Valuation of Capital Structure using Simulation

Techniques

Yevgeny Goncharov∗ and Yaacov Kopeliovich†

August 1, 2006

Valuation of Capital Structure using

Simulation Techniques

Abstract

We propose a simulation based technique for security valuation in anarbitrary capital structure. Extending Merton [M] and Leland [L] wemodel securities in capital structure as derivatives on the asset value of thecompany and we treat all payments made to these securities as a stochasticdividend process. Different seniority coupon paying bonds, warrants, andconvertible bonds are modeled as part of the capital structure. Dilutioneffects on warrants and convertible bonds value are considered. We useMonte Carlo with regression that gives flexibility and simplicity in theimplementation of different stochastic processes.

1 Introduction

In recent years Capital structure valuation and Capital structure arbitragegains an increasing attention among practitioners and Academics. Thisincreased interest has the following reasons:

• Scandalized bankruptcies that hurt many financial institutions( En-ron and MCI) and led to a search for better techniques to estimatethe probability of default.

• Evolution of trading techniques based on correlation between behav-ior of equity and debt instruments belonging to the same corporation

• Credit derivatives - raise the issue of estimating default probabilitiesfor different classes of securities issued by financial corporation.

∗Department of Mathematics, Florida State University, Tallahassee, FL 32306, phone:(850) 645-2481, fax: (850) 644-4053, e-mail: goncharov@math.fsu.edu,

†Meag New York, 540 Madison Avenue, New York, NY 10022

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Additionally Capital structure valuation increased in its importancedue to the wave of buyouts and are not related directly to trading tech-niques. Private Equity investors usually fund their activity of Buyoutstransactions by issuing bonds as well as Warrants with many embeddedoptions in them. These options in bonds usually include but not restrictedto the following:

• Different seniority - Bonds have different stature if the company goesbankrupt or liquidates. The division of Company property will beestablished in certain preference.

• Callability - There are provisions to call the bonds if their priceexceeds certain threshold price

• Puttability - The investor can sell the bonds to the company for acertain fixed price specified in the contract after a certain time.

• Convertibility - The investors have the right to convert the bondsat their disposal into shares based on a fixed ratio of shares α perbond.

• Warrants - The holder of the Warrant has the option to buy certainshares of the company β for a prescribed price K.

• Preferred Shares - The holder is entitled to dividend distributionduring the life of the company.

Private equity investors must report the value of these securities infinancial reporting and taxation purposes. Thus it is important to deter-mine a reasonable price for these securities. These prices should accountthe different options of the issued instruments as well as the more com-mon features like amortization schedule and interest bearing payments forthese bonds. Recently risk management issues of buyouts gain increasedattention and therefore it is important to estimate default probabilities inthis case as well.

One of the main theoretical approaches for Capital structure valuationis the Black Scholes Merton [M] ideas. This relies on the observation thatdifferent securities of the Capital structure can be modeled as options withcertain payoffs on the underlying company. For example let the capitalstructure consist of shares and zero coupon bonds expiring at time T fora total principle L. The equity is a call option on the underlying value ofthe company at time T with strike price L. Applying the Black Scholesformula we can find the value of the equity and the fair price of the zerocoupon bond. The probability of default can be inferred from this modelas well (see [ABZ] for recent application of these ideas).

While this approach is attractive its main setback is that it does nottake into account the real world features of securities issued. For exampleit does not take into account the existence of interest payments of bondsthat are made prior to the expiration of the option. This introduces dif-ferent payoff diagrams not only at its expiry date but also in intermediarytimes. To remedy this deficiency of the Merton model different bonds inthe capital structure are valued individually independent of the overallcapital structure. Assuming a model for the excess spread that is usuallybased on the bond rating (the difference between the bond interest and

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the corresponding treasury bond expiring at the same time T ) and aninterest rate model the bond is valued using the standard tools of interestrate modeling. Another example of this kind of model is the Derman [D]model to value convertible bonds. There the interplay with the companyequity comes into play. However the excess spread of the bond withoutthe convertibility feature is assumed to be given.

In this paper we extend Leland Merton ideas in order to accommodatethe real world features of securities issued on the market. We extend theset up in [Le] and [M] to value arbitrary capital structure with numericalsimulation.1 At the same time our approach is significantly different fromthe previous work on the capital structure. Let us cite an exert fromfootnote 4 in [LeTo]:

As do other studies, we assume that δ2 is not affected by changes inleverage; otherwise, investment would change with capital structure, whichin turn would raise questions which are beyond the scope of this article.In particular, this assumption rules out additional liquidation of assets(raising δ) to met ongoing debt service payment.

