Performance Calibrated LT - Semantic Scholar › 6188 › 20e300d2d68d... · The Performance of the Calibrated Leland-Toft Model Howard Qi,† Sheen X. Liu, and Chunchi Wu First Draft:

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The Performance of the Calibrated Leland-Toft Model

Howard Qi,† Sheen X. Liu, and Chunchi Wu

First Draft: January 2004

This Version: August 26, 2004

Abstract

A common wisdom about term structure models is that they predict much lower spreads than the observed spreads for investment-grade bond, and most of them tend to overpredict spreads for junk bonds. Among them, the Leland-Toft model is perhaps most controversial. Some studies show it always overpredict spreads, in some cases they can be as high as over 5000 bps, yet other studies show it generates extremely low spreads for investment-grade ratings, hence it is just as poor as other models. We introduce an approach for calibration of the model. We particularly stress the importance of carrying out the calibration properly and show what are the cautions one must take for such a calibration to be successful. Using the calibrated model and some actual bonds data as bench mark, we find the Leland-Toft model performs significantly better than reported by some recent studies. Our findings may help to clarify the controversy and serve as a reference for calibration of other term structure models. In the end, we also provide some general thinking on the calibration and applicability of the structural term structure models.

†Howard Qi and Chunchi Wu are at Syracuse University and Sheen X. Liu is at Youngstown State University. Contact address: School of Management, Syracuse University, Syracuse, NY 13244. Email:[email protected]; fax:315-443-5457.

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I. INTRODUCTION

Leland proposed a term structure model with endogenously determined

bankruptcy trigger in 1994 for perpetual bonds. Later the model is extended to finite

maturity bonds by Leland and Toft (1996). The Leland-Toft model considers the firm’s

capital structural in an environment where there are corporate tax benefits and bankruptcy

costs. The equity holders can optimize the capital structure by finding the lowest

bankruptcy trigger such that the equity and firm asset values are maximized at the

expenses of the debt holders. Alternatively, one may think there is an agency cost

associated with the firm, because once the firm is near bankruptcy, the equity holders

have every incentive to take on risky projects. From the options perspective, equity value

is call option on the firm’s asset with an exercise of the debt face value and a time

horizon of bond’s maturity. On the other hand, by issuing debt, the firm enjoys tax

benefits. However, with an increasing financial leverage, the firm is exposed to a growing

probability of default. Once default happens, the equity holders will get nothing and bond

holder will get whatever is left less the cost of financial distress. The cost of bankruptcy

can be a significant portion of the going concern firm value at bankruptcy. Thus, bond

holders have an incentive not to let the bankruptcy happen if a renegotiation can resolve

the distress because of the bankruptcy costs while they prefer to force a bankruptcy to

prevent the bankruptcy trigger going to much lower than the face value of debt. The

Leland-Toft model considers these competing factors and identifies the optimal capital

structure that is a balance of the above-mentioned pros and cons. Hence it falls into the

traditional family of tradeoff theory.

The Leland-Toft model has perhaps the strongest structure component among all

the popular term structure models. It links default probability to firm’s optimal capital

structure characteristics. In this paper, we focus on its ability to predict credit spread.

Most typical structural models for term structure have been criticized for their inability to

explain yield spread in a satisfactory manner. Surveying the literature, we find there is

little consensus on the performance of those models. Some claim that only a small

portion of the yield spread can be explained by structural models. Others show that the

results are mixed. Among those models, perhaps the most “controversial” one is the

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Leland model of infinite maturity and the Leland-Toft model for finite maturity. Huang

and Huang (2002) use the Leland model for perpetual bonds as an approximation for 10-

year bonds to show that the spreads are very small. They conclude that once the version

for finite maturity is used, the LT model would just perform as poorly as other structure

models. However, Eom, Helwege and Huang (2003) find that “the Leland and Toft model

is an exception in that it overpredicts spreads on most bonds”. In their study, the model is

shown to predict a spread as high as 5096 bps while the actual spread is 465 bps.

