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Finally, we add the log price per share of the ﬁrm, PRICE. We expect this variable to be relevant for low prices per share, particularly since both the NYSE and the Nasdaq have a minimum price per share of $1 and commonly delist stocks that fail to meet this minimum (Macey, O’Hara, and Pompilio 2004). Reverse stock splits are sometimes used to keep stock prices away from the $1 minimum level, but these often have negative eﬀects on returns and therefore on market capitalization, suggesting that investors interpret reverse stock splits as a negative signal about company prospects (Woolridge and Chambers 1983, Hwang 1995). Exploratory analysis suggested that price per share is relevant below $15, and so we truncate price per share at this level before taking the log.

All the new variables in our model enter the logit regression with the expected sign and are highly statistically signiﬁcant. After accounting for diﬀerences in the scaling of the variables, there is little eﬀect on the coeﬃcients of the variables already included in the Shumway model, with the important exception of market capitalization. This variable is strongly correlated with log price per share; once price per share is included, market capitalization enters with a weak positive coeﬃcient, probably as an ad hoc correction to the negative eﬀect of price per share.

To get some idea of the relative impact of changes in the diﬀerent variables, we compute the proportional impact on the failure probability of a one-standarddeviation increase in each predictor variable for a ﬁrm that initially has sample mean values of the predictor variables. Such an increase in proﬁtability reduces the probability of failure by 44% of its initial value; the corresponding eﬀects are a 156% increase for leverage, a 28% reduction for past excess return, a 64% increase for volatility, a 17% increase for market capitalization, a 21% reduction for cash holdings, a 9% increase for the market-book ratio, and a 56% reduction for price per share. Thus variations in leverage, volatility, price per share, and proﬁtability are more important for failure risk than movements in market capitalization, cash, or the market-book ratio. These magnitudes roughly line up with the t statistics reported in Table 3.

Our proposed model delivers a noticeable improvement in explanatory power over the Shumway model. We report McFadden’s pseudo R2 coeﬃcient for each speciﬁcation, calculated as 1 − L1 /L0, where L1 is the log likelihood of the estimated model and L0 is the log likelihood of a null model that includes only a constant term. The pseudo R2 coeﬃcient increases from 0.26 to 0.30 in predicting bankruptcies or failures over 1963—1998, and from 0.27 to 0.31 in predicting failures over 1963—2003.

3.1 Forecasting at long horizons

Note that this assumption does not imply a cumulative probability of bankruptcy that is logit. If the probability of bankruptcy in j months did not change with the horizon j, that is if αj = α and β j = β, and if ﬁrms exited the dataset only through bankruptcy, then the cumulative probability of bankruptcy over the next j periods would be given by 1 − (exp (−α − βxi ) /(1 + exp (−α − βxi ))j, which no longer has the logit form. Variation in the parameters with the horizon j, and exit from the dataset through mergers and acquisitions, only make this problem worse. In principle we could compute the cumulative probability of bankruptcy by estimating models for each horizon j and integrating appropriately; or by using our one-period model and making auxiliary assumptions about the time-series evolution of the predictor variables in the manner of Duﬃe and Wang (2003). We do not pursue these possibilities here, concentrating instead on the conditional probabilities of default at particular dates in the future.

As the horizon increases in Table 4, the coeﬃcients, signiﬁcance levels, and overall ﬁt of the logit regression decline as one would expect. Even at three years, however, almost all the variables remain statistically signiﬁcant.

Three predictor variables are particularly important at long horizons. The coeﬃcient and t statistic on volatility SIGMA are almost unchanged as the horizon increases; the coeﬃcient and t statistic on the market-to-book ratio MB increase with the horizon; and the coeﬃcient on relative market capitalization RSIZE switches sign, becoming increasingly signiﬁcant with the expected negative sign as the horizon increases. These variables, market capitalization, market-to-book ratio, and volatility, are persistent attributes of a ﬁrm that become increasingly important measures of ﬁnancial distress at long horizons. Log price per share also switches sign, presumably as a result of the previously noted correlation between this variable and market capitalization.

