13 0.13 −0.15 −0.08 0.02 −0.01 0.21
(−1.36) (−0.48) (−1.23) (−1.07) (1.28) (−0.69) (−1.04) (−0.62) (−1.16) (−0.41)
Panel D: Predicting corporate bond returns with regression-based hedge ratios
Using f2 (XB, XS, ĥ) −4.11 2.15 2.68 1.7 1.75 1.75 3 2.95 4.49 4.74
Panel E: Performance of machine learning High−Low bond portfolio using regression-based hedge ratios
Using f2 (XB, XS, ĥ) 0.18 0.59 0.63 0.52 0.54 0.55 0.80 0.83 0.88 0.83
(1.07) (2.04) (2.23) (2.45) (2.49) (2.47) (2.91) (2.55) (2.61) (3.10)
Using f2 (XB, XS, ĥ) − Using f1 (XB) 0.02 0.12 0.07 0.15 0.25 0.15 0.08 0.16 0.14 0.16
(0.33) (2.35) (1.96) (2.27) (2.45) (2.36) (2.04) (2.54) (2.38) (2.81)
Using f2 (XB, XS, ĥ) − Using f1 (XB, XS) 0.06 0.12 0.11 0.14 0.19 0.14 0.17 0.23 0.25 0.20
(0.74) (2.41) (2.85) (2.18) (2.47) (2.25) (2.28) (2.61) (2.55) (2.46)
Panel F: Predicting corporate bond returns with machine learning-based hedge ratios
Using (XS+XB+ĥ(XB)) −4.37 2.21 2.94 1.89 1.85 1.90 3.03 3.27 5.33 5.31
Panel G: Performance of machine learning High−Low bond portfolio using machine learning-based hedge ratios
Using f3 (XB, XS, ĥ(XB)) 0.16 0.61 0.66 0.53 0.54 0.54 0.82 0.86 0.93 0.86
(0.51) (2.16) (2.29) (2.44) (2.42) (2.47) (2.81) (2.54) (2.94) (4.28)
Using f3 (XB, XS, ĥ(XB)) − Using f1 (XB) 0.00 0.14 0.10 0.16 0.25 0.14 0.10 0.19 0.19 0.19
(0.02) (2.23) (2.01) (2.31) (2.45) (2.52) (1.94) (2.21) (2.44) (2.65)
Using f3 (XB, XS, ĥ(XB)) − Using f1 (XB, XS) 0.04 0.14 0.14 0.15 0.19 0.13 0.19 0.26 0.30 0.23
(0.75) (2.21) (2.33) (2.32) (2.46) (2.41) (2.14) (2.37) (2.69) (2.70)
Using f3 (XB, XS, ĥ(XB)) − Using f2 (XB, XS, ĥ) −0.02 0.02 0.03 0.01 0.00 −0.01 0.02 0.03 0.05 0.03
(−0.18) (0.15) (0.37) (−0.14) (0.02) (-0.19) (0.48) (0.63) (1.01) (0.59)
 Table OA5: Predicting stock returns with stock characteristics
2 , in percentage) for the entire panel of stocks using the 94 stock characteristics.
Panel A of this table reports out-of-sample R-squared (ROS
The models include OLS with all variables (OLS), principal component analysis (PCA), partial least square (PLS), LASSO, Ridge
regression (Ridge), Elastic Net (ENet), Random Forest (RF), feedforward neural network (FFN), long short-term memory neural network
(LSTM), and forecast combination (Combination). The ROS 2 pools prediction errors across firms and over time into a grand panel-level
assessment of each model and is defined as, P 2
2 (it)∈T3 (rit+1 − r̂it+1 )
ROS = 1 − P 2 .
(it)∈T3 rit+1

The full sample covers the periods from July 2002 to December 2017 and is divided into three disjoint time periods i) the training
subsample (the first three years, T1 ) to estimate the model, ii) the validation subsample (the following two years, T2 ) to tune the
hyperparameters, and iii) the test subsample (the rest of the sample, T3 ) used to evaluate a model’s predictive performance. All of the
2
ROS associated with machine learning models from column (2) to column (10) are statistically significant with p-values less than 1%.
Panel B reports the monthly performance of value-weighted stock portfolios (i.e., high minus low return) sorted based on out-of-sample
machine learning return forecasts using the 94 stock characteristics and 43 bond characteristics.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
6