In this paper we drop “no liquidation of assets” assumption. In fact,we assume the opposite: All security payments are financed by selling theassets.3 Additionally, to the best of these authors knowledge, dilutioneffect was not considered up to now. A bond conversion, for example,would require issuing additional shares or selling the assets, thus, leadingto a dilution effect. This, in turn, affects all other securities in the capitalstructure. Another important point is that the exercise of an embeddedoption of a security changes the Capital structure. For example, bondconversion would result in the “exclusion” of such bond from the capitalstructure. In this case options might present a computational problem: insuch event we have to know solution of a (new) Capital structure problemwithout the “excluded” (converted) bond. A number of such “new” prob-lems grows exponentially with the number of securities with embeddedoption in the capital structure. We propose to overcome this complexityby computing the price of exercise at every time t for such options. Then,when the option is optimal to exercise, we assume that the security holderdoes not exercise his option in exchange of the Company payment of thistime-t exercise price.

We treat all the payments made by the Company as dividends. Conse-quently, we formulate the capital structure valuation problem as a “com-plex” derivative valuation on an underlying with a stochastic dividendprocess. This allows us to justify valuation of various securities withdifferent maturities. At the same time this makes incorporation of thedilution effect natural.

The valuation is done backwards from the future to the present. Ateach time step we use regression to compute conditional expectations (see

1Our approach is inspired from these papers as well as oral communication that Lelandhad with one of the authors.

2δ is a constant proportional cash flow generated by the assets and distributed to securitiesholders - G. and K.

3The framework in this paper is very flexible and “no liquidation assumption” can beincorporated as well. We, however, chose to illustrate the situation which was not directlyaddressed in the literature before.

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[Ca] and [TsVR]) and to determine the best possible way to divide theCompany value between different holders of the Capital structure. We dis-tribute the company value at each stage to different securities accordingto their seniority in the Capital Structure. The model values the CapitalStructure all at once based on the Company’s dynamic stochastic behav-ior. As a byproduct we determine the relative value of the security in thecapital structure to each other. Securities with a difficult options to value(like convertible bonds) are a special case of our general approach. Tothe best of these authors knowledge this is the first time in the literaturewhen these hybrid securities are treated uniformly as a part of a broaderCapital structure. Further this uniform treatment provides a possibilityto develop trading strategies and risk management techniques based onthe model for the entire debt and equity structure of companies.

The default and the distribution of recoveries in the presence of claimsfrom different seniority debts with different maturities is modeled with thehelp of a litigation/restructuring argument. In our model a default mightbe announced even if the Company has sufficient funds to meet its juniordebt obligations in the case if a “substantial” senior debt (with paymentsnot currently due) is present.4

The paper is organized as follows. In section 2 we describe the capitalstructure that we consider. We describe our general valuation algorithmin section 3 and illustrate it to the capital structure of section 2. Inparticular, we treat seniority and dilution in the case of bond conversionand warrant exercise. We consider a number of computational techniquesin this section as well. We show the results of the implementation of ourframework in section 5.

2 The Problem Set-up

We develop a framework that handles general capital structure valuation.This includes questions on how to accommodate different seniorities ofbonds and different expiry times (e.g., in the case when the maturity ofa junior bond is shorter than the maturity of a senior bond), Americanfeatures of options (such as convertibility, puttability, callability, etc.),dilution in case of warrant exercise and bond conversion and the interac-tion of option exercise of different securities. To illustrate our framework,we pick the following Capital structure in this paper, which includes allmentioned above features:

1) Senior bonds of value Bt each with the maturity TB . Total numberof bonds is nB . These bonds are scheduled to pay coupons bt and the facevalue bTB

.2) Convertible bonds of value Ct each with the maturity TC . Total

number of bonds is nC . These bonds are scheduled to pay coupons ct andthe face value cTC

. The bond holders have a right to convert their bondsto shares at the conversion ratio α. Since the conversion of these bondsinfluences the price of shares (the Company should provide shares to the

4The specification of default and recovery is not crucial for our algorithm and variousdefault specifications might be incorporated if desired.

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bond holders by printing new ones or giving from the existing), dilutioneffects should be taken into account.

3) The Company has written nW number of warrants of price Wt each.The holder of the warrant has an option to buy the Company’s shares forthe fixed price K at any time up to maturity time TW . As in the case ofthe bond conversion, the exercise of warrants lead to dilution.

4) The Company has nS number of common shares; each of the priceSt.

The asset price of the Company Vt is defined as the total price of all thesecurities in the Capital structure, i.e., Vt = nBBt+nCCt+nW Wt+nSSt.This means, in particular, that the share (equity) price might be written asSt(Vt) = (Vt −nBBt −nCCt −nW Wt)/nS , i.e., the other payoffs influencethe share price. This forces us to consider dilution effects.

In this paper we assume that there is no taxes, bankruptcy costs, andasymmetric information. We assume that the Company does not paydividends to the shareholders.5

3 Preliminary Considerations

The main idea in the Capital Structure Valuation is that we view all thesecurities in the Capital structure as derivatives on the asset Companyvalue Vt. We use derivative valuation theory to price these securities. Weemphasize two major points that follow from this approach:

1. We formulate the payoff of each security in the capital structure interms of the total Company value Vt. For example, the warrantpayoff at time t is expressed in terms of the share price as max(St −K, 0). In our framework this payoff is max(St(Vt) − K, 0), where Vt

is the asset company value at time t and St(Vt) must be defined.6

2. We view Vt as a “stock” on which all the derivatives are written. Anypayment to these derivatives (coupons, warrant exercise and bondconversion) is a “dividend” paid by the “stock” Vt to a hypotheticalsecurity holder. Those dividends are not known apriori (they dependon the current value of the company) and, thus, their process isstochastic.

