Article history:Received 31 August 2011

We present a simple model for risky, corporate debt. Debtholders and equityholders haveincomplete information about the financial state of the debt issuing company. Information

policy and for the credit spreads required by debtholders. Delayed information acceleratesthe equityholders' optimal decision to default. Interestingly, this effect is small, implying

uityholders havetrue state of thelly distributed instructural, and ine with empirical

l debt payments.licy. The delay is

therefore not important for credit spreads either. The degree of information asymmetry, however, i.e., the difference in the

consider noisy information, only delayed information. One can always argue that noisy information in many circumstances

Contents lists available at ScienceDirect

Journal of Economic Dynamics & Control

Journal of Economic Dynamics & Control 39 (2014) 981120165-1889/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jedc.2013.11.006n Corresponding author.E-mail address: snorre.lindset@svt.ntnu.no (S. Lindset).length of the information delay between debtholders and equityholders, is of crucial importance for credit spreads.On a more general level, our paper addresses how incomplete information influences the pricing of bonds. We do notcredit risk and explains the existence of credit spreads.We present a theoretical model of credit spreads for corporate debt, where debtholders and eq

incomplete information about the financial state of the company. The information is incomplete because theissuer is only revealed with a time delay (delayed information). In addition, the information is asymmetricacases where debtholders and equityholders observe the true state with different time delays. The model iscontrast to the seminal structural models, it predicts that also short-term credit spreads can be wide, in linfindings.

A (rational) default policy describes when the equityholders (rationally) choose not to service contractuaWe find that the length of the information delay for equityholders is not important for the default poThe risk of monetary losses due to debt issuers who do not honor contractual debt payments is commonly referred to as31 October 2013Accepted 5 November 2013Available online 15 November 2013

JEL classification:G12G33

Keywords:Default policyCredit spreadsIncomplete information

1. Introductiononly a small impact on credit spreads. Asymmetric information, however, has a majorimpact on credit spreads. Our model predicts high credit spreads for short-term debt, asobserved empirically in credit markets.

& 2013 Elsevier B.V. All rights reserved.Received in revised form is incomplete because it is delayed for all agents, and it is asymmetrically distributedbetween debtholders and equityholders. We solve for the equityholders' optimal defaultCredit risk and asymmetric information:A simplified approach

Snorre Lindset a,n, Arne-Christian Lund b, Svein-Arne Persson c

a Norwegian University of Science and Technology, Department of Economics, Dragvoll NTNU, 7491 Trondheim, Norwayb Norwegian School of Economics, Department of Business and Management Science, Helleveien 30, 5045 Bergen, Norwayc Norwegian School of Economics, Department of Finance, Helleveien 30, 5045 Bergen, Norway

a r t i c l e i n f o a b s t r a c t

journal homepage: www.elsevier.com/locate/jedc

is more common than delayed information. However, our main focus in this paper is on which aspects of informationstructures that are important to obtain realistic models of credit spreads, and not on the realism of different informationstructures.

where all agents are subject to delayed information. We simplify the Duffie and Lando (2001) model by excluding noisy

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112 99(accounting) information, and extend it by explicitly exposing all agents to delayed information.The main merit of our paper is that we identify information asymmetry, and not delay, noise, or other characteristics of

the information, as the important property for a simple and realistic model of corporate credit spreads. This insight alsorefines the results by Choi (2008), who studies some aspects of delayed information (he uses the terminology lagged) in asimilar set-up.

Structural models were pioneered by Merton (1974). Merton models the value of a company's assets by a stochasticprocess and debt and equity are considered as contingent claims on total asset value. Some of the papers in this traditioninclude Black and Cox (1976), Geske (1977), Longstaff and Schwartz (1995), Leland (1994), and Duffie and Lando (2001).In our model the default policy is expressed by an endogenous default barrier. Giesecke (2006) analyzes two classes ofmodels of imperfect information: (1) Models where the bankruptcy barrier is not observable to all agents.1 (2) Models withincomplete information about the value of the company's assets. Our model belongs to his category (2), and according to hisProposition 6.4, a default intensity exists in our model. Default intensities are important for reduced-form models. Thesetypes of models were pioneered by Jarrow and Turnbull (1992), for extensions see e.g., Jarrow and Turnbull (1995), Jarrowet al. (1997), and Schnbucher (1998).2 Papers analyzing technical aspects about credit risk and incomplete informationinclude Coculescu et al. (2008) and Guo et al. (2009). Other issues related to credit risk are analyzed in Rosen and Saunders(2009), Huang and Yu (2010), and Azizpour et al. (2011).

Jarrow and Protter (2004) argue that the difference between structural and reduced form essentially is the assumption ofwhat information the modeler has access to. In their terminology, a model is structural if the modeler can observe the stateof the company, and reduced form if he cannot. They write (page 2): there appears to be no disagreement that the assetvalue process is unobservable by the marketAlthough not well understood in terms of its implications, this consensussupports the usage of reduced form models. Our results indicate that if different groups of agents have access to the sameincomplete information about the process, the error made by using a structural model compared to a reduced form model,interpreted as in Jarrow and Protter (2004), is negligible.

The paper is organized as follows: In Section 2 we present our economic model. Section 3 presents optimal default policyand credit risk valuation. Special cases with numerical examples are presented and analyzed in Section 4. Section 5concludes the paper and gives suggestions for future research.

2. Economic model

This section presents our model of a company with incomplete information about the credit quality of its debt. Becauseour focus is on default policy and debt valuation, we do not address whether debt is issued in an optimal way, i.e., whetherthe capital structure of the issuer is optimal or not. Our model is standard, and we follow closely the set-up by Leland (1994)and Duffie and Lando (2001).