To further explore these controversies, we propose a simple approach to calibrate

the model before one can meaningfully use it to predict anything. We also show that the

performance of the Leland-Toft model can be significantly enhanced after such a

calibration is done. The paper proceeds as follows. Section II lays out the model’s general

settings. Section III explains the ideas about calibration. In particular, we emphasize a

few caveats one must heeds when perform the model calibration. This has a general

meaning for calibrating any structural models. Section IV shows the results from the

calibrated LT model and the analyses. Section V further reviews the calibration concept

in the context of structural term structure models and provides an initiative for a new

direction of comparing model performances. Section VI concludes the paper.

II. THE LELAND-TOFT MODEL

A. The General Setting

As in some previous studies1 the asset of an unleveraged firm, V, follows a

continuous diffusion process process:

( )[ ] dZdttVVdV σδµ +−= , , (1)

where ( )tV ,µ is the total expected rate of return on value V; δ is the payout rate; σ is the

volatility and is a constant. Following Leland and Toft (1996) (hereafter LT), we assume

that once the firm value hits a boundary, VB, the firm defaults the debt.

1 For example, Merton (1974), Black and Cox (1976), Brennan and Schwartz (1978), and Leland and Toft (1996).

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Before calculate the total value of the debt for a firm we consider a single bond

with maturity t first. The bond continuously pays a constant coupon flow c(t) and having

principal ( )tp . Upon bankruptcy, debt holders of ( )td receive a fraction ρ of the asset

value VB, where ρ is assumed to be a constant. Using risk-neutral valuation, the value of

the bond is given by

( ) ( ) ( )[ ] ( ) ( )[ ] ( )dsVVsfeVVVtFetpdsVVsFetctVVd Brst

BBrtt

Brs

B ,,,,1,,1,,00

−−− ∫∫ +−+−= ρ

(2)

( )BVVsF ,, is the cumulative default probability up to time s, and ( )BVVsf ,, is the

incremental default probability from time s to s + ∆s. Leland and Toft gives the following

solution

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )tGrtcVttF

rtctpe

rtctVVd B

rtB ⎥⎦

⎤⎢⎣⎡ −+−⎥⎦

⎤⎢⎣⎡ −+= − ρ1,, , (3)

where F(t) and ( )tG are given by2

( ) ( )[ ] ( )[ ]thNVVthNtF

aB

2

2

1 ⎟⎠⎞

⎜⎝⎛+= (4)

( ) ( )[ ] ( )[ ]tqNVVtqN

VVtG

zaB

zaB

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+−

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= (5)

where ( )⋅N denotes the cumulative standard normal distribution and

( ) ;2

1 ttzbtq

σσ−−

= ( ) ;2

2 ttzbtq

σσ+−

=

( ) ;2

1 ttabth

σσ−−

= ( )t

tabthσ

σ 2

2+−

=

( );2/2

2

σσδ −−

=ra ;ln ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

BVVb 2

2/1222 ]2)[(σ

σσ raz +=

Assume that the firm continuously issues a constant principal amount of new debt

with maturity T and the same amount of debt is retired. Then the debt structure is

2 See Leland and Toft (1996)

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stationary in continuous time,. The value of all outstanding bonds D can be determined

by integrating the debt flow ( )tVVd B ,, over a period of T:

( ) ( )∫=

=T

tBB dttVVdTVVD

0

,,,, , (6)

The complication due to personal tax terms in the denominator prevents us from getting a

closed-form solution. Numerical method will be applied to solve (6).

B. The Optimal Capital Structure

Firms are subject to corporate taxes. However, the interest payment on their debts

can be deducted from their taxable profit. For this reason alone, firms ought to borrow as

much as they can to take advantage of the tax benefit. But as firm’s leverage increases,

the likelihood of bankruptcy increases as well. Bankruptcy does not arrive free. The

financial distress costs make higher leverage unwanted. Thus, how to strike a balance

between the tax benefit and the potential “price” that comes along with this benefit has

been one of the most fundamental questions in corporate finance.

