Bonds vs. Equities:
Information for Investment ∗
Huifeng Chang
†, Adrien d’Avernas
‡, Andrea L. Eisfeldt
§Preliminary and Incomplete: Please do not recirculate.
April 27, 2021
Abstract
We provide robust empirical evidence that uncovers the reason for the observed closer relationship between the bond market versus the equity market and the macroeoconomy. Our results indicate that the tight bond market- macroeconomy link is not due to differences in the investor base, but instead to the unique transformations of asset volatility and leverage that credit spreads and equity volatility represent. We focus on the investment channel. Using firm-level data, we find that the sensitivity of investment to equity volatility is highly significant, but changes sign in the cross section of firms depending on their distance to default. This sign change confounds aggregate inference.
We rationalize these findings using a simple structural model of credit risk and investment with debt overhang.
∗We would like to thank participants in the Macro Finance Lunch at UCLA and Vincent Maurin for helpful comments.
†UCLA Economics, email: huifengchangpku@gmail.com
‡Stockholm School of Economics, email: adrien.davernas@gmail.com
§UCLA Anderson School of Management and NBER, email:
andrea.eisfeldt@anderson.ucla.edu
1 Introduction
Economists and practicioners alike have long argued that there is a tight connec- tion between bond markets and the macroeconomy. Friedman and Kuttner (1992) show that the spread between commercial paper and Treasury bills forecasts reces- sions. Gilchrist and Zakrajˇsek (2012) use firm-level data to construct a credit spread measure with substantial predictive power for consumption, investment, and output.
Philippon (2009) constructs Tobin’s q from bond market data and shows that it outperforms equity-market q in predicting firm-level investment.1
An open question, however, is why bond market data appears to have better fore- casting power for real outcomes (in particular during recessions) than equity market data. Is it because bonds capture downside risk better, while equity prices are more affected by growth options? Or because bond markets have more “smart money”
and better reflects changes in financial conditions? We provide robust evidence and a simple model to rule out the latter explanations and to provide support for the former.
We focus on an investment channel. Bloom (2009) shows that shocks to uncer- tainty measured using implied equity volatility forecast lower investment. However, recent work byGilchrist, Sim, and Zakrajˇsek(2014) shows that controlling for credit spreads substantially reduces the impact of equity volatility on investment. The structural connection between credit spreads and equity volatility has been underes-
1See also the important contributions by Friedman and Kuttner (1998), Bernanke (1990), Gertler and Lown (1999), and Gilchrist et al. (2009), Giesecke et al. (2014), Krishnamurthy and Muir(2017).
timated in the literature using financial market data to forecast economic activity.2 As emphasized in the seminal works of Merton (1974) and Leland (1994), bond spreads and equity volatility are tightly related due to the fact that they both reflect a combination of firms’ asset volatilities and leverage. Indeed,Atkeson, Eisfeldt, and Weill (2017) show that (the inverse of) equity volatility and credit spreads contain very similar information about firms’ financial soundness.
We establish three main facts. First, credit spreads drive out equity volatility in an empirical model of the sensitivity of firm-level investment to equity volatility and credit spreads. Second, this result is due to the heterogeneity of the sensitivity of investment to equity volatility in the cross section of firms. The sensitivity of investment to equity volatility is positive for firms far enough away from default, otherwise it is negative. These different signs in the cross section drive the overall effect to be neutral. Third, both equity volatility and credit spreads are in large part driven by asset volatility and leverage, as predicted by structural models of credit risk. However, credit spreads have higher loadings on leverage, while equity volatility loads more on asset volatility. This is intuitive given the priority of debt versus equity in firms’ capital structures. Our results indicate that the closer relation between bond markets and the macroeconomy is not due to differences in the investor base or the presence of financial frictions, but instead is due to the precise transformation of asset volatility and leverage that they represent. We confirm this finding using credit spreads constructed from equity data alone.
2SeeStock and W Watson(2003) for a comprehensive survey on research using financial markets data to forecast macroeconomic outcomes.
We present VAR evidence which aggregates these findings to the economy-wide level. The aggregate evidence confirms our micro-level results and documents the im- portance of our findings for understanding the role of uncertainty and credit spreads on aggregate activity. The aggregate investment response to a positive shock to asset volatility is positive while the response to a shock to credit spreads (a market proxy for distance-to-default or leverage) is positive. Figure5 provides an intuitive visual representation of what is driving our results. This figure plots the time series and cross section of firms’ elasticity of investment with respect to equity volatility. As shown in the picture, firms with lower credit spreads which are further away from default display a positive elasticity of investment, while firms with higher credit spreads display a negative elasticity. Aggregate effects are driven by the movement of the entire cross section of firms away from and closer to their respective default boundaries. In contrast, Figure 7 shows that the elasticity of investment to credit spreads is negative for all firm quarters.
The starting point for our empirical work is a simple replication of the finding in Gilchrist, Sim, and Zakrajˇsek (2014) showing that (i) individually, the sensitivity of investment to both equity volatility and credit spreads is negative, but that (ii) if both are included the sensitivity of investment to credit spreads is essentially unchanged but the sensitivity of investment to equity volatility is reduced by 70%.
We then show that this is because the sensitivity of investment to equity volatility varies systematically in the cross section of firms with high and low credit spreads.
Firms in the lowest tercile of credit spreads have a statistically and economically significant positive elasticity of investment to equity volatility, while firms in the
highest tercile have a statistically and economically significant negative elasticity.