According to [HP], a contingent claim price is a discounted expecta-tion of the payoff under the risk-neutral probability measure. This is ameasure under which the discounted Vt (viewed as a stock in the stan-dard derivative valuation literature) is a martingale if no “dividends” arepaid by Vt. In the presence of dividends (as in our case since the Com-pany makes payments to the security holders) the process V cannot beviewed as a tradeable (nobody would like to invest in such an asset with-

out receiving the dividend stream). In this case Vt = Vt+“present value

5The dividends might be included in our framework directly or by adding a dividend processas a fictive “junior bond.” In the latter case the share price would be the share price withoutdividends plus the price of this fictive junior bond.

6Note that finding St(Vt) is a part of the whole Capital Structure Valuation problem.

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of all dividends paid”7 can be regarded as a tradeable (see [VeNi] on themodeling a tradeable securities with dividends). Therefore, we can definethe risk-neutral probability measure as a measure under which the dis-counted Vt is a martingale (see [VeNi]). All the securities in the Capitalstructure should be computed under this martingale measure. In our set-up “dividends” (i.e., payments made by Vt) are payments the Companymakes to various security holders in the capital structure. That is we viewthe Capital Structure Valuation problem as a pricing problem of portfo-lio derivatives for a dividend paying stock. We emphasize that here the“dividends” are stochastic since, the payment of the dividends dependson derivatives’ payoffs.

We consider a “Bermudan” approximation of the options/paymentsunder consideration: We assume payments are made and options can beexercised at times t = n△t for n = 1, 2, ..., N only.8 Therefore, for everytime step “n to n − 1,” the options can be considered as European.

The instantaneous risk-free interest rate r is assumed to be constant.We assume the dynamic of Vt (i.e., the asset Company value plus accrued“dividend”/payoffs payments) under the risk-neutral probability to be aGeometrical Brownian motion9

dVt

Vt

= rdt + σdWt, (1)

so that the V e−∫

t

0rtdt is a martingale. Any extension to multi-factor

model will affect only the computation of expectations in subsection 3.2where a multi-dimensional regression will be applied. Addition of a jumpterm to the diffusion changes only dynamics of the Company value (i.e.,the way this value is simulated), but does not change our framework.

Let us look at what dynamics of the Company value Vt the SDE (1)implies. Let δt(Vt−) be an amount the Company pays at time t due toobligations on the securities from the capital structure (e.g., dividends toshare holders, bonds’ coupons, warrants if they are exercised, etc.) DefineVt− , to be the Company value just before distributing the payments tothe securities. Since there are no payment between (n−1)△t and n△t for

n = 1, ..., N , the dynamics of Vt coincides with Vt there (i.e., δt(Vt−) = 0if n△t < t < (n+1)△t). At times n△t, however, the Company value willbe dropped by the amount it pays: Vn△t+ = Vn△t− − δn△t(Vn△t−). Thisis equivalent to the way a binomial tree implements option valuation for adividend paying stock, when the dividends are known: the stock prices onthe tree jumps down on the nodes when the asset pays dividends. (value ofthe jump equals to the dividend amount, otherwise there is an arbitrage.)Therefore, if we know the “dividend” process δt, we know how to simulatethe total Company value under the risk-neutral probability measure.

7Often, this condition is formulated as follows in the case of known dividends: Vt−“presentvalue of all dividends to be paid.” This is equivalent (notice “minus” and “to be paid”, i.e.,in future, and “paid”, i.e., “were paid” in our statement).

8We assume constant time-steps △t for simplicity. The time grid might be adjusted toarbitrary payments schedule if necessary.

9We use this specification for the numerical computations below. In fact, it might be anyMarkovian process or any path-dependent process which is suited for derivative valuation withthe Least Square Monte Carlo method.

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4 Valuation

Let T = max(TB , TC , TW ) be the time horizon for our problem and let

0,△t, 2△t, ..., N△t be a discretization of time. We simulate Vt with anEuler scheme

V mn+1 = V m

n (1+r△t+σ√

△t emn ), n = 0, ..., N −1, m = 1, ..., M, (2)

where M is a number of paths simulated, emn are independent N·M normal

random variables.Valuation is done going backwards from time N to time 0. All values

at times n△t will mean anterior prices unless otherwise stated, i.e., pricesjust before n△t, before the Company makes any payment; that is Wn =Wn− . For posterior prices when the company made its payments, we usethe notation Bn+ , Cn+ , and Wn+ .

We illustrate the computation for a backward step “n to n−1.” Assumethat n△t < TB , TC and TW , i.e., all securities are alive. This is themost complicated case which illustrates dilution effects and interactionof convertible bonds and warrants. The cases in which some securitiesmatured or paid off like warrants or bonds are simpler and can be valuedsimilarly.

