La enseñanza para la comprensión desde el área de filosofía

In order to investigate the impact of using different time windows on the construction of time-varying z-score measures, this study compares z-scores estimated from 3-year (i.e. 12 quarters) to 6-year (i.e. 24 quarters) window lengths for the New Zealand banks. Z-scores are computed using approach Z1. Summary statistics of z-scores using different window lengths are reported in Table 1043, and relevant graphs with 3-year to 6-year windows are shown in Figure 7 (a)-(h). Numbers in bold are statistically significant for the Kolmogorov- Smirnov test at 10% level.

43 _{For easy comparison, Table 10 reports results from the March quarter of 2002 for ANZ NZ, ASB, BNZ, and}

TSB; from the March quarter of 2011 for Kiwibank; and from the December quarter of 2012 for WNZL. In this way, z-scores estimated from different window sizes have the same number of observations.

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(a) Individual z-scores estimated with 3-year rolling windows

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(c) Individual z-scores estimated with 4-year rolling windows

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(e) Individual z-scores estimated with 5-year rolling windows

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(g) Individual z-scores estimated with 6-year rolling windows

(h) Aggregate z-score and minus one z-scores estimated with 6-year rolling windows

Figure 7 – Trends of individual z-scores, aggregate z-scores, and minus one z-scores, with 3-year to 6-year window lengths

This figure shows time series of individual z-scores, aggregate z-scores, and minus one z- scores for the six New Zealand banks, using rolling windows from 3-year to 6-year window lengths. Z-scores are computed using approach Z1.

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Table 10 – Comparison based on 3 to 6 year rolling windows, New Zealand banks, using approach Z1 and quarterly data

This table reports summary statistics of time-varying z-scores computed with rolling windows from 3 to 6 years and on a quarterly basis. Z- scores are computed using approach Z1. Numbers in bold are statistically significant for the Kolmogorov-Smirnov test at 10% level.

ANZ NZ ASB BNZ Kiwi TSB WNZL Aggregate no

ANZ NZ no ASB no BNZ no Kiwi no TSB

no WNZL Panel (a) 3-year window

Mean 40.7 76.6 23.7 31.3 57.6 56.1 45.1 39.9 39.3 52.7 39.6 44.3 47.8 St. dev. 13.4687 42.2402 15.2773 12.2860 20.6216 4.3764 20.1020 20.0347 17.2440 17.5521 15.6587 19.7265 7.8393

Change -11.58% -12.91% 16.86% -2.88% -1.88% -8.21%

K-S p-value 0.087 0.273 0.063 0.840 0.983 0.268

Panel (b) 4-year window

Mean 36.0 68.6 21.7 26.8 53.9 43.5 40.4 36.1 35.1 46.1 31.7 39.6 37.7 St. dev. 10.451 40.6217 13.4338 6.8114 17.4946 5.7403 17.1732 17.9897 14.536 14.0965 13.2988 16.8258 10.6116

Change -10.70% -13.07% 14.17% -2.00% -1.87% -6.97%

K-S p-value 0.029 0.139 0.017 0.781 0.983 0.268

Panel (c) 5-year window

Mean 32.5 62.5 20.3 25.1 51.2 34.6 36.6 32.9 31.7 41.6 25.4 36.0 27.9 St. dev. 8.6364 37.9117 12.5469 3.4076 15.7854 3.4447 15.1205 15.8986 12.4848 13.0685 7.9436 14.8044 8.5758

Change -10.24% -13.44% 13.60% -1.71% -1.87% -6.16%

K-S p-value 0.029 0.214 0.001 0.972 0.945 0.075

Panel (d) 6-year window

Mean 30.2 57.5 19.0 25.6 48.7 29.9 34.2 30.5 29.4 38.9 22.1 33.6 21.9 St. dev. 7.2176 33.6242 11.6116 3.1424 14.9894 1.6475 13.2299 13.7029 10.3077 11.9576 2.2611 12.9639 2.1925

Change -10.92% -14.16% 13.76% -1.84% -1.88% -7.93%

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It is obvious that z-scores estimated with a 3-year window are more volatile. Meanwhile, average values of z-scores, including individual z-scores, aggregate z-scores and minus one z-scores, all decrease in value with longer window lengths, as standard deviations of ROA increase. On the other hand, as window extends, the period with lower values of z-scores during the recession, especially for BNZ, are longer, indicating higher risk, although the extreme values of its z-score are not as low as for shorter windows. With different lengths of window, systemic significance of individual banks generally remains the same. The four largest banks always have greater systemic risk contributions, and the statistical significance of the difference between the distributions of aggregate z-score and minus one z-score also increases with longer window lengths.

