1. GBUS 492 Final Project
The Nonlinear
Relationship between
Default Probability,
Leverage, and
Volatility
Can Hu
2. 1. Numerical Analysis
The following numerical analysis is conducted to investigate the nonlinear relationship between
leverage, volatility (both asset and equity), and default probability.
Inputs are as follows:
Asset Value V=
[100 110 120 130 140 150 160 170 180 190 200];
Book Value Leverage L=
[0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
0.80 0.85 0.90];
Asset Volatility 𝜎 𝑉 =
[0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1 1.10 1.20];
Interest Rate r = 10%, Time to Maturity T = 1;
Strike Price K is defined as the product of Asset Value and Book Value Leverage.
1.1 Vanilla Calls
From Ito’s Lemma, Equity Volatility is defined as follows
𝜎 𝐸 =
𝑉𝑡
𝐸𝑡
𝜎 𝑉 𝑁(𝑑1)
And Default Probability is defined as
𝐷𝑃 = 1 − 𝑁(𝑑2)
Where
𝑑1 =
𝑙𝑛
𝑉𝑡
𝐾 + (𝑟 +
𝜎 𝑉
2
2 )(𝑇 − 𝑡)
𝜎 𝑉 𝑇 − 𝑡
, 𝑑2 =
𝑙𝑛
𝑉𝑡
𝐾 + (𝑟 −
𝜎 𝑉
2
2 )(𝑇 − 𝑡)
𝜎 𝑉 𝑇 − 𝑡
And Equity is calculated via Black-Scholes formula:
1 2( ) ( )rT
t tE V N d Ke N d
As shown in Figure 1, Default Probability is an increasing function of Asset Volatility, when
Leverage is fixed. A meticulous scrutiny of the 3-D surface reveals that for relatively low
Leverage, Default probability is a convex monotonic increasing function of Asset Volatility, while
for relatively high Leverage, Default probability is a concave monotonic increasing function of
Asset Volatility. The critical point is around a Leverage of 50%.
As shown in Figure 2, Default Probability is also an increasing function of Leverage, when Asset
Volatility is fixed. What’s more, Figure 2 also demonstrates the transition from convexity to
concavity. The critical point is around an Asset Volatility of 60% to 70%.
The relationship between Default Probability and Equity Volatility is quite similar that shown in
Figure 1, because Equity is derived from Asset Volatility. The critical point is still around a
Leverage of 50%.However, a 3-D surface cannot be plotted since we have so many different
Equity Volatility Values. Figure 3 gives an illustration using the example: Asset Value=100,
Leverage= 10%, 50%, and 90%.
3. Figure 3 Relationship between Default Probability and Equity Volatility
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Default Probability
Asset Value = 100, Leverage =10%
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Default Probability
Asset Value = 100, Leverage =50%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Default ProbabilityAsset Value = 100, Leverage =90%
4. 1.2 Down-and-Out Calls
Default Probability is defined as
2
1
2( )
2
2 21 ( ) ( )
V
r
H
DP N d N b
V
Where
2
2
ln ( )( )
2
t V
V
V
r T t
Kd
T t
,
22
2
ln ( )( )
2
V
t
V
H
r T t
V K
b
T t
H K , is a predetermined constant.
From Ito’s Lemma, Equity Volatility is defined as follows
t t
E V
t t
V E
E V
Where
2 2 2
1 1 2
1
( ) 2 1 ( ) 2 2 ( )rTt
t
E H H
N d N b Ke N b
V V V V
2
1
2
r
,
22
1
ln ( )( )
2
V
t
V
H
r T t
V K
b
T t
And Equity is calculated via
2 2 2
1 2 1 2( ) ( ) ( ) ( )rT rT
t t t
t t
H H
E V N d Ke N d V N b Ke N b
V V
Generally speaking, Down-and-Out Call model will produce a relatively high Default Probability.
All the results are similar to those discussed in the previous section. When Leverage is fixed, as
shown in Figure 5, for relatively low Leverage, Default probability is a convex monotonic
increasing function of Asset Volatility, while for relatively high Leverage, Default probability is a
concave monotonic increasing function of Asset Volatility. However, the critical point is a little bit
higher, between a Leverage of 65% to 70%.
As shown in Figure 6, Default Probability is also an increasing function of Leverage, when Asset
Volatility is fixed. Figure 2 also demonstrates the transition from convexity to concavity. The
critical point is also a little bit higher, around an Asset Volatility of 80% to 90%.
The relationship between Default Probability and Equity Volatility is quite similar that shown in
Figure 5, because Equity is derived from Asset Volatility. The critical point is still around a
Leverage of 65% to 70%. However, a 3-D surface cannot be plotted since we have so many
different Equity Volatility Values. Figure 7 gives an illustration using the example: Asset
Value=100, Leverage= 10%, 70%, and 90%.