We proceed by backward induction: assume solution for time n△t+,i.e., for every given Company value Vn+ , the (posterior) prices of secu-rities Sn+(Vn+), Bn+(Vn+), Cn+(Vn+), and Wn+(Vn+) are found. Be-cause of limited liability the functions are zero for non-positive valuesof Vn+ ≤ 0. Our target is to determine Sn−1+(Vn−1+), Bn−1+(Vn−1+),Cn−1+(Vn−1+), and Wn−1+(Vn−1+). Then we solve the problem induc-tively from time N to 0. For initialization of the induction we notice thatafter time T all the securities matured thus, CN+(VN+) = 0, WN+(VN+) =0, and SN+(VN+) = VN+ .

Given the anterior securities’ prices, we use a linear regression to es-timate the posterior values of the securities (see [?]). That is the condi-tional expectations at time n△t+ are approximated by a linear regressionof simulated prices at time (n + 1)△t− with a set of K basis functions{φk(V m

n )}Kk=1. At every point n△t in time, we approximate security prices

for all values of V (not only at some specific nodes as in the case of a treevaluation) in the form

∑K

k=1 βnk φk(V ), where βn

k are constants. Therefore,the jump δn can be easily incorporated. Instead of simulating the processVt with jumps δn, we simulate Vt with the diffusion process (1) (i.e., withno jumps) and for every simulated path, at points where the jump is tobe incorporated, we associate a security price given the current Vn minusthe jump δn(Vn). For example for a warrant (if it is not to be exercised)we implement the jump of Vt as

Wn(Vn) = Wn+(Vn − δn(Vn)).

To find the “dividend” process δn itself, we need to know the payoffsof the securities. We use the payoffs to determine anterior prices at timen. We then use these prices to find the posterior securities prices at timen − 1 as European derivatives.

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4.1 Anterior vs posterior prices.

Here we express the anterior prices in terms of (known) posterior prices attime n conditioned on the anterior Company value V . At any time nodethe Company value is split between shares, bonds, and warrants:

V = nSSn(V ) + nBBn(V ) + nCCn(V ) + nW Wn(V ) (3)

The exercise of Warrants and/or convertible bonds lead to dilution, sincethe Company has to finance its obligations and this will affect the shareprice. Clearly, this effect is different for Warrant exercise and bond con-version. For different values of V we will have different action of thesecurities’ holders.

We consider all possible scenarios in turns. For every ith scenario weformulate an appropriate condition which defines the set Hi of V valueswhere this scenario takes place. The sets Hi must cover all possible valuesof the Company values, i.e.,

⋃H1 = (0,∞). To simplify notations, from

now on we assume (without further reminder) that the conditions V 6∈ Hj

for j < i are part of the condition Hi (this guarantees that the sets Hi

are disjoint). For every Hi we define the “dividend” function δ(i)n (V ).10

The general “dividend” process δn(V ) is defined as δ(i)n (V ) if V ∈ Hi.

H1: Default due to senior bonds. If V ≤ nBbn then the senior bondholders take what is left from the Company, i.e., Bn(V ) = V/nB . Theconvertible bonds, warrants, and shares are worthless. The “dividend”process in this case is the asset price of the Company, i.e., δ

(1)n (V ) = V .

Let us make one computational point here. Theoretically, for any Vwe have Bn(V ) ≤ V . However, the computational error in estimatingthe conditional expectations (regression in our case) might introduce anerror which breaks this inequality. This is undesirable since it might leadto negative stock prices, etc. To overcome the problem, we notice thatthe condition V ≤ nBbn is equivalent to V ≤ Bn+(V − nBbn) + nBbn.Indeed, if the former inequality holds, then the latter holds because weadd a non-negative value Bn+(V − nBbn) to the right-hand side. Andreverse, the latter inequality holds only if Bn+(V − nBbn) = 0 since oth-erwise it contradicts with the condition Bn(V ) ≤ V . But the price of thesenior bond is positive for any non-zero Company value V (since there isa probability of getting a non-zero payment). Therefore, we necessarilyhave V ≤ nBbn, which proves the reverse statement.

By using the equivalent condition V − nBbn ≤ Bn+(V − nBbn) wekeep the inequality Bn(V ) ≤ V and remove the problem of negative shareprices.

H2: Default due to convertible bonds. This is the exact case where theseniority must be taken into account.

First, let us consider one simple example which would help us to modeldefault in this case. Assume that the asset Company value is 51 and todaythe Company should make a payment of 50 to a junior bond. Assume also

10Since in our algorithm we need to use some of δ(i)n (V ) functions outside of Hi, functions

δ(i)n (V ) are assumed to be defined for all positive values of V .

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that the Company has a senior debt with a face value 50 which maturesin two years (no coupons). It is intuitively clear that in this case theCompany should go bankrupt. With another hand, if the Company gotsold for 51 and 50 is paid to the junior bond, then this might be viewed bythe senior bond holders as “stealing” of their money. With another hand,if in the same situation the Company value is 100, then the paymentof 50 to the junior bond hardly could be classified this way. How canwe formalize this situation and divide payments between the senior andjunior bonds?

