CreditInCBs

8/3/2019 CreditInCBs

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Credit in Convertible Bonds: A Critical View1

By Zouheir Ben Tamarout, Dexia Asset Management

Convertible Bonds (CBs) are corporate debt securities that confer on the holder the right to exchange them for a

pre-specified number of ordinary shares of the issuer. In theory, a CB may be viewed and valued as a package of

a straight bond and a call on the underlying equity. In practice, the different terms describing a realistic Cb

(callability by the issuer, putability by the holder) make it impossible to decouple the bond from the equity

option.

Convertible bonds, as hybrid products mixing interest rate, equity, volatility and credit factors, are the bestrepresentatives of the intrinsic interrelationship between market and credit risk. Although this lack of separability

is widely recognized, there had been very few published models including (rigorously) default risk in the pricing

of CBs2. To our knowledge, very few articles have discussed the impact of parameters calibration on the Cb

value and sensitivities and on the underlying credit spread curve.

Existing approaches to modeling credit risk may be classified into three groups: structural, reduced-form andhybrid.

The structural approach directly relates default and the firms assets. The firm defaults when its assets fall below

its debt (or an exogenously given level). A major advantage of this approach is that it explicitly links the credit

event to firm-specific variables. It has proven very useful in understanding many empirically observed results

that show that the number of business failure is influenced by macro-economic variables, or that returns on high

yield bonds, compared to investment-grade bonds, are more correlated to equity index returns. Unfortunately,implementing this approach faces practical problems due to the complexity of the firms capital structure, to its

evolution across time (we believe that the capital structure evolves in order to keep a constant equity-debt ratio)

and to the non-respect of priority rules.

In the reduced-form approach (intensity models), default is treated as an unpredictable event governed by a

hazard-rate process. Default events can never be expected (technically speaking, the time of default is an

inaccessible stopping time). The modeled hazard-rate represents the likelihood of the firm defaulting over thenext period. Loss on default is captured by an exogenous (possibly stochastic) recovery rate. The parameters of

these models may be fitted tomarket spread curves. An attractive feature of this approach is its ability to imply

realistic short credit spreads3

Spread curves are not constrained to start at zero and slope upward as with

structural diffusion models.

Hybrid models combine attractive features from both approaches. They aim to provide a structural interpretationto the parameters of intensity models, and to meet empirical observations. Although the hazard rate process (on

which they are based) is not formally linked to the firms assets, it is allowed to depend on macroeconomic and

firm-specific variables.

Here, we explore the use of a simple hybrid model in convertible bond pricing. Through this paper we seek to

demonstrate the important impact of credit risk integration on the value and sensitivities of CBs. We also showthe practical issues raised by model calibration and parameter interpretation. The used model is a simplified

adaptation of the model introduced by Madan and Unal in their article (March 2000) A Two-Factor HazardRate Model for Pricing Risky Debt and the Term Structure of Credit.

The model:

We adopt the mathematical framework of Duffie and Singleton (1994/1999). We take as given an equivalent

risk-neutral measure Q, under which the price process of any contingent claim, discounted with the money

market account (the numeraire), is a martingale. Specifying all the technical assumptions is out of the scope of

this paper. The basic ingredients of our model (like many others) are a stopping time for default and a random

bounded recovery at default. The interpretation of this recovery will be discussed below.

Following Madan and Unal, we suppose that the firm faces multiple random losses at random time. These losses

(adverse movement in financial markets, international crisis, default of a creditor or a customer) are not

necessarily fatal. The firm goes bankrupt only when the loss is serious enough to absorb the equity value. Weassume that a Poisson process (with an intensity ) governs the loss event. Our formulation allows for the


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intensity to be a stochastic process, as far as some non-arbitrage constraints are met. This is referred to as a Cox

Process.

The hazard rate of default is given by:

( ) ( )[ ]SLdtttQdtht += [,[

where denotes the event random time, S the stock price and L the loss amount (by share) experienced at .

This is the instantaneous risk-neutral probability of default at time t, given survival to t.

Assuming that the loss event intensity is independent of the loss magnitude, the hazard rate of default can be

simplified:

][ SLQht =

The hazard rate equals the arrival intensity times the probability that the loss is large enough to destroy the

equity value. One more assumption is needed on the loss distribution in order to calculate this probability.

Many loss distributions commonly used in the literature can be viewed as a mixture of exponential distributions.

The famous Pareto distribution is an example where the gamma function is used as a mixing distribution.