Given the asset value V(t), tax benefits ( )Vh , bankruptcy costs B(V), and the total

outstanding debt D(V, VB, T), the levered firm’s value W(V, VB, T) is given by

( ) ( ) ( )VBVhVTVVW B −+=,, (7)

where the tax benefits h(V) and B(V) are given by (Leland 1994):

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

+zaBC

VV

rCVh 1τ (8)

and

( )za

BB V

VVVB+

⎟⎠⎞

⎜⎝⎛= β (9)

The equity value E(V, VB, T) is given by

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( ) ( ) ( ) ( ) ( ) ( )

( )TVVDVVV

VV

rCV

TVVDVBVhVTVVDTVVWTVVE

B

zaB

B

zaBC

BBBB

,,1

,,,,,,,,

−⎟⎠⎞

⎜⎝⎛−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−+=

−−+=−=++

βτ (10)

As in Ross (1994) and Leland and Toft (1996), we assume a flow-based bankruptcy,

which occurs once the firm can no longer service its bond interest payment. This feature

of debt service distinguishes the model from those with a positive net worth covenant

(e.g., Merton (1974), and Longstaff and Schwartz (1995)). It allows for the firm to

operate with negative net worth as long as there is enough capital raised from equity

holders to meet the debt interest payment. The original LT model (1996) for finite

maturity debt presumes that the tax deductibility of debt is lost when all available cash

flow is needed to pay interest to bond holders. This will occur when the firm value W

falls below the face value of its total debt but still above the bankruptcy trigger, i.e.,

CVWVB =<< δ . This analysis does not allow tax loss carryforwards. As Leland and

Toft (1996) point out, this assumption may overstate the loss of tax shields. Therefore in

our model, we do not include this consideration.3

The default boundary VB can be solved by using the smooth-pasting condition

( ) 0,,=

∂∂

= BVV

B

VTVVE (11)

Without losing generality, we can set the initial asset value V = 100, and impose par-bond

constraint to eliminate p,

( ) ( )TppcVdV

==100

,, . (12)

This leaves us VB as a function of c.4 The smallest coupon density c (i.e., annual coupon

payment TCc /= ) that satisfies par-bond constraint is regarded as the solution to the

optimal leverage problem. Numerical method can be utilized to find the solution. Figure

1 illustrates the existence of such a bankruptcy trigger VB that can indeed maximize the 3 Our study shows that this difference in assumption causes very small variations in the firm value estimates, less than 0.4 percent in all cases we investigated; and about 2 percentage pont underestimation in the model-inferred optimal leverage ratios. In terms of credit spread, the difference is unnoticeable for maturities less than 5 years and can increase by 24 percent for maturity equal to 20 years. But once tax loss carryforwards are allowed, our treatment may happen to be a closer approximation due to the canceling effect of tax loss carryforwards and loss of tax deductibility. 4 See Leland and Toft (1996).

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firm value (and the equity value simultaneously as given by the smooth-pasting

condition). We plot the firm value as a function of VB (in Panel A) and leverage ratio (in

Panel B) for four different maturities. Clearly, for each maturity there is a unique default

boundary VB that maximizes firm value. This optimal VB corresponds to the optimal

leverage ratio that gives the maximum firm value as shown in Panel B. The values of the

parameters are identical to those in Leland and Toft (1996). Two results are worth

mentioning here. First, firm value, optimal leverage and optimal bankruptcy boundary VB

all increase with debt maturity. Second, a comparison between optimal VB and optimal

leverage shows that for shorter maturities (e.g., 6 months and 5 years), the optimal

bankruptcy boundary VB is higher than the total debt value, while VB is set lower than the

total debt value for longer maturities (e.g., 10 and 20 years). These results are similar to

Leland and Toft (1996).

III. CALIBRATION OF THE MODEL

A. Background: Why Calibration is Necessary?

Although most term structure models5 have been criticized for their inability to

explain yield spreads in a satisfactory manner (see Jones et al., 1984; Elton et al., 2001;

Huang and Huang, 2002), there is little consensus on the performance of these models in

the literature. For example, Huang and Huang (2002), and Collin-Dufresne, Goldstein

and Martin (2002) find that structural models explain only a small portion of the yield

spread. On the other hand, Eom, Helwege and Huang (2003) show that the performance

of these term structure models is mixed.6 Among these structural models, the Leland

(1994) model of infinite-maturity debt and the Leland-Toft (1996) model for finite-

maturity debt receive much attention. Using the Leland model for perpetual bonds as an

approximation for 10-year bonds, Huang and Huang (2002) find that the predicted

spreads are fairly small, although they are higher than those predicted by other models.

