Can a Normal Firm Value Diffusion Process Improve the Performance of the Structural
Approach to Pricing Corporate Liabilities?
James Chen1
April 23, 2017
1
Research137 LLC, P.O. Box 3625, Rancho Santa Fe, CA 92067, USA. Tel.: 760-697-5114,
email: [email protected]
1
Can a Normal Firm Value Diffusion Process Improve the Performance of the Structural
Approach to Pricing Corporate Liabilities?
Abstract
We derive the pricing formulas for corporate liabilities by integrating their loss functions
with firm value distributions from a normal and a lognormal firm value diffusion process
(FVDP). By using credit spreads as the input to the pricing formulas, we find that the average
credit spread-implied asset value volatility from the lognormal FVDP is much higher than the
average empirically estimated asset value volatility for investment-grade companies, whereas
the spread-implied asset volatility from the normal FVDP is almost identical to the estimated
asset volatility for companies with different leverage ratios. Consequently, the normal FVDP
can explain (i) the observed level of credit spreads when calibrated to historical default-loss
experience, and (ii) the expected return on the equity market from traded credit spreads.
Keywords: Credit spread puzzle, Market-based equity risk premium for the S&P 500 index,
Credit spreads, Firm value diffusion process, Structural model of credit risk
EFM Classification Code: 140, 310, 370, 520
2
1. Introduction
Well-known structural models of credit risk fail to generate average credit spreads for
investment-grade companies when calibrated to historical default-loss data (Huang and
Huang, 2012). These models, including the original structural model by Merton (1974) and
others, all use a lognormal firm value diffusion process (FVDP hereafter). This detail implies
that the assumption of a lognormal FVDP may be the cause of poor model performances, and
also highlights the absence of a structural model based on a non-lognormal FVDP.
In this paper, we use two different data sets—credit default swap (CDS) spreads and
historical default-loss experience, to show that the structural approach based on diffusive firm
value dynamics alone can explain the observed level of risk premiums for both stocks and
bonds with empirically estimated asset value volatility,1 provided the diffusion process is
normal rather than lognormal.
Specifically, the structural approach with a normal FVDP is shown to succeed in two
key aspects of corporate liability pricing that a lognormal FDVP has failed; They are, (i) the
lognormal FVDP produces credit spread-implied asset value volatility that is not supported
by the empirical evidence, and (ii) the lognormal FVDP is unable to predict the expected
level of risk premiums for both stocks and bonds with the same asset value volatility input.
The assumption of a lognormal FVDP qualitatively seems inadequate because firm
value in theory can be negative. In practice, the firm value is often equated with the asset
value that is the sum of market values of stocks and bonds. And the asset value is always
non-negative due to the limited liability protection given to shareholders. The last point can
be misleading on the lower bound for the firm value. The structural approach models stocks
and bonds as contingent claims on firm value, thus firm value as a modeling concept is the
independent variable that determines the values of stocks and bonds, not the other way
around. Consequently, the range of firm value should not be limited by the constraints of its
dependent variables, such as equity value is always non-negative. In other words, firm value
can be negative.2
1
Throughout the paper, the term asset value volatility is used to represent the firm value
volatility of the firm value dynamics, and we call firm value volatility asset value volatility to
conform with the nomenclature adopted by other papers on the structural approach.
2
The fact that equity investors demand the limited liability protection already implies that the
losses from operating a company could exceed the firm’s total asset; later an empirical
mechanism that can lead to the state of negative firm value is discussed in Section 2 with an
example that negative firm value is expected ex-ante.
3
For investment-grade companies with average leverage ratios ranging from 13% to 40%,
we show that the structural model of credit risk with a normal FVDPencompassing the full
range of firm valueis able to translate the observed credit spreads into asset value volatility
that agrees closely with the estimate of historical asset volatility, such that the percentage
differences between spread-implied and empirically estimated asset value volatility have a
4% mean and 3% standard deviation.3 Because the normal model can relate traded credit
spreads to historical asset value volatility, it is able to predict the observed level of credit
spreads when calibrated to historical default-loss experience. On the contrary, the structural
model with a lognormal FVDP produces credit spread-implied asset value volatility that is
higher than the average long term stock return volatility and exhibits an unreasonable
leverage ratio dependence.
Furthermore, structural models of credit risk should be able to price the equity risk
premium (ERP). This property is explored to differentiate the choice between a lognormal
and a normal FVDP with respect to the pricing of junior part of corporate capital structures.
When credit spread-implied asset value volatility is used as the asset value volatility input to
firm structural models, we find that a normal FVDP, but not a lognormal FVDP, can generate
an ERP value for the S&P 500 Index that is comparable to concurrent ERP survey value over
the period from 2006 to 2014. The agreement between the market- and survey-based ERP
further validates the normal FVDP, at the same time demonstrates that the levels of risk
premium for the equity and credit markets can be related through the structural approach.
Black and Scholes (1973) and Merton (1974) assumed a lognormal FVDP when they
introduced the structural approach to price corporate liabilities. However, two features of a
lognormal firm value distribution are incongruent with the seniorities of debt and equity
within a corporate capital structure. First, the corporate bonds to be valued by the FVDP have
their values significantly below the firm values at the time of valuation; in other words, bond
prices depend on the probability for scenarios where more than 100% of equity market value
is lost. Yet with the empirically estimated asset value volatility, the lognormal firm value
distribution assigns relatively little probability to the region where debt value is most
impacted; in order to generate observed credit spreads for debt, the model volatility input
needs to be significantly higher than the historical asset value volatility. This phenomenon is
3
The estimate of historical asset value volatility (or empirically estimated asset value
volatility) refers to the temporal average of historical asset value volatility estimates as
presented in Schaefer and Strebulaev (2008).
4
more acute for investment-grade debt than for high yield bonds (Jones, Mason, and Rosenfeld,
1984).
Second, a lognormal firm value distribution is asymmetric around current firm value
with a skew toward the downside scenarios, and with relatively high asset value volatility
needed to price credit spreads, the firm value distribution becomes even more asymmetric
around the initial firm value which will cause an overestimation of the ERP value. Overall,
the lognormal structural approach seems unable to price both debt and equity at the same
time, as different parts of the capital structure require different asset value volatility inputs in
order to account for their respective risk premium level.4
In addition, there is no good justification for the FVDP to be lognormal. Specifically,
Black and Scholes (1973) side-stepped the issue of negative firm value by considering the
firm value of Company A that exclusively owns the equity shares of Company B. Hence, by
construction, Company A’s firm value must be greater than or equal to zero, largely
consistent with the range of a lognormal distribution except at the point where firm value is
zero. Merton (1974) rationalized the randomness of firm value dynamics, but not why that
randomness should be proportional to firm value, which leads to a lognormal like FVDP.
Recently, Cheung and Lai (2012) suggested that the FVDP may not always be lognormal
when intertemporal firm profit is maximized.
However, the development of structural models of credit risk since the inception of the
structural approach has consistently used a lognormal FVDP. We think there may be three
reasons for choosing the lognormal FVDP. First, the structural approach has its origin in the
equity option pricing model, which assumes lognormal stock price dynamics. Lognormal
dynamics have the property of constant variance for the percentage return of stock price,
which is reasonable because the value of stock price is somewhat arbitrary and can be
changed by fiat through share split or reverse share split; thus, the return variance should
remain constant regardless of whether stock price is high or low. However, firm value
conceived by the structural approach represents the economic scale of the firm, and thus it
cannot be arbitrarily redefined as the stock price. Moreover, there is a negative correlation
between firm size and asset value volatility, with smaller firms exhibiting higher asset value
4
The quantitative evidence that shows that lognormal firm value dynamics cannot
simultaneously price debt and equity is described in Sections 4 & 5.
5
volatility.5 For these reasons, the lognormal assumption for firm value dynamics does not
have the justification that applies to the lognormal stock price dynamics.
Second, analytical tractability may drive the continued use of a lognormal FVDP. The
firm value dynamics of Merton’s (1974) model contains a firm value drift term that is
proportional to the firm value V. And the closed-form pricing formula for corporate liabilities
is obtainable only if the firm value diffusion term is also proportional to V (see Appendix A),
result in that the FVDP must be lognormal.
Firm leverage ratios are known to be mean-reverting despite firm values changing over
time (Collin-Dufresne and Goldstein, 2001). Hence, when the purpose of modeling firm
value is to price corporate bonds with a targeted leverage ratio, rather than to project future
firm values, the drift term in firm value dynamics should be omitted. Otherwise, firm value
dynamics that have a drift but without commensurate adjustments to outstanding debt and
equity will lead to an effective drift in the leverage ratio as well. On a practical level, driftless
firm value dynamics enable closed-form solutions for corporate liabilities using a nonlognormal FVDP.
Third, for some time the development of the jump-diffusion model has masked the
deficiencies of the lognormal FVDP. However, Huang and Huang (2012) showed that a
lognormal FVDP with jumps cannot explain the observed 4- and 10-year credit spreads with
model parameter choices supported by the empirical evidence.6
More specifically, the jump-diffusion model calibrated to historical default-loss
experience can generate the observed level of credit spreads but they will also produce an
unreasonably high ERP value. Without the complexity of jump-diffusion analytics, we draw a
similar conclusion in Section 5 that lognormal diffusion dynamics grossly overstate the ERP
when credit spread-implied asset value volatility is used as the model volatility input.
In this study, we derive the pricing formulas for corporate liabilities by using a normal
and a lognormal FVDP and use the comparisons of credit spread-implied asset value
volatility with the empirically estimated asset value volatility to differentiate the choice
between a lognormal and a normal diffusion process under the structural approach framework,
before testing the risk premium predictions from those two firm value dynamics against the
5
After controlling for the firm leverage ratios, we observe negative correlations of 32% to
55% between the estimated asset value volatility and firm size for investment-grade
companies.
6
Note that by construction, the volatility of FVDP that is part of a jump-diffusion model
cannot agree with the empirically estimated asset value volatility which already includes
stock price jumps.
6
observed prices across the entire corporate capital structure for investment-grade companies.7
A normal FVDP is chosen because it allows the possibility of negative firm value and has the
property of increasing percentage return variance with decreasing firm size.
Well known empirical tests of structural models of credit risk use comparisons between
time series of model-predicted spreads and those of traded credit spreads (Jones, Mason, and
Rosenfeld, 1984. Eom, Helwege, and Huang, 2004). This testing approach is affected by the
volatility estimate used as the model input. Potential errors in the volatility estimates are
amplified in the model predicted credit spreads.8 Furthermore, the basic assumption of
diffusive firm value dynamics is that the firm value at t+1 is unpredictable by time t. In other
words, the credit spread at time t+1cannot be predicted with the volatility information
collected up to time t. Thus in theory the exercise of estimating the model volatility input
from historical return or volatility data will not be reliable for predicting credit spreads. 9 And
any test on structural models’ performance with estimated volatility as model input is likely
to be biased by their inaccurate volatility estimates.
In this paper we use a novel approach to test the merits of structural models of credit
risk that specifically avoid using estimated volatility as the model input, instead, credit
spreads and other direct observables are used as model inputs, while credit spread-implied
asset value volatility is obtained as the model output, implicitly assuming that the structural
approach framework is correct and only the choice of FVDP is uncertain. Three metrics are
used to evaluate the empirical reasonableness of credit spread-implied asset value volatility:
its value relative to the estimated historical asset value volatility, its leverage ratio
dependence, and its term structure.10
We find that the empirically estimated asset value volatility from 2006 to 2014 for
investment-grade companies that have been members of the Markit CDX North American
index is 20.6%, using the asset value volatility estimation method described in Schaefer and
7
Hereafter, the structural model with a normal FVDP is labeled the normal model, while that
with a lognormal FVDP is labeled the lognormal model.
8
A 10% increase in the model volatility input from 21% to 23.1%, translates into more than
40% and 80% increase in predicted credit spreads from the normal and lognormal models,
respectively.
9
Here the credit spread predictions refer to credit spreads of a single issuer at different points
in time, not average credit spreads of a given issuer or average spreads of many issuers at one
point in time.
10
The current model testing approach has the benefit of comparing two variables—the
average historical asset value volatility and the average model volatility input required to
produce the observed level of credit spreads, both are likely to be constant over the time
period of this study.
7
Strebulaev (2008). While the average credit spread-implied asset value volatility from the
normal FVDP is very close to the estimated historical asset value volatility, at 21.5% and
21.9% for the 5- and 10-year maturities, respectively; whereas the average credit spreadimplied asset value volatility from the lognormal FVDP is unreasonably high, at 40.0% and
36.4% for the 5- and 10-year maturities, respectively. In fact, the lognormal spread-implied
asset value volatility is even higher than the average long-term stock return volatility of
31.6% for the same set of companies.
In addition, the spread-implied asset value volatility from the lognormal model has an
inverted term structure that implying a higher risk at shorter time horizons for high quality
companies, and shows an unexpected leverage ratio dependence, with a decreasing asset
value volatility as the leverage ratio increases.
On the contrary, the credit spread-implied asset value volatility from the normal model
increases slightly as the leverage ratio increases, in close agreement with the empirically
estimated asset value volatility (Figure 5).
Structural models of credit risk can provide a market-based ERP estimation when credit
spread-implied asset value volatility is used as the model volatility input. We find that from
2006 to 2014 the average ERP for the S&P 500 Index is 4.5% and 10.6%, from the normal
and lognormal models, respectively. For the same period, the average ERP value from the
CFO surveys is 3.5% (Graham and Harvey, 2014). This shows that the lognormal model is
unable to produce the observed credit spreads and expected ERP value with the same asset
value volatility input. In contrast, the normal model can describe the expected level of risk
premiums for both stocks and bonds with single asset value volatility input that is very close
to the historical asset value volatility.
Finally, the normal and lognormal models are calibrated to historical default-loss
experience and model-predicted credit spreads are compared directly with historical average
credit spreads for different credit rating categories. We deliberately apply 1973 to 1993
historical default-loss experience, equity premia, and all the other key assumptions used in
Huang and Huang (2012) to calibrate the two structural models presented in this paper—
computing default-loss implied asset value volatility under the real measure and use that as
the model volatility input to predict credit spreads under the risk neutral measure. To our
knowledge, this is the first study of the credit spread puzzle phenomenon (credit risk accounts
for only a small proportion of the observed credit spreads for investment-grade companies)
using a normal FVDP.
8
Three important observations can be made from the default-loss-implied asset value
volatility and model-predicted credit spreads. First, both credit spread predictions and default
loss-implied asset value volatility from the lognormal model are almost identical to those
reported by Huang and Huang (2012) using the Longstaff and Schwarz (1995) model with
lognormal firm value dynamics (Table 10). This finding validates the pricing formula derived
in this study with a lognormal FVDP using the probabilistic approach—integrating loss
functions of corporate liabilities with firm value distributions.11
Second, for investment-grade companies and from the normal model, implied asset
value volatility from 10-year CDS spreads and 10-year historical default-loss data are very
close to each other at 21.9% and 22.5%, respectively; on the contrary, for the lognormal
model, the 10-year spread-implied asset value volatility is much higher than the 10-year
historical default-loss-implied asset value volatility at 36.4% and 30.6%, respectively. This
leads to the qualitative expectation that the normal model can produce close to the full value
of historical average credit spreads, while the lognormal model will underpredict average
credit spreads for investment-grade corporate bonds.
Third, by using lognormal structural models of credit risk, Huang and Huang (2012)
concluded that the credit risk component accounts for 39%, 34%, and 41% of the 10-year
spreads of corporate yields over swap rates for bonds rated Aa, A, and Baa, respectively;
whereas under the same historical default-loss calibration, the normal model produces credit
spreads that are 117%, 80%, and 67% of the 10-year spreads of corporate yields over swap
rates for the Aa, A, and Baa rating categories, respectively.
Our conclusion that the credit spread puzzle phenomenon is a model artifact due to the
assumption of lognormal FVDP shares the same view with that of Feldhutter and Schaefer
(2015), that the structural models of credit risk with diffusive firm value dynamics alone can
explain the observed level of credit spreads. However, these two papers differ in specifics, in
particular we believe that the model calibration with 20 years of default-loss experience is
reliable as its default-loss implied asset value volatility closely corresponds to the empirically
estimated asset value volatility. And because the asset value volatility changes only slightly
among investment-grade companies with different leverage ratios from AAA to BBB
(Schaefer and Strebulaev 2008. And we confirm this observation with our data), little
11
The loss function for a corporate security is its face value minus its payoff function.
9
convexity bias is expected from model generated credit spreads using a representative firm
capital structure within each credit rating category.12
The rest of the paper is organized as follows. Section 2 describes the normal model in
detail and derives its risk premium pricing formula (in Appendix A (B), the normal
(lognormal) model’s pricing formula is modified to include a constant firm value drift term).
Section 3 summarizes the accounting and market data used in this study. Section 4 compares
credit spread-implied asset value volatility from the normal and lognormal models with
empirically estimated asset value volatility. Section 5 uses the credit spread-implied asset
value volatility as the model volatility input to arrive at the market-based ERP predictions for
the S&P 500 Index. Section 6 presents the credit spread predictions when the models are
calibrated to historical default-loss experience. Section 7 concludes.
2. Model and valuation formula
2.1 Model description
Firm value dynamics with a single diffusion process may be written as
dV = a(V,t) dt + b(V,t) dz
(1)
where V denotes firm value, z denotes a standard Wiener process, and t represents time. Both
models presented in this paper assume that a(V,t)=0, instead of a(V,t)=r−, which was
assumed in Merton’s (1974) model, where r is the risk-free interest rate and  is the company
payout rate. This change disallows the drift of the leverage ratio due to the drift of firm value.
Companies generally promise to maintain their financial ratiosincluding the leverage
ratiowithin certain bands when issuing bonds. Rating agencies also assign corporate bond
ratings with the expectation that, even through business cycles, the degree of a firm’s
financial leverage will be maintained. Thus, when pricing corporate bonds investors would
12
The normal model presented in this paper can describe the observed level of credit spreads
for investment-grade bonds with two readily obtainable inputscompany leverage ratio and
empirically estimated asset value volatility, whereas Feldhutter and Schaefer (2015) shows
that the Merton model (very close to the lognormal model used in this paper) can reproduce
the observed Baa-Aaa spreads differential with inputs that are either counterintuitive, such as
the average recovery and default rates from last 90 years of historical data, or inconsistent
with the empirical experience, like assumed Sharpe ratio of 0.22. With average firm specific
equity volatility of 30% and 40% reported in Schaefer and Strebuleav (2008) and used in
Feldhutter and Schaefer (2015), respectively, a Sharpe ratio of 0.22 implies an average equity
risk premium of 6.6% or 8.8% for investment-grade companies, which is clearly larger than
the historical expected return from the equity market.
10
expect the relative mix of debt and equity to be stable over the horizon corresponding to the
bond maturity.
For the normal model, b(V,t) is assumed to be a constant proportion of initial firm value
Vo, and the firm value dynamics reduce to
dV = Vo dz
(2)
where  denotes the volatility of the FVDP, and as the analysis in Section 4 shows that 
numerically agrees with the empirically estimated asset value volatility, and the time
evolution of firm value becomes a driftless arithmetic Brownian motion. Firm value at any
future time t is normally distributed with mean V0 and standard deviation Vo t ,
V(t) ~ V0,Vo t 