In our simplified approach, a single exponential distribution is used:

=m

SSLQ exp][

where m is the (risk-neutral) expected loss.

The probability of exceeding the mean loss decreases exponentially with the loss magnitude.

The survival probability of the firm until a given maturity T, given survival to t, is related to the hazard rate by:

=

T

tt

Q

t

dthETtS exp),( where

=

m

Sh t

t

exp

The expectation is taken under the risk-neutral probability and takes into account all the information available at

time t.

Upon default, bondholders receive a fraction of their investment. This is the recovery rate. Jarrow and Turnbull(1995) define this recovery in terms of an equivalent risk-free bond with an identical cash flow structure. Duffie

and Singleton (1999) define it in terms of the risky bond immediately before the default event occurs.

Unfortunately, these (mathematically convenient) academic definitions are inconsistent with market practice.

The legal documentation specifies recovery in terms of outstanding principal amount plus current accrued

interest. This is the approach we choose here.

For illustrations sake, we assume that the recovery rate and the risk-free interest rate curve are given constants.The assumption of a constant recovery is obviously at odds with reality. There is strong evidence that the

recovery rate depends on the seniority of the issue and exhibits a pronounced state-dependency (business cycle).

Although our model can handle stochastic recovery, we elect to keep it constant for two reasons. In order to

simplify calculations, we actually suppose that recovery is independent (under Q) of interest rates and default

time. This assumption allows us to replace the stochastic recovery by a its expected value (under Q). We alsoshow that estimating the recovery alone is very touchy because of lack of relevant data. Randomness will rather

be introduced through the loss intensity.

In most practical cases, optionality due to stochastic interest rates can be neglected (at a first approximation)

when valuing a CB. This explains the market practice of managing CBs with a single-factor (equity only) model.

According to our interpretation of available literature on the subject, we also suppose that interaction between

interest rates and credit risk can be ignored at this stage. However, extension of this approach to two factors isstraightforward. This issue will be explored in future work.


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Pricing corporate bonds:

Based on previously available results in a more general framework (Jarrow and Turnbull, 2000), the market

value at time t (given survival to t) of a corporate bond with promised coupon payments {Ci} at time {Ti} (i

=1,,n) and redemption value F at T=Tn, is:

( )

+++

++

+=

+=

>

+=

>

dsdthrhTsCFWE

dthrFdthrCETtV

n

tmi

Ti

tTiMax

s

t

ttsii

Q

tt

Ti

t

tt

Ti

t

tt

n

tmi

i

Q

tt

1)( ),1(

1

1)(

)(exp)(1

)(exp)(exp1),(

where is the stopping time for default, r the (stochastic) spot rate, and W the constant recovery rate.

m(t) is the last coupon index. T0 stands for the issue date.

In this formula, we represent the value of the risky bond as the value contingent on survival through maturity

(first term) plus the sum of cash flows generated on default in each of the coupon periods (second term).

Note that this formula is based on the legal recovery approach.

Contrary to the risk-free claim and risky claim cases mentioned above, the risky bond is not strippableunder legal recovery approach (par claim). The bond can not be represented by a sum of discounted risky cash

flows with the same behavior at default. Standard stripping procedures do not apply in our case. In order to

determine the underlying zero-coupon curve, other stripping methods are used.

It is worth noting that recovery introduces a kind of continuous (possibly random) coupon payment given by:

))( )(tmitt TtCFWhc +=

Preliminary tests:

We test our exponential assumption on a real corporate issue: the Adelphia Communication 9.875% 1/3/2007.

This is a senior straight bond noted B+ by S&P and B2 by Moodys. We use historical spreads (versus Libor

Swaps) from Bloomberg, between 1/10/98 and 10/9/2000. The bonds (continuous) credit spread can beapproximated by:

=

T

t

t

Q

t dtWhELntT

Tty )1(exp1

),(

where W denotes the constant recovery rate. This approximation can be justified under the risky claim

recovery approach (Duffie and Singleton), for a zero-coupon, when interest rates are independent from the

hazard rate and the recovery rate. Note that, for bonds trading around par, par claim and risky claim

approaches are close.

Using our calibrated parameters (the calibration procedure is exposed below), we calculate the bonds yield for

each stock level reached on the historical path. Theoretical and historical yields are plotted (versus the stockprice) in Exhibit 1.