OLS PCA PLS LASSO Ridge ENet RF FFN LSTM Combination

2
Panel A: ROS using stock characteristics
Using XS −3.83 0.35 0.44 0.12 0.08 0.12 0.49 0.54 0.56 0.60

Panel B: High−Low stock portfolio return

Using XS 0.48 0.58 0.64 0.48 0.79 0.48 0.89 0.84 1.04 1.10
(1.38) (1.95) (1.92) (2.33) (2.61) (2.33) (2.74) (2.92) (2.32) (2.79)
OA1 Methodology
The excess return of asset i (either a corporate bond or stock) at time t + 1, ri,t+1 , is defined as:

ri,t+1 = Et (ri,t+1 ) + εi,t+1 , (OA1)

where

Et (ri,t+1 ) = g ∗ (zi,t ) (OA2)

is the time-t expected return and g(.) is a flexible function of asset i’s P -dimensional characteristics,
i.e, zi,t = (zi,1,t , · · · , zi,P,t )′ . For ease of exposition, we assume a balanced panel of assets’ returns in
this section. We index assets by i = 1, · · · , N and months by t = 1, · · · , T , where N is the number
of assets at time t.

OA1.1 Linear Regression

The linear prediction regression is perhaps the least complex but most widely-used method in the
literature, which assumes that g ∗ (·) can be approximated by a linear function as:
′
g(zi,t ; θ) = zi,t θ, (OA3)

where θ = (θ1 , · · · , θP )′ can be estimated by the ordinary least squares (OLS) via the following
optimization problem:
N T
1 XX
min L(θ) ≡ (ri,t+1 − g (zi,t ; θ))2 . (OA4)
θ NT
i=1 t=1

The estimate of θ in (OA4) is unbiased and efficient if the number of predictors (P ) is relatively
small, while T is relatively large. In the real world, unfortunately, P is usually comparable with,
or even larger than, T , which raises an overfitting issue and makes the OLS estimate inefficient or
even inconsistent. To deal with large P , in the following we introduce nine representative machine
learning methods that have been recently used in the finance literature.

OA1.2 Penalized Linear Regression: LASSO, Ridge, and Elastic

Net
One of the most commonly used methods for reducing the overfitting issue in (OA4) is to add a
penalty term to the objective function. The penalty is imposed to tradeoff between mechanically
deteriorating in-sample performance of a model and improving its out-of-sample stability. Instead
of (OA4), θ can be estimated via:

min L(θ; .) ≡ L(θ) + ϕ(θ; .), (OA5)

θ

where ϕ(θ; .) is the penalty on θ. Depending on the functional form of ϕ(θ; .), the estimates of some
elements of θ can be regularized and shrunk towards zero.

7
 Specifically, in the machine learning literature, a general penalty function is:
P P
X 1 X 2
ϕ(θ; λ, ρ) = λ(1 − ρ) |θj | + λρ θj , (OA6)
2
j=1 j=1

where λ > 0 is a hyperparameter controlling for the amount of shrinkage; the larger the value of λ,
the greater the amount of shrinkage. The estimation model in (OA5) reduces to the standard OLS
if λ = 0. When ρ = 0, (OA5) corresponds to LASSO, which sets a subset of θ to exactly zero. In
this sense, the LASSO is a sparsity modelling technique and can be used for a variable selection.
When ρ = 1, (OA5) corresponds to the Ridge regression, which shrinks all coefficient estimates
closer to zero but does not impose exact zeros anywhere. Thus, the ridge regression is a dense
modeling technique and prevents coefficients from becoming unduly large in magnitude. Finally,
if ρ has a value between 0 and 1, we have the “elastic net” penalty, representing a compromise
between the Ridge and LASSO. Inheriting from the ridge regression, one advantage of elastic net
is that it can handle highly-correlated characteristics (Diebold and Shin, 2019).