In order to determine the optimal window length for time-varying z-score, this study takes aggregate z-score as an example, and compute aggregate z-scores using rolling windows from 4 quarters. Aggregate z-scores are computed using approach Z1. The mean values of aggregate z-score and the related coefficient of variation are reported in Table 1144. Figure 8 plots the average value of aggregate z-score estimated from rolling windows.

44 _{Values of aggregate z-score are reported till 36 quarters rolling windows. Aggregate z-scores computed from}

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Table 11 – Mean values of aggregate z-score and coefficient of variation, rolling window

This table reports the mean values of aggregate z-score computed with rolling windows from 4 to 36 quarters, and the related coefficient of variation. Aggregate z-scores are computed using approach Z1.

Rolling window

(Quarters) Aggregate z-score Coefficient of variation

4 89.4 93.70% 5 74.6 83.35% 6 64.1 64.06% 7 58.8 59.18% 8 54.3 52.24% 9 51.4 50.33% 10 49.1 48.82% 11 46.9 46.18% 12 45.0 44.07% 13 43.3 41.66% 14 41.7 39.60% 15 40.9 39.88% 16 40.0 40.04% 17 39.1 40.16% 18 38.2 39.89% 19 37.4 39.77% 20 36.7 39.68% 21 36.1 39.57% 22 35.4 39.22% 23 34.8 38.91% 24 34.2 38.61% 25 33.9 38.35% 26 33.9 37.76% 27 33.9 37.41% 28 33.9 37.30% 29 34.0 37.34% 30 34.2 37.34% 31 34.3 37.37% 32 34.2 37.33% 33 34.0 37.20% 34 33.7 36.99% 35 33.5 36.70% 36 33.2 36.45%

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Figure 8 – Mean value of aggregate z-scores with rolling windows, New Zealand banking market

This figure shows how aggregate z-score changes when estimated using rolling windows from 4 quarters. X-axis is the number of quarters used for computation, and y-axis is the mean value of aggregate z-score. Aggregate z-scores are computed using approach Z1.

As indicated in Figure 8, with the increase of window lengths, aggregate z-score first decreases quickly and then levels out. This is also supported by the coefficient of variation in Table 11, which also decreases with the increase of window sizes, meaning that the values of aggregate z-score are becoming less volatile. This indicates that the use of a rolling window to compute time-varying z-score generally requires a relatively long sample period to derive reliable z-score estimates, which is likely to be the reason that some prior z-score related literature computes elements of z-score over the entire sample period.

Aggregate z-score becomes stable at around 33.9 after 25 quarters (approximate 6 years), which has slight fluctuations after 29 quarters (approximate 7 years). This is consistent with an approximate timeline of a CEO turnover, which may change a bank’s strategy and impact on bank performance (Fahlenbrach, Prilmeier, and Stulz, 2012). However, the selection of window size also depends on data availability; data may not be readily available for all countries or banks. Other things being equal, it is suggested a 4-year or 5-year window to be

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used, which generates sufficient observations, but also provides reasonable scope to allow for changes occurring when banks change their risk profiles.

To sum up, the values of z-scores are becoming more stable with a longer window length. For the use of rolling windows, this study finds an optimal window size of 25 quarters (approximate 6 years). This is also consistent with an approximate timeline of CEO turnovers, which may change a bank’s strategy and risk profile. For studies restricted by data availability, it is suggested to use a 4-year or 5-year window, as it provides reasonable scope to allow for changes in a bank’s risk profile.