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

6.50

20.00 30.00 40.00 50.00 60.00 70.00 80.00

Model MarketExhibit 1a Exhibit 2a

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Market Model

Exhibit 2aExhibit 2a


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The exponential shape looks reasonable, but does not capture the variability of the real world. Its failure can be

accentuated through the Exhibit 2a of changes in credit spreads versus changes in stock price. Contrary to the

perfect anti-correlation assumed by our model, changes in spread and stock price are roughly non-correlated.

By introducing a stochastic arrival intensity (as a lognormal process governed by an independent Brownian

motion), we get obviously closer to reality (see Exhibit 2b). Here, we suppose that the arrival intensity has nodrift, a volatility of 35% and a correlation of 0.2 with stock price.

Pricing convertible bonds:

As an equity contingent claim subject to default risk, the CB solves a modified Black-Sholes partial differential

equation (PDE) with specified boundary conditions. This PDE captures the hazard-rate and the recovery.

PhrcStPfS

PSdr

S

PS

t

Pt

)(),,()(2

12

222

+=++

+

+

where S is the stock price, its volatility and d its dividend rate.

F(P,t,S) represents predetermined flows (coupons) paid to the holder.

In solving this PDE, we follow the finite difference approach with a Crank-Nicolson scheme. We dont describe

this method here since it is covered extensively elsewhere. It is important to understand that, through the PDE

mentioned above, we are actually solving a free boundary problem. In other words, our numerical method has to

be accurate enough to deal correctly with the early termination events. Here, we prefer to focus on the available

data and the impact of our calibration choices on the final results.

To demonstrate our approach, we consider a real example: Versatel 4% 30/03/2005 convertible bond. This CBhas been issued in March 2000, is convertible since May 9 th to 16.4582 shares and callable from 30/03/2002 to

maturity. The underlying issuers calls are soft, i.e., they are exercisable when the equity price exceeds some

given triggers. The CB is a non-subordinated issue, rated B- by S&P and B3 by Moodys. Versatel is a

telecommunication company of the Netherlands. It offers basic and business telephone, fast Internet access anddata access services. All the simulations are performed on September 1st. Versatel stock has a price of 32 EUR. It

has no projected dividends, exhibits a volatility of 90% and a repo rate of 1.5% (treated as a continuousdividend).

Calibrating the model:

In our PDE, three parameters should be chosen in order to reflect the available data. These are the intensity, the

mean loss and the recovery rate. Note that the mean loss appears necessarily through its ratio to the share price.

Likewise, the recovery rate appears always through the product W .The lack of default history precludes statistical inference of the hazard rate. Estimating the model parameters

should take advantage of the data at hand: the issuers credit spread curve, recovery historical estimations, and

the historical relationship linking spreads to the stock price.

Spreads are relative to Euro swaps, rather than treasuries. The spreads that we retain correspond to a B gradedissuer (rather than a B-). Spreads and swap rates are graphed in Exhibit 3.

Exhibit 2b

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Market Model

Exhibit 2aExhibit 2a


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Data compiled by Moodys for the period 1974-1997 show that, for senior unsecured bonds for example,

recovery rates range approximately between 30 (25th percentile) and 75 (75th percentile). The median is around

50. Moodys measures the recovery rate as the value of a defaulted bond as a fraction of face value. Given ourdefinition (including accrued interest) and our perception of the Telecom industry in general and Versatelaccounting information in particular, we believe that 30% is a realistic (and perhaps optimistic) expected

recovery rate for our CB. This is our historical (actual) estimation of the recovery rate. It would differ from the

risk-neutral recovery effectively used in the model.

For this exogenously given recovery rate, we attempt to calibrate the model to the initial credit spread term

structure. These spreads are understood to be active for similar non-convertible bonds of the same issuer, i.e.,

straight bonds with a comparable coupon (4%) and redemption value. We put maximal weights on the two pointswe believe the most important for the problem at hand: the CB maturity and the very short term. The former is

fixed at 7.15 and the latter at 6.0. It is easy to see that the very short-term (continuous) spread is given by:

=

m

SWs exp)1(

Relating the long-term spread to the model parameters is more complicated. The calibration on the CBs

maturity is performed numerically. Once the intensity and the mean loss are chosen, the other points of the

spread curve are implied for the same coupon rate (4%). For each maturity, a straight bond (coupon of 4%) is

priced with the model. The retained implied spread is the parallel shift to apply to the risk-free curve in order toget the same price (traditional approach). The theoretical curve (T30) is compared to the market curve in

Exhibit 4. Calibration succeeds to fit the target spreads but at the cost of an unacceptable shape for the spread

curve. We repeat the same operation for different recovery rates. A better fit to the term structure is obtained for

a lower recovery of 10%. The theoretical curves (T10 and T20) are also in Exhibit 4.

