Bankruptcy Prediction Devices

In this section, we shall develop bankruptcy prediction methods for the five models, LLM, SLM, KMV, DAM, and DSM, in sequence.

Using the prospective training sample from the LLM (1), we determine a p^∗ ∈ (0, 1)value to make bankruptcy prediction for the company with predictor values (x0, z0).

By the consistency of its predicted bankruptcy probability ˆp(Y = 1| X = x⁰, Z = z0) derived from (2), if it satisfies

ˆ

p(Y = 1| X = x⁰, Z = z0) > p^∗,

then the company is classified to be in the status of bankruptcy, otherwise it is classified as a healthy company. To decide a proper cut-off point p^∗, usually one would use the training sample to evaluate the performance of the classification scheme. In doing so, there are two types of “in-sample” error rate occurred in this evaluation based on the training sample:

type I error rate αin(p) = Pn

i=1 Yi I{ˆp(Y = 1Pn| X = xⁱ, Z = zi)≤ p}

i=1 Yi

,

and

type II error rate β_in(p) = Pn

i=1 (1− Yⁱ) I{ˆp(Y = 1Pn | X = xⁱ, Z = zi) > p}

i=1 (1− Yⁱ) .

Here I(·) stands for the indicator function. Using the training sample and the cut-off point p, αin(p)is the rate of misclassifying bankrupt company to healthy company, and β_in(p) is the rate of misclassifying healthy company to bankrupt company. To keep these error rates to be as small as possible, we determine a proper cut-off point p^∗ such

that

τin(p^∗) = αin(p^∗) + β_in(p^∗) = min

p∈[0,1], αin(p)≤u{αⁱⁿ(p) + β_in(p)}.

That is to control the in-sample type I error rate αin(p)to be at most u, so that the sum of the two in-sample error rates τin(p) = αin(p) + β_in(p) is minimal. This is essential if the type I error would cause much more severe losses to the investors. On the other hand, if classifying healthy firms as being bankrupt would cause more severe losses to the investors, we might control the in-sample type II error rate β_in(p) to be at most u.

In practice, the value of u ∈ [0, 1] is determined by the investor. If u = 1, then there is no restriction on the magnitude of in-sample type I and II error rates (Altman, 1968;

Ohlson, 1980; Begley, Ming, and Watts, 1996). Since the value of p^∗ depends on that of u, it is also denoted by p^∗(u).

On the other hand, using the case-control training sample from the LLM (1) and treating the sample as if it was a prospective sample from the LLM (1), by the results in Section 1.3, the corresponding predicted bankruptcy probability _1+exp(ˆ^exp(ˆα+ˆ_α+ˆ^{β x}_{β x}^{0+ˆθ z0)}

0+ˆθ z0)

obtained from (2) does not converge to the true bankruptcy probability1+exp(α+β x^{exp(α+β x0+θ z}0+θ z⁰⁾0)

in (1), but approaches _1+exp(α+c^{exp(α+c∗+β x}_{∗+β x}^{0+θ z0)}

0+θ z0). Here c^∗ = log{p(Y = 0)/p(Y = 1)} + log(n1/n0). This drawback is caused by the fact that the resulting maximum likelihood estimates ˆβ and ˆθ of logistic parameters β and θ in the LLM (1) converge to their true values, respectively, but ˆα approaches the quantity α + c^∗, as both sample sizes of control and case data become large. This is the major difference between applying the LLM to case-control data and to prospective data.

Fortunately, we still can use the predicted bankruptcy probability ^{exp(ˆα+ˆβ x0+ˆθ z0)}

1+exp(ˆα+ˆβ x0+ˆθ z0)

obtained by the case-control sample from the LLM (1) to develop a bankruptcy predic-tion device by applying the following simple equivalent inequalities:

exp{α + β x + θ z}

1 + exp{α + β x + θ z} > p,

if and only if

exp{α + c^∗+ β x + θ z}

1 + exp{α + c^∗+ β x + θ z} > p exp(c^∗)

(1− p) + p exp(c^∗) ≡ p^c∗. This result is to say that using the probability exp(α+β x+θ z)

1+exp(α+β x+θ z) to define classification device with cut-off point p is equivalent to using the probability _1+exp(α+c^{exp(α+c∗+β x+θ z)}_{∗+β x+θ z)} to define classification device with cut-off point pc^∗. Hence we may pretend the predicted bankruptcy probability _1+exp(ˆ^exp(ˆα+ˆ_α+ˆ^{β x}_{β x}^{0+ˆθ z0)}

0+ˆθ z0) obtained by the case-control sample from the LLM (1) to be the estimate of the true bankruptcy probability and use it to determine the corresponding proper cut-off point p^∗(u). Then the bankruptcy prediction device for the case-control sample from the LLM (1) can be obtained.

Note that above bankruptcy prediction methods built for the LLM (1) using the two types of data are essentially equivalent. Based on the same arguments, similar bankruptcy prediction devices can be developed directly for the SLM (4) using the two types of data by replacing respectively (2) and ˆα + ˆβ x0 with (5) and ˆH(x0). Hence the bankruptcy prediction methods constructed for the SLM (4) using the two types of data are also essentially equivalent.

Note also that the above method for computing the optimal cutoff value for the LLM (1) can be similarly applied for both the KMV and the DAM by replacing ˆp(Y = 1| X = x0, Z = z0)with πKMV and DF V to derive their optimal cutoff values π^∗_{KM V} and v^∗, respectively in each case. Given the optimal cutoff value π^∗_{KM V}, if πKM V > π^∗_{KM V}, then the company with the probability of default πKM V is classified to be in the status of bankruptcy, otherwise it is classified as a healthy company. Similarly, given the optimal cutoff value v^∗, if DF V > v^∗, then the company with the discriminant function value DF V is classified to be in the status of bankruptcy, otherwise it is classified as a healthy company.

We now give the bankruptcy prediction method for the DSM. Using the discrete-time survival data and following the same arguments of the bankruptcy method based

prediction for the company with predictor values (x0, z0) at the duration time t. By the consistency of its predicted probability of instant bankruptcy h(t, x0, z0; ˆψ), if it satisfies

h(t, x0, z0; ˆψ) > p^∗,

then, at the duration time t, the company is classified to be in the status of bankruptcy, otherwise it is classified as a healthy company. To decide a proper cut-off point p^∗, we use the data (ti, Yi, xi,ti, zi,ti), for i = 1, · · ·, n, to evaluate the performance of the classification scheme. In doing so, there are two types of “in-sample” error rate occurred in this evaluation based on the data (ti, Yi, xi,ti, zi,ti), for i = 1, · · ·, n:

type I error rate αin(p) = Pn

i=1 Yi I{h(tPⁱ, xn i,ti, zi,ti; ˆψ)≤ p}

i=1 Yi

,

and

type II error rate β_in(p) = Pn

i=1 (1− YⁱP) In{h(tⁱ, xi,ti, zi,ti; ˆψ) > p}

i=1 (1− Yⁱ) .

Here I(·) stands for the indicator function. To keep these two error rates to be as small as possible, we determine a proper cut-off point p^∗ = p^∗(u) for the bankruptcy prediction method based on the DSM such that

τin{p^∗(u)} = αⁱⁿ{p^∗(u)} + βin{p^∗(u)} = min

p∈[0,1], αin(p)≤u{αⁱⁿ(p) + β_in(p)}, for each u ∈ [0, 1].