Modeling the value of the firm is one of the more important research topics in finance. The value of an unlevered firm is the value of expected future cash flows discounted at a rate appropriate for an all-equity firm whereas the value of a levered firm is commonly expressed as the value of an unlevered firm plus the gain from leverage due to a tax shield provided by the debt. Including business disruption costs, the optimal capital structure can then be characterized as a trade-off between the interest tax shield and disruption costs. Recent analysis by Hackbarth, Hennessy and Leland [1] extends this line of research by examining an optimal mixture of debt; that is, the optimal mixture of bank debt and market debt (bonds).

Improved models for value of the firm are potentially useful in several contexts. For one example, consider models of credit spreads. Leland and Toft [2] develop an ambitious model of firm value that addresses optimal capital structure, optimal debt maturity, and the term structure of credit spreads. Recently, Qi [3] has modified the Leland and Toft [2] model by setting the lower bankruptcy boundary to be a fraction of bond face value. The importance of good structural models for credit spreads has been enhanced with the growth of credit derivatives and the credit crisis of 2007 and 2008. More specifically, notional amounts of credit derivatives grew by over 100% for every year from 2004 through 2006. At the end of 2006, there was 34.5 trillion outstanding (see Saha-Bubna and Barrett [4]). The weakened credit quality of many financial firms in 2007 and 2008 caused high volatility in equity markets and, also, large changes in the value of credit spreads and credit default swaps.

Our purpose is to derive distributions of whose evolution critically depends on the models for the short term interest rate process,. Models for can be broadly classified as (a) general linear and (b) non linear models. General linear models are also popularly known as affine models. In this context, we refer to Duffie, Filipovic and Schachermayer [5]; Duffie and Singleton [6]; and Lamberton and Lapeyre [7]. In this paper we are particularly interested in a special class of the general linear models called narrow sense linear models described by Arnold [8].

The next section describes the processes for value of the firm and short term interest rates. Next, we discuss a general framework for the solution of the distribution of. Then, we describe solutions in the cases where processes are narrow sense linear. Such processes are quite popular for models of credit risk. The shapes of the distributions are shown to be sensitive to such parameters as the correlation between the and processes. For example, a positive correlation displays a distribution with fatter tails than one with negative correlation.

The time evolution of the value, , of a firm is routinely modeled under the risk neutral measure by a linear, scalar, stochastic differential equation (SDE)

where the instantaneous drift denotes the (exogenous) stochastic variable known as the short-term interest rate process and is a deterministic function representing the instantaneous volatility as in Acharya and Carpenter [9].

This general form of value process has been used in numerous important structural models of credit risk. For example, see Merton [10] and Acharya and Carpenter [9] where any dividends and coupon payments, outflows of γ from the firm to investors, are subtracted from the r_t drift term. Many firms do not pay dividends and our model is one of zero coupon debt so that a γ of zero is reasonable. We note that Longstaff and Schwartz [11] similarly have a zero γ.

The drift of indicates our model is risk neutral. We could assume different firms have different drifts due to such things as different expected returns in their industry as well as different riskiness of assets and future projects. However, such an assumption is arbitrary and yields a model that is not arbitrage free. We believe it is much more theoretically credible to posit a risk neutral, arbitrage free model.

The dynamics of the short term interest rate are modeled, under the same risk neutral measure, by a (scalar) SDE of the type

where the instantaneous drift, and the volatility, are smooth functions. It is further assumed that the Wiener increment processes and are correlated; that is,

with. It is worth noting that in this set up the flow of information is only one way –affects and not vice versa. By combining several well known results from the literature, in this paper we characterize the distribution of the value process for different choices of the processes.

All the known stochastic interest rate models can be broadly classified into two classes—single factor models (SFMs) and multi-factor models (MFMs). We refer to Cairns [12] and Privault [13] for details. In this paper, we are primarily interested in the SFMs. These SFMs can be divided into linear and nonlinear models. Following Arnold [8], linear models can be further subdivided into two subclasses. The SFM in (1.2) is called a narrow sense linear model if

and

A general linear model, on the other hand, has in the form (1.4) and

where and are smooth functions of time. The general linear models are also known as affine models as in Duffie, Filipovic, and Schachermayer [5]; Duffie and Singleton [6]; and Lamberton and Lapeyre [7]. The SFM in (1.2) is called a nonlinear model if either and/or are nonlinear functions of the short rate.

Refer to Tables 1(a)-(c) for examples of these models. The narrow sense linear models of Merton [14], Vasicek [15], Ho and Lee [16], and Hull and White [17] are special cases of the Heath, Jarrow and Morton [18] model and define normal/Gaussian processes.

We first solve the scalar SDE (1.2) for, and using it in (1.1), we then recover. It is well known that is a lognormal process when a constant. See Kloeden and Platen [19]. We extend this result by first showing that also inherits this lognormal distribution where is a normal process defined by the narrow sense linear models in Table 1.

This problem of quantifying the probability distribution of is critical to credit risk analysis. For a review of various approaches to credit risk refer to the books by Duffie and Singleton [6], Bielecki and Rutkowski [20], and Jarrow et al. [21]. Clearly computation of the default probability in structural models requires knowledge of the probability distribution of contingent on the chosen model for the interest rate.

In this section we develop a framework for solving (1.1)- (1.2). Setting and applying Ito’s lemma, equation (1.1) becomes.

See Kloeden and Platen [19].

Setting and

Table 1. Alternative Models of. The single factor model in (1.2) is called a narrow sense linear model if and. In contrast, the model is called an affine model or a general linear model if is of the above form and. The model is called nonlinear if either and/or are nonlinear functions of the short rate. (a) Narrow sense linear models; (b) General linear or affine models; (c) Nonlinear models.