By contrast, the elasticity of investment to credit spreads is always negative. As a result, the elasticity of investment to equity volatility controlling for both credit spreads and the interaction between credit spreads and equity volatility is highly significant and positive. That is, controlling for high-credit-spread firms’ negative elasticity of investment with respect to equity volatility, the relation between equity volatility and investment becomes positive.
Using “fair value spreads” constructed using equity market data alone, we repeath the above analysis and show that the results are virtually identical. These fair value spreads are constructed using the results from structural models of credit risk which derive credit spreads from asset volatility and leverage.3 Thus, the different information in equity and bond data for investment is not due to a difference in investor base or market segmentation.
Since equity volatility is leveraged asset volatility, both are driven by these two factors. We document this using our panel data. We extract asset volatility by de- levering equity volatility. The fractions of variation in equity volatility and credit spreads explained by asset volatility and leverage (including firm and time fixed ef- fects) are 87% and 57%, respectively. In changes, the fractions of variation are 79%
(equity volatility) and 35% (credit spreads). In terms of loadings, these regressions show that the loading for equity volatility on asset volatility is two times the mag- nitude of the loading of equity volatility on leverage. By contrast, the loading for credit spreads have the opposite pattern. For credit spreads, the loading on leverage
3SeeArora et al.(2005) andNazeran and Dwyer(2015).
is three times as large as the loading on asset volatility.
We use this decomposition to show that the reason that equity and bond market data fair differently in explaining investment is due to the fact that asset volatility and leverage have opposite effects on investment. The elasticity of investment to asset volatility is significantly positive for most firms, indicating that the marginal revenue product of capital is a convex function of asset volatility shocks (see Leahy and Whited,1996).4 A one standard deviation increase in asset volatility is associated with a 9% standard deviation increase in firms’ investment rate.
We provide a model to illustrate the structural decomposition and the heterogene- ity in the cross section of the relationship between equity volatility and investment.
Asset volatility has a positive effect on investment unless leverage is sufficiently high enough for the debt overhang problem to dominate. In these high leverage firms, equity holders do not have sufficient property rights over cash flows resulting from investment in all states of the world to induce them to want to invest. For firms with moderate or low leverage, equity volatility is not sensitive to changes in credit risk.
The distortion from default risk is low. As a result, equity volatility mainly reflects asset volatility, which has a positive impact on investment.
4SeeAbel, Eberly, et al. (1994) and Dixit, Dixit, and Pindyck (1994) for models with convex and non-convex costs of adjustment.
2 Empirical Evidence
Data We use a quarterly sample of US firms covered by S&P’s Compustat that runs from 1984:Q1 to 2018:Q4. We exclude firms in the financial sector (SIC code 6000 to 6999) and utility sector (SIC code 4900 to 4949) and observations with negative sales. We use daily returns from the Research in Security Prices (CRSP) data base.
Bond prices come from the Lehman/Warga (1984-2005) and ICE databases (1997- 2018). We require non-missing data for variables we use to construct investment rate, equity volatility, market leverage, and credit spreads and impose a restriction that a firm need to be in the panel for at leat 3 years. This selection criterion yields 1,161 with 54,033 firm-quarter observations. To ensure that our results are not driven by extreme values, we trim the sample by replacing the top and bottom 0.5% of regression variables as missing values.
Equity and Asset Volatility The key variables in our regressions are investment rate, idiosyncratic equity volatility, credit spread, idiosyncratic asset volatility, and market leverage. The first three variables are constructed following Gilchrist, Sim, and Zakrajˇsek (2014). We define investment rate as capital expenditures in quarter t scaled by net property, plant, and equipment in quarter t − 1. Idiosyncratic equity volatility is constructed in two steps. For each firm-fiscal quarter, we first remove the forecastable variation in daily excess returns using the Carhart four-factor model.
Then for each regression we calculate the standard deviation of residuals, and obtain quarterly firm-specific idiosyncratic equity volatility. We only keep observations for
quarters with more than 30 trading days.
We use Merton’s model to derive firm-level implied asset volatility with observa- tion on realized equity volatility, debt, and market capitalization. Realized equity volatility is obtained from historical daily stock returns. The firm’s debt is assumed to be equal to the sum of its current liabilities and one-half of its long-term liabilities.
We implement the iterative procedure proposed by Bharath and Shumway (2008).
We then use the resulting asset values from this procedure to generate times series of daily asset returns. With time series of daily asset returns, we calculate the id- iosyncratic asset volatility using the same methodology used for idiosyncratic equity volatility.
Market leverage Market leverage is defined as the ratio of market value of assets to market value of equity. The market value of assets is built as the book value of assets plus the market value of equity minus the book value of equity.Following Davies, Fama, and French (2000), the book value of equity is defined as the book value of stockholders’ equity, plus balance sheet deferred taxes and investment tax credit, minus the book value of preferred stock. Depending on availability, we use the redemption, liquidation, or par value (in that order) for the book value of preferred stock. If this procedure generates missing values, we measure stockholders’ equity as the book value of common equity plus the par value of preferred stock, or the book value of assets minus total liabilities.
