As can be seen from Equation (2.8), Basel II assumes a downward sloping relationship between the asset correlation and the probability of default. Namely, the correlation between the assets of low-risk firms is assumed to be higher than the ones between high-risk firms. Several studies have results which did not support this relationship (Dietsch and Petey, 2004; D¨ullmann and Scheule, 2003; Hamerle et al., 2003;
R¨osch, 2002). This thesis explored the relationship between asset cor-relation and default probability using Bayesian method. However, the estimation of default probability is not available. Altman (1960,1990), a financial economist, developed a model for predicting the likelihood that a corporation would go bankrupt. This model uses five financial ratios that combine in a specific way to produce a single number. This number, called the Z-score, is a general measure of corporate financial health. Z-score is used as the proxy variable for default probability in this study.
The positive relationship between asset correlation and firm size is generally supported (Lopez, 2004; Dietsch and Petey, 2004; D¨ullmann and Scheule, 2003). This thesis also explored this relationship using Bayesian method. Three different shapes of prior distributions were used to explore the posterior distribution of asset return correlations due to the lack of enough historical data or the experts’ suggestions.
These three priors are ”standard normal”, ”Beta(0.5,0.5)”, and ”Uni-form(0,1)”. The study results from these three priors are very similar.
The results for uniform prior are tabulated in the text and the results of the other two priors are listed in appendices.
4.1 Based on Z-score categories
In this section we choose Altman’s Z-score as a proxy of probabil-ity of default, and classify all firms into three categories. The Z-score formula for predicting bankruptcy is a multivariate formula for a measurement of the financial health of a company and a powerful diagnostic tool that forecasts the probability of a company entering bankruptcy within a 2 year period. Studies measuring the effectiveness of the Z-score have shown the model is often accurate in predicting bankruptcy (72%-80% reliability).
The Z-score bankruptcy predictor combines five common business ra-tios, using a weighting system calculated by (Altman, 1968) to deter-mine the likelihood of a company going bankrupt.
The five ratios are:
. Working capital-to-total assets . Retained earnings-to-total assets
. Earnings before interest and taxes (EBIT)-to-total assets . Market value equity-to-book value of total debt
. Sales-to-total assets
The final discriminant function is as follows:
Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 + 1.0X5 . X1 = working capital/total assets
. X2 = retained earnings/total assets
. X3 = earnings before interest and taxes/total assets . X4 = market value equity/book value of total liabilities . X5 = sales/total assets, and
. Z = overall Index or Score
If a financial risk manager uses a corporate borrower’s accounting ratio weighted by the estimated coefficients in the Z function resulting in a Z-score below a critical value ( for instance, in Altman’s initial study, 1.81 ), this debtor would be sorted as “bad” and the loan may be refused. The detailed boundary could be summarized as follows:
Z-score >2.99 means a healthy company
1.81 < Z-score <2.99 represents it is in the gray zone
Z-score<1.81 implies that the firm is unhealthy
Within the ASRF model framework, we expect to see a negative re-lationship between firms’ asset correlations and their probabilities of default (PD). That is, as a firm’s PD increases due to its worsening condition and approaching possible default, it is reasonable to assume that idiosyncratic factors begin to take on a more important role rel-ative to the common, systematic risk factor.
Table 2 to Table 5present the summary statistics: posterior mean, standard deviation, MC error, median and 95% credible interval, for the distribution of asset correlation by Z-score category for year 2001 to year 2005, respectively.
Table 2: Asset correlation in year 2005 for Uniform prior by z-score category
category mean sd MC error 2.5% median 97.5%
Z-score>2.99 0.0764 0.0035 0.0001 0.0696 0.0764 0.0832 1.81<Z-score<2.99 0.0554 0.0062 0.0001 0.0432 0.0555 0.0673 Z-score<1.81 0.0345 0.0062 0.0001 0.0225 0.0345 0.0466 The sample size of category “Z-score>2.99” is 1422, of category “1.81<Z-score<2.99”
is 454 and of category “Z-score<1.81” is 588.
Table 3: Asset correlation in year 2004 for Uniform prior by z-score category
category mean sd MC error 2.5% median 97.5%
Z-score>2.99 0.0897 0.0028 0.0001 0.08416 0.08966 0.0953 1.81<Z-score<2.99 0.0828 0.0056 0.0001 0.0721 0.0829 0.0937 Z-score<1.81 0.0654 0.0064 0.0002 0.0530 0.0654 0.0781 The sample size of category “Z-score>2.99” is 1499, of category “1.81<Z-score<2.99”
is 520 and of category “Z-score<1.81” is 531.