OA1.3 Dimension Reduction: PCA and PLS

Based on equations (OA1)–(OA3), the excess return can be rewritten as:
′
ri,t+1 = zi,t θ + εi,t+1 . (OA7)

With a matrix notation, we have:

R = Zθ + E, (OA8)

where R is an N T × 1 vector of ri,t+1 , Z is an N T × P matrix of stacked predictors zi,t , and E is

an N T × 1 vector of residuals εi,t+1 .
Since P is relatively large, dimension reduction is an efficient way to attenuate over-fitting by
projecting a large number of characteristics into a small number of factors. Two main dimension
reduction techniques are the principal components analysis (PCA) and the partial least squares
(PLS). Specifically, PCA requires a transformation of a set of asset characteristics into independent
principal components, so that the first one has the largest variance, the second one has the second
largest, and so on. Then, it uses a few leading components to represent all the asset characteristics
and to predict asset returns. Mathematically, the j th principal component, Zwj , can be solved as:

wj = argmaxw Var (Zw) , s.t. w′ w = 1, Cov (Zw, Zwl ) = 0, l = 1, 2, · · · , j − 1. (OA9)

From (OA9), it is apparent that PCA is to maximize the common variation across all the
characteristics and its first K principal components represent the strongest variables that explain
the variations of the P characteristics. However, there is no guarantee that they are close to the
best set of variables that predicts future asset returns. Indeed, this is not surprising since no
information about asset returns is used to find the PCA predictors. In the worst case scenario, if
an individual characteristic has the largest variance albeit little ability to predict asset returns, it
may well be chosen as the first predictor as long as it is uncorrelated with the other characteristics.
Our results indicate that this issue does not occur in our setting with the corporate bond data, but
it seems possible with the equity data as suggested by Gu, Kelly, and Xiu (2020).

8
 In contrast to PCA, PLS is to link the asset characteristics to the asset returns. In our context,
it searches K linear combinations of Z to maximize its covariance with R. The weights wj used to
construct j th PLS component solve for

maxν,wj Cov(Rν, Zwj ), s.t. wj′ wj = 1, Cov (Zwj , Zwl ) = 0, l = 1, 2, · · · , j − 1. (OA10)

OA1.4 Random Forests

Unlike linear models, random forests are fully nonparametric and have different logic from
traditional regressions. A tree “grows” in a sequence of steps. At each step, a new “branch”
separates the data based on one of the return predictors. The final outputs are the average values
of returns in each partition sliced by predictors. Figure OA1 shows an example with two predictors,
“Rating” and “Maturity”. The tree separates to a partition based on characteristic values. First,
observations are sorted by Ratings. Those above the breakpoint of 10 are assigned to Category
3. Those with lower Ratings (investment bonds) are then further sorted by Maturity. Bonds with
lower than one year maturity are assigned to Category 1, while the rest go into Category 2.

Figure OA1: Regression tree example

Mathematically, we can express the expected return approximation function as:

K
X
g(zi,t ; θ, K, L) = θk 1{zi,t ∈Ck (L)} , (OA11)
k=1

where K is the number of “leaves” (terminal nodes), and L is the depth, Ck (L) is one of the K
partitions of the data, 1{·} is an indicator function, and θk is defined to be the sample average of
outcomes within the partition. The prediction equation in Figure OA1 is:

g (zi,t ; θ, 3, 2) = θ1 1{ratingi,t <10} 1{maturity<1} + θ2 1{ratingi,t <10} 1{maturity≥1} + θ3 1{ratingi,t ≥10} .

(OA12)

9
 To estimate θ in (OA11), we follow the algorithm of Breiman, Friedman, Olshen, and Stone
(1984). At each new level, we choose one variable from the set of predictors and the split value
to maximize the discrepancy which we call “impurity” among the average outcome returns in each
bin. Single the tree is subject to potential over-fitting problems, so it is rarely used without some
regulation methods. In our analysis, we build a set of decorrelated trees which are estimated
separately and then averaged out as an “ensemble” tree. Such modeling framework is known as
“random forests”, which is a general procedure known as “bagging” (Breiman, 2001). Through
averaging outcomes, random forests can reduce the overfit in individual bootstrap samples and
make the predictive performance more stable.