Let nBbn + nCcn < V , i.e., formally, the Company has enough valueto make payments to both the senior and the junior (convertible in thepresent set-up) bonds (50 < 51 in the example above). Assume for sim-plicity the absence of payments to the other securities. If the Com-pany makes the bonds payments, the posterior value of the Companybecomes V − nBbn − nCcn and the posterior senior bond price becomesBn+(V − nBbn − nCcn). This is lower than Bn+(V − nBbn), which isthe senior bond price if there were no junior bond payments. That isthe presence of payments to other securities has an adverse effect on thesenior bonds. Therefore, if the senior bond holders had a right to calla default, it would be always optimal for them to exercise this right be-fore a payment to another security is made (i.e., formally, they chooseBn+(V − nBbn) over Bn+(V − nBbn − nCcn)). The bond holders do nothave voting privileges in the Company. However, they implicitly have thisright via the legal system. The bond holders might enforce the defaultthrough a court. The “court” argument is what divides the two situations(with the Company value of 50 or 100) in the example. In our model weassume that the senior bond holders can initiate the Company’s default ifthey have some “sufficient evidence” for such action. Let us formulate acondition for this case and find a recovery value of the senior bond priceexogenously.

After only the senior bond payments nBbn were paid by the Company,the value of the senior bond is Bn+(V − nBbn). The senior bond holdersmight argue that if after these payments the asset Company price minusthe senior bonds price is less than the junior bonds’ scheduled payment,that is in the case of

V − nBbn − nBBn+(V − nBbn) ≤ nCcn, (4)

the Company does not have sufficient funds to make the full paymentto the junior (convertible in our case) bond holders nCcn. Therefore,we assume that in this case the senior bond holders initiate default (say,through a court).

Now, what would be the recovery of the senior and junior bonds inthis case? If the senior bond holders will argue for a payment of morethan bn + Bn+(V − nBbn) each (e.g., all the scheduled payments todayin aggregate) then in this case the convertible bond holders can negotiatea restructure of the payments nCcn with the Company to a later dateleaving the senior bond holders with the “old” bond price Bn+(V −nBbn)(as if there is no junior coupon payment).

To summarize, if the condition (4) holds, then the anterior senior bondprice is Bn(V ) = Bn+(V −nBbn)+bn. The rest of the Company is shared

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between the convertible bonds, i.e., Cn(V ) = (V − nBbn − nBBn+(V −nBbn))/nC , and the other securities are worthless.

As in the previous case, the “dividend” process in this case (default)

is the asset price of the Company too, i.e., δ(2)n (V ) = V .

H3: No default, no conversion, no exercise. The Company makespayments of nBbn +nCcn to the bond holders (i.e., the Company value is

decreased by the amount δ(3)n (V ) = nBbn +nCcn). Therefore, the anterior

prices areBn(V ) = Bn+(V − nBbn − nCcn) + bn, (5)

Cn(V ) = Cn+(V − nBbn − nCcn) + cn, (6)

Wn(V ) = Wn+(V − nBbn − nCcn). (7)

The share price is Sn(V ) = (V − nBBn(V )− nCCn(V )− nW Wn(V ))/nS

and the set of V for this case is determined with the conditions αSn(V ) ≤Cn(V ) and Sn(V )−K ≤ Wn(V ). Using (6) and (7), these inequalities canbe formulated in terms of known posterior prices:

α

nS + αnC

(V −nBbn−nBBn+(V −nBbn−nCcn)−nW Wn+(V −nBbn−nCcn))

≤ Cn+(V − nBbn − nCcn) + cn

V −nBbn−nBBn+(V −nBbn−nCcn)−nCCn+(V −nBbn−nCcn)−nCcn

≤ (nW + nS)Wn+(V − nBbn − nCcn) + nSK

H4: Bond conversion, no warrant exercise. The anterior convertiblebond price in this case is Cn(V ) = αSn(V ), i.e., the bonds are convertedto shares. Using (3), the price of a share can be expressed as

Sn(V ) =V − nBBn(V ) − nW Wn(V )

nS + αnC

. (8)

The Company has to finance the conversion αSn(V ) and the senior bondpayments nBbn. Notice, however, we cannot use, e.g., Bn+(V −αSn(V )−nBbn) because Bn+ (as well as the other posterior prices) was computedconditioned on the “original” Capital structure which included presenceof the convertible bonds, but after the convertible bonds are exercised,there is no other obligations of convertible bonds payments for the Com-pany anymore. One solution would be to compute prices conditioned onthe Capital structure without the convertible bonds which would increasecomputational time. To overcome this complication, we take a differentapproach.