See Appendix for the code of numerical analysis.
5. Figure 7 Relationship between Default Probability and Equity Volatility
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Default Probability
Asset Value = 100, Leverage = 10%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Default Probability
Asset Value = 100 Leverage = 70%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Default Probability
Asset Value = 100 Leverage = 90%
10. 2 Relationship between Leverage, Volatility, and Default Probability
2.1 For Vanilla Calls
Default Probability (DP) = 1-N(d2)
Where
2
2
ln ( )( )
2
t V
V
V
r T t
Kd
T t
2
2
ln ( ) ln ( )
1 1
( )
2 2
t t
V
V V V V
V V
r T t r T t
d T t T tK K
T t T t
2
2
2
2
22
2
2
1
*
2
ln ( )
1 1
0
22
d
V V
t
d
V
dDP
e
V
r T t
T tKe
T t
2
2
( )( )
ln 12( )
V
V V V V
r T t
d L
L T t T t L T t
2 2
2 2
22 2
1 1 1
0
2 2
d d
V
dDP
e e
L L L T t
2
2
2 2
1
2 2
1 1
1
2
1
ln ( )
1
2
ln ( )
( )
( ) 2 ( )
ln ( ) ( )
2 ( )
t
V V
E V E E
t
t t t
t E t t
t
t t
t E t
V
r T t
d T tK
T t
V
r T t
V N d E ET tK
E V N d V N dT t
V
r T t V N d ET tK
E V N dT t
2 2
2 2
122 2
2
1
ln ( )
( )1 1
0
2 ( )2 2
t
d d
t t
E E t E t
V
r T t
V N d EdDP T tKe e
E V N dT t
2.2 For Down-and-Out Calls
Default Probability (DP) is defined as follows:
11. 2
1
2( )
2
2 21 ( ) ( )
V
r
H
DP N d N b
V
Where
2
2
ln ( )( )
2
t V
V
V
r T t
Kd
T t
,
22
2
ln ( )( )
2
V
t
V
H
r T t
V K
b
T t
2 2
2
2
ln ( ) ln ( )
1 1
( )
2 2
t t
V
V V V V
H H
r T t r T t
V K V Kb T t T t
T t T t
2 2
2 2 2 2
2
2
2
2 2
2 2
1 1
2 22 2
23
2
2
2
2
1
2
23
1 4 1
ln ( )
2 2
ln ( )
1 1
22
ln
4 1
ln ( )
2
V V
V
r r
d b
V V V V
t
d
V
r
b
V
d bDP H r H H
e N b e
V V V
V
r T t
T tKe
T t
H
H r H
N b e
V V
2
( )
1
2
t
V
r T t
V K T t
T t
2
2
22
2
( )( )
ln 1 1 12( )
V
V V V V
r T t
b L
L L LT t T t T t L T t
2 2
2 2 2
2 2
2 2 2
2 2
2 2 2
2
1
2 22 2
2
1
2 2
2
1
2 2
1 1
2 2
1 1 1 1
2 2
1 1 1
0
2 2
V
V
V
r
d b
r
d b
V V
r
d b
V
d bDP H
e e
L L V L
H
e e
VL T t L T t
H
e e
VL T t
2
2
2
2
1
2
1
ln ( )
1
2
ln ( )
( )
2 ( )
t V V
E V E E
t t t
t E t
H
r T t
V Kb T t
T t
H
r T t
V K V N d ET t
E V N dT t
12. 2 2
2 2 2 2
2
2
2
2 2
1 1
2 22 2
23
12
2
1
2
1
2
1 4 1
ln ( )
2 2
ln ( )
( )1
2 ( )2
4
ln ( )
V V
V
r r
d b
V
E E V E E
t
d
t t
t E t
r
d bDP H r H H
e N b e
V V V
V
r T t
V N d ET tKe
E V N dT t
H H
N b
V V
2
2 2
2 2
1
2 3
2
2
1
12
2
1
( )
ln ( )
( )1
2 ( )2
V
t
t E
r
b
t t t
t E t
rV N d
E
H
r T t
V K V N d EH T t
e
V E V N dT t
13. 3 Summaries of Papers
a) Do Credit Spreads Reflect Stationary Leverage Ratios? Collin-Dufresne & Goldstein(2001)
Although many debentures have covenants protecting bondholders against adverse changes in
capital structure, firms generally have the legal right to issue additional equal-priority debt. Hence,
in determining the appropriate credit spread for a corporate bond, it’s necessary to account for
both the firm’s current liability structure and its right to alter such structure in the future.
Collin-Dufresne & Goldstein(2001) proposes a structural model of default with stochastic interest
rates and mean-reverting leverage ratios.
Comparison between Traditional Structural Models and CDG: the former preclude firms from
issuing additional debt at intermediate dates, while CDG allows such issuing.