4

4.5

5

5.5

6

6.5

7

7.5

8

Sep-00 Jun-03 Mar-06 Nov-08 Aug-11

Sw ap Rates Credit SpreadsExhibit 3

6

6.2

6.4

6.6

6.8

7

7.27.4

7.6


Market T30 T10 T20

Exhibit 4


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In our approach, fitting any spread curve seems to necessitate a global calibration of the intensity, the mean loss

and the recovery rate. In the risky claim approach, recovery and intensity are completely linked through the

expected loss rate )1( W . Other things being equal, any couple ),( W keeping the expected loss constant

will provide the same curve structure. This structure is of course attainable in our model with W null.

It is worthwhile pointing out that the low-coupon bonds used above, trade far below their face value (under par).

With our legal interpretation of the recovery rate, the quoted credit spread depends on the coupon rate. A higher

coupon bond should trade on a higher spread than a bond with a lower coupon. In fact, at default, both bondsgenerate approximately the same recovery (relatively little differences are due to the accrued interest). It is, thus,

preferable to invest in lower-coupon bonds. In order to quantify this par claim effect, we plot the model

implied spread curve for bonds (with a coupon of 12.8%) trading around par (see Exhibit 5). As expected, the

par claim effect is significant for higher recovery rates. This result would suggest calibrating the model to

bonds trading at par.

The stocks volatility enters the default risk at two levels. First, it increases the hazard rate drift, implies a lower

survival probability and then a higher credit spread. On the other hand it increases the hazard rate volatility,causes a higher survival probability and thus narrower spreads. The latter effect is due to the fact that the risk-

neutral survival probability S(t,T) is an expectation of a convex function (we use Jensen s inequality).

For a given recovery rate (here 20%), we repeat the calibration for different volatility levels. We attaincomparable term structures on the 0-5Y range. For longer maturities, a higher volatility seems to extend the

positive trend in the spread curve slope (see Exhibit 6). The effect of volatility is more obvious when the other

parameters are kept constant (intensity=13.8%, mean loss=50, recovery=20%). The credit curve is steeper (and

becomes positive) for higher volatilities. This effect is shown in Exhibit 7.

6

6.5

7

7.5

8

8.5

Sep-00 Jan-02 Jun-03 Oct-04 Mar-06 Jul-07 Nov-08 Apr-10 Aug-11

T30 T10 T30_AtPar T10_AtParExhibit 5

6

6.2

6.4

6.6

6.8

77.2

7.4

7.6


Market Vol=40 Vol=60 Vol=90Exhibit 6

5

5.5

6

6.5

7

7.5


Vol=30 Vol=60 Vol=90Exhibit 7


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In order to demonstrate the effect of the models parameters, we plot in Exhibit 8 the curve evolution for

different quads (stocks volatility, mean loss, intensity, recovery). Thus, we cover a large spectrum of levels and

shapes.

Credit risk in CBs:

In the traditional approach, CBs are valued and hedged with a deterministic credit spread (possibly a term

structure). Practitioners usually use spreads implied by similar non-convertible issues. They dont take into

account the coupon differential, and unnecessarily penalize the equity upside by discounting it at higher interest

rates. Many alternatives have been used in order to adjust the spread (empirically, or as a function of the delta

and / or the conversion probability) when the Cb gets into the money. These are partial solutions that dont

correctly account for all the cash flows due to coupon payments, and embedded calls and puts. More

importantly, they dont handle credit as a risk factor in its own right.

In our approach, credit spread is linked to the stock price through the mean loss. When the stock falls, the credit

spread deteriorates. The CB doubly suffers. Its conversion value shrinks and its bond floor decreases.We calibrate the model to the initial spread term structure, for a given volatility of 90%. Although making the

arrival intensity depend (deterministically) on time should guarantee a perfect fit to the initial curve, keeping it

constant will not invalid our results.

Our theoretical prices are compared to the traditional approach in Exhibit 9a. Deltas are plotted in Exhibit 10a

and expressed as a percentage of the CB ratio (16.4582). Note that, in these graphs, all the parameters, except the

stock price, are invariable: = 90, = 11.4, m = 52.0, W= 10.