Table 4: Asset correlation in year 2003 for Uniform prior by z-score category
category mean sd MC error 2.5% median 97.5%
Z-score>2.99 0.0852 0.0040 0.0001 0.0774 0.0852 0.0931 1.81<Z-score<2.99 0.0756 0.00624 0.0001 0.0636 0.0756 0.0881 Z-score<1.81 0.0542 0.0096 0.0003 0.0359 0.0540 0.0736 The sample size of category “Z-score>2.99” is 1339, of category “1.81<Z-score<2.99”
is 490 and of category “Z-score<1.81” is 597.
Table 5: Asset correlation in year 2002 for Uniform prior by z-score category
category mean sd MC error 2.5% median 97.5%
Z-score>2.99 0.1479 0.0048 0.0002 0.1386 0.1478 0.1576 1.81<Z-score<2.99 0.1239 0.0066 0.0002 0.1110 0.1239 0.137 Z-score<1.81 0.1066 0.0114 0.0005 0.0866 0.1058 0.1306 The sample size of category “Z-score>2.99” is 1112, of category “1.81<Z-score<2.99”
is 516 and of category “Z-score<1.81” is 783.
Table 6: Asset correlation in year 2001 for Uniform prior by z-score category
category mean sd MC error 2.5% median 97.5%
Z-score>2.99 0.1665 0.0058 0.0002 0.1554 0.1664 0.1786 1.81<Z-score<2.99 0.0799 0.0073 0.0001 0.0656 0.0798 0.0945 Z-score<1.81 0.0944 0.0123 0.0005 0.0726 0.0935 0.1214 The sample size of category “Z-score>2.99” is 1209, of category “1.81<Z-score<2.99”
is 480 and of category “Z-score<1.81” is 607.
From Table 2 to Table 5 we can see that the positive relationship be-tween Z-score and asset correlation exist in year 2002 to year 2005;
however in year 2001, the results do not support the positive relation-ship. We found that asset correlation for category of < 1.81 is larger than that of category 1.81 to 2.99 but the asset correlation of category
> 2.99 is still the largest one among these three categories. All of the
“MC error”s are smaller than 5% of the standard deviation, and it is the thumb rule to assess the accuracy of the posterior estimates.
All of the posterior means from the above tables are displayed in Table 8. We can see that in the same Z-score category the asset correlation is different from year to year.
Table 7: Sample size by Z-score category
year Z-score>2.99 1.81<Z-score<2.99 Z-score<1.81
2005 1422 454 588
2004 1499 520 531
2003 1339 490 597
2002 1112 516 783
2001 1209 480 607
Table 8: Means of asset correlation for year 2001-2005 by Z-score category for Uni-form prior
year Z-score>2.99 1.81<Z-score<2.99 Z-score<1.81
2005 0.0764 0.0554 0.0345
2004 0.0897 0.0828 0.0654
2003 0.0852 0.0756 0.0542
2002 0.1479 0.1239 0.1066
2001 0.1665 0.0799 0.0944
Figure 3 we present the tendency of asset correlation year by year, and we can see a decline tendency roughly. Actually, the downward tendency is most significant in Z-score>2.99 category. Furthermore, from year 2002 to 2005, the range of shift is similar in all of the three groups.
Figure 3: Means of asset correlation during years 2001-2005 by Z-score category
In the end, we will go a step further to exam whether under different prior distribution, the posterior distribution of asset correlation will be similar, if so, we can say that our data set is strong enough and will not affect significantly due to the alteration of prior. All the Monte Carlo Errors (MC errors) are less than 5%. The accuracy of the poste-rior distribution of asset correlation is confirmed and not presented in the following tables. Table 9 is the posterior mean of asset correlation under N ormal(0, 1)I(0, 1) prior, and Table 10 is under Beta(0.5, 0.5) prior. The detailed study results for prior Normal and Beta are given in Appendix A. The study results indicate that no matter under what kinds of prior, the posterior distribution of asset correlation is similar to the one for uniform prior. Then in the next section, we will inves-tigate the effect of different firm size on asset correlation.
Table 9: Means of asset correlation for year 2001-2005 by Z-score category for Nor-mal prior
year Z-score>2.99 1.81<Z-score<2.99 Z-score<1.81
2005 0.0764 0.0552 0.0342
2004 0.0937 0.0825 0.0652
2003 0.0851 0.0755 0.0546
2002 0.1475 0.1239 0.1068
2001 0.1657 0.0799 0.0935
Table 10: Means of asset correlation for year 2001-2005 by Z-score category for Beta prior
year Z-score>2.99 1.81<Z-score<2.99 Z-score<1.81
2005 0.0764 0.0546 0.0341
2004 0.0939 0.0824 0.0648
2003 0.0847 0.0754 0.0532
2002 0.1471 0.1238 0.1057
2001 0.1663 0.0795 0.0927
We list the results of Lopez(2004) in Figure 4 to make a comparision.
