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The relative widths of the individual rating levels here come to A = 1, B = 8, C = 64 and
D = 512. The factor from level to level thus amounts to 23 = 8, as three times fewer levels
occur than in the preceding example.
2.5 AMPLIFICATION OF THE RATING SYSTEM FOR VERY
COMPETITIVE MARKETS
The markets for loans to debtors with very high financial standing, above all, are often extremely
competitive. The rating system presented in Table 2.1 might therefore be insufficiently precise
for such markets. It should therefore be pointed out that the system presented in Table 2.1 may
be further refined. There are three times fewer levels in the simplified system in Table 2.2 than
in the system in Table 2.1. The relative width of the individual levels therefore grows by a factor
of 8 = 23; the exponent 3 being attributable to the number of levels being three times smaller.
Analogously, this factor comes to 21/3, in the case of a system with three times as many levels,
being the third root of 2. By analogy with the method of calculation in the preceding section,
one thus obtains the refined rating system shown in Table 2.3 when the number of levels is
tripled.
Table 2.3 could be developed over all levels to D. This would, however, be superfluous,
as the markets for loans to borrowers of lower financial standing become progressively less
competitive.
It must be noted that " and are in each case identical for the AAA, AA, A and BBB
ratings in Table 2.2, and for the AAA-, AA-, A- and BBB- ratings in Table 2.3.
Table 2.3 Refined rating system
Value of " (%)
" according to according to
Rating from to rating level (%) rating level (rounded)
AAA+ 0.0000 0.0063 0.0063 15755
AAA" 0.0063 0.0143 0.0143 6971
AAA- 0.0143 0.0244 0.0244 4095
AA+ 0.0244 0.0371 0.0371 2694
AA" 0.0371 0.0531 0.0531 1883
AA- 0.0531 0.0733 0.0733 1365
A+ 0.0733 0.0986 0.0986 1014
A" 0.0986 0.1306 0.1306 765
A- 0.1306 0.1709 0.1709 585
BBB+ 0.1709 0.2217 0.2217 451
BBB" 0.2217 0.2857 0.2857 350
BBB- 0.2857 0.3663 0.3663 273
Part II
Mathematical Foundations of the Model
Mathematical Foundations of the Model
Probability model: Development of j
Calculation of the shortfall risk hedging rate in the special case
of shortfall risks being constant
Calculation of the shortfall risk hedging rate in the general case
of variable shortfall risk
Shortfall risk on uncovered loans on the basis of statistics
3
Probability Model: Development of
j
As may be inferred from the basic equation (1.2), determining the probability j of cash flow
C being fulfilled is of decisive importance for the model we are describing. The correlation
j
between the shortfall risk and the survival chance and of the probability of fulfilment j
will be derived in Section 3.1, with the aid of probability calculus.
In Section 3.2 we will show how the shortfall risk and survival chance might be converted
over various terms. Conclusions may be drawn from the results of Section 3.1 for loans that
are unlimited in time and for reasonable terms in relation to the shortfall risk . This will
be presented in Section 3.3. For the sake of clarity the results of Chapter 3 will be presented
again in Section 3.4.
3.1 DETERMINING THE PROBABILITY OF CASH FLOWS
BEING FULFILLED
What we are concerned to do below is develop a model for determining probabilities using
equation (1.2). The components needed for this model are defined as follows (with the verb to
default being used as a synonym for the sentence to no longer be able to meet commitments
to the bank in full ):
n is the number of periods in the term of the loan.
j represents the period concerned: 1 d" j d" n.
j is the probability of the borrower defaulting within period j.
j is the probability of the borrower not defaulting within period j.
j is the probability of the borrower defaulting between the first period and period j.
j is the probability of the borrower not defaulting between the first period and period j.
(n) is the probability of the borrower defaulting at some point during the term of the loan
of n periods.
(n) is the probability of the borrower not defaulting during the whole of the term of the
loan of n periods.
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