Our model consists of two distinct groups of agents, equityholders and debtholders. In general, the two groups do nothave access to the same information and there is no information leakage between the two groups. Equityholders have atleast as much information as debtholders and constitute the group of better informed agents. The debtholders are the lessinformed agents. To rule out the possibility that debtholders extend their information set by buying equity, we (as Duffie andLando, 2001) assume that equity is not traded. Furthermore, we assume that equityholders do not buy debt from thecompany because debtholders could potentially extract information from such transactions. Also, it would alter the

1 See also Giesecke and Goldberg (2004) for more on this.2 Comprehensive treatments of these two approaches can be found in the encyclopedic monograph by Bielecki and Rutkowski (2002) or in the more

accessible monograph by Duffie and Singleton (2003).One assumption of our model is that equityholders are better informed than debtholders. This assumption is based onthe idea that equityholders are closer to the day-to-day operations of the company than the debtholders, and, thus, receiveinformation earlier than debtholders. Although we can visualize cases where this may not be the situation (e.g., a companymay have passive owners), we find it plausible that the best informed equityholder is better informed than the bestinformed debtholder. Debtholders must assess the value of the company based on less information than the equityholders,but they rationally include the observation whether the company is bankrupt or not in their assessment.

In the special case where debtholders and equityholders have complete information, i.e., there is no delay in the flow ofinformation, our model simplifies to the classical Leland (1994) model. The current paper is also inspired by, and closelyrelated to, the seminal Duffie and Lando (2001) model of credit risk. Our model includes continuously observed, but delayedinformation about the state variable, where the true state is immediately, i.e., without a delay, revealed upon bankruptcy.The model of Duffie and Lando (2001) includes noisy (accounting) information released at discrete points in time. Theyassume that equityholders have complete information and, thus, that only debtholders are subject to incomplete (noisy)information. Their model, as ours, includes incomplete and asymmetric information, but does not explicitly cover the case

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112100equityholders' optimization problem that we analyze below. The company is run by the equityholders, i.e., the model doesnot distinguish between owners and management. A decision to stop servicing debt and file for bankruptcy is endogenouslymade by equityholders. Furthermore, all agents in the economy are assumed risk neutral and therefore discount futurecashflows by the constant, continuously compounded risk free interest rate r.

We assume that the only state variable is the stock of assets. It is given as the solution to the stochastic differentialequation

dSt St dtsSt dBt ; S040; 1where or, s, and S0 are constants. Here, the process B fBtgtZ0 is a standard Brownian motion defined on a fixed, filteredprobability space ;F ; F0; P, where F0 fF tgtZ0. The process S fStgtZ0 is known as a geometric Brownian motion and Stis log-normally distributed. Also, P represents the objective probability measure. The information at time t is given by thes-algebra F t . Here F t is generated by the process fSu;0rurtg. Thus, F0 represents the complete information filtration andsatisfies the usual conditions.

The information available at time t for the two groups of agents are given by s-algebras Fmt and F lt . Superscripts m and lsignify more and less information. We denote the starting point in time of our economic analysis by t0. Define real numbers land m such that 0rmr lrt0, and the s-algebras by

Fmt F tm; for all tZt0and

F lt F t l; for all tZt0:Clearly, from this specification the s-algebras can be nested as

F ltDFmt DF t :Thus, in the case m40, also equityholders have incomplete information about the state variable, whereas in the case m0,equityholders have complete information. We interpret m as a measure of information delay and lm as a measure ofinformation asymmetry. Thus, information delay refers to the delay in the information available to the better informedagents, whereas information asymmetry refers to the difference in the information delay between the two groups of agents.The filtration Fmt0 fFmt gtZ t0 represents the development of available information for the equityholders.

An m-delayed stopping time with respect to Fmt0 is a function mt0 : -t0;1 such that fmt0rtgAF tm for all tZt0.

Similarly, in the case of m0, we let t0 be a classical (non-delayed) stopping time. Furthermore, we denote optimalstopping times by m

nt0 and the set off all m-delayed stopping times by T m. As usual, T denotes the set of all classical

stopping times. In our economic model we interpret the optimal stopping time as the default time, i.e., the time ofbankruptcy.

At any time debtholders observe whether a bankruptcy has taken place. Debtholders use this information rationally andformally we define the debtholders' extended information set as

Glt F lt3s1fmn t04ug;urt;where 1fg denotes the usual indicator function. The filtration Glt0 fGltgtZ t0 represents the development of availableinformation for the debtholders.

Consider first an unlevered firm, i.e., a company without debt. The stock of assets generates a continuous stream ofdividend payments to the equityholders. The rate at which dividends are paid at time t is Stm, for some constant 40.Observe that the dividend rate at time t is determined by the delayed value Stm. The time t value of the stock of assets isunobservable to the equityholders if m40. The interpretation of the term stock of assets must be done in a broad sense.In our model it represents the quantity which determines the dividend payments to equityholders. It is therefore anindicator of the cash generating ability of the company's assets. It could depend on a number of factors such as the technicalcondition of the company's production machinery, competence of employees, loyalty of customers, competitors, and marketconditions, as well as other non-financial factors relevant for the company's ability to pay dividends. The stock of assets isusually not readily observable. Equityholders must gather, process, and analyze information in order to assess its value. Thisassessment process takes time and, in practice, the stock of assets is observed with a delay, much in accordance withour model.

The time t present value of all future dividends, Vtm, is

Vmt EZ 1t

e ru tSum dujFmt

Stmr : 2

Observe that the present value Vtmis just a multiple of Stm. Either one of these quantities could therefore be used as the

state variable. The quantity Vtmis sometimes called the unlevered value of the company and in our setting it represents the

equityholders' assessed value of the stream of dividends.The rate of dividends paid at time t, Stm, is observed by debtholders first at time tlm, alternatively, at time t they

observe dividends paid at time tlm. If debtholders could observe dividends at the time they are paid, they couldcalculate the equityholders' assessed value of the company, and thereby eliminate the information asymmetry.