0

2

4

6

8

10

12

14

16

18


Exhibit 8

750.0

850.0

950.0

1050.0

1150.0

1250.0

1350.0

1450.0

1550.0

1650.0

1.00 21.00 41.00 61.00 81.00

Flat Spread ModelExhibit 9a

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

1.00 21.00 41.00 61.00 81.00

Flat Spread ModelExhibit 10a


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The evolution of the hedge ratio may look reasonable at first glance. Due to credit risk, the models hedge ratio

exceeds the traditional approach recommendation.

In order to demonstrate the importance of parameters choice, we calibrate the model to the markets spread

curve for a lower stock volatility: = 42, = 23.0, m = 22.6, W= 0. Results are represented in Exhibit 9b(CB price versus stock price) and Exhibit 10b (hedge ratio versus stock price).

As the stock price decreases, the CBs delta decreases first (conversion effect) before going up (credit effect).

The attained levels, however, seem too high. Two plausible reasons for this:

- The model compensates for other missing factors (interest rates, arrival intensity).- Keeping the models parameters constant when the stock falls is a hazardous assumption. In such a scenario,

the firm could be downgraded. Its spread curve and mean loss parameter should be reviewed in order to

reflect its new situation.

Our approach raises another debatable point. Given the deterministic dependence of the hazard rate on the stockprice, the model infers a spread curve scenario for each possible future value of the state variable (stock price).

Some practitioners build their own scenarios of credit spread relative to stock price. This is comparable, in spirit,

to pricing an option on volatility by guessing its future evolution (as a function of the stock price). But does this

approach imply a realistic dynamics for the volatility?

Conclusion:

The key ingredient to the model we are seeking is its ability to capture the spreads dynamics, and to correctly

price and hedge CBs, as well as corporate debt. This is a very ambitious target. We believe that measuring the

hedging power of this model would be the unique way to judge its ability to capture the relevant risks. The

purposes of this paper, however, are modest. They can be summarized as follow:

- To critically introduce the models behavior and recommendations when used to trade CBs.

- To demystify credit risk by integrating it with market risk, following Jarrow and Turnbull. Our modelsuggests five determinants for credit spread. These are the stocks price and volatility, the mean loss, thearrival intensity and the recovery rate.

- To warn against black boxes for credit risk pricing. Different interpretations of available data could generateimportant bias. The same data can be used differently (calibration process) and lead to unexpected effects.

- To show that choosing / including a credit risk model is a critical task. Its impact is important enough to behandled with a great caution.

The present model should be viewed as an illustrative example, nothing more.During the last years, we saw a proliferation of credit risk models. Pragmatic ideas are forcing their way.

Ben Jonson said: True happiness consists not in the multitude of friends, but in the worth and choice.

Selected References:

D. Duffie, D. Lando, June 1997, Term Structures of Credit Spreads with Incomplete Accounting InformationD Duffie, K. Singleton, 1999, Modeling Term Structures of Defaultable Bonds Review of Financial Studies 12

400.0

600.0

800.0

1000.0

1200.0

1400.0

1600.0

1.00 21.00 41.00 61.00 81.00

Flat Spread ModelExhibit 9b

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

120.0%

1.00 21.00 41.00 61.00 81.00

Flat Spread ModelExhibit 10b


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R.A. Jarrow, S.M. Turnbull, 1995, Pricing Options on Financial Securities Subject to Credit Risk, Journal of

Finance 50 (1)

R.A. Jarrow, S.M. Turnbull, 2000, The Intersection of Market and Credit Risk, Journal of Banking & Finance

24 (2000)

C.L. Keatinge, Working paper, Modeling Losses with the Mixed Exponential DistributionD. Lando, Working paper, 1994/1997, On Cox Process and Credit Risky Securities

Moodys Investors Services, January 1996, Corporate Bond Defaults and Default rates 1938-1995)C. Zhou, March 1997, A Jump-Diffusion Approach to Modeling Credit Risk and Valuing Defaultable

Securities

1 Implementation has been achieved by Olivier Clapt (Financial Programmer)

We wish to acknowledge the comments and suggestions of all our Alternative Investmentteams, particularly

Convertible Bonds and High Yield managers.

2 K. Tsiveriotis and C. Fernandes proposed (see Valuaing Convertible Bonds with Credit Risk) a rigorous

framework with two coupled Black-Scholes PDEs in order to correctly introduce the credit spread.

3 Zhu introduced a jump-diffusion process (in a structural approach) in order to capture the basic features of

credit spread structures. Duffie and Lando studied the implication of imperfect accounting information andshowed that imperfect observation of firms assets is consistent with a hazard-rate approach.

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