Comparison between Exogenous Barrier Models and CDG: the Former presume default barrier
to be a monotonic function of the level of outstanding debt, thus leverage ratio declining
exponentially over time. While in practice, leverage ratios to be stationary, always staying within a
fairly narrow band. On the contrary, CDG presumes a mean-reverting leverage ratio.
Comparison between Other Stochastic Default Boundaries Models and CDG: The source of the
randomness of the boundary in the former models is tied to the interest rate process and will not
lead to stationary leverage ratios. Moreover, these models also assume the default probability
possesses a drift that increases linearly with the spot rate and thus causes the default probability,
and in turn the credit spreads, to be independent of the level of the spot rate. However, CDG uses
a Cox-Ingersoll-Ross short rate model that depends on the current rate.
i. An Illustrative Example: Generalizing Merton(1974)
They compare the credit spreads between firms that cannot change debt structure and that can
issue bonds with the same maturity as the previously issued debt and reset the leverage back to its
initial. The proceeds of the new debt issuance are used to repurchase existing equity, leaving firm
value unchanged. The figure 1 above demonstrates significant increase in credit spreads generated
by equity’s right to a one-time increase in debt levels. Permitting the firm to issue new debts
several times would increase credit spreads even further.
ii. A Model of Credit Spreads with Stationary Leverage
Under Q measure, define log-firm-value
14. 2
( )
2
Q
t tdy r dt dz
μ is the expected drift under P measure, σ is the volatility and δ is the payout rate. Default is
triggered the first time firm value reaches some exogenously specified threshold.
Define log-default threshold
( )t t tdk y v k dt
When kt is less than (yt-v), the firm acts to increase kt, and vice-versa. This model captures the fact
that firms tend to issue debt when their leverage ratio falls below some target, and are more
hesitant to replace maturing debt when their leverage ratio is above that target
Define the log-leverage
t t tl k y
From Ito’s lemma
( )Q Q
t t tdl l l dt dz
Where
2
2Q
r
l
Define as the random time at which l(t) reaches zero for the first time. Assume that a risky
discount bond with maturity T receives one dollar at T if > T, or (1-w) at time T if ≤T. The
price of this risky discount bond can be written as
0 ( ) ( ) 0( ) 1 (1 )1 1 ( , )T rT Q rT
T TP l e E w e wQ l T
Here 𝑄(𝑙0, 𝑇) is the risk-neutral probability that default occurs before time T given the leverage
ratio is l0 at time 0. The closed-form solution is listed in page 1936-1937. They also derive the
price formula for coupon bond with promised coupon payment C at time tj, 𝑗 ∈ (𝑖, 𝑁),tN=T
0 0 0
1
( ) (1 ( , )) (1 ( , ))j
N
rtT rT
coup j
j
P l Ce w Q l t e wQ l T
The yield to maturity for this coupon bond YT
is defined through the equation
0
1
( )
TT
j
N
Y tT Y T
j
P l e C e
Finally, the credit spread CS(T) is defined via
( ) T
CS T Y r
The following figure 2 shows the comparison between credit spread predictions of the constant
boundary model with the predictions of the stationary leverage model, both on a AAA investment-
grade bond. It is well documented that structural models predict negligible credit spreads for very
short maturities and underestimate credit spreads for long maturities, inconsistent with empirical
findings. On the other hand, the CDG model performs considerably better for longer maturities.
15. The following figure 3 shows the comparison between credit spread predictions of the constant
boundary model with the predictions of the stationary leverage model, both on a speculative-grade
bond. In contrast to the traditional constant default-boundary model, the CDG model predicts an
upward-sloping yield curve for reasonable parameter choice, a conclusion consistent with the
empirical findings of Helwege and Turner (1999). Also notable is that our model predictions from
CDG model are much less sensitive to changes in leverage than those from the constant default
boundary model. (The initial leverage is 15% for the chart above and 65% for the chart below).
It’s apparent that the traditional model predicts for all maturities counter-factually low credit
spreads for low leverage firms and high credit spreads for speculative grade debt. In contrast,
CDG model with mean-reverting leverage ratio improves the predictions of structural models.
iii. Credit Spreads with Stochastic Interest Rates
Longstaff and Schwartz(1995) and Duffee(1998) find that credit spreads are a decreasing function
of interest rates. Further, as documented by Malitz(1994), firms tend to issue less debt when
interest rates are high. Hence, stochastic interest rate is relevant in determining credit spreads.
16. Under measure Q, define log-firm value and short rate as follows
2
1
2
( ) ( )
2
( ) ( )
Q
t t
Q
t t
dy r dt dz t
dr k r dt dz t
With 1 2( ) ( )Q Q
dz t dz t dt . Default is triggered when asset value falls below some exogenously
specified threshold. Define the log-default threshold as
[ ( ) ]t t t tdk y v r k dt
The drift of the log-default threshold is a decreasing function of the spot rate, consistent with the
findings of Malitz(1994): debt issuances dropped dramatically during the high interest rate period
of the early 1980s.