At the time of bankruptcy, mn, the value of the company is immediately revealed and publicly available. We define the

liquidation value of the company as

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112 101V mn V0m

n S

mn

r ;

i.e., as the unlevered value of the company given full information at the time of bankruptcy. This value does not take anyeffects of possibly optimally restructured debt at bankruptcy into account. Furthermore, at the time of bankruptcy, abankruptcy cost Sm

n=r, A 0;1, proportional to the liquidation value of the company occurs.

As in Leland (1994), we assume that the company has issued perpetual debt with face value D. The debt is serviced by aconstant rate of coupon payments C. These payments are tax deductible (only interest is paid on perpetual debt). The taxbenefit rate is C, where is the tax rate. For a levered firm, i.e., a company with debt, the time t net dividend rate to equityholders is Stm1C, and can take both positive and negative values.

3. Optimal default policy and credit risk valuation

3.1. Bankruptcy wild card

In the event of bankruptcy, the debtholders claim the amount identical to the face value of the debt D. According toabsolute priority, debtholders' claims have priority over equityholders' claims.

Due to the information delay, there is a positive probability that the liquidation value of the company is more thansufficient to cover debt and bankruptcy costs, i.e., the event

V mnV m

nD40

has positive probability. Any positive amount, in excess of debt and bankruptcy costs, is the property of the equityholders.By deciding to file for bankruptcy at time m

n, the equityholders obtain a bankruptcy wild card3 with payoff

1=rSmnD . This payoff is similar to the payoff of a European call option on 1=r units of the stock

of assets. It is straight forward to calculate its value using valuation theory for stock options.From standard properties of geometric Brownian motions, the complete-information value of the stock of assets at the

bankruptcy time mn, Sm

n, expressed as a function of the delayed value of the stock of assets observed by equityholders, Sm

nm,

is given by

Smn Sm

nme

1=2s2msBm Bmn

m 3

and is log-normally distributed. The time mnvalue of the bankruptcy wild card is expressed in closed form in Proposition 1.

Proposition 1. Given default at time mn, the time m

nvalue of the bankruptcy wild card is

Smnm

E 1r Smn D

Fmmn

1r e

mSmnmN z DN zs

m

p ; 4

where

zln

1Smnm

rD

1

2s2

m

sm

p

and N is the cumulative standard normal probability distribution function.

Proof. The result follows from the standard BlackScholesMerton formula for a European call option, but withoutdiscounting because the payoff is received instantaneously at the time of bankruptcy.

3.2. Optimal default policy

The decision to file for bankruptcy is taken by the equityholders. Equityholders maximize the value of the equity bydetermining when to default on the debt payments.

We now conjecture that the optimal stopping time is a first hitting time for a constant Wm, i.e., the company is bankruptand liquidated the first time Stm Wm. Observe that Wm is defined relative to the state variable, i.e., the stock of assets.

The information delay is illustrated in Fig. 1. The black line shows observed asset values given complete information. Thegrey line shows asset values as observed by equityholders. In particular, observe that the equityholders make the

3 This wild card has some resemblance to the wild card play that is present when trading the CBOT Treasury bond futures, see e.g., Hull (2012, p. 135).

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112102bankruptcy decision at time mnwhen they observe an asset value equal toWm. With no delay, the equityholders would have

defaulted earlier, i.e., at time n when the asset value equals W (assuming that W Wm).At any time tZt0 (assuming that the company is not bankrupt at time t) the equityholders face the optimal stopping

problem:

Stm supmtAT m

EZ mtt

e rv tSvm1C dve rmt tSmmjFmt" #

: 5

The first term inside the expectation operator in expression (5) is the discounted value of the dividends, net of after-taxcoupon payments. The second term is the present value of the bankruptcy wild card. There are three differences betweenthe optimization problem in expression (5) and the standard complete information optimization problem, see e.g., Duffie(2001), Chapter 11.C. The first is the inclusion of the bankruptcy wild card in the optimization problem. Second, the delayedstate variable Stm enters, and third, the optimization is based on less information (Fmt ) than the standard case withcomplete information (F t).

The optimal stopping problemwith delayed information can be transformed into an optimal stopping problemwith non-delayed information (see ksendal, 2005). The relationship between the optimal stopping time n from the non-delayedproblem and the optimal stopping time m

nfrom the delayed problem is given by m

n nm. We can then write

Stm suptmAT

E Z tm

tme rvtmSv1C dv:

e rtm tmSjF tm: 6

By substituting u tm in the problem (6) we obtain

Su supuAT

EZ uu

e rvuSv1C dve ruuSjFu

; 7

for uZt0m. We recognize expression (7) as a standard optimal stopping problem. To summarize, the equityholders'optimization problem with delayed information has been transformed into an equivalent optimization problem with non-delayed information. The latter problem can be solved using standard methods.

The solution to problem (7) satisfies the HamiltonJacobiBellman equation

ss12 s2s2ssrs 1 C 0; 8where s S , and subscripts denote partial derivatives, i.e., s=s, 2s=s2, together with the boundary

Fig. 1. Illustration of information delay. Black line shows observed asset values given complete information, and grey line shows observed asset valuesgiven delayed information.t s ss

conditions

Wm Wm 9and

sWm Wm; 10where is given in expression (4) and Wm denotes the derivative of with respect to the state variable, evaluated atthe point Wm. Eqs. (9) and (10) are known as the value matching and the high contact (or smooth pasting) conditions,respectively.