3)
Normal firm value dynamics have increasing volatility for the percentage return of firm
value as firm value decreases, in agreement with the empirical observation of higher firm
value volatility for a smaller firm.
The normal firm value distribution also permits negative firm value; thus, the
relationship between firm value and the market value of debt and equity should be considered
for the full range of firm value. Specifically,
If V(t)  0, then V(t) = E(t) + D(t)


If V(t)  0, then E(t)= 0, and D(t)= 0



a)
(4b)
where E(t) and D(t) denote the market values of equity and debt, respectively. Equation (4b)
states that scenarios of negative firm value correspond to scenarios where both debt and
equity investment have experienced total losses. Equation (4a) and (4b) together define the
relationship between the model concept of firm value within the structural approach and the
values of corporate liabilities.
Figure 1 shows the impact on the asset value distribution when firm value dynamics
change from lognormal to normal. The normal firm value distribution allows asset value to be
zero with a finite probability, and its likelihood is further augmented by the cumulative
probability of negative firm value scenarios. The resulting asset value distribution resembles
a diffusion process plus a jump to the state asset value is zero, a feature that intuitively
associates with the abrupt decline in asset value observed shortly before and through the
bankruptcy process.13
The magnitude of the “jump” is implicit in the asset value volatility for the FVDP, and need
not be separately modeled.
13
11
For comparison purposes, an asset value distribution from a lognormal firm value
distribution with the same standard deviation (asset value volatility) as that of the normal firm
value distribution is also shown in Figure 1. At 22% asset value volatility, close to the
average historical asset value volatility for investment-grade companies, the lognormal firm
value distribution assigns less than 0.6% probability to scenarios where debt value is
impacted, and it is also asymmetric around the initial firm value Vo, with 60/40 split between
the downside and upside scenarios.
[Insert Fig. 1 here]
Negative firm value may be regarded as an externality of the firm’s operations; when
this occurs, losses are imposed on society or, sometimes directly on the government. For
example, the explosion of Fukushima nuclear power plant operated by Tokyo Electric, caused
an estimated damage of USD 200 billion, which is more than Tokyo Electric’s total asset
value before the accident. Had the Japanese government not bailed out the company, equity
investors would have experienced 100% losses and senior debtholders would have suffered
near complete principal losses.
The Tokyo Electric example illustrates that negative firm value can arise through the
creation of new claims on a company’s assets. Moreover, when the value of new claims
exceeds total asset value, existing claims will be backed by negative firm value that equals to
the total asset value less the value of new claims. These new claims are likely to appear
suddenly, however, the new claims’ creation process could also take place over time, as in the
case of fraud, or falling coal prices over time that resulting in total claims on coal companies
exceed firm’s asset value.
Further, Equation (4b) implicitly assumes that new claims are senior to existing debt
claims. Thus, when total assets cannot satisfy all new claims, existing debt value equals zero.
This is an approximation, because debtholders are senior claimants on a company’s assets
and may be ranked pari-passu with new claims, in which case debt value is low but not
zeroed. Because negative firm value is a low probability event, the impact of Equation (4b)’s
approximation is likely to be small.
Finally, deposit losses due to failing banks are examples of negative firm values being
anticipated ex-ante by the market. The firm value of a bank is significantly smaller than its
total portfolio value which in large part is funded by deposits. It is possible that portfolio
losses may exceed the total market value of bank stocks and bonds, thus leading to the
scenarios of deposit losses. By Equation (4b) these are also scenarios of negative firm value.
Suffice it to say the deposit insurance premium predicted by the normal FVDP for large U.S.
12
banks agrees well with the actual deposit insurance premium assessed by the Federal Deposit
Insurance Corporation for large and complex financial institutions (Chen 2016).
2.2 Valuation formula
We derive the pricing formula for corporate liabilities through a probabilistic approach that
weights a given security’s loss function by the firm value distribution, while the loss function
is defined by security’s seniority and in terms of initial firm value.
Bond holders generally do not have the right to liquidate a firm in time when firm value
falls below the face value of debt, and there is no ex-ante consensus among investors on the
firm value (as a fraction of debt face value) which the event of default will take place.
Therefore it is not realistic to assume that the FVDP would stop at a predefined firm value,
and the expected bond principal losses (expected losses) should be modeled to bond
maturity.14
Corporate liabilities of most companies consist of two classes of securitiessenior
unsecured debt and common equity. Each class of corporate liabilities is defined by its
boundaries K1 and K2 within the capital structure. K1 and K2 are expressed as the percentages
of firm value at the time of valuation, the initial firm value Vo (Figure 2). Below the lower
boundary K1 are securities subordinated to the class, and in practice K1 is also called
attachment point. For example, K1 for common equity is zero because common equity is the
junior most security within the capital structure. By construction K2 is always greater than K1
and is called detachment point.
[Insert Fig. 2 here]
The notation described above also applies to more complex capital structures that may
have debt and equity of different seniorities, such as senior unsecured and subordinated debt
as well as preferred and common stocks. The pricing formula to be developed uses the K1 and
K2 values of a given security as the input irrespective of the complexity of capital structure.
Altogether, the key assumptions used for the valuation formula of the normal model are:
i) Firm value dynamics are described by dV = Vodz.
ii) The corporate capital structure is defined by the market value of equity and book
value of long- and short-term debt at the time of valuation.
14
In other words, the pricing formulas presented in this paper do not assume there is a
particular default boundary that the firm ceases to exist when the firm value falls below,
instead, the expected losses due to the FVDP are calculated to the bond maturity, like the
Merton (1974) model.
13
iii) FVDP is not terminated until security’s maturity date, in other words, no default
boundary is assumed that will stop the FVDP before the maturity date.
iv) The perfect priority rule is followed when distributing liquidation repayments to the
corporate liabilities of different seniorities.
v) The fixed rate r of an interest rate swap with tenor T is the risk-free interest rate for
time T, where T denotes the maturity of the corporate liability to be valued. The term
structure of the risk-free interest rate is flat.
Define the time now as 0. At time t>0, the expected losses for a given security is the
integration of firm value losses that exceed the security’s subordination amount K1Vo, with
the firm value distribution at time t (Figure 3):
Vo  K1Vo
EL(t )  
Vo  K 2Vo
Vo  K 2Vo
f (V , t )(Vo  V  K1Vo)dV  