Again, define the log-leverage
t t tl k y
From Ito’s lemma
1( ( ) ) ( )Q Q
t t tdl l r l dt dz t
Where
2
12( ) ( )Q
t tl r r
This equation implies the risk-neutral target leverage ratio is a decreasing function of the current
interest rate. The price for a risky discount bond is as follows
0
0 0 ( ) 0 0 0( , ) (1 1 ) ( ) 1 ( , , )
T
sdsrT Q T T
TP r l E e w D r wQ r l T
Where 𝐷 𝑇
𝑟0 = 𝑒 𝐴 𝑇 −𝑟0 𝐵 𝑘(𝑇)
is the Vasicek risk-free bond price with deterministic function A
and B. Closed-form solutions for 𝑄 𝑇
(𝑟0, 𝑙0, 𝑇) can be found in page 1942.
The yield to maturity for this coupon bond YT
is defined through the equation
0 0
1
( , )
TT
j
N
Y tT Y T
j
P r l e Ce
Finally, the credit spread CS(T) is defined via
( ) T T
CS T Y R
And RT is defined implicitly via
1
( 0, 0)
T T
j
N
R tT R T
C
i
D r l Ce e
The following figure 4 shows the comparison between credit spread predictions of the constant
boundary model with the predictions of the stationary leverage model, both on a investment-grade
bond. Both models predict increasing term structures of credit spreads with counter-factually low
credit spreads at the short end. At the long end of the term structure, however, the constant
17. boundary model predicts spreads of about 8 bp whereas the CDG model predicts more realistic
spreads, around 60 bp. Modeling mean reversion in leverage appears to solve the problem of
structural models at fitting long-term credit spreads for low-leverage firms. Comparing figure 2
and figure 4, we can conclude that CDG with stochastic r is in better agreement with empirical
findings to predict higher credit spreads for all maturities than CDG with constant r.
The following figure 5 shows the comparison between credit spread predictions of the constant
boundary model with the predictions of the stationary leverage model, both on a speculative-grade
bond. The constant boundary model exhibits a decreasing term structure of credit spreads from 5
to 20 years, whereas the CDG model exhibits an increasing term structure of credit spreads, more
in line with the empirical finding by Helwege and Turner (1999). Comparing figure 2 and figure 4,
we can find that CDG with stochastic r predicts a even more upward-sloping credit spreads term
structure.
The following figure 6 shows that an increase in the level of the short-term rate lowers credit
spreads in CDG model. It’s probably because that the firm value is an increasing function of
18. interest rate while default barrier is a decreasing function of interest rate, and therefore the bond
price will increase, leading to a yield decrease.
The following figure 7 shows the effect of correlation between firm value and interest rate on
credit spreads. A negative correlation implies that a decrease in interest rates will have two
countervailing effects on credit spreads: 1. credit spreads may increase due to the decrease in the
drift of asset value;2. a decrease in interest rates will typically be associated with an increase in
underlying asset value. Thus, credit spreads may decrease. Conversely, when the correlation is
positive, the two effects work in the same direction, and hence higher credit spreads.
iv. Conclusion
Corporating a firm’s ability to control its level of outstanding debt has a significant impact on
credit spread predictions. The CDG model predicts the term structure of credit spreads of
speculative-grade debt to be upward sloping. Moreover, it predicts that the sensitivity of credit
spreads to changes in leverage is much lower. Also, They documents a negative correlation
between credit spreads and interest rates for all reasonable parameter values.
19. b) Leverage Expectations and Credit Spreads Flannery, Nikolova and Oztekin(2012)
Using a quarterly sample of 394 U.S. corporations’ credit premia for the period 1986–1998,
Flannery, Nikolova and Oztekin(FNO)(2012) test whether proxies for investors’ expectations
about the firm’s future leverage can affect bond credit spreads to a statistically and economically
significant extent. They document that changes in leverage expectations do have a positive impact
on credit-spread changes beyond the effect of contemporaneous leverage. More specifically, a
1-standard-deviation increase in leverage expectations will cause the issuer’s bond credit spread to
widen by almost 100 bp. This finding is also robust to alternative leverage definitions and
alternative methods of forming expectations proxies.
i. Bond Credit Spreads and Corporate Leverage: Theory
Following Black and Scholes’(1973) and Merton’s(1974) theory that defaults occurs when firm
value falls below a default threshold, generally expressed in terms of its outstanding debt
obligations, FNO establish a linear relationship between leverage and credit spreads as follows
, , , , 1 , ,( )i j t j t t j t t i j tCS LEV E LEV Z w
CSi,j,t is the difference between the yield to maturity on bond i of firm j and the yield on a similar-
maturity T-bond at the end of quarter t; LEVj,t is the ratio of debt to (debt + equity) of firm j at the
end of quarter t; Zt is a vector of control variables motivated by structural models. It’s apparent
that credit spreads at time t reflect the latest available information about the firm’s default
probability, which depends on current leverage and investors’ expectations of future leverage.