Before we introduce two technical conditions, we first define

x x

xC1

1r e

mN z 1 1C x

D1C N zsm

p ;

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112 103where

zln

1xrD

1

2s2

m

sm

p

and o0 is a constant defined below. Here, x is essentially a weighted sum of the value of the bankruptcy wild card fromexpression (4), , and its derivative.

Condition 1. Assume that 1emNzo j1j and that Wmo=r1=r.

Condition 2. Assume that 1emNz4 j1j and that Wm4=r1=r.Loosely speaking, Condition 1 is relevant for (relatively) small values of the delay m, whereas Condition 2 may be

relevant for larger values of m.In Proposition 2 we state the solution to problem (7).

Proposition 2. Assume that either Condition (1) or (2) is satisfied. Problem (7) has the following solution: the optimal stoppingtime is given by

nu infftZu : StrWmgwith respect to the filtration Fu. The value of equity is

s s

r Wm

rs

Wm

1 C

r1 s

Wm

Wm s

Wm

for sZWm;

0 for soWm;

8>: 11

where

12s2

1

2s2

22rs2

s

s2o0;

is the negative root of the quadratic equation

12 s

2 1 r 0:The constant default barrier Wm is implicitly given by the equation

Wm 1 C

r Wm

r 1 Wm ; 12

where the function is given in expression (4).

Proof. See Appendix B.

We remark that the connection between the optimal stopping time n from the non-delayed problem given inProposition 2 and the corresponding optimal delayed stopping time m

n, observable by the equityholders, is m

n nm.

As usual, the factor s=Wm is interpreted as the present value of 1 payable upon default, given current state s. The totalvalue of equity can therefore nicely be interpreted as the present value of dividends until default, from which the after taxpresent value of coupons until default is deducted, and the present value of the bankruptcy wild card is added. The onlydifference between the above solution and the classical solution is the addition of the bankruptcy wild card, whichcomplicates the solution so that, in general, no explicit solution for Wm is available. However, if there exists a solution toEq. (12), it is easy to find it numerically.

When m0, the bankruptcy wild card has no value and the expression for the default barrier W simplifies to

W 1

r

1Cr

: 13

3.3. Credit risk valuation

The only credit sensitive asset issued by the company is a loan. The debtholders belong to the less informed group ofagents and assess the fair terms of the loan. The loan is securitized into a continuum of zero-coupon bonds, where thecontinuum is with respect to the time to maturity. Duffie and Lando (2001) explain the connection between perpetual debtand a continuum of zero-coupon bonds with finite maturities and an example of such a connection is further elaborated in

Appendix C. Throughout the paper we analyze one such zero-coupon bond with fixed maturity. The bond matures at time T,with recovery function Rm

n; T in the case of default at time m

noT . The price of the bond consists of two parts:

1. The discounted value of the principal paid at maturity.2. The discounted value of the recovery payment in case of default.

ny observed defaults prior to, and including, time t.rocess instead of a jump-diffusion process. Few companies jump

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112104into default when their bonds are rated investment grade. However, it should be pointed out that many bonds jump several categories down to junk ratingfrom investment grade and default shortly thereafter. This happened to Enron who defaulted from a junk rating, but who was rated investment grade 2days before it defaulted.

6 In the special case considered by Leland (1994), W 1C=r0:5s2.4 By the definition of Glt , the survival probability includes information regarding a5 This observation also gives some justification for using a model with a diffusion pMore formally, the time t price of a bond maturing at time TZt is the conditional expected discounted payoff, i.e.,

t; T Ee rT t1fmnt4Tge rm tRm; T1fm

ntrTgjGlt

e rT tPmnt4T jGlt1; 14

where we interpret Pmnt4TjGlt as the survival probability4 of the company until time T. In the second equality, we have

used the recovery function Ru; T 1e rTu, uAt; T, i.e., the same recovery function as in Duffie and Lando (2001).This recovery function facilitates analytical solutions of bond prices and is therefore used throughout the paper.

For a credit risky zero-coupon bond, the credit spread l;m is defined as the excess yield compared to the yield on ariskfree zero-coupon bond. From expression (14) we have that

er l;mT t e rT tPm

nt4T jGlt1;

so

l;m lnPmnt4T jGlt1Tt : 15

Notice that the credit spread vanishes as -0 and tightens as the survival probability Pmnt4T jGlt increases.

4. Special cases and numerical examples

In this section we look at four special cases and provide some numerical examples. Condition 1 is satisfied in all theexamples. We start with the simplest case, i.e., the base-case with complete information. Here we have no informationasymmetry, lm 0, and no information delay, m0. This case serves as a benchmark case. We then look at the mostgeneral case with information asymmetry, lm40, and information delay, m40. We end this section by looking at thecases with (1) information asymmetry, but no information delay, lm40, m0, and (2) no information asymmetry, butinformation delay, lm 0, m40. The numerical examples in this section are based on the parameter values in Table 1. Forthese parameter values we are able to solve Eq. (12) for delays mo15:4 years.

In the 12 year period from 1995 to 2007, the average recovery rate for senior secured loans in the US was 72.3%, while thecorresponding recovery rate for high yield bonds was 42.2%, cf. Altman et al. (2004). Our model does not include anexogenously specified recovery rate parameter. Instead, the use of a bankruptcy cost parameter induces different(expected) recovery rates, depending on the numerical values of the other parameters that are used. A tax rate of 30% seemsreasonable for many companies. A volatility of 30% means that the instantaneous standard deviation of the log-returns ofthe stock of assets is 30% (with the base-case parameters, also the instantaneous standard deviation of the log-return of V is30%). Some of the parameters are altered in the numerical examples.

