UNIENDO LAS PIEZAS DEL ROMPECABEZAS

A firm’s term structure of credit spreads is the set of credit spreads, observed jointly at time-t, for bonds issued by the same firm with different remaining maturities. The Merton model predicts that a bond’s credit spread at time-t, is a function of its term to maturity, T₋t, the log-solvency ratio of the issuing firm, x(t), the firm’s asset return volatility, σv, the risk-free rate, r, and the net payout to other claimants, δ. and By varying the remaining maturity, holding all other parameters constant, the Merton model predicts a theoretical term structure for the firm. In Figure 2.1(a), we plot the predicted

0 20 40 60 80 100 120 140 160 180 200 0 5 10 15 20 25 30

Credit Spread (basis points)

Term to Maturity (years) x(t)=1.44 x(t)=0.775 x(t)=0.367

(a) Credit spreads

0 10 20 30 40 50 60 0 5 10 15 20 25 30 Default Probability (%)

Term to Maturity (years) x(t)=1.44

x(t)=0.775 x(t)=0.367

(b) Risk-neutral default probability

40 50 60 70 80 90 100 0 5 10 15 20 25 30

Expected Recovery Rate (%)

Term to Maturity (years) x(t)=1.44 x(t)=0.775 x(t)=0.367

(c) Risk-neutral expected recovery rate

Figure 2.1: Shown in figure (a) is the credit spread term structure predicted by the Merton model at different initial levels of log-solvency, x(t) =lnV(t)/K. The risk-neutral default probability is shown in (b), and the risk-neutral expected recovery rate is shown in (c). Initial log-solvency levels of 1.44 (low risk), 0.775 (medium risk), and 0.367 (high risk), are equal to the sample quartile levels of the log of the market solvency ratio. The market solvency ratio is calculated from CRSP and COMPUSTAT data as the sum of the firm equity value and book debt over book debt. Other parameters areσv= 25 percent,r=6 percent, andδ=5

percent.

credit spread term structures, assuming different levels of initial solvency, for a repre- sentative firm. Figure 2.1(b) shows the implied risk-neutral probabilities of default, and Figure 2.1(c) shows the risk-neutral expected recovery rates, corresponding to the model

parameters and solvency levels. The initial level of unobserved log-solvency,x(t), is set equal to the quartiles of our sample of observed log-solvency ratios, measured across trade dates and firms. For illustration only, we use the observed log-solvency ratio as a proxy for the unobserved latent firm solvency,x(t). Table 3.5 reports the market credit spreads observed in our sample, pooled across time and firms, by quartiles of the market log-solvency ratio.

The observed log-solvency ratio is calculated as the firm’s market value of equity plus book value of debt, divided by the book value of debt. Computation of the observed solvency ratio is explained further in Section 3.2.6 where we use it to drive our initial firm-specific estimate of the latent log-solvency ratio, and to derive a firm-specific initial estimate of asset volatility. Other parameters are chosen for illustration.

As shown in Figure 2.1, the shape of the term structure of credit spreads is a function of the term structures of the default probability, expected recovery rates, and the time value of money. For the safest firms, at the upper quartile of solvency atx(t)=1.44, credit spreads are low and increase monotonically with maturity from zero at time-t to approximately 50 basis points at(T₋t)=30 years. The default probability is the lowest for the safest firms and rises approximately linearly with respect to maturity from zero at time-0, to approximately 25 percent at(T₋t)=30 years . The expected recovery rate for the safest firms is the highest, beginning at 100 percent at time-t, and decreases with maturity to approximately 50 percent at(T₋t)=30 years. Thus, the combined effect of increasing default probabilities and decreasing recovery rates, causes the credit spread to increase with maturity. The negative correlation between default and recovery with maturity, is caused by the increasing volatility of the firm’s future asset value over time. As maturity is lengthened, the range of possible asset values that may result at maturity increases, causing a greater chance of asset values ending below the default boundary. If the firm defaults, the expected asset value available to pay bondholders in recovery is also lower.

For the highest risk firms, illustrated in Figure 2.1(a) by the lower quartile of sol- vency ratios, the credit spread term structure initially increases rapidly caused by the very steep rise in default rate as shown in Figure 2.1(b). The default probabilities are the highest for this quartile, rising from zero at time-t to approximately 60 percent at (T₋t)=30 years, and the recovery rates, as shown in Figure 2.1(c), are the lowest falling to approximately 40 percent at(T₋t)=30 years. Higher default risk and lower recov- ery is due to the initial firm value starting close to the default boundary. The default probability rises with maturity but at a decreasing rate. The resulting flattening of the default probability results in a falling credit spread for medium and long-term bonds due to the influence of discounting; the present value cost of default outweighs the marginal increase in default probability. Figure 2.1(a) shows the peak credit spread for the highest risk quartile of firms to be at approximately(T₋t)=5 years. Thus, the Merton model,

and structural models generally, predict a characteristic ‘hump-shape’ to credit spread term structures of high risk firms. Empirically there is mixed support for this predic- tion. Sarig & Warga (1989) average credit spreads across firms and time and report downward sloping term structures for firms rated B and C. However, Helwege & Turner (1999) show that this does not hold when term structure is measured from bonds issued by the same firm on the same date. They conclude that most speculative grade firms exhibit positively sloped credit spread term structures.

The Merton model, like all structural models, is a dynamic model of the credit spread term structure. The term structure, at a given time-t, is a result of the model’s predic- tion at time-t by equation (2.8) for for different remaining maturities as shown in Fig- ure 2.1(a). The term structure will also vary in time as a function of the path taken by the firm’s log-solvency ratio as modelled by equation (2.7). For a given maturity, the credit spread is expected to change over a finite time step as a function of the firm’s expected future log-solvency ratio. Knowledge of a firm’s model parameters, therefore facilitates prediction of a credit spread term structure and the likely transition of the credit spreads between time periods. We refer to the former model property as a cross- sectional constraint and the latter as a time-series constraint. The distinction is important when implicitly estimating parameters from term structure models. If we were to aver- age spreads over time, or find the best fitting parameters to match a firm’s term structure at a single point in time, then parameters are fitted using only cross-sectional informa- tion without regard to how well the time-series of observed credit spread movements is explained by the model. If we were to fit the time-variation of constant maturity bond spreads then we would select parameters that are maximised to only explain the time series transition of credit spreads without regard to how well the model jointly fits bonds of different maturity. Geyer & Pichler (1999) show that in the context of risk-free term structure modelling, the state-space estimation method, ‘simultaneously integrates time series and cross-sectional aspects of the model. Since this approach is consistent with the underlying economic model, and can utilise all available information, it provides a powerful test’, (Geyer & Pichler 1999, p.1). Therefore, in order to provide the strongest test of model specification, we transform the credit models into state-space form as de- scribed further in Section 3.2, and estimate the models on panel data.