However, this specification may yield spurious results if CS is not stationary (i.e., it has a unit root)
Instead, they employ a difference specification in most of the analysis
, , , , 1 , ,( )i j t j t t j t t i j tCS LEV E LEV Z , , , ,i j t i j tw
They also employ different measures of leverage, including both book leverage and market
leverage. Book leverage is defined as
𝐵𝐿𝐸𝑉 =
𝐿𝑜𝑛𝑔 − 𝑇𝑒𝑟𝑚 𝐷𝑒𝑏𝑡 + 𝑆𝑜𝑟𝑡 − 𝑇𝑒𝑟𝑚 𝐷𝑒𝑏𝑡
𝐿𝑜𝑛𝑔 − 𝑇𝑒𝑟𝑚 𝐷𝑒𝑏𝑡 + 𝑆𝑜𝑟𝑡 − 𝑇𝑒𝑟𝑚 𝑑𝑒𝑏𝑡 + 𝐶𝑜𝑚𝑚𝑜𝑛 𝐸𝑞𝑢𝑖𝑡𝑦
Market leverage is defined as
𝑀𝐿𝐸𝑉 =
𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐷𝑒𝑏𝑡
𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐷𝑒𝑏𝑡 + 𝑀𝑎𝑟𝑘𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑡𝑦
They estimate market value of debt in two ways. The first method is to model it as risk free debt
short a put option on the underlying market value of the firm’s assets with a strike price equal to
the face value of the firm’s debt. The second method is to extrapolate the market value of debt
from the subset of outstanding bonds with quoted prices in the Warga-Lehman Brothers Fixed
Income Database.
Control variables includes:
Rt
10
: the 10-year, constant-maturity nominal T-bond rate at the end of quarter t
SLOPEt: the difference between the 10- and 2-year Treasury yields at the end of quarter t
VIXt: the implied volatility of the Standard & Poor’s (S&P) 100 Index, calculated by the Chicago
Board Options Exchange on the basis of historical data on the S&P 100 Index options
RS&P
: the return on the S&P 500 Index for the quarter ending at t
JUMPt: the slope of the “smirk” of implied volatilities from options on S&P 500 Index futures,
calculated as described in Collin-Dufresne et al. (2001), using option and futures prices obtained
20. from the Chicago Mercantile Exchange
CRPREMt: the difference between Moody’s average yield on Baa- and Aaa-rated bonds, as a
measure of market aversion to default risk.
ii. Credit Spreads and Realized Future Leverage
They first use the following equation to test whether there is a connection between spread changes
and actual future leverage innovations.
, , , , ,i j t j t t i j tCS LEV Z , , , 1 , ,= ( )i j t t j t i j tE L E V
For the residual, their original hypothesis states that a change in expected future leverage affects
contemporaneous credit spreads implies that γ > 0. Therefore, If investors’ expectations are
rational, a positive residual in the above specification should be followed by higher leverage, and
a negative residual should be followed by lower leverage. Another hypothesis postulates that a
reduction (increase) in credit spreads encourages the firm to seek more (less) debt financing
because debt has become relatively cheap (expensive). In this case, γ should be negative, so
positive (negative) residuals should be followed by lower (higher) leverage.
They hence separate observations into one group with positive residuals and another group with
negative residuals and find that positive regression residuals are indeed followed by significantly
higher leverage changes, and vice versa. This finding suggest that credit spreads reflect (rather
than cause) future leverage changes. They further examine the residuals to determine which
specific components of future leverage are reflected in current bond prices. (Table 2, 697)
d
d s
Dp
MLEV
Dp Sp
And 2
1 2 3 4
( )
s d s d s d d s
d s
Sp p D Sp D p Dp p S Dp S p
MLEV
Dp Sp
Where D is the nominal value of outstanding debt, S is the number of outstanding shares, pd is
the market price of a $1 bond, and ps is the market price of a share of stock. Δ1 = the change
in leverage due to a change in outstanding debt, Δ2 = the change in leverage due to a change in
the price of debt, Δ3 = the change in leverage due to a change in outstanding shares, and Δ4 =
the change in leverage due to a change in share price.
Since market efficiency implies that investors should be unable to forecast bond or share price
movements, bond yield spreads are not expected to be affected by price-driven changes, i.e. Δ2
andΔ4, in future leverage. Rather, the future leverage changes anticipated by investors should
reflect only Δ1andΔ3, the components under management’s direct control. They find thatΔ1
and Δ3 are never the same between observations with positive versus negative residuals, whileΔ
2 and Δ4 never differ between the 2 groups. They finally decide that book leverage might be a
more appropriate basis for constructing measures of leverage expectation. (Table 3, 698)
iii. Credit Spreads and Expected Future Leverage
They construct estimates of investors’ future leverage expectations under three theories.