f (V , t )( K 2Vo  K1Vo)dV (5)
and f(V,t) is the firm value distribution from Equation (3), specifically
f (V , t ) 
1
Vo t 2
[V Vo ]2
2
2
e 2 Vo t
For scenarios where V > Vo−K1Vo, the firm value has not decreased by more than the
subordination amount of K1Vo, there are no losses to the security; thus, the expected losses
calculation does not include the region where Vo−K1Vo < V < ∞.
The first term in Equation (5) corresponds to the losses allocated to the security when
firm value V declines from its initial value Vo to values between Vo−K1Vo and Vo−K2Vo, and
the security experiences a partial principal loss of an amount equal to the total firm value
losses of Vo−V less the subordination amount of K1Vo. The second term represents the
security’s maximum loss when firm value V decreases more than K2Vo. It’s clear that the
expected losses from Equation (5) in all cases are a positive value between zero and the
security’s face value of (K2–K1)Vo.
[Insert Fig. 3 here]
By integrating Equation (5) and dividing by (K2–K1)Vo to normalize the expected losses
at time t as a percentage of the security’s face value, we have
 K2
  K1
 t
2
 2 2t
EL(t ) 
 e 2 t
e
( K 2  K1 ) 2 

2
2

 K1 N (Vo  K1Vo)  N (Vo  K 2Vo)
 N (Vo  K 2Vo)

K 2  K1


(6)
14
where N(Vo-K1Vo) and N(Vo-K2Vo) are cumulative normal distribution functions with mean
Vo and standard deviation Vo t 
The present value of total expected losses (PVEL) is calculated as the sum of the
marginal normalized expected losses from the time of valuation to the security’s maturity T,
and discounted back by the risk-free discount factor. Hence,
PVEL 
i T /