(Shreve [22]) we can rewrite the pair of equations (1.2) and (2.1) as

and

where and are two independent Wiener increment processes and

Integrating (2.3), we obtain

where

and

From the properties of the Ito integral, Mikosch [30] and Shreve [22], it follows that

where

Since and are independent Wiener processes, it readily follows that . Thus, the distribution of and hence of critically depend on the properties of the process in (2.2).

In closing this section consider the special case when, a constant. Then,

Further, when, we obtain

4. Narrow Sense Linear Models for r_t

Setting

in (2.2), we get a narrow sense (time varying) linear model known as the generalized Hull and White [17] model given by

Since all the other narrow sense linear models in Table 1 are special cases of (3.2), we first concentrate on solving (3.2). Defining

we get

This is known as the fundamental solution of (3.2). Hence the solution of (3.2) is given by (Arnold [8], Gard [31], Kuo [32], Lamberton and Lapeyre [7])

where

and

Hence,

where

Now combining (3.5)-(3.10), it follows that

Applying integration by parts to the second integral on the right hand side of (3.11) and using (3.7), it follows that

Is also a Gaussian process with mean zero and variance given by

Substituting (3.11)-(3.12) in (2.5), we get

where

and

Substituting (2.6), (3.12) and (2.7) in (3.16), the latter becomes

Since and are independent, it follows that where

where

and

.

Combining (3.15)-(3.18) with (3.14), we finally obtain

where given by (3.16) and given by (3.19).

We summarize the above developments in the following:

Theorem 3.1: Let the interest rate evolve according to a narrow sense linear, scalar, SDE of the type (3.2). Then, is a Gaussian process and consequently in (2.5) is also a Gaussian process given by (3.20).

Since, from (3.14)-(3.20), we get

where

and

The following corollary is immediate.

Corollary 3.2: Since,

is a lognormal process whose probability density function, as a function of time, is given by

It can be verified (Johnson et al. [33]) that the time evolution of the mean and variance of the value process

are given by

and

We now enlist a number of nested corollaries by considering special cases of interest rate models.

Case 1: Let, a constant. Then

, and is given by

(3.8). From (3.15) and (3.20), the mean is

where is given in (3.6). From (3.19)-(3.20), the variance is

Case 2: Hull and White [17] model: In this model, and where and are constants. Thus, , and

Hence the mean is

and the variance is

Case 3: Ho-Lee [16] model: In this model, and. Then

The mean is

and the variance is

Case 4: Vasicek [15] model: In this model, , and. Then, and

Hence, the mean is

and the variance is

Case 5: Merton [14]: In this case, , and. Then and

Hence, the mean is

and the variance is

Case 6: Let, a constant and. Then, and. The mean

and the variance

We now provide sample plots of the distribution, when follows the Vasicek [15] model, for three different values of the correlation (, 0.9 and −0.9) in Figures 1-3 respectively. In each case the distribution of for and 20 are given. From these figures it follows that as increases both the mean and variance of increases. Further, comparing Figures 1 and 2, it follows that the effect of the positive correlation () is to reduce the peak while making the tails fatter compared to the case when. Similarly from Figures 1 and 3, we readily see the negative correlation has the opposite effect of increased peak and thinner tails compared to.

The primary motivation for characterizing the distribution of is to compute the probability of default. Within the framework of structural models, there has been an evolution of the definition of default. In the now classic paper, Merton [10] defines default as the event bond with maturity. Using the results described above, we could readily compute the probability default according to this classical definition1.

However, Longstaff and Schwartz [11] define default by the event. Recently, Giesecke [34] has expanded on this theme and has defined the default by the compound event for.

Recall that the probability of these later events can be readily calculated using the “reflection principle” if is a standard Wiener process or by using the Girsanov theorem if is a Wiener process with a drift. (See Elliott and Kopp [35] and Giesecke [34]). To enable computation of default probability according to Giesecke [34], in the following, we seek to express in (3.14) as the sum of a drift term and a (time changed) Wiener process.

To this end recall that every Ito integral is equivalent to a time changed Wiener process. (See Shiryaev [36], Oksendal [37], Karatzas and Shreve [25]). Accordingly, from (3.17) we obtain

where and are two independent Wiener process with

,

and and are given in (3.19). Since and are independent, there exists a Wiener process such that

where

as given by (3.18)-(3.19).

Combining (3.43) with (3.14), it follows that

where as in (3.20).

5. Conclusions

We have analyzed the impact of on when evolves according to a narrow sense linear model in Table 1(a). Consider the case when evolves according to a general linear model, such as for example, the Brennan-Schwartz [38] model in Table 1(b). In this case the explicit form of the solution for is well known (Arnold [8], Gard [15], Lamberton and Lapeyre [7]) and is given by

where the process is given by

with. Hence,

involves a process

which is an integral of the exponential functionals of the Wiener process. Processes of the type routinely arise in the evaluation of Asian type options (Vorst [39]). By relating to a Bessel process, Yor [40] and Geman and Yor [41] have provided a complete characterization of the distribution of the process. Combining these results with (2.5) to derive the distribution of is an interesting open problem. Similarly, computing the distribution when evolves according to the nonlinear models is Table 1(c) is also wide open. Solutions to these problem will shed further light on the impact of the choice of interest rate models on default probability and hence on credit risk analysis.

6. Acknowledgements

We are grateful to Robert J. Elliott (University of Calgary) and to Luciano Campi (Universite Paris Dauphine) for their interest and comments that improved the presentation.

REFERENCES