Credit Spreads We follow Gilchrist and Zakrajˇsek (2012) to compute bond-level credit spreads. First, we construct a theoretical risk-free bond that replicates exactly the promised cash flows. The price of this risk-free bond is calculated by discounting the promised cash flows using continuously-compounded zero-coupon Treasury yields fromG¨urkaynak, Sack, and Wright (2007). The credit spread of an individual bond is difference between the yield of the actual bond and the yield of the corresponding risk-free bond. We then define the credit spread of a firm as the quarterly average of the month-end credit spreads of all bonds issued by that firm.
Fair Value Spreads We also use a proprietary data set from Moody’s on its Public Firm Expected Default Frequency (EDF) Metric, which is an equity-based measure of firm’s probability of default. The core model used to generate the EDF metric belongs to the class of option-pricing based, structural credit risk models pioneered byBlack and Scholes (1973) and Merton (1974). A modified model, called Vasicek-Kealhofer (VK) model, summarizes information on asset volatility, market value of assets, and the default point into one metric, Distance-to-Default (DD), and then maps the DD to obtain the EDF metric. The DD-to-EDF mapping step is empirical and utilizes the empirical distribution of DD and frequency of realized defaults. Nazeran and Dwyer(2015) provide a detailed description of their methodology. Most importantly for our purpose, the EDF credit risk measure relies only on equity market inputs and does not contain bond market information.
Using the EDF credit risk measure, we construct a cumulative EDF (CEDF) over T years by assuming a flat term structure, that is, CEDFT = 1−(1−EDF )T. Then, we
convert our physical measure of default probabilities (CEDF) to risk-neutral default probabilities (CQDF) using the following equation:
CQDFT = Nh
N−1(CEDFT) + λρ√ Ti
,
where N is the cumulative distribution function for the standard normal distribution, λ is the market Sharpe ratio and ρ is the correlation between the underlying asset returns and market returns. Given this risk-neutral default probability measure, the spread of a zero-coupon bond with duration T can be computed as:
ˆ s = −1
T log(1 − CQDFT · LGD),
where LGD stands for the risk-neutral expected loss given default. We follow Moody’s convention and set T = 5, LGD = 60%, λ = 0.546, and ρ = √
0.3 to build our “fair value spread” measure ˆs. We successfully match 39, 925 fair value spreads with our firm-quarter observations.
Empirical Analysis We first replicate the results inGilchrist, Sim, and Zakrajˇsek (2014) that the adverse effect of idiosyncratic equity volatility on investment rate is dampened when controlling for credit spreads. Specifically, we estimate the following empirical investment equation:
log[I/K]it = β1log σite + β2log sit+ ηi+ λt+ it,
where log[I/K]it denote the log of investment rate of firm i in period t, log σite is the log of idiosyncratic equity volatility, and log sit is the log of credit spread. We control for the firm fixed effects and time fixed effects by including ηi and λt. Table 1presents regression results for this specification.
[Table 1 about here.]
As shown in columns 1-3 of Table1, the coefficient on idiosyncratic equity volatility and credit spread are statistically significant and economically important on their own (columns 1-2). However, when both measures are included in the regression, the coefficient on equity volatility is substantially reduced both in terms of magnitude and statistical significance while the coefficient on credit spread is unaffected (column 3).
To see why bond spreads can drive out equity volatility, we split the sample by credit spread and run the same regression. The tercile cutoffs in credit spread are 147 and 326 basis points.5 The results are reported in column 4-6 of Table1. Note that the coefficient on equity volatility becomes significantly positive among firms with low credit spread. The last column shows results from the regression with an interaction term, and confirms our findings from columns 4-6. In Section C, we show that regressions using lags of the independent variables generate similar results, which highlights the predictive power of these measures on investment.
[Table 2 about here.]
5Quarter-specific cutoffs lead to similar results.
In Table 2, we replace credit spreads with fair value spreads. The results are qualitatively identical to Table1. The coefficient on equity volatility goes from sig- nificantly positive to significantly negative as firm’s credit spread goes up, while the coefficient on the fair value spread remains significantly negative across the sub- groups. In Section C, we run additional robustness checks to address concerns over sample selection bias. As the fair value spreads are constructed with only equity market information and does not contain bond market information, the results from Table 3 cannot be driven by differences in the investor base or information about financial frictions only reflected in credit spreads.
[Table 3 about here.]
Equity volatility can be decomposed into asset volatility (derived from Merton’s model) and market leverage. In Table3, we run the same regression but we replace idiosyncratic equity volatility σite with idiosyncratic asset volatility σit. The coefficient on asset volatility is always positive and statistically significant in the full sample and in all subgroups.
[Table 4 about here.]
Given the decomposition, a natural question is whether the coefficient on asset volatility is also positive when asset volatility and market leverage are used together as explanatory variables for investment. We present the results in table 4. The results are inconclusive. The coefficient on asset volatility is not always positive and there is no clear monotonicity. We will rationalize this in the model section.
[Table 5 about here.]
Asset volatility and leverage are also important drivers for bond spreads. To understand why there is no such sign flip for credit spreads, we consider the loadings of credit spreads and equity volatility on asset volatility and leverage and estimate the following equation:
log yit= β1log σit+ β2log[M A/M E]it+ ηi+ λt+ it,
where yit is either the equity volatility (σeit) or the credit spread (sit), and M A/M Eit is market leverage. We estimate the equation both in levels and in first differences.