Collin-Dufresne et al. (2010) find that since 1937 only four companies have defaulted on bonds with an investment graderating from Moody's. This empirical observation suggests that realistic values of the information asymmetry lm is not toolarge. With a sufficiently large information asymmetry, even a company with bonds rated investment grade may have timeto move into default.5

4.1. The base-case (lm 0;m 0)

In this case there is neither information asymmetry, nor information delay, both debtholders and equityholders haveperfect information. Thus, Glt Fmt F t . If r and 0, this case corresponds to the model by Leland (1994). Also, withr4 and 40, this is a well-known case and serves well as a benchmark case when delayed information is included in thesubsequent subsections.

Equityholders are faced with the same optimal stopping problem as in expression (5), but the bankruptcy wild card is notpresent (it has zero value). The value matching and the high contact conditions therefore become W 0 and sW 0.The solution for W is given in expression (13).6 For the base-case parameter values, W65.

0.175

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112 1050.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

0.025

0.050

0.075

0.100

0.125

0.150

Time to maturity (in years)Table 1Base-case parameters.

St0 m 100 Initial value of stock of assets 0.035 Fraction of stock of assets paid as dividendr 0.08 Riskfree interest rate 0.045 Growth rate of stock of assetss 0.3 Volatility of stock of assets 0.3 Tax rate 0.5 Bankruptcy cost parameterC 13 Coupon paymentD 90 Face value of debt

SpreadBond prices are calculated by expression (14) with m l 0. Thus,

t; T Pnt4T jF t Tt; lnW=St;

where an analytical expression for ; is given in expression (A.1) in Appendix A.In Fig. 2 we plot the credit spreads for bonds with maturities between 0 and 3 years for three different levels of the tax

rate : 20%, 30%, and 40%. The widest credit spreads are required for the lowest tax rate and the tightest spreads for thehighest tax rate. The explanation for this observation is that, ceteris paribus, lending money is less risky when the tax rate ishigh because the value of the tax-shield from interest payments is worth more to equityholders and they will therefore waitlonger, i.e., accept lower dividend payments before they default on their loan payments. The survival probability isincreasing in the tax rate.

Observe how the credit spreads vanish as the time to maturity approaches zero, a typical property of structural models ofcredit risk, but it contradicts empirical observations in credit markets.

4.2. The general case (lm40;m40)

We now assume that both information asymmetry and information delay are present, i.e., both groups of agents haveincomplete information about the value of the state variable, and the information is asymmetrically distributed between thetwo groups. More formally, Glt Fmt F t .

The bankruptcy barrier is not a function of the current value of the state variable. Thus, even though equityholders arebetter informed than debtholders, also debtholders can calculate Wm. Denote the minimum value of the process S over aperiod u; v by Mu;v, i.e.,

Mu;v minfSt ;urtrvg:

The survival probability, as seen from the debtholders' point of view, is given by

Pmnt4T jGlt Pmn t4T jMt l;tm4Wm;F lt: 16

Fig. 2. Credit spreads base case with complete information. The figure shows credit spreads, expression (15), for zero-coupon bonds with up to 3 years tomaturity. The tax rates are 20% (widest spreads), 30%, and 40% (tightest spreads).

Using Baye's rule, expression (16) can be written as

P mnt 4TjGlt

P

mnt4T4T \ Mt l;tm4WmjF lt

PMt l;tm4WmjF lt

PMt l;Tm4WmjF lt

PMt l;tm4WmjF lt

Tmt l; lnWm=St l

lm; lnWm=St l; 17

where the expression for ; is given in expression (A.1) in Appendix A.Assume the same parameter values as in Table 1. In addition, l0.4. Fig. 3 shows the credit spreads for m0.1 (widest

spreads), m0.2, and m0.3 (tightest spreads with solid line, the dotted line represents the complete-information case(base-case)). For all cases, Wm65 (with at least 4 zeros after the decimal point). Note in particular how asymmetricinformation leads to wider credit spreads for short-term bonds.

It may at first seem counter intuitive that the spreads decrease as equityholders become less informed (m increases).However, recall that the degree of asymmetric information between debtholders and equityholders, lm, decreases as mincreases. Thus, the decrease in credit spreads is a result of a decreased degree of asymmetric information.

In Fig. 4 we check how sensitive the default barrier and the value of the bankruptcy wild card are to the delay. In the

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112106figure we vary the delay m from 0 to 4 years. Corporations usually announce financial results quarterly, so realistic values ofthe delay should probably be less than 1 year.

From Fig. 4, (a) and (b), we see that even a delay of 1 year has only a negligible effect on the optimal default barrier Wm.The graphs are increasing in the delay, but they are rather flat for reasonable values of the delays, in particular for 0:5,suggesting that the delay has only a low influence on the optimal default barrier. A higher volatility leads to a lower defaultbarrier, but volatility does not seem to be important for the slope of the graphs.

The bankruptcy wild card is essentially a call option and its value is therefore non-decreasing in volatility, cf. Fig. 4(c) and (d).The probability that the bankruptcy wild card matures in-the-money decreases in the bankruptcy cost parameter . The value ofthe bankruptcy wild card is therefore decreasing in . This fact also explains why the default barrier is more sensitive to thedelay when is low: the more economically lucrative the alternative to still run the company is, the earlier will theequityholders default on the debt payments.

4.3. DuffieLando (lm40;m 0)

In this case there is information asymmetry, but no information delay. Similar to the model by Duffie and Lando (2001),the information structure is as follows: Glt Fmt F t .

When no information delay is present, i.e.,m0, the bankruptcy wild card is not present and the default barrier isW, thesame as in the case of complete information (see Section 4.1). Thus, the survival probability is given in expression (17) withm0.