Using the LT model-inferred parameters in their base model, they show that those 5 For example, Merton (1974), Longstaff and Schwartz (1995), Collin-Dufresne and Goldtein (2001), and Leland and Toft (1996). 6 They report that “most structural models predict spreads that are too high on average except for the Merton model. In particular, they tend to severely overstate the credit risk of the firm with high leverage or volatility and yet suffer from a spread underprediction problem with safer bonds”.

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parameters can only generate 8.9 bps and 12.4 bps for Aaa and Aa-rated 10-year bonds.

Based on this finding, they conclude that if the finite-maturity version is used, the LT

model would just perform as poorly as other structural models. In contrast, Eom,

Helwege and Huang (2003) find that the L-T model is an exception to most structural

models in that it overpredicts spreads on most bonds.7 A potential source of these

conflicting findings about the performance of the LT model hinges on calibration and the

way it is carried out. To evaluate the model performance, one need to estimate how much

of the observed yield spreads are explained by the model given that the model-implied

default probability is reasonably close to the observed one. Thus, a proper calibration is

necessary. A structural model imposes links among a few variables.

B. Caveats in the Calibration Process

The essence of calibration is to tune some unobserved variables such that the

model generates the rest of the variables, among which we want as many observable

variables as possible to match the historical data (e.g., default rate). It must be brought to

our attention that some variables generated by the model can never sensibly match the

observed ones. For example, it is well known that models on capital structure cannot

accommodate the actual observed leverage ratios across ratings.8 Therefore, to calibrate

a model like the LT model, one must be aware that leverage is not a variable to

“calibrate” to match the observed value. However, this does not mean that the model fails.

It may imply that there exist many firms operating under suboptimal capital structures. In

predicting credit spreads for different bond ratings, this would not be an issue because

what ultimately determines the rating boils down to the default probability and recovery

ratio once bankruptcy occurs. Thus, to calibrate the LT model, it is necessary to match

the observed default probability and recovery ratio. We elaborate on this just to

emphasize the importance in choosing “model-appropriate” targets for a proper

calibration.

7 In their study, the LT model is shown to predict a spread as high as 5096 bps while the actual spread is 465 bps. 8 This is supported by a vast volume of literature, including both theoretical and empirical studies, as well as actual industry surveys. For example, see Byoun and Rhim (2003), Myers (1993), Titman and Wessels (1988), Sunder and Myers (1999), Pinegar and Wibricht (1989), Hittle, Haddad and Gitman (1992), Liesz (2003), and Megginson (1997).

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For example, our calibration procedure involves two major parts. First, we select

a few target parameters to which our model must conform. These parameters must be

observable. Second, because we are often facing more target parameter choices than the

model can accomodate,9 we need to identify which are the most important ones10 to

choose. For example, compared to other structural models (e.g., Longstaff and Schwartz,

1995; Collin-Dufresne and Goldstein, 2001), the LT model has a more sophisticated

structure, which links key financial variables closely together. A proper calibration must

ensure that this link is always maintained.11

C. Targets for the Calibration

Based on the structure of the present model, we choose three targets for

calibration: default probability, equity risk premium, and recovery ratio.12 This allows us

to generate all other variables in the model. One of these unknown variables is yield

spreads. Therefore, to use the model properly for predicting yield spreads, we need to

adjust some unobserved parameters in the model such that the model would conform to

the three target parameters. Theoretically, unobserved asset volatility and observable

equity volatility have a deterministic relationship linked by leverage. Shiller (1981) has

shown that firms’ equity return volatility may be too high to be justified by the

fluctuation of the firm’s fundamental asset values. Therefore, we choose to adjust asset

volatility σ so that the model variables conform to the targets.