e
 r ti
i 1
 EL(t i )  EL(t i1 ) 
(7)
where yearsis the time increment between ti and ti+1, t1=0.25 years, and EL(t0)=0. The
first term outside the curly bracket is the risk-free discount factor; the difference inside the
curly bracket represents the marginal normalized expected losses.15
The market convention is to express the price of a corporate bond in terms of a spread to
the risk-free interest rate, which requires the conversion of PVEL to an equivalent number of
basis points accruing on the security’s loss adjusted principal value, namely,
Credit Spread = S =
PVEL
PV 01
(8)
where PV01 is the present value of a 1 basis point (i.e., 10-4) accrual on the loss-adjusted
principal value from the time of valuation to maturity, and is given by
PV 01 
i T / 
 e  rt  1 EL(ti ) 10  4
i
(9)
i 1
Inside the curly bracket is the original normalized principal value (100% or 1) adjusted by the
normalized expected losses.16
The pricing formula derived above does not exclude the possibility that once firm value
is negative (at one point in time), it may become positive again (at a later point in time). This
property of the pricing formula contradicts to the intuition that once corporate securities are
worthless (because firm value is negative) they cannot subsequently have positive values.
This apparent shortcoming of the normal model pricing formula may not be too important for
two reasons. First, the probability of negative firm value occurring is small, and within that
15
Equation (7) verifies that the total normalized expected losses calculated at a quarterly time
interval are 99.8% of the total normalized expected losses calculated at a weekly time interval
for a 5-year security with K1 =60%, K2=100%, and =20%. This shows that the quarterly
frequency for the marginal expected losses calculation is a good approximation of the
continuous time marginal expected losses calculation.
16
Using a quarterly loss-adjusted principal value to compute the PV01 in effect compensates
bondholders for the risk that any declines in firm value more than the subordination amount
before the bond maturity date.
15
subset, the probability of negative firm value becoming positive later is even smaller. Second,
as the example of Tokyo Electric illustrates, the state of negative firm value could include
scenarios which negative firm value can be cured via a government bailout.17
In sum, the pricing formula derived here computes the expected losses of a corporate
security to its maturity using a normal FVDP. And the essential difference between the
normal model and the Merton (1974) model is their respective firm value dynamics.
3. Data
We choose to work with companies that are members of on-the-run Markit CDX North
American Investment-grade index (CDX NA IG) and Markit CDX North American High
Yield index (CDX NA HY), because their CDS are traded daily, and their financial reports
for the leverage ratio calculation are updated quarterly. Altogether, 80 companies
representing a diverse industry groups had a continuous presence in those two indexes from
June 2006 to June 2014.18 Sixty of the 80 companies are investment-grade rated and the
remaining 20 companies are high yield (Tables 1 and 2). These companies have most of their
debt liabilities as senior unsecured, for which credit risks are referenced by their CDS.19 The
starting date for the data set is purposely chosen to precede the financial crisis of 2008–2009.
From June 2006 to June 2014, total of 5,120 CDS spreads on the last business day of
each calendar quarter are gathered from Bloomberg, in addition to other market observables
and accounting records, including risk-free interest rates, stock price return volatilities, and
17
Technically, when the firm value hits a negative value, the debt and equity values should
be zero permanently from that point on. In reality, firm value is not observable, and with
examples like that of Tokyo Electric, it may be argued that debt and equity values will not be
zeroed immediately after the firm value is negative. Hence, with both technical and real
economic considerations, Equation (4b) becomes
𝐼𝑓 𝑉(𝑡) ≤ x, 𝑡ℎ𝑒𝑛 𝐸(𝜏) = 0 𝑎𝑛𝑑 𝐷(𝜏) = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜏 ≥ 𝑡, and x <0
(4b)'
For x = −0.1Vo and x = −0.2Vo, and assuming the normal firm value dynamics of Equation
(2), the Monte Carlo simulated credit spreads are 103% and 101% of calculated credit
spreads obtained from the pricing formula for the normal model at the 5-year maturity, for a
representative capital structure of investment-grade rated companies (i.e., leverage =24% and
asset value volatility = 21%). At the 10-year maturity, the simulated credit spreads are 117%
and 108% of calculated spreads, for scenarios of x = −0.1Vo and x = −0.2Vo, respectively.
18
Both the CDX NA IG and the CDX NA HY indexes are updated semi-annually in March
and September to ensure the members of new index series continue to meet the respective
index requirements, of which high trading volume is the key criterion for admission into the
indexes. Companies that belong to the utility sector are excluded because they are regulated
entities so that their credit spreads and stock return volatilities are not entirely market driven.
19
For senior unsecured debt, K1= 1– L and K2 =100%, where L denotes the leverage ratio.
16
the amount of outstanding short- and long-term debt for each company. These data and their
sources are described next.
3.1 CDS spreads
The source of 5-year and 10-year CDS spreads is Credit Market Analysis Limited. The values
of these spreads represent the composite views of market participants and they are used by
buy-side firms to mark their portfolios to market.
As the data show that unlike CDX indexes, most corporate bonds and their issuers’ CDS
do not trade every day. As of early 2014, the average daily trading volume for U.S. corporate
bonds (thousands of bonds in number) was USD 20 billion, whereas the daily average traded
notional of the on-the-run CDX NA IG 5Y (125 member companies) and CDX NA HY 5Y
(100 member companies) was USD 24 billion and USD 8.5 billion, respectively.20 Clearly
CDX members’ credit spreads are more accurate representations of credit risk than those of
non-members as credit spreads of index members are from actual trades of meaningful
volume rather than from traders’ marks or matrix pricing (bond pricing by key bond attributes
rather than actual traded bond prices).
3.2 Firm leverage
We define the leverage ratio as
Book value of long - and short - term debt
Market value of equity + Book value of long - and short - term debt
because the denominator is equal to the firm value at the time of valuation, and the leverage
ratio definition is consistent with the essence of the structural approach—the firm value
dynamics determine the prices of corporate liabilities. The book value of long- and short-term
debt are taken from companies’ quarterly financial statements.
The average leverage ratio for each company between June 2006 and June 2014 is
tabulated in Tables 1 and 2 together with the standard deviation of leverage ratios. The same
set of statistics is also included for the sub-period after the 2008–2009 financial crisis from
June 2010 to June 2014. The sub-period has almost the same average leverage ratio as that of
the entire 8-year period, showing that the mean-reversion of the leverage ratio takes place in a
time span much shorter than four year span of the sub-period.
The 80 companies are sorted by their average leverage ratios into seven almost evenly
spaced leverage groups (Table 3). As the structural approach principally relates three
variablesthe firm leverage ratio, asset value volatility, and the risk premiumto each other,
20
The CDX and U.S. corporate bond trading volume estimates were from Clarus Financial
Technology, www.clarusft.com, and Sifma, respectively for January 2014.
17
it is natural to organize the data set by the leverage ratio in order to analyze implied asset
value volatility when the risk premium is given, and to compute the market-based ERP when
credit spread implied asset value volatility is used as the model volatility input.
[Insert Tables 1–3 here]
3.3 S&P 500 Index leverage distribution
The S&P 500 companies are sorted quarterly by the same leverage groups used to organize
the 80-company set.21 The result is the index leverage distribution. The average leverage
distribution over the 8-year period ending in June 2014 is given in Table 4. The leverage
distribution is used in conjunction with the ERPs from the 80-company set to arrive at the
market-based ERP prediction for the S&P 500 Index. The quarterly average index leverage
and the normal model predicted ERP for the index are plotted in Figure 4.
[Insert Table 4 here]
[Insert Fig. 4 here]
3.4 Stock price return volatility
Stock return volatilities that correspond to six different time intervals (30, 90, 150, 250, 750,
and 1,000 trading days) immediately preceding the quarter-end date, are obtained from
Bloomberg. The average stock return volatility of companies (belonging to the same leverage
group) increases as the leverage ratio increases and the firm size decreases. Moreover, stock
return volatility generally has a higher value for a longer measurement time interval, such
that average stock return volatility from the 30 trading-days interval has the lowest value,
whereas that from the 1,000 trading-days interval has the highest. 22
3.5 Risk-free interest rate
The New York closing value of 5- and 10-year fixed rates of USD interest rate swaps are
used as the risk-free interest rates to calculate the 5- and 10-year spread-implied asset value
volatility and the value of ERP, respectively.
21
On average, leverage ratios for 98.7% of all members of the S&P 500 Index were available
at the end of each calendar quarter from June 2006 to June 2014.
22
This observation is likely due to the fact that over the time period of present study (2006 to
2014), a longer measurement period for the stock return volatility contains more repeated
sampling of the trading days during the 2008-2009 crisis period, thus result in higher average
stock return volatility.
18
4. Credit spread-implied asset value volatility
We use the pricing formulas derived in Section 2.2 and Appendix B for the normal and
lognormal models, respectively, and CDS spreads and leverage ratios as the known quantities
in these pricing formulas to solve for the unknown variable of asset value volatility
iteratively.23 The solutions are credit spread-implied asset value volatility, and they are
compared to the empirically estimated asset value volatility obtained from stock return
volatility and the leverage ratio with additional adjustments to account for the volatility of
debt and debt/equity return correlations. An advantage of the current approach for solving
spread-implied asset value volatility is that all model inputs are either market observables
(CDS spreads and risk-free interest rates) or accounting data (company leverage ratios), thus
that implied volatility is calculated without any bias from either assumed or modeled
parameters.
The comparison of average spread-implied and empirically estimated asset value
volatility is like the default-loss calibration approach (Huang and Huang 2012), in that both
methods are devised to test the ability of structural models of credit risk to explain the
observed level of credit spreads. In our case, the model that can produce the average spreadimplied asset value volatility agreeing with the empirically estimated asset value volatility is
the model that can explain historical average credit spreads. Furthermore, both implied and
estimated asset value volatility should also have same leverage ratio dependence; while the
term structure of implied asset value volatility can be used to further evaluate the
reasonableness of chosen FVDP.
First, we assess the value of credit spread-implied asset value volatility relative to both
the plausible range and the empirical estimate of historical asset value volatility. The upper
bound of historical asset value volatility is approximated by the average of long-term stock
return volatility (the average of stock return volatility from 1,000 trading-days for companies
of a given leverage group), implicitly assuming that debt volatility has the same value as that
of the equity volatility and the two asset classes are perfectly correlated. The lower bound of
historical asset value volatility is given by the product of (1−the average leverage ratio) and
the average of short-term stock return volatility (the average of stock return volatility from 30
trading-days), implicitly assuming that debt has zero volatility (Table 3).
23
Only numerical solutions for spread-implied asset volatility are obtained because the
expression of asset volatility as a function of credit spreads, leverage ratio, and risk free
interest rate is not available.
19
We can safely conclude that the average spread-implied asset value volatility of 40%
from the lognormal model for investment-grade companies is both outside the plausible range
and unreasonably high in value, because it is significantly higher than the 31.6% estimate of
the upper bound for asset value volatility (Table 5). Consequently, the lognormal model will
underpredict the level of credit spreads when estimated historical asset value volatility is used
as the model volatility input.
[Insert Table 5 here]
On the contrary, with respect to the investment-grade companies, the 21.5% credit
spread-implied asset value volatility from the normal model is both inside the plausible range
and is slightly higher than the estimated lower bound of historical asset value volatility of
20.4%, suggesting that investment-grade companies with an average leverage ratio of 24%
have a low and stable debt value volatility in the vicinity of 4% per annum, in agreement with
the empirical evidence on the price volatility of investment-grade corporate bonds.
Furthermore, the 21.5% spread-implied asset value volatility from the normal model is
close to the empirically estimated asset value volatility of 20.7% (Table 6), which means
when the average of historical asset value volatility estimates is used as the model volatility
input, the credit spread produced by the normal model is expected to agree with the observed
credit spreads average.
The empirically estimated asset value volatility is obtained from the relationship
between asset value volatility and leverage-adjusted stock return volatility, and is expressed
as a multiple of E (1-L) where E denotes the stock return volatility derived from monthly
stock price returns over a 3-year period. For companies with different leverage ratios, their
respective multiples can be inferred from Table 7 of Schaefer and Strebulaev (2008),
[Insert Table 6 here]
For the 60 investment-grade companies used in the current study, the estimated
historical asset value volatility of 20.7% from 2006 to 2014 is also close to the earlier
estimate of 21.8% for a much larger set of investment-grade companies from 1996 to 2003
(the Merrill Lynch Corporate Master index used by Schaefer and Strebulaev, 2008),
demonstrating that the present 60 investment-grade company set is representative of the
investment-grade company universe. The small difference between these two historical asset
value volatility estimates is likely to be caused by the average market capitalization of the
companies used in the present study is somewhat larger than that of the Merrill Lynch
Corporate Master index.
20
Second, the leverage ratio dependence of credit spread-implied asset value volatility is
an important metric that can differentiate the choice between a normal and a lognormal
FVDP at a more granular level within the investment-grade bond universe. Specifically, as
the average leverage ratio increases from 12.7% to 40.1%, credit spread-implied asset value
volatility from the normal model rises slightly from 21.1% to 23.5%, suggesting a small
negative impact on firm risk among investment-grade companies as their leverage ratios
increase. These observations agree closely with the leverage ratio dependence of empirically
estimated asset value volatility, such that the average percentage differences between implied
and empirically estimated asset value volatility is only 4%, with 3% standard deviation for
the percentage differences (see Figure 5 and Table 6).
The close numerical agreement shown in Figure 5 implies that different levels of credit
spreads for different leverage groups within the investment-grade bond universe can be
derived from their respective historical asset value volatility estimates.
[Insert Fig. 5 here]
On the contrary, from the lognormal model, credit spread-implied asset value volatility
decreases significantly from 47.1% to 34.3% as the leverage ratio increases, suggesting a
decrease in firm value risk as the leverage ratio rises, contradicts to our expectation. Further,
the average percentage difference between the credit-spread-implied asset value volatility and
the corresponding estimate of historical asset value volatility is 75%, such that the lognormal
model is clearly unable to explain the average credit spreads for any of the leverage groups
using the historical asset value volatility.
The leverage ratio dependence of spread-implied asset value volatility from the
lognormal model suggests that its underprediction of credit spreads becomes less severe as a
firm’s leverage ratio increases, agreeing with the observation that credit spread predictions
using Merton’s (1974) model for high yield companies are closer to traded credit spreads than
those for investment-grade companies (Jones, Mason, and Rosenfeld 1984).
Third, the term structures of credit spread-implied asset value volatility from the normal
and lognormal models have divergent slopes. For investment-grade companies, from the
normal model, the observed term structure of spread-implied asset value volatility between
the 5- and 10-year maturities is relatively flat with a slightly upward slope (across the four
different leverage groups, there is an average 0.4% increase in asset value volatility as
maturity lengthens). By contrast, spread-implied asset value volatility from the lognormal
model has a negative slope between those two tenors, with average values of 40.0% and
21
36.4% for the 5- and 10-year maturities, respectively, resulting in a 3.6% decrease in
volatility as maturity lengthens.
[Insert Fig. 6 here]
The observations made on both the term structure and leverage ratio dependence for
spread-implied asset value volatility from the lognormal model are not limited to the
investment-grade companies studied herein. The historical default-loss-implied asset value
volatility reported by Huang and Huang (2012) from calibrating a group of structural models
with a lognormal FVDP also has an inverted term structure with a 4.3% drop in asset value
volatility between the 4- and 10-year maturities. Moreover, for the 10-year maturity Huang
and Huang (2012) also shows a steady decline in default-loss implied asset value volatility
from 32.1% to 25.8% as the average leverage ratio increases from 13.1% to 43.3% for
investment-grade companies.
Lastly, we digress briefly to discuss a shortcoming of the driftless normal firm value
dynamics with respect to high yield companies. The term structure of spread-implied asset
value volatility for high yield companies shows a steep increase (16.3%) between 5 and 10
years, such that 10-year spread-implied asset value volatility is 6.7% higher than long-term
stock return volatility. The unreasonably high spread-implied asset value volatility seen from
the normal model for high yield companies is partially a model artifact. The normal firm
value distribution extends equally to both positive and negative firm value directions with
increasing volatility, thus could misrepresent an increase in the downside risk associated with
high asset value volatility as an equal increase in the upside potential.24 One way to mitigate
this shortcoming is to introduce additional terms to the firm value dynamics, such that in the
high volatility regime the probability for the down side scenarios will increase more than the
probability assigned to the upside scenarios.
We leave the exercise of refining the normal model for long-dated high yield bonds to
future research as the present focus is on examining whether the structural approach can
explain the observed level of credit spreads for investment-grade companies, which have
relatively low and stable asset value volatility and thus are less affected by the shortcoming
24
Owing to this shortcoming of the normal model, credit spread-implied asset value volatility
is not always solvable, especially when CDS spreads are very high. Although this is not a
problem for investment-grade companies, for the high yield companies studied in this paper,
at 5- and 10-year maturities, credit spread-implied asset value volatility is found for 97% and