If we use the base case parameters and plot credit spreads for l0.1, l0.2, and l0.3, we get a figure identical to Fig. 3.The explanation is that the information asymmetry lm is the same as in the previous subsection, i.e., these values of l arethe same as the values of lm in previous subsection. In addition, the value of the bankruptcy wild card is not significant forthe base case parameters. We show later that in cases where the delay m is sufficiently long, the optimal default barrier andcredit spreads in the present case and in the general case in Section 4.2 differ.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

0.02

0.04

0.06

0.08

0.10

Spread

Time to maturity (in years)

Fig. 3. Credit spreads general case. The figure shows credit spreads, expression (15), for zero-coupon bonds with up to 3 years to maturity. The informationdelays m are 0.1 (widest spreads), 0.2, and 0.3 (tightest spreads) and l0.4. The lower, dotted line represents the complete-information case.

80 8080

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112 1070.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

10203040506070

m

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0(W m)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

10203040506070

m

10203040

506070

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 (Wm)90100 W

m

90100 W

m

90100 W4.4. Case of symmetrically delayed information (lm 0;m40)

The final case includes delayed information, but with no information asymmetry. The information is symmetricallydistributed between debtholders and equityholders, i.e., lm;m40.

Bonds are priced using the survival probability in expression (17). Because lm, the expression simplifies to

Pmnt4T jGlt Pmn t4T jFmt Tt; lnWm=Stm; 18

where the expression for ; is given in expression (A.1) in Appendix A. By comparing this expression with thecorresponding expression for the case of full information, the only way symmetric, but delayed information, can affect creditspreads is through a change in the default barrier Wm, i.e., if WmaW .

In Fig. 5, the credit spreads for the four cases are plotted for three different assumptions about the lengths of the delays.In parts (a)(d) the credit spreads for the cases complete information and symmetrically delayed information are notdistinguishable. The same is true for the cases DuffieLando and the general case. Comparing the plots (c) and (d) to theplots (a) and (b), we clearly see that a higher degree of asymmetric information leads to wider credit spreads. Also, bycomparing the plots (b) and (d) to the plots (a) and (c), it is clear that a lower bankruptcy cost parameter tightens creditspreads, cf. the definition of credit spreads in expression (15).

To visualize different credit spreads for the four cases, we must increase the information delay significantly (to m2 inthe example), see parts (e) and (f) in Fig. 5. The reason for different spreads in the different cases is that the delay m is solong that Wm is sufficiently different from W to also affect credit spreads. Note in particular in part (f) of the figure howcredit spreads in the symmetrically delayed case are tighter than the spreads in the DuffieLando case formo0:5 and widerfor m40:5.

5. Conclusions and suggestions for future research

In this paper we have proposed a new model for incorporating delayed and asymmetric information betweendebtholders and equityholders. We articulate the effect of delayed information on shareholders' endogenous decision to

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0m

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0m

Fig. 4. Optimal default barrier Wm and the value of the bankruptcy wild card Wm for different levels of information delay m and volatility s for twodifferent levels of the bankruptcy parameter . The s values range from 0.2 to 0.7 with intervals of 0.1. (a) The optimal default barrierWm as a function ofmfor 0:5. Lower graph is for higher volatility. (b) The optimal default barrierWm as a function of m for 0:3. Lower graph is for higher volatility. (c) Thevalue of the bankruptcy wild card Wm as a function ofm for 0:5. Higher graph is for higher volatility. (d) The value of the bankruptcy wild card Wmas a function of m for 0:3. Higher graph is for higher volatility.

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 981121080.02

0.04

0.06

0.08

0.10

0.12 Spread

0.02

0.04

0.06

0.08

0.10

0.12 Spreaddefault. In particular, for realistic parameter values we show that incomplete information to equityholders about the truestock of asset value only has a small effect on their decision to default on the loan payments. Any effect is likely to acceleratea default. The decision to default is accelerated because by defaulting, equityholders receive a bankruptcy wild card withnon-negative value. This wild card gives the equityholders a valuable alternative to continued operation of the company.If both debtholders and equityholders have access to the same delayed information, the only reason for changed creditspreads is a potential change in equityholders' optimal default policy, compared to the complete-information case. Forrealistic parameter values, these changes are small. Furthermore, we find that asymmetric information between debtholdersand equityholders is important for credit spreads, far more important than delayed, symmetrically distributed information.Increased information asymmetry leads to wider credit spreads. Our model produces short-term credit spreads more in linewith empirical observations than most standard structural models of credit risk.

The results in this paper have empirical testable implications: Do companies where there is likely to be more asymmetricinformation between debtholders and equityholders pay higher interest rates on their loans? Do companies where there is

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00Time to maturity (in years)

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00Time to maturity (in years)

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

0.02

0.04

0.06

0.08

0.10

0.12 Spread

Time to maturity (in years)0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

0.02

0.04

0.06

0.08

0.10

0.12 Spread

Time to maturity (in years)

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

0.02

0.04

0.06

0.08

0.10

0.12 Spread

Time to maturity (in years)0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

0.02

0.04

0.06

0.08

0.10

0.12 Spread

Time to maturity (in years)

Fig. 5. Examples of credit spreads for the four cases. In plots (a)(d), the widest spreads are for the two cases with asymmetric information. In plots (e) and(f) (for mo0:5), the widest spreads are for the general case, followed by the DuffieLando case, the symmetrically delayed information case, and thetightest spreads for the case with complete information. (a) No information delay, moderate information asymmetry, s 0:3, lm 0:2, m0, 0:5.(b) No information delay, moderate information asymmetry, s 0:3, lm 0:2, m0, 0:3. (c) Moderate information delay, moderate informationasymmetry, s 0:3, lm 0:4, m0.2, 0:5. (d) Moderate information delay, moderate information asymmetry, s 0:3, lm 0:4, m0.2, 0:3.(e) Long information delay, moderate information asymmetry, s 0:3, lm 0:2, m2, 0:5. (f) Long information delay, moderate informationasymmetry, s 0:3, lm 0:2, m2, 0:3.