Lopez(2004) used data from year-end 2000 and invested the
relation-Figure 4: Results of Lopez(2004) for EDF category
ship of EDF, asset size and asset correlation. His results presented that the asset correlation in the U.S. is between 0.15 and 0.225 based on EDF category. Our results from year 2001 to 2005 are smaller than Lopez’s results, but the negative relation between defalt probability and asset correlation are similar.
4.2 Based on firm size categories
The second part of our analysis focuses on the relationship between firm size and asset correlation, and we use the book value of firm’s equity to estimate the firm’s size. As same as previous section, we group the firm size into three categories, the cut-off points follow the one used in Lopez’s paper (2004).
In the general theory of portfolio diversification, as the number of different securities within a portfolio increases, the portfolio becomes more diversified, and the idiosyncratic element of the portfolio’s return becomes less important. An analogous view could be taken with re-spect to a firm’s asset size; that is, as a firm becomes larger and comes to contain more assets, its risk and return characteristics should more closely resemble the overall asset market and be less dependent on the idiosyncratic elements of the individual business lines. Within the ASRF model framework, this intuition suggests that a firm’s asset correlation should increases as its asset size increases. From Table 11 to Table 17, the study results confirm the aforementioned intuition.
The MC errors are less than 5% and confirm the accuracy of posterior distribution of asset correlation.
Table 11: Asset correlation in year 2005 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%
> $1b 0.1385 0.0047 0.0001 0.1292 0.1386 0.1477 ($100m,$1b) 0.1310 0.0038 0.0001 0.1235 0.1310 0.1382 ($0m,$100m) 0.0187 0.0054 0.0001 0.0082 0.0188 0.0290
The sample size of category “> $1b” is 567, of category “($100m,$1b)” is 1025 and of category “($0m,$100m)” is 867.
Table 12: Asset correlation in year 2004 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%
> $1b 0.1758 0.0045 0.0001 0.1669 0.1759 0.1843 ($100m,$1b) 0.1642 0.0038 0.0001 0.1567 0.1642 0.1717 ($0m,$100m) 0.0388 0.0056 0.0001 0.0277 0.0390 0.0498
The sample size of category “> $1b” is 601, of category “($100m,$1b)” is 1041 and of category “($0m,$100m)” is 901.
Table 13: Asset correlation in year 2003 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%
> $1b 0.2136 0.0046 0.0001 0.2046 0.2137 0.2225 ($100m,$1b) 0.1459 0.0041 0.0002 0.1378 0.1459 0.1544 ($0m,$100m) 0.0295 0.0062 0.0001 0.0181 0.0294 0.0419
The sample size of category “> $1b” is 556, of category “($100m,$1b)” is 989 and of category “($0m,$100m)” is 874.
Table 14: Asset correlation in year 2002 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%
> $1b 0.2771 0.0055 0.0002 0.2666 0.2772 0.2879 ($100m,$1b) 0.2115 0.0049 0.0002 0.2020 0.2115 0.2213 ($0m,$100m) 0.0482 0.0078 0.0002 0.0334 0.0480 0.0639
The sample size of category “> $1b” is 481, of category “($100m,$1b)” is 958 and of category “($0m,$100m)” is 958.
Table 15: Asset correlation in year 2001 for Uniform prior by firm size category firm size mean sd MC error 2.5% median 97.5%
> $1b 0.2767 0.0057 0.0002 0.2656 0.2767 0.2881 ($100m,$1b) 0.1920 0.0061 0.0003 0.1808 0.1917 0.2053 ($0m,$100m) 0.0645 0.0089 0.0003 0.0484 0.0640 0.0829
The sample size of category “> $1b” is 455, of category “($100m,$1b)” is 934 and of category “($0m,$100m)” is 895.
Unlike the results of classification by Z-score in year 2001 that the cat-egory of Z-score under 1.81 is bigger than catcat-egory of Z-score between 18.1 and 2.99, the results by firm size categories conclude that firm’s asset correlation increases as its asset size increases for year 2001 to year 2005. Besides, comparing the results under different prior from Table 17, Table 18 and Table 19, the results are very similar for three different priors. Therefore, our models by firm size category pass the sensitivity analysis and form a strong likelihood belief.
