Stock’s Default Probability and FF Factors

In June of every year, we form seven portfolios by sorting according to the predicted default probability obtained from Discrete-Time Hazard model. The portfolio stock composition is kept for one year (from July to the following June) with monthly rebalancing the portfolio weights. These seven portfolios are obtained by sorting the out-sample estimates of the probability of default. These predicted default probabilities are determined from accounting and market information since the end of the previous year.

We are going to test the hypothesis that the FF factors are related to a default risk measure.

In particular, we try to establish a connection between returns and the probability of default, and between the probability of default and factor loadings. We anticipate that returns and default risk are related. A high probability of default should cause a stock to have high loading on the SMB and HML factors, thus delivering high returns which compensate the investor for holding default risk.

Table 9a reports the average characteristics of seven portfolios using the probabilities obtained from Discrete-Time Hazard Model. We can find that, generally, the portfolios with higher default probability have relatively higher returns. The average size (stock price times number of shares) is almost monotonically decreasing

Table 9a Characteristics of Default Portfolios

The table shows reports average characteristics of seven portfolios when using the probability obtained from Discrete-Time Hazard Model. The average value-weighted returns are reported in percentage terms. The average size of each portfolio is reported in billions of NT dollars.

Low 2 3 4 5 6 High

ret 1.9328 2.4728 3.0523 2.0030 2.1799 3.1359 2.7590

size 28390.22 10158.65 13915.43 10299.67 4944.04 3932.51 2410.34 /

BM ME 0.5362 0.7692 0.8112 0.9352 1.0223 1.1005 1.1611

For each portfolios, we run a time-series regression of the value-weighted returns on a four-factor model. The table reports loading estimates along with robust Newey and West t-statistics. ** and * indicate statistically significance at the 1% and 5% statistical level, respectively.

Low 2 3 4 5 6 High

Adjusted R² 0.9273 0.8520 0.8689 0.7908 0.8241 0.7563 0.8220

Table 9b Four-Factor Model on Default Portfolios

along the default dimension. And the average of book-to-market is increasing in the default measure. Firms with high default probabilities have lower size (2410.34) and higher book-to-market (1.1611) on average, while firms with low default probabilities have relatively high average size (28390.22) and low book-to-market (0.5362).

For each portfolios, we run a time-series regression of the value-weighted returns on a four-factor model (FF factors plus a momentum factor).

( )

, , , ,

it ft i MKT i mt ft SMB i t HML i t MOM i t it

R −R =α β+ R −R +β SMB +β HML +β MOM +ε

Table 9b represents the estimates of the factor loadings along with robust Newey and West (1987) t-statistics. We can find that all β_MKT are statistically significant, but they don’t have a trend. If SMB and HML are related to financial distress, we would expect the portfolio loadings to increase along the dimension of default risk. The β_SMBs are not strongly increasing along the default measure, but generally they seem increasing. The result is much stronger for HML than SMB. The β_HML almost monotonically increase along the default measure (except second portfolio). In general, this result is consistent with our anticipation that a high probability of default should cause a stock to have high loading on the SMB and HML factors, thus delivering high returns which compensate the investor for holding default risk.

We also investigate how individual stocks (as opposed to portfolios) relate to default risk.

If the factors are pricing systematic distress risk, it should be that individual firm loadings on the factors are related to their estimated default probability. Every month we estimate individual stock loading for four-factor model using a 60-month window and requiring that any stock has at least 36 monthly observations in every window. In June of each year, we sort the stocks into seven portfolios based on the default probability. Hence, for each portfolio, we compute the average loading for the stock in that portfolio in July of each year of the out-sample. The result is reported in Table 10. As we can see from the table, the average

loadings of SMB and HML are increasing along the default measure. The average firm in the low default probability has a negative loading on HML and that in the high default probability has a positive loading. This result is similar to result reported in Table 9b and conforms our anticipation.

Table 10 Average Individual Stocks Loadings on Factors

Low 2 3 4 5 6 High

βSMB 0.4543 0.5903 0.6661 0.6354 0.7097 0.7604 1.0232 βHML -0.0462 0.0487 0.0426 0.0832 0.1561 0.1853 0.2125

A more rigorous test of the relation between default probability and factor loading at the individual stock level involves a two-stage Fama and MacBeth (1973) procedure mentioned in section 3.7.

Table 11 Fama-MacBeth Procedure

Model(1) Model(2) Model(3)

Table 11 reports the results. Model (1) gives the estimates of the model which uses only size and book-to-market as opposed to the factor loadings. The base case regression Model (2), relates the SMB and HML loadings to the default probability and highlights a positive and statistically significant relationship. We re-estimate Model (2) by including size and book-to-market as control variables. If the FF factor loadings are related to the default probability, we expect that the parameter estimates retain statistically significance. In Model (3) of Table 11, we find that both of λ_SMB and λ_HML maintain their signs and significance, although the significance of λ_HML reduces. Therefore, we consider that SMB and HML are related to default risk.