Applying It's lemma to qt;St, writing s St , we get

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112 109dq t; s e rtt0 m ss12 s2s2ssrs1C dtsss dBt: B:1

The dt-term in the above expression is zero from the HamiltonJacobiBellman equation (8) for sZWm. We now showthat both conditions (1) and (2) imply that the dt-term of Eq. (B.1) is non-positive for soWm. In this case s ss 0.The dt-term of Eq. (B.1) simplifies to s1C. Now, s1Cr0 for all soWm if Wm1Cr0. InsertingWm fromexpression (12) and simplifying we get the sufficient condition

r 1C W

m 1r

Wm

r1: B:2more uncertainty about asset values, i.e., a higher degree of incomplete information, default earlier than other companies?One indication that may lead to a confirmative answer to the last question is if equityholders tend to receive payments fromthe bankruptcy wild card more often than equityholders of companies with a lower degree of incomplete information. Ourmodel also predicts that companies with high bankruptcy costs, for instance because of relatively illiquid assets, wait longerbefore they default. A typical reason for illiquid assets is a high degree of asset specificity.

For future extensions of the results in this paper, it would be interesting to include management as a third group ofagents. Management would always belong to the better informed group. With three groups of agents, we could for instanceassume that debtholders and equityholders both belong to the less informed group of agents. With this assumption, wecould extend the analysis to also include companies whose equity is traded in a financial market. Unfortunately, thisassumption makes the model harder to solve.

Acknowledgement

The authors would like to thank Fred Espen Benth, Carl Chiarella, Darrel Duffie, Hans Marius Eikseth, Steinar Ekern, ChrisFlorackis, Nadine Gatzert, Kay Giesecke, Jrgen Haug, Kristian Miltersen, Aksel Mjs, Jril Mland, yvind Norli, andPer stberg. In particular, thorough comments from anonymous referees have substantially improved the paper. Earlierversions have been presented at faculty seminars at Trondheim Business School, the Norwegian School of Economics,Princeton University, the University of Stavanger, Norwegian University of Science and Technology, Department ofEconomics, European Financial Management Association Annual Meeting in Athens 2008, European Group of Risk andInsurance Economists Meeting in Toulouse 2008, and at Workshop on Innovations in Stochastic Analysis and MathematicalFinance Norwegian School of Economics 2013.

Appendix A. Survival probability

Consider a geometric Brownian motion with dynamics as in expression (1) with initial value S0, and a barrier sboS0.Consider also the arithmetic process with dynamics dXt 12 s2

dts dBt , starting at X0 0. The first time the process S

hits sB is equivalent to the first time Xt hits x lnsB=S0. The probability for the process S of not crossing the barrier sb in atime period of length v is identical to the probability for the process X fXtgtZ0 of not crossing the barrier x lnsB=S0 in atime period of length v and is

v; x N xvs

v

p

e2x=s2N xvs

v

p

; A:1

where 12 s2, see e.g., Musiela and Rutkowski (1997, Corollary B.3.4) .

Appendix B. Proof of Proposition 2

To prove that the solution of expression (11) is optimal, we follow the approach in Duffie and Lando (2001).The function s in expression (11) (for sZWm), with Wm implicitly given in expression (12), satisfies Eq. (8) and the

boundary conditions (9) and (10).We now consider as a function of s, where s has dynamics given in expression (1). As Duffie and Lando (2001), we

apply It's lemma to s to getd s ss12 s2s2ss

dtsss dBt :

We define

qt;St e rtt0mStZ tt0m

e rvt0 mSv1C dv:

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112110Assume first that Wm=o j1j=r. Then (B.2) can be written as

r 1

r ZWm

W

m1C W

m ;where is defined in Section 3.2. Assume now that Wm=4 j1j=r. Then (B.2) can be written as

r 1

r rWm

W

m1C W

m :Either of these conditions, called conditions 1 and 2 in Section 3.2, is sufficient to ensure that the dt-term of expression (B.1)is non-positive.

To show that the dBt-term of expression (B.1) is a martingale, we show that Yt R tt0sStsSt dBt defines a martingale

with respect to the filtration Ft0m. From expression (11) we calculate sSt and find thatsStSt AStBSt ;

where the constants A =r and B Wm=r. Here, Yt is a martingale if

EZ Tt0

AStBSt 2 dt

A2EZ Tt0

S2t dt

2ABEZ Tt0

S1 t dt

B2EZ Tt0

S2t dt

o1: B:3

It is well known that ER Tt0 X2t dto1 for Xt log-normally distributed. Here, St,S1 t

q, and St are all log-normally distributed.

Hence, each of the three expectations on the right-hand side of expression (B.3) is well defined and Yt defines a martingalewith respect to the filtration Ft0m.

From the above arguments it follows that qt; is a super-martingale with respect to Ft0m, i.e.,qt0m; ZEqU; jF t0 m, for any stopping time UAT . Recall that T is the set of all Ft0m stopping times.