90%, respectively, of the total 33 calendar quarters.
22
of a driftless normal FVDP that understates the downside risks in high asset value volatility
scenarios. 25
5. Market-based equity risk premium predictions
Market-based ERP predictions—ERP value obtained from the normal and lognormal models
with credit spread-implied asset value volatility as the model asset value volatility input —
test the FVDP assumption with respect to the pricing of equity tranches within corporate
capital structures. The results are especially interesting for the normal model because in the
preceding section it has been shown to be able to account for the investment-grade corporate
bond pricings with empirically estimated asset value volatility.
Furthermore, it is almost a necessary condition for the correct firm value dynamics to be
able to explain the level of credit spreads as well as the expected value of ERP, because the
structural approach assumes that corporate liabilities, both equity and debt, are contingent
claims on the same firm value, and the practice of capital structure arbitrage further ensures
that equity and credit markets are sufficiently integrated.
For the ERP calculation, an equity investment is approximated by a 10 year bond. A 10year maturity is chosen because equity as a perpetual instrument has a duration slightly
longer than 10 years. Thus, the risk premium of a 10-year bond with zero subordination
(K1=0) and K2=1–L is used here to approximate the value of ERP.
The average predicted ERPs for companies of different leverage groups are presented in
Table 7. As expected, the ERPs predicted by both models increase with rising leverage ratios.
Qualitatively, the magnitude of predicted ERP from the lognormal model seems too high
(ranging from 8.9% to 14.7%),26 whereas the ERP predictions from the normal model for
investment-grade companies ranging from 3.4% to 6.6% are more comparable to the
historical ERP estimates reported in Huang and Huang (2012) ranging from 5.35% to 6.55%.
[Insert Table 7 here]
Quantitatively, we can compare market-based ERP predictions for the S&P 500 Index
with concurrent survey values and the long-term ERP estimate from historical asset return
25
As of Q1 2016, investment-grade corporate bonds make up 84% of the total corporate bond
universe.
26
The unreasonably high ERP values predicted by the lognormal FVDP may be the reason
for not using the structural approach to estimate the expected return on the equity market in
Merton (1980).
23
data.27 The model predicted ERP for the index is obtained as the weighted sum of average
ERPs of different leverage groups with the weights given by the index leverage distribution.
The results from the normal model are plotted in Figure 4 together with the time series
data for the index’s average leverage ratio. As expected that market-based ERP obtained from
the structural approach seems correlated with the temporal variations of the index’s leverage
ratio because leverage ratio in addition to asset value volatility are two primary determinants
for the value of ERP.
The values of model-predicted ERP as well as the CFO ERP survey values are shown
together in Figure 7. From June 2006 to June 2014, the average ERP for the S&P 500 Index
predicted by the normal model is 4.5%, whereas the average value from the lognormal model
is 10.6%. For the same time period, the average CFO survey value for the ERP of the same
index is 3.5% (Graham and Harvey, 2014) and the ERP estimated from U.S. stock market
returns from 1928 to 2012 is 4.2% (Damodaran, 2013). Both model-predicted ERP and ERP
survey values share the mean-reversion feature with comparable variability.
[Insert Fig. 7 here]
The poor performance of the lognormal model in predicting the ERP by using credit
spread-implied asset value volatility as the model volatility input is unsurprising. As noted in
the Introduction, in order to match the observed level of credit spreads, relatively high asset
value volatility is needed to shift more probability density of the lognormal firm value
distribution toward the senior part of the capital structure or downside scenarios; and this
directly results in overstating the ERP value.
The market-based ERP from the normal model is expected to be somewhat different and
perhaps even higher than the ERP survey values, as the former may be considered the hard
data (derived from prices of traded securities) and the latter is soft data from CFO surveys,
and because the normal model uses driftless firm value dynamics that does not take into
account the impact of firm value drift on the value of ERP. Hence, a normal FVDP with drift
(Appendix A) will be a better normal model for market-based ERP predictions. Nevertheless,
the ability of the normal model to quantitatively relate the levels of risk premium in the credit
and equity markets, already represents a significant improvement over the lognormal model
that is only able to characterize the co-movement of returns in those two markets (Schaefer
and Strebulaev 2008. Bao and Pan 2012).
27
Note that approximately 90% of the index is investment-grade rated, thus both the ERP
predictions for the index and credit spread-implied asset value volatility discussed in Section
3 are applicable to companies with similar leverage characteristics.
24
In sum, the analyses presented in Sections 4 and 5 show that expected additional returns
over the risk free interest rate from stocks and bonds can be explained by a normal FVDP
with historical asset value volatility, thus meaningful real time asset value volatility may be
inferred from traded credit spreads via the structural approach with a normal FVDP.
6. Is credit spread puzzle a model artifact?
The manifestation of credit spread puzzle is consistent across many economic considerations
within the structural approach framework using a lognormal FVDP when models are
calibrated to historical default-loss data (Huang and Huang, 2012). Illiquidity premia for
trading credit, tax benefits for bond issuers, and other factors have been suggested as noncredit variables contributing to the observed credit spreads that cannot be explained by credit
risk.
However, since the development of the CDS market, the empirical data show that CDS
spreads, without the effect of tax considerations and with average bid-offer spreads often
narrower than a few basis points, track closely to bond yield spreads to swap rates, such that
the CDS-bond basis on average only amounts to a small percentage of total credit spreads.
This observation strongly suggests that credit risk is responsible for most of the observed
credit spreads.
Already the conclusions of Section 4, derived from CDS data and the firm value
dynamics in the risk-neutral measure, imply that credit spread puzzle is an artifact of
lognormal FVDP. Here we apply default-loss experience to calibrate the firm value dynamics
in the real measure and find that the normal model can generate over 80% of observed level
of credit spreads, whereas the lognormal model reproduces the credit spread puzzle
abnormality as observed by Huang and Huang (2012).
Specifically, we calibrate the normal and lognormal models to the historical default-loss
data from 1973 to 1993, and the other assumptions used in Huang and Huang (2012),
including those related to the leverage ratios, equity premia, and the r− value. The pricing
formulas for both models with a drift term for firm value dynamics are used to incorporate
the asset premium and r− assumptions under the real probability measure (see Appendices
A and B). Further, the expected asset premium is the weighted sum of the targeted equity
premia and the observed level of credit spreads, with the weights given by (1−L) and the
leverage ratio L, respectively.
25
Notice that the normalized expected losses EL(ti) in Equation (9) can be interpreted as
the cumulative default probability, because 1−EL(ti) is used as the survival probability to find
the loss-adjusted principal value at time ti. For this reason, we match EL(T) to the historical
default probability over time T. The implied asset value volatility that matches EL(T) to the
desired default probability with the corresponding assumptions of the asset premium,
r−and the leverage ratio are then used to calculate the credit spread in the risk-neutral
measure with both asset premium and r−set to zero and a recovery rate of 51.3%.28 The
predicted credit spreads from the normal and lognormal models are summarized in Tables 8
and 9 for the 10- and 4-year maturities, respectively, together with the base case results from
Huang and Huang (2012).
[Insert Tables 8 and 9 here]
We adopt the same table format that was used by Huang and Huang (2012) for ease of
comparison, but focus on two new observations from the present study. First, for investmentgrade companies and from the normal model, the historical default-loss-implied asset value
volatility and the credit spread-implied asset value volatility (albeit from a more recent time
period) have similar term structure and comparable values. Specifically, the average defaultloss-implied asset value volatilities are 20.1% and 22.5% for the 4- and 10-year maturities,
respectively, which compare closely to the average credit spread-implied asset value
volatilities of 21.5% and 21.9% for the 5- and 10-year maturities.
On the contrary, from the lognormal model, the average default-loss-implied asset value
volatilities are 31.1% and 30.6% for the 4- and 10-year maturities, respectively. These values
are less than the average credit spread-implied asset value volatilities of 40.0% and 36.4% for
the 5- and 10-year maturities from the lognormal model.
Because spread-implied asset value volatility from the normal model is shown to agree
with the empirically estimated asset value volatility in Section 4, thus by extension, for
investment-grade companies, the default-loss-implied asset value volatility from the normal
model also agrees well with the empirically estimated asset value volatility. This strongly
supports that the model calibration using 20-year default-loss data is valid, contrary to the
claim that much longer time series of default data is required for model implementation
(Feldhutter and Schaefer 2015).
28
When interpreting EL(t) as the cumulative default probability, we can incorporate the
51.3% historical recovery rate by multiplying EL(t) by 1-51.3% to obtain the expected losses
that match the historical default-loss experience, and to model credit spreads in the riskneutral measure.
26
Furthermore, the observed difference between the default-loss-implied asset value
volatility from the lognormal model (Tables 8 and 9 Panel B Column 6) and the empirically
estimated asset value volatility (Table 6 Column 7), is similar to the difference seen from the
Table 2 of Huang and Huang (2012) and the Table 7 of Schaefer and Strebulaev (2008).
Second, the increase in credit spreads predicted by the normal model relative to the base
model used by Huang and Huang is 30.2 bps, 31.5 bps, 33.7 bps, and 38 bps for Aaa, Aa, A
and Baa rating categories (Table 8 Panel A and C Column 7), such that with the same 10-year
swap spread used in Huang and Huang (2012), the credit spreads from the normal model are
117%, 80%, and 67% of the historical 10-year spreads of corporate yields over swap rates for
Aa-, A-, and Baa-rated corporate bonds, respectively.29
Why did not the normal model predict closer to the full level of observed credit spreads
when its default-loss-implied asset value volatility is so close to the spread-implied asset
value volatility? We think the reason may be that the corporate yield data used by Huang and
Huang (2012) to compute historical credit spreads include bonds that have call options, and
some bond prices are derived from matrix pricing, both which are likely to introduce errors to
the calculated corporate-Treasury yield spreads. Furthermore, the corporate-Treasury yield
spread data cover the period from 1973 to 1993, whereas the swap spread data run from 1988
to 1995. This mismatch in time periods would have led to additional errors in the conversion
of corporate-Treasury yield spreads to the spreads of corporate yields over swap rates.
Lastly, the results in Tables 8 and 9 show that even the lognormal model presented in
this paper predicts slightly higher credit spreads than those of the base model (the Longstaff
and Schwartz 1995 model) from Huang and Huang (2012). This difference is likely due to the
driftless firm value dynamics assumption used by the lognormal model in this paper, but not
in the base model used by Huang and Huang (2012). As shown in Table 10, the predicted
credit spreads by the lognormal model, when calibrated to the r−assumptionagree
more closely with those from Huang and Huang (2012) for both 4- and 10-year maturities,
29
The10-year swap spread estimated by Huang and Huang (2012) is 52 bps, with that Huang
and Huang (2012) concluded that the credit risk component accounts for 39%, 34%, and 41%
of the 10-year spreads of corporate yields over swap rates for bonds rated Aa, A, and Baa,
respectively. Similar comparisons of model-predicted spreads with the spreads of corporate
yields over swap rates for the 4-year maturity are not possible because no 4-year swap spread
data are available from 1973 to 1993.
27
because all models are effectively using driftless lognormal firm value dynamics in this
scenario.30
[Insert Table 10 here]
The incremental effect on model-predicted credit spreads when taking into account the
constancy of the leverage ratio with a lognormal firm value dynamics was also noted in
Huang and Huang (2012), where the Collin-Dufresne and Goldstein (2001) model with a
mean-reverting leverage ratio was found to produce credit spreads that are larger than but
similar to the predictions of the Longstaff and Schwartz (1995) model.
7. Conclusion
In this paper, we showed that average asset value volatility inferred from credit spreads
using the structural approach with a normal FVDP is surprisingly close to the empirically
estimated asset value volatility for investment-grade companies with a wide range of leverage
ratios. This finding suggests that for a given leverage ratio and maturity, the observed level of
credit spreads is largely determined by historical asset value volatility, hence credit spreads
are almost entirely due to credit risk as modeled by the structural approach based on normal
diffusive firm value dynamics without jumps.
In addition, the average market-based ERP value obtained from the normal firm value
dynamics is comparable to the average S&P 500 ERP survey value over the 8-year period
ending in June 2014, demonstrating that the normal model can explain the expected level of
risk premiums for both debt and equity with single asset value volatility input that is
supported by the empirical evidence. Finally, the conclusion that credit risk is responsible for
the majority of observed levels of credit spreads for different rating categories is separately
verified by the credit spread predictions from the normal model when calibrated to historical
default-loss experience.
On a practical level, the connection between the normal model and the empirically
estimated asset value volatility demonstrated in this paper will enhance the usefulness of the
structural approach to the trading and investing of corporate securities.
For the real probability measure, r− should be 2% rather than 0% from 1973 to 1993.
Thus, the calibration exercise using r−is solely to show that the lognormal model
derived in this paper is pari-passu to the base model studied by Huang and Huang (2012).
30
28
Figure legend
Figure 1. Comparison of asset value distributions from a normal and a lognormal
FVDP. The horizontal axis represents firm value V as well as asset value of the firm, and
decreases to the left. Both normal and lognormal distributions are based on a 5-year time
horizon and 22% asset value volatility. Vo denotes the initial firm value, the numerical
values of K1 and K2 are 0.75 and 1, respectively, and they are the starting and ending points
of the debt class within the capital structure.
Figure 2. Schematic illustration of a firm’s capital structure. The capital structure
consists of two classes of corporate liabilities: debt is the senior class and equity is the junior
class, and K1 and K2 denote debt tranche’s subordination and detachment point, respectively.
Figure 3. Debt security loss function and firm value distribution. A normal firm value
distribution is superimposed on the normalized debt loss function for a bond with K1=75%
and K2=100%, where normalized loss function =min[K2Vo- K1Vo,max(0, Vo-VK1Vo)]/[ K2Vo- K1Vo]. The horizontal axis represents the firm value V; it decreases toward
the left and can be negative. Vertical axis on the left hand side is the probability for a given
value of V by Equation (3). Initial firm value is denoted as Vo and is both the center and the
peak of the firm value distribution for a driftless normal FVDP. The debt security starts to
experience a principal loss when firm value falls more than its subordination K1Vo, and losses
become 100% when firm value falls more than K2Vo. The illustrated firm value distribution
is for T=5 years and 
Figure 4. Time series of market ERP predicted by the normal model with concurrent
average leverage ratio of the S&P 500 Index. The S&P 500 Index average leverage ratio at
the end of each calendar quarter between June 2006 and June 2014 (left axis), and the equity
risk premium for the Index predicted by the normal model (right axis).
Figure 5. Comparison of spread-implied asset value volatility with estimated historical
asset value volatility for investment-grade companies. All but one investment-grade
companies are sorted into 4 leverage groups based on each company’s average leverage ratio
from June 2006 to June 2014*. The average 5-year CDS spread-implied asset value volatility
of each leverage group from the normal model closely agrees with the average historical asset
29
value volatility estimated from the empirical relationship inferred from Schaefer and
Strebulaev, 2008), on the other hand, the 5-year CDS spread-implied asset value volatility
from the lognormal model is significantly higher than historical asset value volatility
estimation and has a counter-intuitive trend of decreasing asset value volatility with
increasing leverage ratio.
*Deere & Co is excluded from the group of 60 investment-grade companies in this graph
because its leverage ratio is distorted by company’s large current receivables, which is almost
80% of company’s long term debt. Consequently the company’s average leverage ratio of
48% is the highest among all 60 investment-grade companies and is overstating its leverage
relative to other companies.
Figure 6a. 5- and 10-year spread-implied asset value volatility derived from the normal
model for investment-grade companies by leverage groups. The spread-implied asset
value volatility term structure is either flat or has a slightly upward slope that suggests
business risks are little changed or slightly increasing for investment-grade companies as time
horizon lengthens from 5 to 10 years.
Figure 6b.
5- and 10-year spread-implied asset value volatility derived from the
lognormal model for investment-grade companies by leverage groups. The spreadimplied asset value volatility term structure is either flat or has an apparent downward slope
that suggests a lower risk at longer time horizon for investment-grade companies.
Figure 7. Time series of ERP for the S&P 500 Index predicted by the normal and
lognormal models, compared with concurrent CFO ERP survey values.
Appendix
Appendix A: The valuation formula for the normal model with a constant drift rate as
part of the firm value dynamics
Although the normal model presented in this paper does not have a firm value drift term for
the reason given in Section 1, in order to consider firm value dynamics in the real probability
measure where asset value is assumed to increase at the rate of expected asset premium, a
valuation formula with a constant firm value drift term is needed. This valuation formula is
used in Section 5 to obtain the historical default-loss-implied asset value volatility, which is
then used as the model’s asset value volatility input in the risk-neutral pricing formula
derived in Section 2 to predict the level of credit spreads.
30
The normal firm value dynamics with a constant drift can be written as
dV =  Vo dt + Vo dz
(A1)
where V denotes the firm value at time t, z denotes a standard Wiener process,  is a constant
and denotes the volatility of FVDP, and Vo denotes the initial firm value. We have chosen
the constant drift rate to be proportional to Vo rather than V so that the integration of the
expected losses calculation can be solved analytically.
The firm value at any future time t will be normally distributed with mean V0 + Vo t
and standard deviation Vo t ,
V(t) ~  V0+V0t,Vo t 