Table 16: Sample size–classified according to firm size year > $1b ($100m,$1b) ($0m,$100m)
2005 567 1025 867
2004 601 1041 901
2003 556 989 874
2002 481 958 958
2001 455 934 895
Table 17: Means of asset correlation for year 2001-2005 by firm size category for Uniform prior
year > $1b ($100m,$1b) ($0m,$100m)
2005 0.1385 0.1310 0.0187
2004 0.1758 0.1642 0.0388
2003 0.2136 0.1459 0.0295
2002 0.2771 0.2115 0.0482
2001 0.2767 0.1920 0.0645
Table 18: Means of asset correlation for year 2001-2005 by firm size category for Normal prior
year > $1b ($100m,$1b) ($0m,$100m)
2005 0.1385 0.1309 0.0187
2004 0.1757 0.1640 0.0387
2003 0.2138 0.1461 0.0297
2002 0.2775 0.2111 0.0484
2001 0.2766 0.1920 0.0651
To see the yearly trend classified according to firm size in Figure 5, we find a decline tendency, which is the same as classifying according to Z-score. However, what different from the results of classifying ac-cording to Z-score is that the category of “($0m,$100m)” has a big gap with categories of “($100m,$1b)” and “(> $1b)”. In “(> $1b)” cate-gory, the downward tendency is the most significant one, and for the analysis based on the Z-score category, the “Z-score>2.99” category is also the most significant in all of the three groups. That means, the firms in a healthy financial state or the firms with large asset size have a more notable decreasing trend in asset correlation comparing with the other two categories.
Table 19: Means of asset correlation for year 2001-2005 by firm size category for Beta prior
year > $1b ($100m,$1b) ($0m,$100m)
2005 0.1386 0.1309 0.0181
2004 0.1755 0.1635 0.0383
2003 0.2137 0.1456 0.0291
2002 0.2774 0.2116 0.0474
2001 0.2765 0.1907 0.0645
Figure 5: Means of asset correlation during years 2001-2005 by firm size category
The results of Lopez(2004) in Figure 6 to make a comparision. The
Figure 6: Results of Lopez(2004) for size category
data from year-end 2000 and invested the relationship of asset size and asset correlation. His results presented that the asset correlation in the U.S. is between 0.1 and 0.3 basedon asset size categories. Our results are smaller than the results of Lopez’s in asset size categories and EDF categories.
5 Conclusion
Asset correlation is the key factor for determining the risk weight in capital requirement of Basel II. There are two findings in Quantita-tive impact study 3 technical guidance (QIS-3) based on the formula for asset correlation in Basel II; negative relationship between asset correlation and default probability, and positive relationship between asset correlation and firm size. Some studies support these findings but some studies do not support them. The problem is that there is no theoretical background for asset correlation formula in Basel II. This study applies Baysian method to the asymptotic single risk factor (ASRF) to investigate the relationship among asset correlation, de-fault probability, and firm size. This approach incorporates the prior information (historical data) and the variability (or precision) of asset correlation into ASRF model. This method provides the theoretical basis for discussing the aforementioned relationship. The data we used in this study include the listed firms in both CRSP and COMPUSTAT databases during years 2001-2005. Equity correlation was used as the proxy for asset correlation since it has been industry practice. Due to the lack of data for probability of default, Alman’s Z-score was used as the proxy of default probability. The relationship was discussed based on the ordered categories of Z-score and firm size, individually.
There are three main findings in this study.
First, the asset correlation is positively correlated with Z-score. This indicates that the negative relationship between average asset correla-tion and probability of default. The value of total assets are controlled by a systematic factor and an idiosyncratic one. The negative rela-tionship implies that the higher probability of default, the lower the
systematic risk components of a firm. The systematic risk is tied to the general economic environment summarized by the common factor.
Secondly, the study results further indicate that average asset corre-lation is increasing in firm size. That is, as firms increase the book value of their assets, they become more correlated with the general economic environment.
This result is intuitive in the sense that larger firms can generally be viewed as portfolios of smaller firms, and such portfolios would be relatively more sensitive to systematic risks than to idiosyncratic risks.
Finally, the asset correlation is different from year to year, so there may be some other important factors that affect the asset correlation and were not considered in the formula of asset correlation.
There are three limitations of this study. Firstly, we used proxy vari-ables to make comparisons. Secondly, the trend over years should be investigated further to find other important factor for determin-ing asset correlation, third, we use only the firms in the U.S., and as Lopez(2004) mentioned the country effect may be a concern.
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