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B - B C - C = B C (8.10)
Multiplying out gives:
2 2 2 2 2 2
B C - B C + C B + B C = B C (8.11)
Which leads to:
1 = B + C (8.12)
r may thus only assume the value -1 if the sum of both individual risks is equal to 1. This
means, on the basis of equation (8.6), that b = 0 in this case.
Maximum
Maximum values for r and B )" C are reached by the consideration that such is the case if
collateral falling short leads to default on the loan, i.e. B )" C = C. The frequency distribution
96 Risk-adjusted Lending Conditions
The default occurrence has a value of 1 .
collateral
The no default occurrence has a value of 0 .
1 0 a
0 b d
borrower
0 1
Figure 8.3
can then be portrayed as follows and illustrated in Figure 8.3.
B )" C =
B = + d
(8.13)
C =
1 = + b + d
As d 0 applies as probability value for d, this means that B > C. The shortfall risk
of the borrower is thus greater or at least equal to the shortfall risk of the collateral and to
the combined shortfall risk respectively. This makes sense, as the bank would of course not
otherwise fall back on collateral at the time of granting the loan. One obtains the following
[KREY91, S. 304] for the correlation of this frequency distribution:
- ( + d)
rmax = (8.14)
[( + d) - ( + d)2] [ - 2]
After insertion of the shortfall risks according to equation (8.12), one obtains:
C - B C
rmax = (8.15)
2 2
B - B C - C
According to statistical theory, the highest value that a correlation coefficient may assume
is +1 [BOHL92, S.235]. We intend therefore to investigate under what assumptions this value
may be reached.
C - B C
1 = (8.16)
2 2
B - B C - C
Conversion results in:
2 2
B - B C - C = (C - B C)2 (8.17)
Multiplying out gives:
2 2 2 2 2 2 2 2
B C - B C - C B + B C = C - 2 B C + B C (8.18)
Loans Covered against Shortfall Risk 97
Which leads to:
2
B - B - C + B C = 0 (8.19)
Taking out of brackets and abbreviation gives:
B = C (8.20)
r may thus only assume the value +1 if both individual risks are equally large. This means,
on the basis of equation (8.13), that d = 0. Borrower default and collateral fall short therefore
always occur either simultaneously or not at all.
8.3 SHORTFALL RISK OF THE COVERED LOAN
This section is concerned with bringing together the individual elements that have been calcu-
lated to date. Equation (8.5) details the risk that both the borrower defaults and the collateral
falls short, which is indeed a prerequisite for default occurring on a covered loan. This proba-
bility must be multiplied by the risk of loss (1 - bC), in order to obtain the shortfall risk of the
covered loan. In proceeding thus it is implicitly assumed that only the realisation of the col-
lateral makes breakdown distributions possible, but not the realisation of the borrower s other
asset values. This does, however, make sense and is in line with the principle of conservatism.
(B )" C)" = (B )" C) (1 - bC) (8.21)
By insertion of equations (8.5) and (7.41) one obtains:
C
"
2 2
(B )" C)" = B C + r B - B C - C (8.22)
C
In the special case r = 0, i.e. in which there is no correlation at all, equation (8.20) reduces
itself to:
"
(B )" C)" = B C if r = 0 (8.23)
In the special case of the maximum correlation, equation (8.22) reduces itself, after insertion
of equation (8.15) to:
"
(B )" C)" = C if r = rmax (8.24)
B must be determined according to the rules from Section 7.4 for applying equations (8.22)
and (8.23).
In practice the challenge consists above all in determining the correlation coefficient r for
the various loan transactions empirically, using statistical methods. Here one may put forward
the supposition that the correlation coefficient for many loan transactions lies either close to
zero (for instance, in the case of financing owner-occupied houses) or close to the maximum
(for example, in the case of loans secured against collateral in the form of securities), and the
application of equations (8.23) and (8.24) is therefore permissible. [ Pobierz całość w formacie PDF ]
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B - B C - C = B C (8.10)
Multiplying out gives:
2 2 2 2 2 2
B C - B C + C B + B C = B C (8.11)
Which leads to:
1 = B + C (8.12)
r may thus only assume the value -1 if the sum of both individual risks is equal to 1. This
means, on the basis of equation (8.6), that b = 0 in this case.