2)
From Figure 3, at time t the expected loss of a debt security is,
EL(t )  
Vo  K1Vo
Vo  K 2Vo
f (V , t )(Vo  K1Vo  V )dV  
Vo  K 2Vo

f (V , t )( K 2Vo  K1Vo)dV
(A3)
where f(V,t) is the firm value distribution at time t, and from Equation (A2) f(V,t) is
f (V , t ) 
1
Vo t 2
[V (Vo  Vot )]2
2 2Vo 2t
e
Integrating Equation (A3) and dividing by (K2 –K1)Vo to normalize the expected losses
as a percentage of the security’s principal value:
( t  K 2 )
 ( t  K1 )
2
2

2

t
EL(t ) 
e
 e 2 t

( K 2  K1 ) 2 

 t
2
2

 ( K1  t )N (Vo  K1Vo)  N (Vo  K 2Vo)
 
K 2  K1

 N (Vo  K 2Vo)
(A4)
where N(Vo−K1Vo) and N(Vo−K2Vo) are cumulative normal distribution functions with V0
+ Vo t mean and Vo t standard deviation
Follow the same procedure described in Section 2.2 from Equation (7) to (9) to obtain
the pricing formula for credit spreads from Equation (A4).
In the real probability measure, such as when calibrating the normal model to the
historical default-loss data,
  r  
where r is the risk-free interest rate, is the firm’s payout rate, and  is the asset risk
premium. The asset risk premium is the weighted sum of observed credit spreads and equity
31
risk premia, with the leverage ratio as the weight for credit spreads and 1−L as the weight for
the equity risk premium.
In the risk-neutral measure, such as when using the pricing formula of the normal model
to calculate credit spreads and equity risk premia, and to solve for credit spread-implied asset
value volatility, =0.
Appendix B: The valuation formula for the lognormal model with a constant drift rate
as part of the firm value dynamics
The lognormal model studied in this paper assumes a zero drift term in the firm value
dynamics. However, in order to calibrate the model in the real probability measure, a
valuation formula with a constant firm value drift term is needed. This valuation formula is
used in Section 5 to obtain the historical default-loss-implied asset value volatility, which is
then used as the asset value volatility input to the risk-neutral pricing formula (i.e., the pricing
formula with ) to predict the level of credit spreads. For the analyses of credit spreadimplied asset value volatility and the equity risk premium calculation in Sections 3 and 4,
respectively, the constant drift rate is set to zero.
The change in firm value following a geometric Brownian motion with a constant drift
can be written as
dV =  V dt + Vdz
(B1)
where V denotes the firm value at time t, z denotes a standard Wiener process, and and
denote the constant drift rate and the volatility of FVDP, respectively.
From Equation (B1), it follows that the firm value at time t is
[(  
V (t )  Voe
2
2
)t  t  ]
(B2)
where Vo denotes the initial firm value at t=0, and is a standard normal variable with
mean=0 and variance=1.
From Figure 3, at time t the expected loss of a debt security is,
Vo  K1Vo
EL(t )  
Vo  K 2Vo
Vo  K 2Vo
g (V , t )(Vo  K1Vo  V )dV  
0
32
g (V , t )( K 2Vo  K1Vo)dV (B3)
where g(V,t) is the firm value distribution at time t, and from Equation (B2) g(V,t) is a
1


lognormal distribution with mean ln Vo      2 t , and standard deviation 
2


g (V , t ) 
1
 tV 2
t 
1
[ln V (ln Vo     2 )t )]2
2
2
2

t
e
Integrating Equation (B3) and dividing by (K2 –K1)Vo to normalize the expected losses
as a percentage of the security’s face value:
1  K1   ln( b)  m 
 ln( a)  m  e
EL(t ) 
  
 

K 2  K1    t 
  t 
m
2
2
t
     ln( a)  m 
( K 2  K1 )


 t


(B4)
where  denotes the cumulative standard normal function, and
m  ln Vo  (  
2
2
)t
a  Vo  K 2Vo
b  Vo  K1Vo