Table5summarizes the results. Columns 1-2 show how the levels of equity volatility and credit spread load on levels of asset volatility and leverage, and columns 3-4 show how the changes load on corresponding changes. Both specifications imply that bond spreads are mainly driven by leverage, while equity volatility is mainly driven by asset volatility. Since shocks to asset volatility and leverage impact investment differently, bond spreads and equity volatility contain different information for investment.
[Figures 5, 6, and 7about here.]
To understand the implications of our findings for time series, we plot the elasticity of investment rate with respect to equity volatility, asset volatility, and credit spread across time and across firms using the estimates from the regressions with interaction terms. In Figure 5, we compute the overall coefficient on equity volatility at each
credit spread level using estimates on equity volatility (log σite) and the interaction term (log σeit×log sit) reported in the last column of Table1. We repeat the procedure for asset volatility and credit spreads in Figures 6 and 7.6 Figure 5 shows that the cross section of elasticities of investment with respect to equity volatility varies a lot over time. In particular, this coefficient is negative for the whole cross-section of firms during the Great Recession, while it is mainly positive in the late 1980s. By contrast in Figures 6 and 7, the elasticity of investment to asset volatility remains positive and the elasticity to credit spread remains negative, both in the cross-section and over time.
[Figure 8about here.]
Using an identified vector autoregression (VAR) framework, we confirm that our micro-level results—asset volatility has positive impact on investment—still holds at the macro-level. We aggregate the variables in our sample and estimate a simple VAR consisting of the three endogeneous variables: the log of idiosyncratic asset volatility (log σt), the log of credit spread (log st), and the log of investment rate (log[I/K]t).7 We employ a standard recursive ordering technique and consider two identification schemes, one in which credit spread has an immediate impact on asset volatility and the other where asset volatility has immediate impact on credit spread. Figure 8 reports the impulse responses of investment rate to credit spread and asset volatility
6We present the elasticity of investment with respect to credit spread using estimates on credit spread and its interaction with asset volatility from Table3 in SectionC.
7We use the value-weighted average of σit, sit and [I/K]it to generate the corresponding ag- gregate time series. We seasonally adjust the investment rate time series by using its four-quarter moving average. All variables are detrended using the HP filter with weight 1600.
using the two specifications. Credit spread has a negative impact on investment while asset volatility has a positive impact.
3 Investment Decisions with Debt Overhang
In this section, we develop a simple but general credit risk model to analyze the investment choices of a firm with outstanding debt already in place. Two forces drive the investment decision: debt overhang and the option value of equity. We demonstrate that credit spreads and asset volatility are jointly unambiguous signals of these two forces. However, the signals provided by leverage and asset volatility or credit spreads and equity volatility are ambiguous and can change in the cross- section. All proofs are relegated to Section D.
Consider a firm that has risky assets in place and has funded itself partly with debt. In the first period, shareholders choose how much to invest. After observing the payoff of their investment, shareholders decide whether to file for bankruptcy or not. For our basic argument, we make the following assumptions regarding the firm and its investments.
Assumption 1 (Assets in Place). The firm has existing real assets in place with a final value of Y (ι, z), which is a function of investment ι and a random productivity shock z realized in the future. The assets are normalized to have an initial value of one. That is, E [Y (0, z)] = 1.
Assumption 2 (Firm Liabilities). The firm is funded by equity, together with a debt
claim with total face value b that is due in the second period when the asset returns are realized. In the second period, shareholders decide whether to default. Upon bankruptcy, the entirety of the firm’s value is lost. Furthermore, shareholders cannot liquidate the firm (ι ≥ 0).
Assumption 3 (Pricing). All securities are traded in perfect Walrasian markets.
We normalize the risk-free interest rate to zero and set prices of securities equal to their expected payoff with respect to a risk-neutral distribution F (z) of firm’s asset productivity z, and Y (ι, z) with full support on [0, ∞).
Given our assumptions about payouts and pricing, it follows that the value of equity E and debt D are given by:
E(b, ι, z, σ) = Z ∞
z
(Y (ι, z) − b)dF (z; σ) − ι, D(b, z, σ) = (1 − F (z; σ))b,
where ι and z are such that:8
Z ∞ z
Yι(ι, z)dF (z; σ) = 1, Y (ι, z) = b.
Credit spreads are given by: cs(z, σ) = F (z; σ). Given the normalization from As- sumption 1, leverage is simply the face value of debt b. To streamline our analysis,
8For ease of notation, we write ∂f (x)∂x = fx.
we also make assumptions on the risk distribution.
Assumption 4 (Investment Returns). The investment return function Y (ι, z) is continuous in both ι and z, positive, homogeneous of degree 1 in z, strictly increasing in ι, strictly concave in ι, and normalized to z when ι = 0. That is,
Y (ι, z) = k(ι)z ≥ 0, k(0) = 1, kι(ι) > 0, kιι(ι) < 0.
Furthermore, the standard deviation σ of z is a finite moment of the distribution F . Assumption 5 (Vega). The distribution of the productivity F (z; σ) is log-concave and such that vega is always positive:
ν(z, σ) = ∂
∂σE(z − z)+ > 0.
Assumption 4 imposes restrictions common in models with investment. The ho- mogeneity of degree 1 in z allows us to disentangle the effect of investment i and risk z on the investment returns Y (ι, z). Assumption 5 is always satisfied with the Black–Scholes–Merton model or the unimodal risk distributions usually considered in finance.