Leverage and past excess stock returns have coeﬃcients that decay particularly rapidly with the horizon, suggesting that these are primarily short-term signals of ﬁnancial distress. Proﬁtability and cash holdings are intermediate, with eﬀects that decay more slowly.

In Table 4 the number of observations and number of failures vary with the horizon, because increasing the horizon forces us to drop observations at both the beginning and end of the dataset. Failures that occur within the ﬁrst j months of the sample cannot be related to the condition of the ﬁrm j months previously, and the last j months of the sample cannot be used to predict failures that may occur after the end of the sample. Also, many ﬁrms exit the dataset for other reasons between dates t−1 and t − 1 + j. On the other hand, as we lengthen the horizon we can include failures that are immediately preceded by missing data. We have run the same regressions for a subset of ﬁrms for which data are available at all the diﬀerent horizons. This allows us to compare R2 statistics directly across horizons. We obtain very similar results to those reported in Table 4, suggesting that variation in the available data is not responsible for our ﬁndings.

**3.2 Comparison with distance to default**

A leading alternative to the reduced-form econometric approach we have implemented in this paper is the structural approach of Moody’s KMV (Crosbie and Bohn 2001), based on the structural default model of Merton (1974). This approach uses the Merton model to construct “distance to default”, DD, a measure of the diﬀerence between the asset value of the ﬁrm and the face value of its debt, scaled by the standard deviation of the ﬁrm’s asset value. Taken literally, the Merton model implies a deterministic relationship between DD and the probability of default, but in practice this relationship is estimated by a nonparametric regression of a bankruptcy or failure indicator on DD. That is, the historical frequency of bankruptcy is calculated for ﬁrms with diﬀerent levels of DD, and this historical frequency is used as an estimate of the probability of bankruptcy going forward.

** To implement the structural approach, we calculate DD in the manner of Hil-**

legeist, Keating, Cram, and Lunstedt (2004) by solving a system of two nonlinear equations. The details of the calculation are described in the Appendix. Table 5 compares the predictive power of the structural model with that of our best reducedform model. The top panel reports the coeﬃcients on DD in a simple regression of our failure indicator on DD, and in a multiple regression on DD and the variables included in our reduced-form model. DD enters with the expected negative sign and is highly signiﬁcant in the simple regression. In the multiple regression, however, it enters with a perverse positive sign at a short horizon, presumably because the reduced-form model already includes volatility and leverage, which are the two main inputs to the calculation of DD. The coeﬃcient on DD only becomes negative and signiﬁcant when the horizon is extended to one or three years.

The bottom panel of Table 5 reports the pseudo R2 statistics for these regressions.

While the structural model achieves a respectable R2 of 16% for short-term failure prediction, our reduced-form model almost doubles this number. Adding DD to the reduced-form model has very little eﬀect on the R2, which is to be expected given the presence of volatility and leverage in the reduced-form model. These results hold both when we calculate R2 in-sample, using coeﬃcients estimated over the entire period 1963-2003, and when we calculate it out-of-sample, using coeﬃcients each year from 1981 onwards that were estimated over the period up to but not including that year. The two sets of R2 are very similar because most failures occur towards the end of the dataset, when the full-sample model and the rolling model have very similar coeﬃcients.

The structural approach is designed to forecast default at a horizon of one year.

This suggests that it might perform relatively better as we forecast failure further into the future. It is true that DD enters our model signiﬁcantly with the correct sign at longer horizons, but Table 5 shows that the relative performance of DD and our econometric model is relatively constant across forecast horizons.