9 In other words, the number of constraints (i.e., targets) cannot exceed the number of unknowns (parameters to be calibrated). 10 For example, tradeoff-type models are not suitable to explain observed ratings-leverage relationship because most large profitable firms exhibit strong pecking-order-theory behavior, while some classes of smaller firms show tradeoff-theory behavior. Thus if we use data classified by ratings, then the average leverage is not a proper target for our model because there will be too much “noise” from firms who follow different capital structure policies. However, rating-related default probability is a good target because firms with the same rating will most likely to have similar default probabilities. Same may be said of recovery rate. 11 For other models mentioned in this paper, this does not seem to be a problem because of their weaker structure. But for the LT model, this issue calls for sound judgment in choosing parameters. 12 Actually, if we added one more target, the structure of the model would be broken, i.e., the link among some variables specified by the model would not be maintained in all cases. In other words, it is “over-specified”.

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Based on previous studies,13 we choose two target values for recovery ratio, 80

percent and 85 percent. We borrow the 10-year bond data for the period of 1973-1993

from Huang and Huang (2002). Their data come from various sources. The data used in

this study include spreads on 10-year investment grade bonds based on the Lehman bond

index from 1973-1993. Historical junk bond yield spreads were taken the estimates of

Caouette, Altman, and Narayanan (1998). The default frequencies were based on

Moody’s reports for the period of 1973-1998 and the equity premium was based on

regression estimates in Bhandari (1988). Table 1 shows the selected target parameters

and historical average yield spreads, which will be used as the benchmark to evaluate the

model performance.

D. Calibration Procedure

The first step is to input target recovery ratio and choose an initial estimate of

asset volatility σ. Based on these parameters, the model will generate the optimal

leverage lo and other variables including yield spread, the values of debt D, equity E,

coupon C, principal P, and the firm value W.14 The second step is to use the model-

inferred optimal leverage lo and target equity risk premium to calculate asset risk

premium. Based on the MM’s model for equity return rE,15 we derive the following

approximation formula 16 to link equity risk premium fE rr − to asset risk premium

fA rr − :

( ) ( ) ( )

( )o

oC

o

fDoCfE

fA

ll

lrrl

rrrr

−×−+

−

−×−+−

≈−

111

11

τ

τ, (13)

13 Andrade and Kaplan (1998) suggest that the costs of financial distress are likely in the range of 15% to 20%, while Alderson and Betker (1996) report that liquidation values are no less than 63 percent. Our test shows the difference caused by the variation in recovery rate is fairly small for investment grade bonds after calibration. 14 This must be the first step of calibration because all these parameters can be determined by the model under the risk neutral probability measure, therefore we do not have to worry about asset risk premium at this stage. 15 For example, see Brealey and Myers (1996, page 536). 16 The approximation comes from treating the tax shield as independent of default risk.

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where rA is the expected asset return or the opportunity cost of capital, rf is the riskfree

rate, τC is the corporate tax rate, and Dr is the cost of debt capital, which equals pc / for

a par bond. The third step is to use this calculated asset premium, the initial asset

volatility estimate σ (used in the first step) and equation (5) to determine the cumulative

default probability under the true measure. The fourth step is to check whether this

model-inferred true default probability agrees with the historical default data. If not, we

restart the first step with a different trial asset volatility σ and repeat this process until the

two default probabilities agree. In the end, all target parameters are matched and the

structure of the model is perfectly preserved. We note that with this particular order of

calibration, at each step there is only one parameter to calibrate, which guarantees that we

would not run into a situation where two model-inferred parameters have to be adjusted

to match the observed data. This avoids the potential problem of indetermination.

IV. IMPLEMENTATION OF MODEL CALIBRATION

Table 2 reports the credit spreads predicted by the model after calibration. We

choose different interest rates and payouts. Our results show that a properly calibrated LT

model has much better performance than reported by previous studies. For example, the

estimated credit spread is in the range of 15 – 22 bps for the AAA bonds, and for BBB

bonds, it increases to a range of 52 – 68 bps, which are significantly higher than 8.9 bps

and 12.4 bps, respectively, reported by Huang and Huang (2002). Another significant

improvement is that the model does not generate excessively high spreads as documented

by Eom et al. (2003). Our results show that the calibrated LT model generates much more

reasonable spreads, although they are still lower than the historical spreads for all

bonds.17

We note that the differences in spreads are quite small between cases of recovery

ratio =−= βρ 1 80 and 85 percent, especially for investment grade ratings. Thus we

17 A model that generates lower credit spreads than observed is much more acceptable than one that generates higher credit spreads than observed, because the total observed spreads include premiums due to taxes, liquidity, etc. that are not accounted for by credit premium.