In the risk neutral world the convertible bond holders are indifferentif they exercise their bonds or if they receive “the option price of exerciseat this date” and do not exercise this option until the next time (n +1)△t. In the latter case the Company pays the option (conversion) pricenC(αSn(V )−Cn+(V −nBbn−nCcn)−cn) to the convertible bond holders(i.e., the difference of the payoff and the bond price if not converted) tobond holders and, in return, the bond holders do not convert their bonds,

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i.e., the Company pays additionally nCcn. Thus, totally the Companyvalue is decreased by

δ(4)n (V ) := αnCSn(V ) − nCCn+(V − δ(3)

n ) − nBbn. (9)

Thus, the anterior warrant price is

Wn(V ) = Wn+(V − δ(4)n (V )) (10)

and the anterior senior bond price is

Bn(V ) = bn + Bn+(V − δ(4)n (V )). (11)

As we see, we have a system of nonlinear equations for unknowns Bn,Wn, and Sn: Sn depends on Bn and Wn in (8) and Bn and Wn dependson Sn in (11), (10) via the definition of δn(V ) in (9). We solve this systemwith iterations. We pick an initial guess S0

n = Sn+ . Then B0n and W 0

n arecomputed with the help of (11) and (10). The next iteration S1

n is com-puted with the help of (8), etc. We proceed until maxV |Si

n(V )−Si−1n (V )|

is less then some specified tolerance. The convergence is fast and less thenthree iterations are needed in general.11 As the result, we get the share andwarrant prices in the case of the “forced” bond conversion and “forbidden”warrant exercise. The interval, where this action is financially optimal,is found with the help of the inequalities αSn(V ) > Cn+(V − δ

(3)n ) + cn

and Sn(V ) − K ≤ Wn(V ), where Sn(V ) and Wn(V ) are solutions of thesystem (8) and (10).

H5: No bond conversion, warrant exercise. The anterior warrant priceis Wn(V ) = Sn(V ) − K, i.e., the warrants are exercised. Taking this and(3) into account, the price of a share in this case is

Sn(V ) =V − nBBn(V ) − nCCn(V ) + nW K

nS + nW

. (12)

The Company has to finance the exercise nW (Sn(V )−K) and the bondpayments nBbn + nCcn. Notice, as in the previous case, we cannot use,e.g., Bn+(V − αSn(V ) − nBbn) because Bn+ (as well as the other poste-rior prices) was computed conditioned on the “original” Capital structurewhich included presence of the convertible bonds, but after the convert-ible bonds are exercised, there is no other obligations of convertible bondspayments for the Company anymore.

Similarly to the previous case, in the risk neutral world (in which weare valuating the Capital Structure), the warrant holders are indifferent ifthey exercise their warrants or if they receive “the option price of exerciseat this date” and do not exercise this option until the next time (n+1)△t.In the latter case the Company pays the option price nW (Sn(V ) − K) −nW Wn+(V − nBbn − nCcn) (i.e., the difference of the payoff and thewarrant price if not exercised) to warrant holders and, in return, thewarrant holders do not exercise their warrants. Thus, the asset Companyvalue is decreased by

δ(5)n (V ) := nW (Sn(V ) − K) − nW Wn+(V − δ(3)

n ) + nBbn + nCcn. (13)

11Acceleration techniques might be used if more iterations are required.

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Thus, the anterior convertible bond price is

Cn(V ) = cn + Cn+(V − δ(5)n (V )) (14)

and the anterior senior bond price is

Bn(V ) = bn + Bn+(V − δ(5)n (V )). (15)

Now we have a system of nonlinear equations (12), (14), and (15) forunknowns Sn, Cn, and Bn. We solve this system with iterations simi-larly to the previous case. We pick an initial guess S0

n = Sn+ . ThenB0

n and C0n are computed with the help of (15) and (14). The next it-

eration S1n is computed with the help of (12), etc. We proceed until

maxV |Sin(V ) − Si−1

n (V )| is less then some specified tolerance. The con-vergence is fast and less then three iterations are needed in general. Asthe result, we get the share and warrant prices in the case of the “forced”warrant exercise and “forbidden” bond conversion. The interval, wherethis action is financially optimal, is found with the help of the inequalitiesαSn(V ) ≤ Cn(V ) and Sn(V ) − K > Wn+(V − δ

(3)n ), where Sn(V ) and

Cn(V ) are solutions of the system (12) and (14).

H6: Bond conversion, warrant exercise. This is the case for all valuesof V which are not covered by the previous cases. In this case the shareprice is

Sn(V ) =V − nBBn(V ) + nW K

nS + nC + nW

. (16)

The convertible bond and the warrant price are given by Wn(V ) =Sn(V ) − K and Cn(V ) = αSn(V ). Theoretically, Bn(V ) should be in-fluenced by the convertible bonds and warrants payments. In particular(similarly to the cases 4 and 5) the senior bond Bn(V ) should be evaluatedconditioned on the absence of the warrants and the convertible bonds. Wewill deal with this situation in the following paragraphs (again, similarlyto the cases H4 and H5, where a non-linear system were solved with thehelp of iterations). However, first let us note, that this region (wherethe conversion and warrant exercise are profitable) implies high value ofthe share price and, consequently, low probability of default. Therefore,Bn(V ) ≈ Bn+(V ) (that is the bond is close to a “risk-free” bond) and wemight be able to simplify our computation by taking Bn(V ) := Bn+(V ).This approximation of Bn(V ) removes necessity of solving the system (16),(17) because the share price is expressed with known quantities through(16). The error incurred by using this simplification is negligible (lowerthen the numerical error of the procedure and possible error of model anddata specification) in our numerical examples below. This observationis reflected in the numerical examples with the fact that convergence isachieved in one iteration.