The model has two free parameters, leverage b and asset volatility σ, and two endogenous decision variables, investment ι and default threshold z. We are now interested in studying the behavior of investment following changes in the different observable variables from our empirical section: asset volatility σ, leverage b, credit spreads cs, and equity volatility σe.
0 0.2 0.4 0.6 0.8 1 0
0.5 1 1.5 2 2.5 3
0 0.2 0.4 0.6 0.8 1
0 1 2 3 4
Figure 1: Optimal investment with normal distribution.
Proposition 3.1 (Credit Spread and Asset Volatility). Holding asset volatility con- stant, the partial derivative of investment with respect to credit spread is given by:
∂ι
∂cs = kι(ι) kιι(ι)
z
µ(z, σ) < 0, (1)
where µ(z, σ) = E [z|z ≥ z] P [z ≥ z]. Holding credit spreads constant, the partial derivative of investment with respect to volatility is given by:
∂ι
∂σ = −kι(ι) kιι(ι)
ν(z, σ) µ(z, σ) > 0.
In Proposition 3.1, we provide the elasticities of investment when observing asset volatility and credit spreads. Given Assumptions1-5, the sign of these partial deriva- tives match our empirical results. An increase in credit spreads contracts investment as the debt-overhang problem intensifies when the firm gets closer to default. As the default boundary z increases, expected returns µ(z, σ) decreases and shareholders have less incentives to invest. Contrarily, investment reacts positively to an increase
0 0.2 0.4 0.6 0.8 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
0 0.2 0.4 0.6 0.8 1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Figure 2: Optimal investment with log-normal distribution.
in volatility as the payout to shareholders is non-linear with limited downside and unlimited upside, that is, vega ν(z, σ) is positive.
In both cases, the ratio of the marginal investment return Yι to investment return concavity Yιι determines the strength of the investment response. If the marginal investment return kι is large or the marginal productivity does not fall too fast (low
|kιι|), then the investment response to a change in volatility or credit spreads is stronger.
We us this unambiguous case, where changes in credit spread cs signal changes in the debt-overhang effect and changes in asset volatility σ signal changes in the option value of equity, as our benchmark. In Figures Figure 1 and Figure 2, we show the optimal investment function for the normal and log-normal distribution, respectively.
Proposition 3.2 (Leverage and Asset Volatility). Holding asset volatility constant,
the partial derivative of investment with respect to leverage is given by:
∂ι
∂b = kι(ι) kιι(ι)
z
µ(z, σ)ξb|σ(z, σ) < 0, (2)
where
ξb|σ(z, σ) ≡ f (z; σ)
k(ι) ϕ(z, σ) > 0, ϕ(z, σ) ≡
1 + kι(ι)2z2f (z; σ) k(ι)kιι(ι)µ(z; σ)
−1
> 0.
Holding leverage constant, the partial derivative of investment with respect to volatil- ity is given by:
∂ι
∂σ = −kι(ι) kιι(ι)
ν(z, σ)
µ(z; σ)ξσ|b(z, σ), (3)
where
ξσ|b(z, σ) ≡
1 −zFσ(z; σ) ν(z, σ)
ϕ(z, σ).
As shown in Proposition 3.2, if instead of controlling for credit spreads cs, we ob- serve leverage b, the elasticities of investment become more intricate. we characterize the differences with Proposition3.1with the wedges ξb|σ and ξσ|b. The first additional force stems from ϕ. First, note that ϕ is always positive at a maxima as imposed by the second-order conditions. Second, ϕ represents the impact of the change in the shareholders’ default decision following a change in investment returns. Indeed, following a decrease in investment, shareholders also default more often as output
0 0.2 0.4 0.6 0.8 1 0.18
0.2 0.22 0.24 0.26 0.28 0.3 0.32
0 0.2 0.4 0.6 0.8 1
0.2 0.25 0.3 0.35 0.4
Figure 3: Optimal investment with normal distribution.
decreases and thus incentives to pay back the debt also decrease. That additional force was not present in Proposition3.1, as changing credit spreads F (z; σ) controls for the default decision z, instead of Y (ι, z)—a function of ι and z.
While ϕ is always positive, the trade-off between ν(z, σ) and zFσ(z; σ) makes the sign of the partial derivative in equation (3) ambiguous. When the increase in investment returns lost to default zFσ(z; σ) dominates the increase in the option value ν(z, σ), shareholders reduce investment following an increase in volatility. Whether one effect dominates the other is highly dependent on the distribution. In ??, we show that the optimal investment decision as a function of asset volatility σ when holding leverage b constant with two distribution: normal and log-normal.
We are now in a position to study the changes in investment when observing credit spreads and equity volatility. First, we define equity volatility as measured in the
0 0.2 0.4 0.6 0.8 1 1.2 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
0 0.5 1 1.5 2
0.1 0.15 0.2 0.25 0.3
Figure 4: Optimal investment with log-normal distribution.
data as:9
σe(z, σ) = σ µ(z, σ)s ,
where µ(z, σ) = E [(z − z)|z ≥ z] P [z ≥ z] is the unconditional left-truncated ex-s pectation of returns above the default threshold. The denominator represents the impact of leverage on equity volatility. If the debt burden b increases, then the de- fault threshold z increases as well and µ(z, σ) decreases. Conversely, if the firm iss funded entirely by equity (b = 0), then z is equal to zero—the lower bound of the support. In that case, equity volatility is equal to asset volatility (σe(z, σ) = σ) since sµ(0, σ) = 1.