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only plot the results for 80=ρ percent (i.e., bankruptcy cost 20=β percent) in Figure 2.

The central dark column in each rating group is the actual observed spread and other

columns are for model-inferred spreads representing cases of various interest rates and

payouts. After calibration, the model performance is not sensitive to the choice of these

parameters. This is not much of surprise because the model is calibrated to match the

same observed default probability which is one of the key determinants of yield spreads.

Our results show that a proper calibrated LT model has much improved

performances that those reported by previous studies. For example, in various scenarios

with different interest rate and payouts, for the AAA rating, the credit spread is in the

range of 15 – 22 bps, but for BBB rating, the credit spread jumps to a range of 52 – 68

bps, which are significantly higher than 8.9 bps and 12.4 bps, respectively, estimated by

Huang and Huang (2003). Another improvement is in sharp contrast to studies by Eom et

al (2003) that “the LT model overpredicts spreads on most bonds”, in some cases as high

5096 bps. Our results show that the calibrated LT model predicts reasonable credit

spreads, although lower than the observed values for all ratings.18 We do not question the

validity of other studies. We only intend to show that in our framework the importance of

a proper calibration and that the calibrated LT model seems to perform reasonably better.

In summary, calibration is a vital step in applying a structural term structure

model. The actual procedure of calibration must be appropriate for the specific model

under study in that one must choose the targets that are right for the given framework.

For example, as we argued earlier, leverage cannot be chosen as a calibration target for

the LT model. On the other hand, for models like Collin-Dufresne and Goldstein (2001),

leverage can be used to calibrate the model because it is an exogenous process. Finally, a

properly calibrated LT model has better performance.

18 A model that generates lower credit spreads than observed is much more acceptable than one that generates higher credit spreads than observed, because the total observed spreads include premiums due to taxes, liquidity, etc. that are not accounted for by credit premium.

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V. COMMENTS ON STRUCTURAL MODELS AND CALIBRATION

It is well known that structural term structure model tend to generate corporate

bond spreads that are too low. To further complicate the issue, as mentioned earlier in

this paper, there is no consensus on the performances of different structural term structure

models. The most recent comparisons done by Huang and Huang (2003) conclude that all

the structural models perform about equally poorly. Although our calibrated model

performs noticeably better, this difference is probably due to our different treatment of

calibration procedure. As we emphasized in Section III, a proper calibration requires

sound judgment in choosing the targets. For example, leverage ratio should not (and

actually cannot) be chosen as one of the target for tradeoff models. Second, Huang and

Huang (2003) use the perpetual bond case to approximate for the 10-year bond case. We

use the true finite maturity framework for the 10-year bond. When we use the same

model for perpetual bond, we were still unable to obtain the same results. We are not

questioning the validity of their interesting studies because their exact or detailed

calibration procedure about the LT model is not clearly given. We use this case to show

that calibration is not a trivial issue as it appears to be.

Nevertheless, that all the structural models they studied just perform about equally

poorly is not a surprise. Because if the calibration is about pushing the model-induced

default probability as close to the real one as possible, then all those models should

perform similarly in predicting yield spreads.

Indeed, to compare the performances of the structural term structure models, one

should only compare the model-induced default probabilities, see how close the can be

“calibrated” to the observed one. The reason not to compare their predicted credit spreads

has two reasons. First, it is simply a by-product of model-induced default probability.

Second, notion much more profound yet even more controversial, given the default

probability, it is not necessarily true that even under the risk neutral measure, investors

view the cash flow as the expected value. Because for the expectation to be meaningful,

it implies that investors could repeat the process many times. Since for each holding

period, there is only one chance, investors may not price the bond as equation (2) predicts.