Maximum
Maximum values for r and B )" C are reached by the consideration that such is the case if
collateral falling short leads to default on the loan, i.e. B )" C = C. The frequency distribution
96 Risk-adjusted Lending Conditions
The default occurrence has a value of 1 .
collateral
The no default occurrence has a value of 0 .
1 0 a
0 b d
borrower
0 1
Figure 8.3
can then be portrayed as follows and illustrated in Figure 8.3.
B )" C =
B = + d
(8.13)
C =
1 = + b + d
As d 0 applies as probability value for d, this means that B > C. The shortfall risk
of the borrower is thus greater or at least equal to the shortfall risk of the collateral and to
the combined shortfall risk respectively. This makes sense, as the bank would of course not
otherwise fall back on collateral at the time of granting the loan. One obtains the following
[KREY91, S. 304] for the correlation of this frequency distribution:
- ( + d)
rmax = (8.14)
[( + d) - ( + d)2] [ - 2]
After insertion of the shortfall risks according to equation (8.12), one obtains:
C - B C
rmax = (8.15)
2 2
B - B C - C
According to statistical theory, the highest value that a correlation coefficient may assume
is +1 [BOHL92, S.235]. We intend therefore to investigate under what assumptions this value
may be reached.
C - B C
1 = (8.16)
2 2
B - B C - C
Conversion results in:
2 2
B - B C - C = (C - B C)2 (8.17)
Multiplying out gives:
2 2 2 2 2 2 2 2
B C - B C - C B + B C = C - 2 B C + B C (8.18)
Loans Covered against Shortfall Risk 97
Which leads to:
2
B - B - C + B C = 0 (8.19)
Taking out of brackets and abbreviation gives:
B = C (8.20)
r may thus only assume the value +1 if both individual risks are equally large. This means,
on the basis of equation (8.13), that d = 0. Borrower default and collateral fall short therefore
always occur either simultaneously or not at all.
8.3 SHORTFALL RISK OF THE COVERED LOAN
This section is concerned with bringing together the individual elements that have been calcu-
lated to date. Equation (8.5) details the risk that both the borrower defaults and the collateral
falls short, which is indeed a prerequisite for default occurring on a covered loan. This proba-
bility must be multiplied by the risk of loss (1 - bC), in order to obtain the shortfall risk of the
covered loan. In proceeding thus it is implicitly assumed that only the realisation of the col-
lateral makes breakdown distributions possible, but not the realisation of the borrower s other
asset values. This does, however, make sense and is in line with the principle of conservatism.
(B )" C)" = (B )" C) (1 - bC) (8.21)
By insertion of equations (8.5) and (7.41) one obtains:
C
"
2 2
(B )" C)" = B C + r B - B C - C (8.22)
C
In the special case r = 0, i.e. in which there is no correlation at all, equation (8.20) reduces
itself to:
"
(B )" C)" = B C if r = 0 (8.23)
In the special case of the maximum correlation, equation (8.22) reduces itself, after insertion
of equation (8.15) to:
"
(B )" C)" = C if r = rmax (8.24)
B must be determined according to the rules from Section 7.4 for applying equations (8.22)
and (8.23).
In practice the challenge consists above all in determining the correlation coefficient r for
the various loan transactions empirically, using statistical methods. Here one may put forward
the supposition that the correlation coefficient for many loan transactions lies either close to
zero (for instance, in the case of financing owner-occupied houses) or close to the maximum
(for example, in the case of loans secured against collateral in the form of securities), and the
application of equations (8.23) and (8.24) is therefore permissible. [ Pobierz całość w formacie PDF ]