We conclude that the structural approach captures important aspects of the process determining corporate failure. The predictive power of DD is quite impressive given the tight restrictions on functional form imposed by the Merton model. If one’s goal is to predict failures, however, it is clearly better to use a reduced-form econometric approach that allows volatility and leverage to enter with free coeﬃcients and that includes other relevant variables. Bharath and Shumway (2004), in independent recent work, reach a similar conclusion.

** **

3.3 Other time-series and cross-sectional eﬀects

As we noted in our discussion of Table 1, there is considerable variation in the failure rate over time. We now ask how well our model ﬁts this pattern. We ﬁrst calculate the ﬁtted probability of failure for each company in our dataset using the coeﬃcients from our best reduced-form model. We then average over all the predicted probabilities to obtain a prediction of the aggregate failure rate among companies with data available for failure prediction.

Figure 1 shows annual averages of predicted and realized failures, expressed as a fraction of the companies with available data.7 Our model captures much of the broad variation in corporate failures over time, including the strong and long-lasting increase in the 1980’s and cyclical spikes in the early 1990’s and early 2000’s. However it somewhat overpredicts failures in 1974-5, underpredicts for much of the 1980’s, and then overpredicts in the early 1990’s.

We have explored the possibility that there are industry eﬀects on bankruptcy and failure risk. The Shumway (2001) and Chava-Jarrow (2004) speciﬁcation appears to behave somewhat diﬀerently in the ﬁnance, insurance, and real estate (FIRE) sector.

That sector has a lower intercept and a more negative coeﬃcient on proﬁtability.

However there is no strong evidence of sector eﬀects in our best model, which relies more heavily on equity market data.

We have also used market capitalization and leverage as interaction variables, to test the hypotheses that other explanatory variables enter diﬀerently for small or highly indebted ﬁrms than for other ﬁrms. We have found no clear evidence that such interactions are important.

The realized failure rate among these companies is slightly diﬀerent from the failure rate reported in Table 1, which includes all failures and all active companies, not just those with data available for failure prediction.

4 Risks and average returns on distressed stocks We now turn our attention to the asset pricing implications of our failure model.

Recent work on the distress premium has tended to use either traditional risk indices such as the Altman Z-score or Ohlson O-score (Dichev 1998, Griﬃn and Lemmon 2002, Ferguson and Shockley 2003) or the distance to default measure of KMV (Vassalou and Xing 2004, Da and Gao 2004). To the extent that our reduced-form model more accurately measures the risk of failure at short and long horizons, we can more accurately measure the premium that investors receive for holding distressed stocks.

Before presenting the results, we ask what results we should expect to ﬁnd. On the one hand, if investors accurately perceive the risk of failure they may demand a premium for bearing it. The frequency of failure shows strong variation over time, as illustrated in Figure 1; even though much of this time-variation is explained by timevariation in our ﬁrm-level predictive variables, it still generates common movement in stock returns that might command a premium.

Of course, a risk can be pervasive and still be unpriced. If the standard implementation of the CAPM is exactly correct, for example, then each ﬁrm’s risk is fully captured by its covariation with the market portfolio of equities, and distress risk is unpriced to the extent that it is uncorrelated with that portfolio. However it seems plausible that corporate failures may be correlated with declines in unmeasured components of wealth such as human capital (Fama and French 1996) or debt securities (Ferguson and Shockley 2003), in which case distress risk will carry a positive risk premium.8 This expectation is consistent with the high failure risk of small ﬁrms that have depressed market values, since small value stocks are well known to deliver high average returns.

Fama and French (1996) state the idea particularly clearly: “Why is relative distress a state variable of special hedging concern to investors? One possible explanation is linked to human capital, an important asset for most investors. Consider an investor with specialized human capital tied to a growth ﬁrm (or industry or technology). A negative shock to the ﬁrm’s prospects probably does not reduce the value of the investor’s human capital; it may just mean that employment in the ﬁrm will grow less rapidly. In contrast, a negative shock to a distressed ﬁrm more likely implies a negative shock to the value of human capital since employment in the ﬁrm is more likely to contract.