ln( a)  (m   2 t )
 t
ln( b)  (m   2 t )
 t
Follow the same procedure described in Section 2.2 from Equation (7) to (9) to obtain
the pricing formula for credit spreads from Equation (B4).
33
References
Bao, J and Pan, J (2012) “Relating Equity and Credit Markets through Structural Models:
Evidence from Volatilities.” Working paper, Ohio State University
Berndt, A (2015) “A Credit Spread Puzzle for Reduced-Form Models” Review of Asset
Pricing Studies 5:48-91.
Black, F and Scholes, M (1973) “The Pricing of Options and Corporate Liabilities” Journal
of Political Economy 81, 637-659.
Chen, J (2016) “Estimating the Deposit Insurance Premium from Bank CDS Spreads: An
Application of Normal Firm Value Diffusion Process” SSRN working paper.
Cheung, AMK and Lai, VS (2012) “On the Consistency of a Firm’s Value with a Lognormal
Diffusion Process” Journal of Mathematical Finance 2, 31-37.
Collin-Dufresne, P and Goldstein, RS (2001) “Do Credit Spreads Reflect Stationary Leverage
Ratios?” Journal of Finance 56, 1929-1957.
Damodaran, A (2013) Equity Risk Premium (ERP): Determinants, Estimation, and
Implications – the 2013 edition. New York, New York University.
Eom, YH, Helwege, J, and Huang, JZ (2004) “Structural Models of Corporate Bond Pricing:
an Empirical Analysis,” The Review of Financial Studies 17, 499-544.
Feldhutter, P and Schaefer, S (2015) “The Credit Spread Puzzle – Myth or Reality?”,
working paper , London Business School.
Graham, JR and Harvey, CR (2014) “The Equity Risk Premium in 2014.” Working paper,
Duke University.
Huang, JZ and Huang, M (2012) “How Much of the Corporate-Treasury Yield Spread is Due
to Credit Risk?” Review of Asset Pricing Studies 2(2), 153–202.
Jones, EP, Mason SP, Rosenfeld E (1984) “Contingent Claims Analysis of Corporate Capital
Structures: an Empirical Investigation,” Journal of Finance 39, 611-625.
Keenan, SC, Shtogrin I, and Sobehart, J (1999) Historical Default Rates of Corporate Bond
Issuers, 1920-1998. New York, Moody’s Investor’s Service.
Longstaff, FA and Schwartz, ES (1995) “A Simple Approach to Valuing Risky Fixed and
Floating Rate Debt” Journal of Finance 50, 789-820.
Merton, RC (1974) “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,”
Journal of Finance 29, 449-470.
Merton, RC (1980) “On Estimating the Expected Return on the Market,” Journal of
Financial Economics 8, 323-361.
Schaefer, SM and Strebulaev, IA (2008) “Structural Models of Credit Risk are Useful:
Evidence from Hedge Ratios on Corporate Bonds,” Journal of Financial Economics 90, 1-19.
34
Tables
Table 1: Average leverage ratios for investment-grade companies that have been members of
on-the-run CDX NA IG index through out the eight-year period from June 2006 to June 2014
From June 2006 to June
2014
From June 2010 to June
2014
Excluding 2008-2009 financial
crisis
Company name
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
21st Century Fox America Inc
AutoZone Inc
Carnival Corp
CBS Corp
Comcast Cable Communications LLC
Marriott International Inc/DE
McDonald's Corp
Normalell Rubbermaid Inc
Nordstrom Inc
Omnicom Group Inc
Target Corp
Time Warner Inc
Walt Disney Co/The
Whirlpool Corp
Altria Group Inc
Campbell Soup Co
ConAgra Foods Inc
CVS Health Corp
General Mills Inc
Kroger Co/The
Mondelez International Inc
Safeway Inc
Wal-Mart Stores Inc
Anadarko Petroleum Corp
ConocoPhillips
Devon Energy Corp
Halliburton Co
Transocean Inc
Valero Energy Corp
Aetna Inc
Amgen Inc
Baxter International Inc
Bristol-Myers Squibb Co
Cardinal Health Inc
McKesson Corp
Pfizer Inc
Boeing Capital Corp
Caterpillar Inc
CSX Corp
Avg.
leverage
ratio
1 std. of
quarterly
calculated
leverage ratio
Avg.
leverage
ratio
1 std. of
quarterly
calculated
leverage ratio
23%
21%
25%
30%
33%
19%
13%
31%
21%
20%
29%
33%
17%
30%
16%
20%
27%
16%
24%
32%
29%
39%
18%
32%
21%
23%
12%
32%
29%
21%
20%
12%
11%
16%
12%
16%
16%
41%
28%
7%
2%
5%
14%
6%
6%
2%
11%
10%
4%
7%
6%
5%
9%
7%
3%
7%
3%
3%
4%
7%
9%
2%
10%
5%
8%
4%
12%
11%
7%
6%
4%
2%
5%
4%
7%
5%
8%
5%
22%
21%
25%
23%
32%
18%
12%
28%
22%
22%
31%
30%
14%
27%
18%
22%
27%
16%
22%
34%
30%
45%
20%
27%
21%
27%
13%
40%
31%
25%
24%
15%
10%
15%
13%
19%
16%
38%
27%
4%
1%
2%
8%
5%
2%
1%
7%
2%
3%
2%
4%
3%
7%
2%
2%
9%
3%
1%
4%
5%
10%
2%
5%
3%
8%
3%
5%
9%
3%
4%
3%
2%
1%
3%
4%
5%
4%
3%
35
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Deere & Co
Honeywell International Inc
Ingersoll-Rand Co
Lockheed Martin Corp
Norfolk Southern Corp
Northrop Grumman Corp
Raytheon Co
Southwest Airlines Co
Union Pacific Corp
Computer Sciences Corp
Hewlett-Packard Co
International Business Machines Corp
Motorola Solutions Inc
Alcoa Inc
Dow Chemical Co/The
Eastman Chemical Co
EI du Pont de Nemours & Co
International Paper Co
Sherwin-Williams Co/The
AT&T Inc
Verizon Communications Inc
48%
16%
23%
14%
25%
19%
14%
25%
18%
30%
21%
15%
14%
38%
33%
27%
22%
42%
11%
27%
33%
7%
5%
11%
4%
3%
3%
4%
9%
5%
10%
14%
3%
5%
12%
11%
6%
5%
12%
3%
4%
7%
48%
14%
21%
16%
25%
20%
17%
27%
15%
34%
32%
14%
13%
44%
34%
26%
22%
39%
10%
28%
34%
4%
3%
4%
3%
3%
1%
3%
7%
3%
7%
11%
2%
2%
7%
6%
6%
3%
6%
2%
3%
5%
Average
24%
6%
24%
4%
Table 1 lists the companies that have been members of on-the-run CDX NA IG index from June 2006 to June
2014, excluding the companies from the utility sector. The leverage ratios are computed as the ratio of (A) book
value of short-term plus long-term debt to (B) the sum of book value of short-term and long-term debt plus the
market value of equity, as of the last trading day of each calendar quarter.
Table 2: Average leverage ratios for high-yield companies that have been members of onthe-run CDX NA HY index through out the eight-year period from June 2006 to June 2014
From June 2006 to June
2014
From June 2010 to June
2014
Excluding 2008-2009 financial
crisis
Company name
1
2
3
4
5
6
Advanced Micro Devices Inc
American Axle & Manufacturing Inc
Chesapeake Energy Corp
Community Health Systems Inc
DISH Network
Forest Oil Corp
Avg.
leverage
ratio
1 std. of
quarterly
calculated
leverage
ratio
Avg.
leverage
ratio
1 std. of
quarterly
calculated
leverage
ratio
42%
58%
43%
70%
37%
50%
17%
15%
8%
16%
7%
17%
38%
58%
44%
76%
39%
58%
8%
6%
7%
5%
4%
16%
36
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Frontier Communications Corp
Goodyear Tire & Rubber Co/The
HCA Inc
KB Home
Kinder Morgan Inc
Level 3 Communications Inc
MGM Resorts International
Meritor Inc
Royal Caribbean Cruises Ltd
Standard Pacific Corp
Tenet Healthcare Corp
Tesoro Corp
Unisys Corp
United Rentals North America Inc
60%
60%
64%
60%
45%
63%
62%
58%
52%
66%
64%
33%
35%
67%
8%
10%
7%
9%
5%
12%
15%
14%
11%
16%
10%
12%
19%
13%
61%
61%
64%
63%
45%
65%
65%
55%
52%
67%
62%
33%
26%
63%
7%
6%
7%
7%
5%
11%
8%
12%
8%
12%
10%
9%
12%
11%
Average
54%
12%
55%
9%
Table 2 lists the companies that have been members of on-the-run CDX NA HY index from June 2006 to June
2014, excluding the companies from the utility sector. The leverage ratios are computed as the ratio of (A) book
value of short-term plus long-term debt to (B) the sum of book value of short-term and long-term debt plus the
market value of equity, as of the last trading day of each calendar quarter.
Table 3: Average historical stock return volatilities for the 80 companies listed in Tables 1
and 2
AVG. STOCK RETURN VOL. FROM JUNE 2006 TO JUNE 2014
Avg. Leverage (L)
Nr. Of
Company
Avg. Market
Cap.
(million USD)
L<=15%
15% < L <=25%
25% < L <=35%
35% < L <=45%
45% < L <=55%
55% < L <=65%
65% < L <=75%
9
26
21
9
3
9
3
50,676
79,600
51,460
27,191
15,825
9,069
5,617
30 DAYS
30D
30
23.9%
25.3%
30.3%
36.8%
42.9%
49.1%
51.7%
90 DAYS 150 DAYS 250 DAYS 750 DAYS 1,000 DAYS
90D
90
25.6%
26.6%
31.7%
39.5%
44.4%
53.0%
54.1%
150D
150
250D
250
750D
750
1000D
1000
26.2%
27.0%
32.2%
40.3%
44.6%
54.2%
54.9%
26.9%
27.7%
32.9%
41.5%
45.4%
55.6%
56.3%
28.2%
29.0%
34.3%
43.1%
46.6%
58.9%
57.8%
29.1%
29.6%
34.9%
43.6%
46.5%
59.7%
57.8%
Table 3 shows the distribution of the 80 companies by each company’s average leverage ratio from June 2006 to
June 2014 in seven different leverage groups. And the average quarter-end stock return volatilities over
different time intervals for companies of each leverage group are presented. For example, the 30-day interval
stock return volatility equals to the annualized standard deviation of the daily closing price change for the 30
trading days immediately before each calendar quarter-end.
37
Table 4: The average S&P 500 Index leverage distribution from June 2006 to June 2014
Leverage Groups
Avg. Leverage within Each
Leverage Group
Pct. Of S&P 500 Index in Each
Leverage Group
6.9%
19.6%
29.7%
39.7%
49.5%
59.5%
74.8%
40.0%
21.2%
13.9%
10.6%
6.9%
3.7%
3.7%
L<=15%
15% < L <=25%
25% < L <=35%
35% < L <=45%
45% < L <=55%
55% < L <=65%
65% < L <=75%
23.6%
Avg. Leverage for S&P 500
Table 4 reports the average leverage distribution for the S&P 500 Index from 33 S&P 500 Index leverage
distributions on the last business day of each calendar quarter from June 2006 to June 2014.
Table 5: Comparisons of average credit spread-implied asset value volatility with the
average of empirically estimated historical asset value volatility
Company Credit
Grade
Avg.
Avg. 5Y CDS
Leverage
Spread
(bp)
Investment-Grade
High Yield
23.9%
54.4%
77
555
Historical Asset Value Vol.
5Y Spread Implied Asset Value Vol.
Lower Bound Upper Bound
on Asset
on Asset
Value Vol.
Value Vol.
(1-L)30D
1,000D
From the Normal
From the
Model
Lognormal Model
20.4%
21.2%
31.6%
55.5%
21.5%
45.9%
40.0%
47.2%
Historical Asset Value Vol. Estimate
Based on Schaefer and Strebulaev
(2008)
20.7%
29.6%
Table 5 shows the average spread-implied asset value volatility from 5-year CDS spreads, and the expected
range for historical asset volatility from June 2006 to June 2014. The 5-year rather than 10-year CDS spreadimplied asset volatility is used for comparison because the empirical relationships used to estimate the historical
asset value volatility are derived from the Merrill Lynch Corporate Master index that represents the investmentgrade corporate bond universe, and the average maturity of investment-grade corporate bond universe is close to
5 years.
38
Table 6: Comparisons of spread-implied asset value volatility with estimated historical asset value
volatility from June 2006 to June 2014 for investment-grade companies
Using Schaefer From the From the
and Strebuleav normal lognormal
(2008)
model
model
Leverage (L)
Groups
Nr. Of
Com pany
Avg. L
E
(1-L)E
Adjustm ent
Factor (f)
L<=15%
15% < L <=25%
25% < L <=35%
35% < L <=45%
45% < L <=55%
9
26
20
4
1
12.7%
19.7%
29.9%
40.1%
48.7%
23.9%
24.4%
29.3%
34.2%
32.0%
20.9%
19.6%
20.6%
20.5%
16.4%
1.00
1.00
1.05
1.10
1.20
Historical Asset
Value Volatility
Estim ate
(1-L) Ef
weighted average
5Y Spread- 5Y SpreadIm plied
Im plied
Asset
Asset
Value
Value
Volatility
Volatility
20.9%
19.6%
21.6%
22.5%
20.7%
21.1%
21.1%
21.9%
23.5%
15.9%
21.5%
47.1%
41.4%
36.9%
34.3%
22.6%
40.3%
Table 6 gives the comparisons of spread-implied asset value volatility from the normal and lognormal models
with empirically estimated asset value volatility using the relationships between asset value volatility, stock
return volatility, and leverage ratio inferred from Schaefer and Strebulaev (2008).
Table 7: Average equity risk premium from June 2006 to June 2014 for both investmentgrade and high-yield companies using the normal and lognormal models
Average Firm Equity Risk Premium
Leverage (L)
Groups
Average
Leverage
Nr. Of
Companies
Average 10Y
CDS
from the normal
model
from the
lognormal model
3.44%
3.78%
4.60%
6.60%
6.79%
8.96%
9.27%
9.28%
8.94%
9.58%
14.71%
13.52%
20.33%
19.35%
(bp)
L<=15%
15% < L <=25%
25% < L <=35%
35% < L <=45%
45% < L <=55%
55% < L <=65%
65% < L <=75%
12.7%
19.7%
30.0%
40.3%
50.2%
60.9%
67.3%
9
26
21
9
3
9
3
73
87
134
382
329
647
572
Table 7 shows average market-based ERP derived from the normal and lognormal models. ERP is calculated
quarterly for each company. And the 80 companies are stratified into seven leverage groups by their average
leverage ratios for the eight-year period from June 2006 to June 2014; ERPs of companies within the same
leverage group are averaged and presented here.
39
Table 8: Calculated credit spreads from the normal and lognormal models when calibrated to
average 10-year historical default loss rates from Moody’s
Panel A: From the Normal Model that is Calibrated to Historical Default Loss; Maturity = 10 years
Target
Implied
Cumulative
Credit Leverage Equity
Default
Rating Ratio
Premium Prob.
Asset
Risk
Asset
Premium Vol.
Aaa
Aa
A
Baa
Ba
B
4.76%
4.61%
4.47%
4.55%
5.11%
6.09%
13.1%
21.2%
32.0%
43.3%
53.5%
65.7%
5.38%
5.60%
5.99%
6.55%
7.30%
8.76%
0.77%
0.99%
1.55%
4.39%
20.63%
43.91%
21.0%
21.1%
21.6%
26.4%
60.5%
849.0%
Calc
Credit
Spread
(bps)
40.2
45.7
57.0
94.5
233.4
460.8
Avg
Yield
% of Spread
Spread due to
(bps) Default
63
64%
91
50%
123
46%
194
49%
320
73%
470
98%
Panel B: From the Lognormal Model that is Calibrated to Historical Default Loss; Maturity = 10 years
Target
Implied
Cumulative
Credit Leverage Equity
Default
Rating Ratio
Premium Prob.
Asset
Risk
Asset
Premium Vol.
Aaa
Aa
A
Baa
Ba
B
4.76%
4.61%
4.47%
4.55%
5.11%
6.09%
13.1%
21.2%
32.0%
43.3%
53.5%
65.7%
5.38%
5.60%
5.99%
6.55%
7.30%
8.76%
0.77%
0.99%
1.55%
4.39%
20.63%
43.91%
34.4%
30.5%
27.8%
29.8%
43.7%
60.8%
Calc
Credit
Spread
(bps)
15.8
21.6
33.2
70.5
215.5
457.7
Avg
Yield
% of Spread
Spread due to
(bps) Default
63
25%
91
24%
123
27%
194
36%
320
67%
470
97%
Panel C: From Huang and Huang (2012) Base Model; Maturity = 10 years
Target
Implied
Cumulative
Credit Leverage Equity
Default
Rating Ratio
Premium Prob.
Asset
Risk
Asset
Premium Vol.
Aaa
Aa
A
Baa
Ba
B
4.96%
4.91%
4.89%
5.01%
5.48%
6.46%
13.1%
21.2%
32.0%
43.3%
53.5%
65.7%
5.38%
5.60%
5.99%
6.55%
7.30%
8.76%
0.77%
0.99%
1.55%
4.39%
20.63%
43.91%
32.1%
28.4%
25.6%
25.8%
32.4%
39.5%
Calc
Credit
Spread
(bps)
10.0
14.2
23.3
56.5
192.3
387.8
Avg
Yield
% of Spread
Spread due to
(bps) Default
63
16%
91
16%
123
19%
194
29%
320
60%
470
83%
Table 8 shows 10-year credit spreads that are predicted by the normal model and the lognormal model. The
average yield spreads correspond to the corporate-Treasury yield spreads. The credit spread benchmarks are 52
basis points lower than the average yield spreads. The model calibration is done with average 10-year historical
default loss rates (from Moody’s report by Keenan, Shtogrin, and Sobehart 1999), and the corresponding
leverage ratios, equity premia, and r−risk-free interest rate − the payout rate) used by Huang and Huang
(2012). A default boundary at 100% of the bond face value and a 51.13% recovery rate are used in both models
to calculate credit spreads.
40
Table 9: Calculated credit spreads from the normal and lognormal models when calibrated to
average 4-year historical default loss rates from Moody’s
Panel A: From the Normal Model that is Calibrated to Hisotrical Default Loss; Maturity = 4 years
Target
Credit
Rating
Aaa
Aa
A
Baa
Ba
B
Leverage
Ratio
Equity
Premium
13.1%
21.2%
32.0%
43.3%
53.5%
65.7%
5.38%
5.60%
5.99%
6.55%
7.30%
8.76%
Implied
Cumulative
Default
Prob.
0.04%
0.23%
0.35%
1.24%
8.51%
23.32%
Asset
Risk
Premium
4.75%
4.55%
4.38%
4.40%
5.11%
6.09%
Asset
Vol.
17.9%
20.2%
19.8%
22.5%
36.3%
67.7%
Avg
Calc Credit Yield
Spread
Spread
(bps)
(bps)
5.7
55
17.5
65
23.8
96
57.8
158
215.6
320
490.9
470
% of
Spread due
to
Default
10%
27%
25%
37%
67%
104%
Panel B: From the Lognormal Model that is Calibrated to Historical Default Loss; Maturity = 4 years
Target
Credit
Rating
Aaa
Aa
A
Baa
Ba
B
Leverage
Ratio
Equity
Premium
13.1%
21.2%
32.0%
43.3%
53.5%
65.7%
5.38%
5.60%
5.99%
6.55%
7.30%
8.76%
Implied
Cumulative
Default
Prob.
0.04%
0.23%
0.35%
1.24%
8.51%
23.32%
Asset
Risk
Premium
4.75%
4.55%
4.38%
4.40%
5.11%
6.09%
Asset
Vol.
34.7%
33.4%
28.2%
27.9%
39.4%
54.5%
Avg
Calc Credit Yield
Spread
Spread
(bps)
(bps)
1.6
55
7.7
65
12.5
96
37.6
158
182.1
320
466.7
470
% of
Spread due
to
Default
3%
12%
13%
24%
57%
99%
Panel C: From Huang and Huang (2012) Base Model; Maturity = 4 years
Target
Credit
Rating
Aaa
Aa
A
Baa
Ba
B
Leverage
Ratio
Equity
Premium
13.1%
21.2%
32.0%
43.3%
53.5%
65.7%
5.38%
5.60%
5.99%
6.55%
7.30%
8.76%
Implied
Cumulative
Default
Prob.
0.04%
0.23%
0.35%
1.24%
8.51%
23.32%
Asset
Risk
Premium
4.96%
4.91%
4.89%
5.01%
5.48%
6.46%
Asset
Vol.
36.2%
34.4%
29.8%
28.9%
34.3%
39.6%
Avg
Calc Credit Yield
Spread
Spread
(bps)
(bps)
1.1
55
6.0
65
9.9
96
32.0
158
172.3
320
445.7
470
% of
Spread due
to
Default
2%
10%
12%
23%
60%
109%
Table 9 shows 4-year credit spreads predicted by the normal model and the lognormal model. For investmentgrade companies, the default-loss implied asset value volatility is used to compare with the CDS spread-implied
asset value volatility. The model calibration is done with average 4-year historical default loss rates (from
Moody’s report by Keenan, Shtogrin, and Sobehart 1999), and corresponding leverage ratios, equity premia,
and r−risk-free interest rate − the payout rate) that have been used by Huang and Huang (2012). A
default boundary at 100% of the bond face value and a 51.13% recovery rate are used in both models to
calculate credit spreads.
41
Table 10: Percentage of observed corporate-Treasury yield spreads due to default risk with r-

