The Model

We start from the structural model of Merton (1974) and take the dynamics of the asset cash flows of the firm as a primitive.5 The main departure from the Merton (1974) model is the introduction of heterogeneous investors who disagree about the growth rate of future cash flows. Investors are identical in all other aspects, such as preferences, endowments, and risk aversion. The firm has a simple capital structure consisting of debt and equity, where debt is the sum of two defaultable zero bonds with different seniority and identical maturity. Since cash flows are stochastic, the firm may default on its obligations. It follows that in equilibrium the value of the debt depends on the probability that the firm will be able to generate enough cash flows to cover its liabilities. Let the assets’ cash flows A(t) follow the process:

dlogA(t) = µA(t)dt+σAdWA(t),

dµA(t) = (a0A+a1AµA(t))dt+σµAdWµA(t),

where µA(t) is the cash flow’s expected growth rate, σA>0 its volatility, a0A∈R the growth rate of expected cash flow growth, a1A < 0 its mean-reversion parameter and

σµA > 0 the volatility. The vector composed ofWA(t) and WµA(t) is assumed to be a

standard two–dimensional Brownian motion.

The cash flow process A(t) is observable by all investors in the economy, but the ex- pected growth rateµA(t) is unknown and must be estimated. The volatility parameter

σµA measures the objective uncertainty of the expected future growth of firm cash flows.

It is linked to the subjective degree of uncertainty among investors via their Bayesian updating rules and the observed realizations of A(t). Denote by mA(t) the estimated value of the true growth rateµA(t). In practice, estimating future earnings or cash flows is the goal of financial analysts. These forecasts inherently display some degree of sub- jective uncertainty about firms’ future earnings. Thus, agents may eventually disagree and have different values for mi_A(t), fori= 1,2. This apparently innocuous departure from the basic Merton model conveys some important implications. First, whereas in Merton’s model the value of assets can be taken as exogenous, in our model it depends on agents’ relative demands, which are functions of their subjective beliefs. This simple

5_{Merton’s (1974) model of credit risk assumes an exogenous firm value process with constant volatility.}

Even if we treat the firm value as endogenous, the predictions of our model for the case in which there is no disagreement are identical to those of Merton (1974).

feature makes the equilibrium value of the firm a function of the degree of uncertainty and belief disagreement in the economy. Second, due to market incompleteness, contin- gent claims on the firm value cannot be priced by standard replication arguments: A dynamic portfolio investing in the firm assets and the risk-free bond does not replicate the payoffs of equity and corporate bonds. Additional financial assets are needed to complete the market and their prices depend on equilibrium portfolio demands.

To reproduce a common business cycle component of belief disagreement, we assume that the expected growth of firm cash flows is linked to a market-wide risk factor related to the business cycle and the competitive landscape.6 _{We model this feature by a signal}

z(t) that analysts can use to improve the estimation of the cash-flow growth rate. The dynamics of z(t) follows a Gaussian process with a drift that is related to the drift of firm cash flows:

dz(t) = (αµA(t) +βµz(t))dt+σzdWz(t),

dµz(t) = (a0z+a1zµz(t))dt+σµzdWµz(t),

where σz > 0 is the volatility of the signal, a0z ∈ R the long-term growth rate of expected signal growth, a1z <0 the mean-reversion parameter and σµz >0 the volatil-

ity. (Wz(t), Wµz(t))0 is a standard two-dimensional Brownian motion independent of

(WA(t), WµA(t))0. Investors use observations of both A(t) and z(t) to make inferences

about µA(t). When β = 0, the expected change in z(t) is perfectly correlated with

µA(t) and z(t) contains pure information about the expected growth rate of firm’s cash flows. When β 6= 0, the expected change in z(t) is a mixture of µA(t) and the growth rate µz(t) of another systematic risk factor, which can be linked, e.g., to market–wide information about the state of the economy. In this way one can interpretµz(t) as the part of aggregate expected growth rate in the economy that is orthogonal to the firm

6_{Therefore, it is optimal for analysts to use information that goes beyond simple evidence on historical}

firm-specific cash flows (Buyd, Hu, and Jagannathan, 2005 and Beber and Brandt, 2006). Malmendier and Tate (2005) find that especially during the new economy boom in 2000 the accounting values of some companies were not very reliable. An imprecisely observed firm value is treated in Duffie and Lando (2001), who show that the quality of the firm’s information disclosure can affect the term structure of corporate bond yields. Yu (2005) finds empirically that firms with higher disclosure rankings tend to have lower credit spreads. An imperfectly observed firm value is also modeled by C¸ etin, Jarrow, Protter, and Yildirim (2004), who assume that investors can access only a coarsened subset of the manager’s information set. Giesecke (2004) develops a model with an imperfectly observed default boundary. Collin-Dufresne, Goldstein, and Helwege (2003) assume that firm values are observed with one time-lag. An industry implementation of these academic concepts is presented in CreditGradesTM_{, as described}

in the RiskMetrics (2002) technical document, which models the unobservable distance to default by a latent process explicitly.

specific expected growth rate µA(t). The volatility parameter σµz then measures the

aggregate uncertainty associated with this market–wide component.