1) Trade-off Theory
Under the trade-off theory, firms currently operating below their leverage target would prefer
to issue debt rather than equity in an attempt to reach their optimal leverage level.
∆Et(BLEV*
j,t+1) = the change in firm j’s expected target leverage during the quarter ending at t
21. + 1, conditional on information available at the end of quarter t. The target leverage ratio
(BLEV*
) is computed from a partial adjustment model estimated via generalized method of
moments.
2) Pecking-order theory
The pecking order theory predicts that firms generally prefer to issue debt rather than equity
when they need to raise external capital. Therefore, a positive expected financing deficit
implies an expected increase in leverage, and vice versa.
∆Et(FINDEFA*
j,t+1)= the change in firm j’s expected financing deficit in quarter t + 1,
conditional on information available at the end of quarter t.
3) Kisgen(2006)
Kisgen (2006) argues that firms with “minus” ratings wish to avoid falling to the next lower
letter category, and firms with “plus” ratings generally wish to raise themselves to the next
higher letter category, and consequently firms with a plus or minus credit rating are less likely
to issue debt for at least a year.
CRPOMj,t=1 if firm j’s credit rating in quarter t includes a plus or minus, and 0 otherwise.
After regression the original difference equation, they find contemporaneous leverage is positively
related to credit spreads, and this effect appears to be economically large. As for the expected-
leverage proxies, no matter added to the equation individually or jointly, all of them carry
statistically and economically significant coefficient. (Table 4,701)
They further split the sample into subsets of firms grouped according to their leverage, size,
profitability and credit rating to separate one theory of leverage determination from another.
Under the trade-off theory, leverage targets is more informative about future leverage for firms
more likely to incur bankruptcy costs or to value tax-shield benefits. Pecking order theory implies
for firms facing low asymmetric information costs, preference for debt might be less conspicuous
in forming expectations about future leverage changes. Lemmon and Zender (2010) argue that
firms subject to high default risk is limited in their ability to borrow funds. These
debt-capacity-constrained firms might deviate from the predictions of the pecking order theory,
making their financing deficit a less relevant basis for predicting future leverage.
Leverage proxies significantly affect contemporaneous credit spreads for nearly all subsamples
and the relative size of the coefficients seems intuitively reasonable. For example, the trade-off
proxy coefficient for high-leverage firms exceeds that for low leverage, low profitability exceeds
high profitability, and junk (far) exceeds investment grade.(Table 5, 703)
iv. Robustness
1) Modeling Credit-Spread Levels
Instead of regressing the differenced equation, they estimate the following
, , , , 1 , ,( )i j t j t t j t t i j tCS LEV E LEV Z w
and find contemporaneous leverage still has significantly positive coefficients, as do the tradd-
off and pecking order proxies for expected future leverage. (Table 6, 704)
2) Explaining Returns Rather Than Yield Changes
Instead of using simple credit spread as the dependent variable, they estimate the following
, , , , , , 1 , ,( )M
i j t i j t j t t j t t i j tD CS LEV E LEV Z
where D is the modified duration. The results does not alter their earlier conclusion.
22. 3) Nonlinear Specification
They regress the nonlinear model as
, , 1
1 , ,
2 2
, , , , 1 1 ,
2 2
, 1 ,
( ) ( ( ) ( ))
( ( ) ( ))
j t j t
t t i j t
i j t j t t j t t j t
NL
j t j t t
CS LEV LEV LEV E LEV E LEV
E LEV E LEV Z
The results indicate that the relationship between changes in credit spreads and changes in
leverage is nonlinear in the tradeoff proxy but not in the pecking order proxy. However, allowing
for nonlinearity does not alter the conclusion that expected leverage affects contemporaneous
credit spreads.
4) Alternative Targets
They also estimated models using a 1- or 3-year trailing average of past leverage to proxy for
a firm’s leverage target. These alternative proxies for future leverage carry even larger coefficients
than those reported in Table 4.
5) Asymmetric Response to Expected-Leverage Increase Versus Decrease
The original model constrains firms to have a symmetrical response to future leverage changes,
whether intending to raise or lower their leverage ratios. However, it may be systematically easier
for firms to increase their debt level than to decrease it, or vice versa. FNO regress the following
model instead
, ,
, , , , 1 , 1 , 1
, 1 , 1
( ) ( ) _
( ) _ i j t
i j t j t t j t t j t j t
ASYM
t j t j t t
CS LEV E LEV E LEV EXP DECR
E LEV EXP INCR Z
the coefficients on the trade-off proxies weakly suggest that firms above their targets are expected
to adjust more slowly than firms below their targets. The estimated coefficients on the pecking
order proxies indicate statistically and economically similar effects of expected future leverage on
credit spreads. All in all, the previous findings are not systematically different for firms expected
to increase versus decrease their leverage.