In a more realistic case when the Company has several seniority levelsand large number of bonds as well as other debt instruments, this sim-plification might not be adequate12 and we have to actually compute the

12In the next section we show how a presence of a junior bond with shorter maturity cansignificantly lower the price of a senior bond. The presence of several bonds might magnifythe effect which, in turn, might make Bn(V ) sensible to change of V even in the region ofwarrant exercise and bond conversion.

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change of the senior bond price due to the jump of the Company value inthe case of the conversion and the exercise.

As in the previous cases H4 and H5, we assume that the Companypays the option price of bond conversion and warrant exercise to “post-pone” the action to the next time. Let us compute these option prices ofbond conversion and warrant exercise at time n△t:

Cn(V ) = αSn(V ) − Cn+(V − δ(5)n (V )),

i.e., the difference of the conversion payoff and the price of the convertiblebond if not converted;

Wn(V ) = Sn(V ) − K − Wn+(V − δ(4)n (V )).

i.e., the difference of the exercise payoff and the price of the warrant ifnot exercised. The “dividend” payment in this case is δ

(6)n (V ) := nBbn +

nCCn(V ) + nW Wn(V ) and the anterior senior bond price is

Bn(V ) = Bn+(V − δ(6)n (V )). (17)

Here we have a system of non-linear equations (16) and (17) for Bn andSn. This system can be solved with iterations as above. The bond and thewarrant price are given by Wn(V ) = Sn(V ) − K and Cn(V ) = αSn(V ),where Sn is a solution of the system.

4.2 Computation

The posterior prices at time (n − 1)△t are expectations of discountedprices at time n△t because no action is taken in the time interval ((n −1)△t, n△t):

Bn−1+(V ) = e−r△tE[Bn(Vn)|Vn−1 = V ],

Cn−1+(V ) = e−r△tE[Cn(Vn)|Vn−1 = V ],

Wn−1+(V ) = e−r△tE[Wn(Vn)|Vn−1 = V ],

The share price is defined as

Sn−1+(V ) =V − nBBn−1+(V ) − nCCn−1+(V ) − nW Wn−1+(V )

nS

.

We approximate the expectations with a linear regression: given theCompany values V m

n−1 for different scenarios m = 1, ..., M and their evo-lutions V m

n to time n△t, we fit the linear combination∑K

k=1 βkφk(V mn−1)

to data Bn(V mn ), Cn(V m

n ) and Wn(V mn ). That is in the case of warrants,

for example, we find−→β W = (βW

1 , ..., βWK ) such that it minimizes the mean

square error, i.e.,

−→β W = arg min

−→β

∑

m

(Wn(V m

n ) −K∑

k=1

βkφk(V mn−1)

)2

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The discounted linear combinations of φk(V ) with the appropriate−→β

are taken as the posterior prices at time (n − 1)△t conditioned on theCompany value V :

Bn−1+(V ) := e−r△t

K∑

k=1

βBk φk(V ),

Cn−1+(V ) := e−r△t

K∑

k=1

βCk φk(V ),

Wn−1+(V ) := e−r△t

K∑

k=1

βWk φk(V ).

Now we can repeat the procedure for the other time steps until wearrive to the initial time.

5 Numerical Results

We consider the Company with one senior bond with the face value 50,one convertible bond with the face value 50 and the conversion ratio 0.5,and one warrant with the strike price 50. The interest rate is constant 5%,the volatility of the Company value is 20%, the bonds pay 5% coupons“continuously.”

For the capital structure evaluation in our framework all the payoffsshould be formulated in terms of V . Figure 1 shows what are these payoffs(anterior prices) at maturity (if maturities of all securities are the same)as functions of V . This graph might be split to the five intervals, whichcorrespond to the scenarios we considered: H1 on (0, 50), H2 on (50, 100),H3 on (100, 150), H5 on (150, 250), and H6 on (250,∞). The share pricefunction is a piecewise linear function. It is zero (i.e., default) on H1

and H2, then it grows with the slope one on H3, on H4 the grows fallsdue to the warrant dilution, and on H6 the grows falls even more due tothe warrant and convertible bond dilution. On this picture we can seethe effect of the dilution on the convertible bonds and the warrants. Forexample, the slope of the “in-the-money” warrant payoff is different forthe company value on the intervals (150, 250) and (250,∞). The decreaseof the slope on (250,∞) is due to the convertible bond dilution.