Panel A: Maturity = 10 years and r-=0%
Credit
Rating
Aaa
Aa
A
Baa
Ba
B
Implied Asset Vol.
The
The
Normal
Lognormal Huang and
Leverage Structrual Structural Huang
Ratio
Model
Model
(2012)
13.1%
21.2%
32.0%
43.3%
53.5%
65.7%
18.4%
18.3%
18.7%
22.6%
47.8%
265.0%
32.5%
28.4%
25.5%
27.1%
40.6%
50.6%
32.1%
28.4%
25.6%
25.8%
32.4%
39.5%
% of Spread due to Default
The
The
Normal
Lognormal Huang and
Structrual Structural Huang
Model
Model
(2012)
42.9%
34.1%
32.5%
37.7%
61.7%
87.1%
17.8%
16.7%
19.3%
27.9%
58.3%
88.8%
16.7%
16.6%
20.5%
31.3%
62.0%
83.6%
Panel B: Maturity = 4 years and r-=0%
Credit
Rating
Aaa
Aa
A
Baa
Ba
B
Implied Asset Vol.
The
The
Normal
Lognormal Huang and
Leverage Structrual Structural Huang
Ratio
Model
Model
(2012)
13.1%
21.2%
32.0%
43.3%
53.5%
65.7%
16.8%
18.8%
18.3%
20.5%
33.3%
62.1%
34.7%
33.4%
28.2%
27.9%
39.4%
54.5%
36.2%
34.4%
29.8%
28.9%
34.3%
39.6%
% of Spread due to Default
The
The
Normal
Lognormal Huang and
Structrual Structural Huang
Model
Model
(2012)
6.4%
17.5%
16.3%
26.1%
58.1%
97.2%
2.0%
8.9%
9.9%
18.7%
50.4%
91.9%
2.1%
9.4%
10.6%
20.9%
55.0%
96.2%

Table 10 reports implied asset volatility from calibrating the normal model and the lognormal model to average
10- and 4-year historical default loss rates (from Moody’s report by Keenan, Shtogrin, and Sobehart 1999), and
the corresponding leverage ratios, equity premia, and r- 0%. Because both models presented in this paper
assume a driftless firm value dynamics, the calibration exercise of r- 0% enables the calculated credit spreads
from the lognormal model to be directly comparable to the sensitivity analysis reported in Table 10 of Huang
and Huang (2012) using the Longstaff and Schwartz (1995) model.
42
Figures
Figure 1
2.5%
Normal Firm Value Distribution
2.0%
Asset Value Distribution Derived from the
Normal Firm Value Distribution
1.5%
Lognormal Firm/Asset Value Distribution
1.0%
0.5%
0.0%
o
5V
2.
43
Vo
25
2.
Equity
o
2V
Debt
K1
Vo
75
1.
K2
o
5V
1.
Vo
25
1.
Vo
Vo
75
0.
o
5V
0.
Vo
1*
-K
Vo
=0
Vo
2*
-K
Vo
o
5V
.2
-0
Vo
.5
-0
Figure 2
Figure 3
1.0%
200%
0.9%
Firm Value Distribution
0.8%
180%
160%
Debt Loss Function
0.7%
140%
0.6%
120%
0.5%
100%
0.4%
80%
0.3%
60%
0.2%
40%
0.1%
Debt
20%
Equity
0.0%
0%
Figure 4
35%
6%
30%
5%
25%
4%
20%
3%
15%
10%
5%
S&P 500 Index Average Leverage Ratio
(left axis)
S&P 500 Index Equity Risk Premium
Predicted by the Normal Model (right
axis)
0%
2%
1%
0%
14
nJu 3
-1
ec
D 3
1
nJu 2
-1
ec
D 2
1
nJu 1
-1
ec
D 1
1
nJu 0
-1
ec
D 0
1
nJu 9
-0
ec
D 9
0
nJu 8
-0
ec
D 8
0
nJu 7
-0
ec
D 7
0
nJu 6
-0
ec
D 6
0
nJu
44
Figure 5
50%
45%
Asset Value Volatility
40%
35%
Spread-implied asset value
volatility from the lognormal
model
Spread-implied asset value
volatility from the normal model
30%
25%
20%
Historical asset value volatility
estimate using Schaefer and
Strebuleav (2008)
15%
10%
5%
0%
0%
10%
20%
30%
40%
50%
Firm Leverage Ratio
Figure 6a
Normal Model Spread-Implied Asset Value Volatility for Investment-Grade Corp.
Spread-Implied Asset Value Vol.
30%
28%
26%
24%
22%
20%
18%
L<=15%
16%
15% < L <=25%
14%
25% < L <=35%
12%
35% < L <=45%
10%
5
6
7
8
Tenor (Years)
45
9
10
Figure 6b
Lognormal Model Spread-Implied Asset Value Volatility for Investment-Grade Corp.
Spread-Implied Asset Value Vol.
50%
L<=15%
48%
15% < L <=25%
46%
25% < L <=35%
44%
35% < L <=45%
42%
40%
38%
36%
34%
32%
30%
5
6
7
8
9
10
Tenor (Years)
Figure 7
15%
10%
5%
0%
06
n-
14
nJu
13
cDe
13
nJu
12
cDe
12
nJu
11
cDe
11
nJu
10
cDe
10
nJu
09
cDe
09
nJu
08
cDe
08
nJu
07
cDe
07
nJu
06
cDe
Ju
S&P 500 Index Equity Risk Premium Predicted by the Lognormal Model
S&P 500 Index Equity Risk Premium Predicted by the Normal Model
S&P 500 Index Equity Risk Premium from the CFO Survey
46