Mathematical and Statistical Methods for Actuarial Sciences and Finance

Solvency evaluation of the guaranty fund at a large financial cooperative 299

to the historical data. With these distributions on hand, one now needs to evaluate the
probability that the demand for subsidies is inferior to the current level of capital.
It must be remembered that historical values have occurred in the interval ranging
from 0 to 20.8 basis points. On the other hand the current level of capital was 66.2
bp. Thus one must extrapolate the distributions to a point that is more than three
times bigger than the biggest value ever observed. With the awareness of this fact, we
proceeded to evaluate the three estimated distributions at the point corresponding to
the current level of capital. Table 3 presents the results that were obtained together
with the implied rating according to the scales of S&P and Moody’s.

Ta b le 2 .Three estimated distributions of the aggregate demand for subsidies

Distribution Weibull Gamma Log-normal

Parameter 1 α= 0.742 α= 0.527 μ=− 8. 95

Parameter 2 β= 0.000225 β= 0.000616 σ=1.36

Mean value 0.03245 0.03245 0.03245

R^2 99.14 % 98.37 % 99.06 %

Kolmogorov-Smirnov Test 0.943 0.964 0.964

Chi squared test 0.139 0.505 0.405

Chi squared test w/o the greatest contribution 0.964 0.948 0.954

Ta b le 3 .Solvency estimates of the guaranty fund using statistical distributions

Distribution Probability Probability of S&P rating Moody’s rating

of solvency default (in bp)

Weibull 99.9995 % 0.05 AAA Aaa

Gamma 99.9996 % 0.04 AAA Aaa

Log-normal 99.8142 % 18.58 BBB Baa2

As can be observed, the Weibull and the Gamma distributions give very similar
results, whereas the log-normal distribution points to a higher probability of default
and accordingly a lower credit risk rating. Under the first two distributions, the guar-
anty fund achieves the implied triple A rating very easily because a probability of
default of less than 1 basis point is enough to obtain this rating.
Thus, the aggregated approach has allowed the estimation of the solvency of the
fund. Accepting their admittedly strong assumptions, two out of three theoretical
distributions lead to a very strong evaluation of the solvency of the fund, the third
distribution showing a somewhat weaker position of the fund.