6) Adding Accounting Measures as Explanatory Variables
Since accounting variables may forecast default because they proxy for expected future financing
decisions, FNO estimates the following equation
, ,, , , , i j t
ACCT
i j t j t j t tCS LEV A Z
where the A vector contains O-SCORE, Z-SCORE,QUICK RATIO, CASH AVAILABILITY and
so on. Regardless of the combination of accounting variables used, future leverage proxies are
always strongly significant.
v. Conclusion
FNO construct 3 proxies for leverage changes, based respectively on the trade-off theory, the
pecking order theory, and the Kisgen’s (2006) hypothesis that firms enjoy discrete benefits from
moving to a higher (letter) credit rating. They use all 3 proxies for investors’ leverage expectations
to explain credit-spread changes in a sample of 1,243 bonds over the period 1986–1998. Their
findings are: 1. bondholders predict future leverage changes resulting from changes in outstanding
debt and equity, but not from fluctuations in stock or bond prices.; 2. investors’ expectations about
future leverage changes do affect credit spreads and this effect is above and beyond the effect of
contemporaneous leverage changes; 3. Estimating the model for firm subgroups indicates that the
trade-off proxy for future leverage is more robust than the other two.
23. Appendix MATLAB Code for Numerical Analysis
For Vanilla Call
r = 0.1;
T = 1;
%generate the Asset Value vector A
for i = 1:11
A(i) = 100+10*(i-1);
end
%generate the Leverage vector L
for i = 1:17
L(i) = 0.1+0.05*(i-1);
end
%generate the Asset Volatility vector SigmaA
for i = 1:11
SigmaA(i) = 0.2+0.1*(i-1);
end
%compute all the possible combinations of the three parameters
[X,Y,Z] = meshgrid(A,L,SigmaA);
Combinations = [X(:) Y(:) Z(:)];
N = size(Combinations,1);
%BlackScholesInputs is a matrix whose 1st column is Asset Value, 2nd column
%is Strike Price, 3rd column is Asset Volality
for i = 1:N
BlackScholesInputs(i,1)=Combinations(i,1); %Asset Value
BlackScholesInputs(i,2)=Combinations(i,1)*Combinations(i,2);%Strike
Price
BlackScholesInputs(i,3)=Combinations(i,3); %Asset Volatility
end
%Numerical Analysis
for i = 1:N
[Call, Put] =
blsprice(BlackScholesInputs(i,1),BlackScholesInputs(i,2),r,T,BlackSch
olesInputs(i,3));
Equity(i,1) = Call;
LeverageRatio(i,1) = BlackScholesInputs(i,1)/Call;
d1(i,1) =
(log(BlackScholesInputs(i,1)/BlackScholesInputs(i,2))+(r+BlackScholes
24. Inputs(i,3)^2/2)*T)/(BlackScholesInputs(i,3)*sqrt(T));
d2(i,1) = d1(i,1)-BlackScholesInputs(i,3)*sqrt(T);
SigmaE(i,1) =
BlackScholesInputs(i,1)/Call*BlackScholesInputs(i,3)*normcdf(d1(i,1))
;
DefaultProbability(i,1) = 1-normcdf(d2(i,1));
end
FinalResults = [BlackScholesInputs Equity
LeverageRatioSigmaEDefaultProbability d1 d2];
col_header = {'Asset
Value','StrikePrice','AssetVolatility','Equity','LeverageRatio','Equi
tyVolatility','Default Probobility','d1','d2'};
xlswrite('Final Results Vanilla Call.xlsx',col_header,'Sheet1','A1');
xlswrite('Final Results Vanilla
Call.xlsx',FinalResults,'Sheet1','A2');
For Down-and-Out Calls
r = 0.1;
T = 1;
pho = 0.6;
%generate the Asset Value Vector A
for i = 1:11
A(i) = 100 + (i-1)*10;
end
%generate the Leverage Vector L
for i = 1:17
L(i) = 0.1 + (i-1)*0.05;
end
%generate the Asset Volatility Vector SigmaA
for i = 1:11
SigmaA(i) = 0.2 + (i-1)*0.1;
end
%generate all the possible combination of three parameters
[X,Y,Z] = meshgrid(A,L,SigmaA);
Combinations = [X(:) Y(:) Z(:)];
N = size(Combinations,1);
%DOC is a matrix whose 1st column is the Asset Value, 2nd column is
%the Strike Price, 3rd column is the Asset Volatility.
