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Based on Section 4.1, some operational conclusions are derived in Section 4.6. Section 4.7
summarises the most important results again, and these are illustrated in Section 4.8 by means
of an example.
4.1 FIXED ADVANCE WITHOUT REPAYMENTS
Loans of this kind take the form, at first, of a cash flow from the bank to the borrower, i.e. the
total of the loan is paid out. There then follow several cash flows from the borrower to the bank,
covering the regular interest payments and the repayment of the loan at the end of the term. In
the event that the borrower defaults, the breakdown distributions concerned take the place of
repayment. If the loan is covered, the equivalent values should count as being included in the
collateral.
The following thus applies, according to equations (1.2), (2.4) and (3.19):
n
j j-1
n
Ç · i · L Çn · L Ç · Á · b · L · (1 + i)
› = + + (4.1)
(1 + is)n
(1 + is)j (1 + is)j
j=1
j=1
› : loan market value
34 Risk-adjusted Lending Conditions
L represents the amount of loan paid out. The first summand (in the first set of brackets)
represents the sum of the discounted expectation values of the interest payments. The second
summand represents the expectation value of the discounted loan repayment.
The third summand (in the second set of brackets) represents the sum of the expectation
values of the breakdown distributions for each individual period. Here it has been assumed
that the breakdown distributions will be paid out at the end of the period in which the borrower
defaulted and will always be of the same size, irrespective of the period concerned. The factor
j-1
Ç represents the probability that default on the loan will not have occurred in the first j - 1
periods.
The factor Á represents the probability that default on the loan does occur in period j.
L(1 + i) represents the lender s demands, where for the sake of simplicity it is assumed that
the full amount of interest for the period j will become due in the case of bankruptcy. It
is thus being assumed, for the sake of simplicity and without departing much from real-
life circumstances, that if bankruptcy occurs at all it occurs precisely at the end of the pe-
riod concerned. Under our assumptions interest was in practice paid during the first j - 1
periods.
Comparison with the study of Fooladi, Roberts and Skimmer is interesting at this juncture.
Their thesis is indeed the duration of obligations in cases where credit-worthiness is at risk
[FRSK97]; they did, however, also have to establish a correlation between loan interest,
risk-free interest and shortfall risk. Their starting equation is therefore very similar. They
do, however, deal with the general case right from the beginning (see later in Chapter 5).
When a loan is paid out, its market value must correspond at least to its nominal value:
› e" L. The bank would otherwise be already accepting a loss at that point. In what follows it
is intended that r is calculated in such a way that the market value and nominal value of the
loan are identical when it is paid out: › = L (cf. Section 1.6).
The result reads as follows (see Appendix 5 for derivation):
Á"
r = · (1 + is) (4.14)
1 - Á"
In this way the final result for the shortfall risk hedging rate r is, following from the assumption
that shortfall risks are constant, independent of the number of periods in the term of the
loan!
This comes about because the revenue arising from the shortfall risk hedging rate in the
case of interest being paid in each period precisely covers the rise in the shortfall risk of the
loan repayment. Seen mathematically, it is a consequence of the properties of geometrical
series.
The correlation between r, is, and Á" and b is portrayed graphically in Figures 4.1 and 4.2.
The higher the risk-free rate of interest is, the higher will also be the risk premium r in
the case of the same credit shortfall risk Á", as interest payments do indeed also have to be
 insured from the point of view of risk.
If a 100% breakdown distribution were assumed in an extreme case, then the risk of default
on the loan would be zero, and with it the risk premium too.
Constant Shortfall Risk 35
Figure 4.1
r
1%
0.5%
b
0%
0% 50% 100%
r = 1%, is = 3%
Figure 4.2 [ Pobierz całość w formacie PDF ]