9Defining equity volatility as:
σe(z, σ) = pV ar [E(Y (ι, z), ι, z, σ)]
E(Y (ι, z), ι, z, σ) = sσ(z, σ) µ(z, σ) − ι/k(ι)s ,
whereσ(z, σ) =s pV ar [(z − z)|z ≥ z] P [z ≥ z] makes the analysis untractable and is further away from how equity volatility is measured as (i) the truncation of the volatility is not reflected in the measurement of equity volatility unless default occurred and (ii) investment, as measured in Compustat, does not varies over a quarter and cannot impact changes in equity volatility.
Proposition 3.3 (Credit Spread and Equity Volatility). Holding equity volatility constant, the partial derivative of investment with respect to credit spread is given by:
∂ι
∂cs = kι(ι) kιι(ι)
z
µ(z, σ)ξcs|σe(z, σ),
where
ξcs|σe(z, σ)(z, σ) ≡ 1 + σez(z, σ)ν(z, σ)
zf (z) ξσe|cs(z, σ).
Holding equity volatility constant, the partial derivative of investment with respect to credit spread is given by:
∂ι
∂σe = −kι(ι) kιι(ι)
ν(z)
µ(z, σ)ξσe|cs(z, σ),
where
ξσe|cs(z, σ) ≡
σσe(z, σ) − Fσ(z; σ)
f (z; σ) σze(z, σ)
−1
.
4 Conclusion
We establish three main facts. First, credit spreads drive out equity volatility in an empirical model of the sensitivity of firm-level investment to equity volatility and credit spreads. Second, this result is due to the heterogeneity of the sensitivity
of investment to equity volatility in the cross section of firms. The sensitivity of investment to equity volatility is positive for firms far enough away from default, otherwise it is negative. These different signs in the cross section drive the overall effect to be neutral. Third, both equity volatility and credit spreads are in large part driven by asset volatility and leverage, as predicted by structural models of credit risk. However, credit spreads have higher loadings on leverage, while equity volatility loads more on asset volatility. This is intuitive given the priority of debt versus equity in firms’ capital structures. Our results indicate that the closer relation between bond markets and the macroeconomy is not due to differences in the investor base or the presence of financial frictions, but instead is due to the precise transformation of asset volatility and leverage that they represent. We confirm this finding using credit spreads constructed from equity data alone. We interpret these facts in the context of a structural model. Our model and evidence provide support for the idea that the close connection between bond markets and the macroeconomy is due to the unique non-linear transformation of asset volatility and leverage that credit spreads represent.
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Appendices
A Figures
Figure 5: This figure presents the elasticity of investment with respect to equity volatility across time and across firms using the estimates from the regressions with interaction terms. In each quarter we generate five cutoffs in the cross-section of log credit spread: {p10, p30, p50, p70, p90}.
Using the estimates in column 7 of Table1on
log[I/K]it= β1log σeit+ β2log sit+ γ log σite × log sit+ ηi+ λt+ it, the elasticity at each cutoff point is computed as β1+ γpn, n = 10, 30, 50, 70, 90.
.
1985 1990 1995 2000 2005 2010 2015
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
10th pctile of cs 30th pctile of cs 50th pctile of cs 70th pctile of cs 90th pctile of cs
Figure 6: This figure presents the elasticity of investment with respect to asset volatility across time and across firms using the estimates from the regressions with interaction terms. In each quarter we generate five cutoffs in the cross-section of log credit spread: {p10, p30, p50, p70, p90}.
Using the estimates in column 5 of Table3on
log[I/K]it= β1log σit+ β2log sit+ γ log σit× log sit+ ηi+ λt+ it, the elasticity at each cutoff point is computed as β1+ γpn, n = 10, 30, 50, 70, 90.
1985 1990 1995 2000 2005 2010 2015
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
10th pctile of cs 30th pctile of cs 50th pctile of cs 70th pctile of cs 90th pctile of cs
Figure 7: This figure presents the elasticity of investment with respect to credit spread across time and across firms using the estimates from the regressions with interaction terms. In each quarter we generate five cutoffs in the cross-section of log equity volatility: {p10, p30, p50, p70, p90}. Using the estimates in column 7 of Table1 on
log[I/K]it= β1log σeit+ β2log sit+ γ log σite × log sit+ ηi+ λt+ it, the elasticity at each cutoff point is computed as β2+ γpn, n = 10, 30, 50, 70, 90.
1985 1990 1995 2000 2005 2010 2015
-0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05
10th pctile of equity vol 30th pctile of equity vol 50th pctile of equity vol 70th pctile of equity vol 90th pctile of equity vol
Figure 8: This figure plots the impulse responses of investment to an orthogonalized 1 standard deviation shock to asset volatility and credit spread. The VAR is estimated using four lags of each endogenous variable. Subfigures (a) and (b) correspond to the recursive ordering: (s, σ, I/K). Subfigures (c) and (d) correspond to the recursive ordering: (σ, s, I/K). The shaded bands represent the 95% confidence interval.
