A flow-based corporate credit model

Abstract

The main purpose of this paper is to develop a flow-based corporate credit model. This model can concurrently and endogenously generate a firm’s multi-period probabilities of liquidity crunch and expected liquidity shortfalls. This study builds a state-dependent internal liquidity model that incorporates both systematic and idiosyncratic shocks into corporate internal liquidity dynamics. The flow-based credit model differs from structural form credit models in that it considers a flow-based insolvency rather than a stock-based one, and has a potential to capture short-term credit information. Additionally, it differs from both reduced form and traditional accounting-based bankruptcy prediction models in that it is able to provide multi-period expected liquidity shortfalls endogenously.

Appendices

Appendix I

1.1 The discussion of state-dependent internal liquidity model

Based on the above contents, the state-dependent internal liquidity model is constructed by the Eqs. 10 and 11. It is a general setting for the potential state factors and the idiosyncratic innovations. By totally differencing Eq. 10 and introducing Eq. 11, we can summarize the preliminary stochastic differential equation (S.D.E) of internal liquidity as Eq. 12.

$$ L_{t} = E\left( L \right) + \sum\limits_{j = 1}^{k} {\alpha_{j} F_{jt} } + \xi_{t} \quad \xi_{t} \sim N\left( {0,\sqrt {1 - h}_{t} } \right) $$

(10)

$$ {\text{d}}F_{jt} = a_{{F_{jt} }} [b_{{F_{jt} }} - F_{jt} ]{\text{d}}t + \sigma_{{F_{jt} }} {\text{d}}z_{j} $$

(11)

$$ \begin{aligned} {\text{d}}L_{t} & =& \sum\limits_{j = 1}^{k} {\alpha_{j} {\text{d}}F_{jt} + {\text{d}}\xi_{t} } \\ & = \sum\limits_{j = 1}^{k} {\alpha_{j} \left[ {a_{{F_{jt} }} \left( {b_{{F_{jt} }} - F_{jt} } \right){\text{d}}t + \sigma_{{F_{jt} }} {\text{d}}z_{j} } \right]} + {\text{d}}\xi_{t} \\ & = \sum\limits_{j = 1}^{k} {\alpha_{j} a_{{F_{jt} }} \left( {b_{{F_{jt} }} - F_{jt} } \right){\text{d}}t + \sum\limits_{j = 1}^{k} {\alpha_{j} \sigma_{{F_{jt} }} {\text{d}}z_{j} + {\text{d}}\xi_{t} } } \\ & = \sum\limits_{j = 1}^{k} {\alpha_{j} a_{{F_{jt} }} b_{{F_{jt} }} {\text{d}}t - \sum\limits_{j = 1}^{k} {\alpha_{j} a_{{F_{jt} }} F_{jt} {\text{d}}t} + \sum\limits_{j = 1}^{k} {\alpha_{j} \sigma_{{F_{jt} }} {\text{d}}z_{j} + {\text{d}}\xi_{t} } } \\ \end{aligned} $$

(12)

For the special case that the number of state factor equals one (k = 1), we can reorganize the internal liquidity’s S.D.E. and show as Eq. 13.

$$ \begin{aligned} {\text{d}}L_{t} & =& \alpha_{1} a_{{F_{1t} }} b_{{F_{1t} }} {\text{d}}t - \alpha_{1} a_{{F_{1t} }} F_{1t} {\text{d}}t + \alpha_{1} \sigma_{{F_{1t} }} {\text{d}}z_{1} + {\text{d}}\xi_{t} \\ & = \alpha_{1} a_{{F_{1t} }} b_{{F_{1t} }} {\text{d}}t - a_{{F_{1t} }} \left( {L_{t} - E\left( L \right) - \xi_{t} } \right){\text{d}}t + \alpha_{1} \sigma_{{F_{1t} }} {\text{d}}z_{1} + {\text{d}}\xi_{t} \\ & = a_{{F_{1t} }} \left( {\left( {E\left( L \right) + \alpha_{1} b_{{F_{1t} }} + \xi_{t} } \right) - L_{t} } \right){\text{d}}t + \alpha_{1} \sigma_{{F_{1t} }} {\text{d}}z_{1} + {\text{d}}\xi_{t} \\ \end{aligned} $$

(13)

Next, based on the some assumptions, the parameters of internal liquidity’s S.D.E can be simplified as Eq. 14. a _Lt indicates the mean-reverting speed of L _t at time t; b _Lt is the long-term average level of L _t at time t; σ _Lt indicates the standard deviation of the term variation of L _t at time t, and dz _L is a wiener process. ${\text{assume}}\;a_{{F_{1t} }} = a_{Lt} ,$ ${\text{d}}z_{1} = {\frac{1}{{\sqrt {1 - h_{t} } }}} {\text{d}}\xi_{t} = {\text{d}}z_{L}$ for the S.D.E.: ${\text{d}}L_{t} = a_{Lt} \left( {b_{Lt} - L_{t} } \right){\text{d}}t + \sigma_{Lt} {\text{d}}z_{L}$

$$ b_{Lt} = E\left( L \right) + \alpha_{1} b_{{F_{1t} }} + \xi_{t} ,\sigma_{Lt} = \left( {\alpha_{1} \sigma_{{F_{1t} }} + \sqrt {1 - h_{t} } } \right) $$

(14)

Appendix II

2.1 The parameter estimation results of the sample firms used in empirical examinations correlation of long- and short-term ratings

This section includes two tables. Table 4 presents the parameter estimation results of the cyclicality factor extracted by the change rates of industrial productions and GDP by MLE method. Table 5 shows the distributions of the two parameters, the long-term average internal liquidity level and the loadings corresponding to the cyclicality factor, of all sample firms in 95th, 75th, 50th, 25th, 5th percentiles.

Table 4 The parameter estimation results of the cyclicality factor by MLE method

Table 5 The distributions of the two parameters, E(L) and α, of all sample firms