OA1.5 Feed-Forward Neural Network

Figure OA2: Feed-forward neural network with three hidden layers

Input Hidden Hidden Hidden Output

layer layer 1 layer 2 layer 3 layer

z1,t

z2,t r̂t+1

z3,t

As a typical neural network, feed-forward neural networks (FFN), as shown in Figure OA2,
include an “input layer” of raw predictors, one or more “hidden layers” that interact and nonlinearly
transform the predictors, and an “output layer” that aggregates hidden layers into outcome
prediction. The information that flows from input layer to hidden layers is transformed and
aggregated into an output at the output layer. The model becomes more flexible by adding hidden
layers between the inputs and output. Each hidden layer takes the output from the previous layer
and transforms it into an output as:
l
= g bl−1 + (z l−1 )′ W l−1 ,


zK (OA13)

where g(·) is the nonlinear “activation function” to its aggregated signal before sending its output
to the next layer. The final output is:

G(z, b, W ) = bL−1 + (z L−1 )′ W L−1 , (OA14)

where G(·) is the linear form of drawing information from the last hidden layer output.
There are many choices for the nonlinear activation functions, and we adopt the most commonly

10
used rectified linear unit (ReLU), which is defined as:

ReLU(zk ) = max(zk , 0). (OA15)

In this paper, we apply one hidden layer for FFN, with the idea of achieving better performance
about shallow learning as in Gu, Kelly, and Xiu (2020).

OA1.6 Long Short-Term Memory Neural Network

In financial markets, many predictors have long-term effects on stock returns. For example, return
volatility is known to have a long memory effect (see, e.g., Lo, 1991), and using only one lag of
volatility is unlikely to bring volatility to its full potential when predicting future returns. To deal
with this type of long-term dependencies, we use a more complex LSTM model (Hochreiter and
Schmidhuber, 1997), which transform a sequence of input variables to another output sequence,
with the same set of parameters at each step.

Figure OA3: Long Short-Term Memory Networks

ht

ct−1 ct
× +

tanh
×
ft it c̃t ot ×
σ σ tanh σ
ht−1 ht

zt

Specifically, the LSTM model takes the current input variable zt and the previous hidden state
ht−1 and performs a non-linear transformation to get the current state ht :
 
(c) (c)
ht = g Wh ht−1 + Wz(c) zt + w0 . (OA16)

This type of structure is powerful if only the immediate past is relevant, but is not suited if the
time series dynamics are driven by events that occur in the distant past. We can think of an LSTM
as a flexible hidden state space model for a large dimensional system. The LSTM is composed of a
cell (the memory part of the LSTM unit) and three “regulators” of the flow of information inside
the LSTM unit: an input gate, a forget gate, and an output gate.

11
 The memory cell is created with current input zt and previous hidden state ht−1 :
 
(c) (c)
c̃t = tanh Wh ht−1 + Wz(c) zt + w0 . (OA17)

The input and forget gates control for the memory cell, and the output gate controls for the amount
of information stored in the hidden state:
 
(i) (i)
inputt = g Wh ht−1 + Wz(i) zt + w0 ,
 
(f ) (f ) (f )
forgett = g Wh ht−1 + Wz zt + w0 , (OA18)
 
(o) (o)
outt = g Wh ht−1 + Wz(o) zt + w0 .

Define the element-wise product by ⊙, the final memory cell and hidden state are given by:

ct = forgett ⊙ ct−1 + inputt ⊙ c̃t (OA19)

ht = outt ⊙ tanh(ct )

Figure OA3 presents the diagram of a long short-term memory network. We consider one hidden
layer LSTM method for our model comparison.

OA1.7 Forecast Combination

(m)
Let r̂i,t+1 be asset i’s expected return estimated with method m (m = 1, · · · , M ) and M = 8 be
the number of methods, consisting of PCA, PLS, LASSO, Ridge, Enet, RF, FFN, and LSTM. We
(m)
equally combine r̂i,t+1 to obtain a new prediction of expected return:

M
1 X (m)
r̂i,t+1 = r̂i,t+1 . (OA20)
M
m=1

(m)
The idea behind equation (OA20) is that some of the expected return predictions, r̂i,t+1 ,
may have a high variance, equally weighting them can reduce the overall variance dramatically,
although it may increase the bias to some extent. The literature provides evidence that the forecast
combination method works well for return predictability (e.g., Rapach, Strauss, and Zhou, 2010;
Chen, Pelger, and Zhu, 2019).

12
 :

Swiss Finance Institute

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research, doctoral training, knowledge exchange, and continuing
education in the fields of banking and finance. SFI’s mission is to
grow knowledge capital for the Swiss financial marketplace. Created
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Swiss Confederation.

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T +41 22 379 84 71, rps@sfi.ch, www.sfi.ch
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