25. for i = 1:N
DOC(i,1) = Combinations(i,1);%Asset Value
DOC(i,2) = Combinations(i,2)*Combinations(i,1);%Strike Price
DOC(i,3) = Combinations(i,3);%Asset Volatility
end
%Numerical Analysis
for i = 1:N
d1 =
(log(DOC(i,1)/DOC(i,2))+(r+DOC(i,3)^2/2*T))/(DOC(i,3)*sqrt(T));
d2 = d1 - DOC(i,3)*sqrt(T);
eta = r/DOC(i,3)^2 + 0.5;
H = pho*DOC(i,2);
b1 =
(2*log(H)-log(DOC(i,1))-log(DOC(i,2))+(r+DOC(i,3)^2/2)*T)/(DOC(i,3)*s
qrt(T));
b2 =
(2*log(H)-log(DOC(i,1))-log(DOC(i,2))+(r-DOC(i,3)^2/2)*T)/(DOC(i,3)*s
qrt(T));
b2 = b1 - DOC(i,3)*sqrt(T);
Equity(i,1) =
DOC(i,1)*normcdf(d1)-DOC(i,2)*exp(-r*T)*normcdf(d2)-DOC(i,1)*(H/DOC(i
,1))^(2*eta)*normcdf(b1)+DOC(i,2)*exp(-r*T)*(H/DOC(i,1))^(2*eta-2)*no
rmcdf(b2);
LeverageRatio(i,1) = DOC(i,1)/Equity(i,1);
delta = normcdf(d1) + (H/DOC(i,1))^(2*eta)*(2*eta-1)*normcdf(b1) +
DOC(i,2)*exp(-r*T)/DOC(i,1)*(H/DOC(i,1))^(2*eta-2)*(2-2*eta)*normcdf(
b2);
SigmaE(i,1) = DOC(i,3)*DOC(i,1)/Equity(i,1)*delta;
DefaultProbability(i,1) = 1 -
normcdf((log(DOC(i,1)/(DOC(i,2)))+(r-DOC(i,3)^2/2)*T)/(DOC(i,3)*sqrt(
T)))+(H/DOC(i,1))^(2*(r/DOC(i,3)^2-0.5))*normcdf(b2);
end
FinalResults = [DOC Equity LeverageRatioSigmaEDefaultProbability];
col_header = {'Asset
Value','StrikePrice','AssetVolatility','Equity','LeverageRatio','Equi
tyVolatility','DefaultProbobility'};
xlswrite('Final Results DOC.xlsx',col_header,'Sheet1','A1');
xlswrite('Final Results DOC.xlsx',FinalResults,'Sheet1','A2');
![1. Numerical Analysis
The following numerical analysis is conducted to investigate the nonlinear relationship between
leverage, volatility (both asset and equity), and default probability.
Inputs are as follows:
Asset Value V=
[100 110 120 130 140 150 160 170 180 190 200];
Book Value Leverage L=
[0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
0.80 0.85 0.90];
Asset Volatility 𝜎 𝑉 =
[0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1 1.10 1.20];
Interest Rate r = 10%, Time to Maturity T = 1;
Strike Price K is defined as the product of Asset Value and Book Value Leverage.
1.1 Vanilla Calls
From Ito’s Lemma, Equity Volatility is defined as follows
𝜎 𝐸 =
𝑉𝑡
𝐸𝑡
𝜎 𝑉 𝑁(𝑑1)
And Default Probability is defined as
𝐷𝑃 = 1 − 𝑁(𝑑2)
Where
𝑑1 =
𝑙𝑛
𝑉𝑡
𝐾 + (𝑟 +
𝜎 𝑉
2
2 )(𝑇 − 𝑡)
𝜎 𝑉 𝑇 − 𝑡
, 𝑑2 =
𝑙𝑛
𝑉𝑡
𝐾 + (𝑟 −
𝜎 𝑉
2
2 )(𝑇 − 𝑡)
𝜎 𝑉 𝑇 − 𝑡
And Equity is calculated via Black-Scholes formula:
1 2( ) ( )rT
t tE V N d Ke N d
As shown in Figure 1, Default Probability is an increasing function of Asset Volatility, when
Leverage is fixed. A meticulous scrutiny of the 3-D surface reveals that for relatively low
Leverage, Default probability is a convex monotonic increasing function of Asset Volatility, while
for relatively high Leverage, Default probability is a concave monotonic increasing function of
Asset Volatility. The critical point is around a Leverage of 50%.
As shown in Figure 2, Default Probability is also an increasing function of Leverage, when Asset
Volatility is fixed. What’s more, Figure 2 also demonstrates the transition from convexity to
concavity. The critical point is around an Asset Volatility of 60% to 70%.
The relationship between Default Probability and Equity Volatility is quite similar that shown in
Figure 1, because Equity is derived from Asset Volatility. The critical point is still around a
Leverage of 50%.However, a 3-D surface cannot be plotted since we have so many different
Equity Volatility Values. Figure 3 gives an illustration using the example: Asset Value=100,
Leverage= 10%, 50%, and 90%.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)