(a) Asset volatility
-2-1.5-1-.50.511.52percent
0 4 8 12 16
quarters
(b) Credit spread
-2-1.5-1-.50.511.52percent
0 4 8 12 16
quarters
(c) Asset volatility
-2-1.5-1-.50.511.52percent
0 4 8 12 16
quarters
(d) Credit spread
-2-1.5-1-.50.511.52percent
0 4 8 12 16
quarters
B Tables
Table 1: This table documents the relationship between equity volatility, credit spread and in- vestment. Investment rate is regressed on all the regressors on the right-hand side in column 7, on equity volatility and credit spread spread in columns 3-6, on credit spread in column 2 and on equity volatility in column 1. Columns 4-6 use subsamples sorted by terciles on credit spread. Each observation is a firm-quarter. Coefficients are reported with t-statistics in parentheses. ***, **, and * indicate significance at 1%, 5% and 10% levels and standard errors are clustered at the firm level.
log[I/K]it= β1log σite + β2log sit+ γ log σite × log sit+ ηi+ λt+ it
(1) (2) (3) (4) (5) (6) (7)
all all all low cs mid cs high cs all
log σite -0.167*** -0.051*** 0.070*** -0.048** -0.098*** 0.801***
(-9.00) (-3.37) (3.79) (-2.37) (-4.17) (9.78)
log sit -0.285*** -0.268*** -0.103*** -0.289*** -0.479*** -0.462***
(-13.34) (-12.85) (-3.38) (-7.97) (-12.15) (-16.61)
log σite × log sit -0.152***
(-10.21)
Firm FE X X X X X X X
Time FE X X X X X X X
Observations 52897 52900 52414 17556 17536 17322 52414
R-squared 0.101 0.125 0.126 0.141 0.111 0.121 0.134
Table 2: This table documents the relationship between equity volatility, fair value spread and investment. Investment rate is regressed on all the regressors on the right-hand side in column 7, on equity volatility and fair value spread in columns 3-6, on fair value spread in column 2 and on equity volatility in column 1. Columns 4-6 use subsamples sorted by terciles on credit spread. Each observation is a firm-quarter. Coefficients are reported with t-statistics in parentheses. ***, **, and * indicate significance at 1%, 5% and 10% levels and standard errors are clustered at the firm level.
log[I/K]it= β1log σite + β2log ˆsit+ γ log σite × log ˆsit+ ηi+ λt+ it
(1) (2) (3) (4) (5) (6) (7)
all all all low cs mid cs high cs all
log σite -0.154*** -0.008 0.095*** 0.005 -0.067*** 0.298***
(-7.92) (-0.54) (4.58) (0.24) (-2.84) (7.00)
log ˆsit -0.152*** -0.148*** -0.080*** -0.112*** -0.166*** -0.232***
(-13.96) (-13.88) (-4.41) (-7.53) (-10.69) (-14.42)
log σite × log ˆsit -0.069***
(-7.19)
Firm FE X X X X X X X
Time FE X X X X X X X
Observations 39925 40331 39925 13224 13241 13130 39925
R-squared 0.108 0.140 0.139 0.159 0.121 0.130 0.144
Notes: Each observation is a firm-quarter. Coefficients are reported with t-statistics in parenthese. ***, **, and * indicate significance at 1%, 5% and 10% levels and standard errors are clustered at the firm level.
Table 3: This table documents the relationship between asset volatility and investment, controlling for credit spread. The regression in column 5 includes all the regressors in the estimation equation, and the regressions in columns 1-4 drop the interaction term. Columns 2-4 use subsamples sorted by terciles on credit spread. Each observation is a firm-quarter. Coefficients are reported with t- statistics in parentheses. ***, **, and * indicate significance at 1%, 5% and 10% levels and standard errors are clustered at the firm level.
log[I/K]it= β1log σit+ β2log sit+ γ log σit× log sit+ ηi+ λt+ it
(1) (2) (3) (4) (5)
all low cs mid cs high cs all
log σit 0.090∗∗∗ 0.099∗∗∗ 0.067∗∗∗ 0.053∗∗ 0.683∗∗∗
(6.33) (5.62) (3.49) (2.49) (7.45)
log sit -0.280∗∗∗ -0.097∗∗∗ -0.288∗∗∗ -0.531∗∗∗ -0.457∗∗∗
(-12.61) (-3.02) (-7.70) (-12.38) (-13.44)
log σit× log sit -0.107∗∗∗
(-6.53)
Firm FE X X X X X
Time FE X X X X X
Observations 47384 16067 15878 15439 47384
R-squared 0.125 0.145 0.110 0.122 0.128
Table 4: This table documents the relationship between asset volatility and investment, controlling for market leverage. The regression in column 5 includes all the regressors in the estimation equa- tion, and the regressions in columns 1-4 drop the interaction term. Columns 2-4 use subsamples sorted by terciles on market leverage. Each observation is a firm-quarter. Coefficients are reported with t-statistics in parentheses. ***, **, and * indicate significance at 1%, 5% and 10% levels and standard errors are clustered at the firm level.