Prices of the securities 5 years before the maturity given on the Fig.2. Notice, that the share price is not a convex function of the Companyvalue V as it would be if the dilution would not be taken into account.

Now, consider the same set-up with the only difference that the con-vertible bond matures in 2.5 years earlier than the other securities. Thenthe prices/payoffs of the securities at the maturity of the convertible bond(i.e., 2.5 years before maturity of the other securities) are given in Fig. 3.

Notice the jump in the senior and share price. The region to the leftof the jump corresponds to default induced by the senior bond holder. Inthis case the senior bond holder collects the present value of his/her bond,the rest goes to the convertible (junior) bond. Seniority is accounted forin this case by allowing the senior bond holder to induce the default even

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0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

160

Figure 1: Payoffs of the convertible bond (blue line), senior bond (green line),warrant (red line), and share price (black line) as functions of the Companyvalue.

though the company has “enough” funds to make junior bond payment(e.g., if the Company asset value was 55, it formally would be able topay 50 to the junior bond). Notice, that in the case of default the seniorbond does not collect the whole face value (or the whole Company if itsvalue is less). If the senior bond holder would insist on this outcome, thenthe junior bond holder would restructure his bond in such a way that theprospective payments were not in the conflict with the senior payments(e.g., “moving” the face value payment to or beyond the maturity of thesenior bond). In this case the price of the senior bond would be givenexactly by the left-side of the green line (i.e., as in the absence of thejunior bond).

If, however, there was no default (e.g., the Company value is 101),then the Company makes a full payment (of the face value 50) to theconvertible bond and the asset Company value loses this amount. Theprobability of default in future in this case becomes greater than in thecase of the absence of this payment (e.g., the part of the green line to theleft of the jump) and, consequently, the price of the senior bond is less.Prices of the securities 2.5 years before the maturity of the convertiblebond (i.e., 5 years before the maturity of the other securities) in this caseare given on the Fig. 4.

Let us illustrate the effect of the presence of a junior bond on thesenior bond price. Assume that the Company is financed with the seniorand junior bond only. Assume that the senior bond has the face valueof 50, makes 5% coupon payments continuously, and matures in 5 years.Now we compare the price of the senior bond in the following cases:

1. The junior bond is scheduled to pay 15 units in one, two, and threeyears only.

2. The junior bond’s maturity is 5 years (or more).

Notice that in the second case we did not specify the face value. This is

15

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

160

Figure 2: Prices of the convertible bond (blue line), senior bond (green line),warrant (red line), and share price (black line) as functions of the Companyvalue.

to underly that in this case the junior bond does not have influence onthe senior bond price. On Fig. 5 we compare the senior bond prices inthis two cases.

References

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[Ca] Carriere, J., Valuation of Early-Exercise Price of Options Using Sim-ulations and Nonparametric Regression, Insurance: Mathematics andEconomic, 19, pp. 19–30, 1996.

[D] Derman E. Valuing Convertible bonds as derivatives, November 1994,http://www.ederman.com/new/index.html

[Ha] Haug E. The Complete Guide to Option Pricing Formulas McgrawHill

[Hu] Hull J. Options Futures and Other Derivatives Prentice Hall UpperSaddle River NJ

[M] Merton R. On the Pricing of Corporate Debt: The Risk Structure ofInterest Rates Journal of Finance (June 1974) pp. 449-470.

[Le] Leland H. Corporate Debt Value, Bond Covenants and Optimal Cap-ital Structure Journal of Finance (September 1994) pp. 1213-1243

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0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

160

Figure 3: Prices of the convertible bond (blue line), senior bond (green line),warrant (red line), and share price (black line) as functions of the Companyvalue at the maturity of the convertible bond.

[LeTo] Leland H., Toft K. Optimal Capital Structure, EndegoneousBankruptct and the Term Structure of Credit Spreads Journal of Fi-nance (July 1996) pp. 987-1019

[HP] Harrison, J. M. and S.R. Pliska: “Martingales and Stochastic In-tegrals in the Theory of Continuous Trading.” Stochastic Processesand their Applications 11, pp. 215–260, 1981.

[TsVR] Tsitsiklis, J., and Van Roy, B., Optimal Stopping of Markov Pro-cess: Hilbert Space Theory, Approximation Algorithms, and an Ap-plication to Pricing High-Dimensional Financial Derivatives, IEEETransactions on Automatic Control, 44(10), pp. 1840–1851, 1999.

[VeNi] Vellekoop, M. and H. Nieuwenhuis: ”Modeling of Tradeable Se-curities with Dividends.” Working Paper, University of Twente, TheNetherlands, 2006.

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0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

Figure 4: Prices of the convertible bond (blue line), senior bond (green line),warrant (red line), and share price as functions of the Company value.

0 50 100 150 200 2500

10

20

30

40

50

60

70

80

90

100

Figure 5: Prices of the senior bond if 1) the junior bond has payments before thesenior maturity (blue line); 2) the junior bond is scheduled to pay on or afterthe senior maturity (red line). The dashed line shows the price of a risklessbond of the same as senior terms.

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