One would naturally think of risk aversion. But risk aversion is irrelevant under the risk

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neutral measure. This issue goes beyond asset pricing arena. Rabin (2000) shows for any

concave utility function, the expected utility maximization would lead to unrealistic risk

aversion. One possible reason is that the use of expectation is questionable. This is

beyond the topic of this paper. We simply wish to show that moving from default

probability to pricing asset implicitly brings in extra complication. Thus, summing up the

two reasons given above, we believe to compare the model-induced default probability is

enough and appropriate for comparing different structural term structure models.

However, we recognize that this perhaps is practically more difficult because the default

data is not as readily and widely available as spreads data.

Next precaution one should take in calibration is about applicable range. Previous

studies (Huang and Huang, 2003; Eom et al, 2003) have applied the models to either

different bond ratings or “firms with simple capital structure”. But this practice is

potentially questionable. For example, it is well known that most firms do not behave as

the tradeoff theory predicts in choosing a (target) optimal leverage. Pecking order theory

by Myers (e.g., see Myers, 1993; Brealy and Myers, 1996) tries to explain why the

observed trend in leverage as compared against the firm’s profitability is opposite of what

the tradeoff theory predicts. Therefore to apply any tradeoff model such as the one by

Leland (1994) or the by Leland and Toft (1996), the basic assumption in the model has

already been violated. Thereby, even if the model generates “good” results, the results do

not mean anything because at most they are just accidental. Similar arguments apply to

models like Collin-Dufresne (2001, hereafter CG) and Longstaff and Schwarz (1995,

hereafter LS). Since these models explicitly rely on drastically different ways the capital

structure is decided by the firm, then a simple brute force application to any data at hand

without checking whether he basic assumption is still valid or not can only lead to

spurious comparisons. One may raise the question that in reality, the capital structure

information is not available. Even so, it should not be the excuse to ignore the

fundamental difference in the assumptions upon which different models rely on. However,

in practice, it may be possible to collect the relevant data before one even starts the model

calibration. For example, Pinegar and Wilbricht (1989) surveyed the Fortune 500 firms

and find 31 percent use target (i.e., optimal) capital structure. Hittle et al (1992) surveyed

500 large OTC firms and find only 11 percent use optimal target capital structure.

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Obviously these firms understandably behave quite differently than the LS model would

predict. It would be meaningless to apply the LS model to price these firms’ debts in the

first place. Therefore, before starting calibrating models for applications, it is important

and necessary to verify the data suitability. Only after then and the verification does not

(strongly) indicate otherwise can one proceed meaningfully to the next step.

VI. CONCLUSIONS

In this paper, we have shown how a proper calibration is carried out in detail. This

gives future study a more clear line to follow and check our results. We particularly wish

to stress the importance of a proper calibration of the model. With fewer targets available,

the model is under-calibrated with some parameters floating. An under-calibrated model

will give less reliable results or none. A proper calibration must ensure all the internal

links specified by the model are preserved throughout the calibration. When we are facing

more target candidates, our choice of targets must be in line with the structure of the

original model, otherwise a spurious calibration is carried out and the structure of the

model is implicitly broken.

Furthermore, using our calibrated model, we find the LT model performs

significantly better than what has been reported in different studies. We hope our findings

can add some contribution to the literature in clarifying the controversy over the LT

model’s performance.

Finally, we lay out a few caveats in model calibration. We propose a more

appropriate way to compare different structural term structure models, i.e., comparing

model-induced default probability instead of credit spreads. We also give our general

thinking on how to apply these models in a more meaningful sense.