We calculate

EZ nt0mt0m

e rvt0 mSv1C dve rnt0m t0 mSnt0mjF t0m

St0mr

Wm

rSt0mWm

1 C

r1 St0m

Wm

Wm St0m

Wm

St0 m:

The first equality follows from the definition of nt0m in Proposition 2 and calculations.At any point in time u, the continuation value Su must be at least as high as the value of stopping at u, Su, so

SuZSu: B:4Finally, we have for any stopping time U that

St0m qt0m;St0mZEqU;SUjF t0 m

EZ Ut0m

e rvt0mSv1C dve rUt0mSUjF t0m

ZEZ Ut0 m

e rvt0mSv1C dve rUt0mSUjF t0m

:

The two equalities follow from the definitions of q; and St0m in expression (11). The first inequality is due to the factthat qt; is a super-martingale with respect to Ft0m. The second inequality follows from the inequality (B.4).

Thus, we have verified that the candidate solution (11) is optimal.

Appendix C. Connection between perpetual debt and zero-coupon bonds

In this appendix we assume complete information and show one example of a portfolio of a continuum of zero-couponbonds which has the same value as a perpetual debt contract.

Consider perpetual debt with coupon rate C to be paid until the company defaults. The company defaults on its debtpayments the first time the stock of assets in Eq. (1) hits the default barrierW from above. The time of default is given by thestopping time

n inftZ0

ft : StrWg;

defined with respect to F0 and where W is given in expression (13).

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112 111The parameter determines bankruptcy costs, i.e., bankruptcy costs are =rW . The value of the debt at time 0 canbe calculated as

EZ n0

Ce rs dse rn 1r W

C:1

with solution

Cr C

r 1

r W

S0W

;

where is given in Proposition 2, cf. Black and Cox (1976).Observe that we can also write expression (C.1) as

EZ n0

Cr 1r W

e rs ds

1

r W : C:2

The natural interpretation of expression (C.1) is the sum of the present value of coupon payments until default and thepresent value of the recovery amount 1=rW upon default.

Expression (C.2) suggests that the recovery amount 1=rW may instead be paid at time 0, but where interestpayment for this amount has to be deducted from the coupon C until default, without changing the overall value of the debt.

Our next step is to securitize the perpetual debt into a continuum of zero-coupon bonds. Let NT be the number of zero-coupon bonds maturing at time T, and let NT N for all TA 0;1. The time 0 value of a continuum of N zero-coupon bondswith expiration at time T and with general recovery function Rn; v if ov is

0N EZ n0

Ne rt dt

NE e rnZ 1n

Rn; t dt

:

The first term represents the present value of the zero-coupon bonds which expire before default. The second termrepresents the present value of the recovery amounts of the unexpired bonds upon default.

Consider now the particular recovery function used in this paper, Rn; v 1e rv n. In this case the aboveexpression simplifies to

0 N EZ n0

Ne rt dt

N 1 EZ 1n

e rt dt

EZ n0

Ne rt dt

N 1r

E e rn

EZ n0

Ne rt dt

N 1r

: C:3

Consider now a portfolio composed of two parts, a continuum of N Cr1=rW= zero-coupon bonds and atime 0 bank deposit (equivalently, the bank deposit can be zero-coupon bonds maturing at time 0) of1=rW1=C=r1=rW. The time 0 value of this portfolio follows from expression (C.3) and is

0 EZ n0

Cr 1r W

e rt dt

1

Cr 1

r W

1r W

1

Cr 1

r W

EZ n0

Cr 1r W

e rt dt

1

r W ;

which is identical to the time 0 value of perpetual debt from expression (C.2).

References

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Huang, S.J., Yu, J., 2010. Bayesian analysis of structural credit risk models with microstructure noises. J. Econ. Dyn. Control 34 (11), 22592272.Hull, J.C., 2012. Options, Futures and other Derivatives, 8th ed. Pearson Essex, CM20 2JE, England.Jarrow, R., Lando, D., Turnbull, S., 1997. A markov model for the term structure of credit risk spreads. Rev. Financ. Stud. 10 (2), 481523.Jarrow, R., Protter, P., 2004. Structural versus reduced form models: a new information based perspective. J. Invest. Manag. 2 (2), 110.Jarrow, R., Turnbull, S., 1992. Credit risk: drawing the analogy. Risk Mag. 5 (9), 5385.Jarrow, R., Turnbull, S., 1995. Pricing derivatives on financial securities subject to credit risk. J. Financ. 50 (1), 5385.Leland, H.E., 1994. Corporate debt value, bond coventnats, and optimal capital structure. J. Financ. 49 (4), 12131252.Longstaff, F.A., Schwartz, E., 1995. A simple approach to valuing risky debt. J. Financ. 50 (3), 789821.Merton, R.C., 1974. On the pricing of corporate debt: the risk structure of interest rates. J. Financ. 29 (2), 449470.Musiela, M., Rutkowski, M., 1997. Martingale Methods in Financial Modeling. Springer Verlag, Berlin Heidelberg.ksendal, B., 2005. Optimal stopping with delayed information. Stoch. Dyn. 5 (2), 271280.Rosen, D., Saunders, D., 2009. Analytical methods for hedging systematic credit risk with linear factor portfolios. J. Econ. Dyn. Control 33 (1), 3752.Schnbucher, P., 1998. Term structure modelling of defaultable bonds. Rev. Deriv. Res. 2 (23), 161192.

S. Lindset et al. / Journal of Economic Dynamics & Control 39 (2014) 98112112

Credit risk and asymmetric information: A simplified approachIntroductionEconomic modelOptimal default policy and credit risk valuationBankruptcy wild cardOptimal default policyCredit risk valuation

Special cases and numerical examplesThe base-case (lminusmequal0,mequal0)The general case (lminusmgt0,mgt0)DuffieLando (lminusmgt0,mequal0)Case of symmetrically delayed information (lminusmequal0,mgt0)

Conclusions and suggestions for future researchAcknowledgementSurvival probabilityProof of Proposition 2Connection between perpetual debt and zero-coupon bondsReferences