log[I/K]it= β1log σit+ β2log[M A/M E]it+ γ log σit× log[M A/M E]it+ ηi+ λt+ it
(1) (2) (3) (4) (5)
all low lev mid lev high lev all
log σit 0.018 0.014 0.048∗∗ -0.001 0.063∗∗∗
(1.23) (0.67) (2.44) (-0.03) (2.97)
log[M A/M E]it -0.460∗∗∗ -1.024∗∗∗ -0.506∗∗∗ -0.355∗∗∗ -0.543∗∗∗
(-19.38) (-8.19) (-7.46) (-11.50) (-14.65)
log σit× log[M A/M E]it -0.053∗∗
(-2.56)
Firm FE X X X X X
Time FE X X X X X
Observations 47400 16066 15920 15414 47400
R-squared 0.149 0.134 0.112 0.128 0.149
Table 5: This table presents the loadings of equity volatility and credit spread on asset volatility and market leverage. The dependent variable log yitdenotes either equity volatility log σite or credit spread log sit. We report results for estimations in levels in Panel A and results for estimations in first differences in Panel B. Each observation is a firm-quarter. Coefficients are reported with t- statistics in parentheses. ***, **, and * indicate significance at 1%, 5% and 10% levels and standard errors are clustered at the firm level.
log yit= β1log σit+ β2log[M A/M E]it+ ηi+ λt+ it
Panel A: Levels Panel B: Changes
(1) (2) (3) (4)
log σite log sit ∆ log σite ∆ log sit
log σit 0.783*** 0.180*** ∆ log σit 0.780*** 0.015***
(90.77) (15.91) (83.77) (5.00)
log[M A/M E]it 0.447*** 0.613*** ∆ log[M A/M E]it 0.244*** 0.248***
(56.20) (32.17) (25.66) (24.37)
Firm FE X X Firm FE X X
Time FE X X Time FE X X
Observations 47327 47250 Observations 44706 44640
R-squared 0.865 0.571 R-squared 0.794 0.345
C Robustness Checks
In this appendix, we provide several robustness checks for the results discussed above.
In particular, in Table A1 - Table A3, we show that using lagged values of equity volatility, asset volatility and credit spread generates similar results, which highlights the predictive power of these measures on investment. In TableA4through TableA7, we present the regression results replicating those in Table 1to Table 5 in the main text. The coefficients can be interpreted as the move in the dependent variable scaled by its standard deviation associated with one standard deviation increase in the explanatory variable. These results help us interpret the economic sigificance of the coefficients on equity volatility and credit spread. Also, the split-sample results confirm that our cross-sectional findings are not sensitive to different dispersion of the variables in different subgroups. In Table A8, we report results for regressions that estimate the same spefication as in Table 1 while using the same sample as used to generate Table2. Comparing TableA8with Table2indicates that fair value spread behaves similarly to credit spread in our investment regressions, and there are no concerns over the sample selection since we are using exactly the same sample.
In Figure A1, we show the elasticity of investment with respect to credit spread using asset volatility as the moderator variable and find very similar results those in Figure7 in the main text.
Table A1: This table replicates Table1 using lagged values. Investment rate is regressed on all the regressors on the right-hand side in column 7, on equity volatility and credit spread spread in columns 3-6, on credit spread in column 2 and on equity volatility in column 1. Columns 4-6 use subsamples sorted by terciles on lagged values of credit spreads log si,t−1. Each observation is a firm-quarter. Coefficients are reported with t-statistics in parentheses. ***, **, and * indicate significance at 1%, 5% and 10% levels and standard errors are clustered at the firm level.
log[I/K]it= β1log σi,t−1e + β2log si,t−1+ γ log σi,t−1e × log si,t−1+ ηi+ λt+ it
(1) (2) (3) (4) (5) (6) (7)
all all all low cs mid cs high cs all
log σi,t−1e -0.182*** -0.065*** 0.038** -0.052*** -0.105*** 0.830***
(-9.89) (-4.44) (2.02) (-2.60) (-4.54) (9.85)
log si,t−1 -0.293*** -0.272*** -0.115*** -0.286*** -0.513*** -0.477***
(-13.44) (-12.80) (-3.70) (-8.04) (-12.46) (-16.56)
log σi,t−1e × log si,t−1 -0.160***
(-10.51)
Firm FE X X X X X X X
Time FE X X X X X X X
Observations 51228 51220 50820 16991 17007 16822 50820
R-squared 0.101 0.126 0.126 0.136 0.100 0.119 0.135
Table A2: This table replicate Table 2 using lagged values. Investment rate is regressed on all the regressors on the right-hand side in column 7, on equity volatility and fair value spread in columns 3-6, on fair value spread in column 2 and on equity volatility in column 1. Columns 4-6 use subsamples sorted by terciles on lagged values of credit spreads log si,t−1. Each observation is a firm-quarter. Coefficients are reported with t-statistics in parentheses. ***, **, and * indicate significance at 1%, 5% and 10% levels and standard errors are clustered at the firm level.
log[I/K]it= β1log σi,t−1e + β2log ˆsi,t−1+ γ log σi,t−1e × log ˆsi,t−1+ ηi+ λt+ it
(1) (2) (3) (4) (5) (6) (7)
all all all low cs mid cs high cs all
log σi,t−1e -0.161*** -0.003 0.075*** 0.010 -0.066*** 0.334***
(-8.30) (-0.21) (3.63) (0.48) (-2.86) (7.51)
log ˆsi,t−1 -0.161*** -0.159*** -0.095*** -0.124*** -0.181*** -0.254***
(-14.29) (-14.34) (-5.16) (-7.97) (-11.27) (-14.82)
log σi,t−1e × log ˆsi,t−1 -0.076***
(-7.73)
Firm FE X X X X X X X
Time FE X X X X X X X
Observations 38740 37935 37600 12564 12427 12323 37600
R-squared 0.107 0.143 0.141 0.153 0.110 0.134 0.148