16

0 10 20 30 40 50 60 70 80100

102

104

106

108

110

112

114

116

(B) Firm value versus leverage (%)

Bankruptcy Boundary VB ($)

Leverage (%)

Maturity of Newly Issued Debt (years): 0.5, 5, 10, 20

0 10 20 30 40 50 60 70 80100

102

104

106

108

110

112

114

116

(A) Firm value versus VB ($)

Figure 1. Firm value as a function of bankruptcy boundary and leverage. Personal tax rate is assumed to be zero. The plots show the value of the firm as a function of bankruptcy boundary VB (Panel A) and leverage (Panel B) for firms issuing debt with maturity T equal to 6 months (thick solid line), 5 years (thin solid line), 10 years (thick dashed line) and 20 years (thin solid line with open circles). The parameter choices are the same as in Leland and Toft (1996), i.e., before-tax riskfree interest rate r = 7.5 percent, the firm’s payout rate δ = 7 percent, the firm’s asset volatility σ = 20 percent, the bankruptcy costs β = 50 percent, and the corporate tax rate =Cτ 35 percent. The value of the firm’s underlying assets V = 100, and the bankruptcy trigger VB is determined endogenously.

17

AAA AA A BBB BB B0

50

100

150

200

250

300

350

400

450

500AAA AA A BBB BB B

0

50

100

150

200

250

300

350

400

450

500

Yie

ld S

prea

d (b

p)

Bond Rating

Model Calibrated without Personal Taxes:

(1) r = 4.5%, δ = 4.0%(2) r = 6.0%, δ = 4.0%(3) Observed Spreads(4) r = 6.0%, δ = 5.5%(5) r = 7.5%, δ = 7.0%(6) r = 9.0%, δ = 7.0%

Bankruptcy cost β = 20 percent

Figure 2. Yield spread generated by the calibrated model. It is assumed that the bankruptcy costs β = 20 percent and the corporate tax rate =Cτ 35 percent. The value of the firm’s underlying assets V = 100, and the bankruptcy trigger VB is determined endogenously. In addition to the observed spread (third column in each group), five cases are presented to represent various interests and payouts. The results given by the calibrated model are fairly insensitive to the choices of the interest rate and payout. We note that the model performance is enhanced after calibration compared to the uncalibrated model which predicts 114 bps across all ratings (see Table 2).

18

Table 1. Target parameters for model calibration This table lists the target parameters that our model will conform to after calibration. The fifth column shows the observed average yield spread. The data presented here are taken from Huang and Huang (2003) for the period of 1973 – 1993.

Target Parameters

Credit Rating

Recovery Rate*

(%) Equity Premium

(%)

Cumulative. Default Probability.

(%)

Average Yield Spread

(bps)

AAA

80 and 85

5.38

0.77

63

AA

80 and 85

5.60

0.99

91

A

80 and 85

5.99

1.55

123

BBB

80 and 85

6.55

4.39

194

BB

80 and 85

7.30

20.63

320

B

80 and 85

8.76

43.91

470 * According to studies by Andrade and Kaplan (1998), the cost of financial distress β is likely in the range of 15 percent to 20 percent. We choose both 80 percent and 85 percent as the recovery target for every case we study.

19

Table 2. Credit Spreads Predicted by the Calibrated LT Model This table reports the credit spreads predicted by the LT model after calibration. Since the capital structure is optimized without personal tax in this case, our model converges to the LT model. Cases with different interest rates and payouts are tested. The corporate tax rate τC is 35 percent. The bankruptcy trigger VB is determined endogenously, and financial distress cost ratio β is set to 15% and 20%, respectively, based on Andrade and Kaplan (1998). Notice that a calibrated LT model has much improved performances in predicting credit spreads, especially for investment grade bonds, and β is not a sensitive factor for investment grade bonds. * The average yield spreads on 10-year bonds are taken from Huang and Huang (2003) for the period of 1973 – 1993.

β = 15%

β = 20%

AAA

AA

A

BBB

BB

B

AAA

AA

A

BBB

BB

B

Average Yield Spread (bps) * 63 91 123 194 320 470 63 91 123 194 320 470

r = 4.5%; δ = 4.0%

21

24

31

52

124

241

22

26

33

56

140

287

r = 6.0%; δ = 4.0%

15

19

30

64

142

265

17

22

34

68

156

306

r = 6.0%; δ = 5.5%

24

27

34

57

135

260

25

29

36

61

151

302

r = 7.5%; δ = 7.0%

25

28

36

60

145

276

26

30

38

64

161

316

Model Inferred Credit

Spread (CS) (bps)

r = 9.0%; δ = 7.0%

15

19

30

63

156

295

16

22

34

67

171

334

20

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