download; ebook; do ÂściÂągnięcia; pobieranie; pdf
Pokrewne
- Start
- Colin Forbes Stacja nr.5
- Superpamić™ć‡
- czarny gerard may k.
- 49 Pan Samochodzik i Pruska Korona Sebastian Miernicki
- Min Anchee Dziki imbir
- Forgotten Realms Anthology 02 Realms of Infamy
- Bram Stoker Dracula
- KAREL APEK Povdky z Jedn Kapsy
- Magia Ceremonialna Aleister Crowley 58 stron oUtLaWzZz
- Zajdel_Janusz_A_ _Prawo_do_powrotu
- zanotowane.pl
- doc.pisz.pl
- pdf.pisz.pl
- klimatyzatory.htw.pl
[ Pobierz całość w formacie PDF ]
(1 + is)
In the case of the second summand, we are likewise concerned with a geometric series with:
Á · b · (1 + i) Ç
a1 = Á = (4.4)
(1 + is) (1 + is)
So Á is identical in both geometric series and only a1 is different.
By using the overall equation for geometric series and substitution by equation (4.3), equation
(4.2) is rewritten as follows:
i · Á · (1 - Án) Á · b · (1 + i) · (1 - Án)
1 = + Án + (4.5)
(1 - Á) (1 + is) · (1 - Á)
Taking the mean summand Án on the left-hand side and extending by (1 - Á) results in:
Á · b · (1 + i)
(1 - Á) · (1 - Án) = i · Á · (1 - Án) + · (1 - Án) (4.6)
(1 + is)
The fact that the equation may be shortened at this point by (1 - Án), is important. This now
results in:
Á · b · (1 + i)
(1 - Á) = i · Á + (4.7)
(1 + is)
156 Risk-adjustedLendingConditions
Sothepathof thetermwiththetimeelement of thenumber of periods n has beenshortened
as a consequence of the assumption that shortfall risks are constant. The key result for the
shortfall risk hedging rate r comes thus to be independent of the termof the loan! Tasking i
out of brackets results in:
Á · b Á · b
(1-Á) =i · Á + + (4.8)
(1+is) (1+is)
Reverse substitutionof Á andreplacement of i by(is +r) accordingtoequations (1.1) and
(1.3) results in:
Ç Ç +Á · b Á · b
1- =(is +r) · + (4.9)
(1+is) 1+is (1+is)
Multiplicationby(is +r) andreplacingÇ by(1-Á) results in:
is +Á =(is +r) · (1-Á +Á · b) +Á · b(4.10)
Multiplyingout andinsertionof (1-Á +Á · b) =(1-Á") (see equation(2.6)) results in:
is +Á =is · (1-Á") +r · (1-Á") +Á · b(4.11)
Solutionbyr results in:
is +Á -is · (1-Á") -(Á · b)
r = (4.12)
(1-Á")
is +Á" -is +is · Á"
r = (4.13)
(1-Á")
Á"
r = · (1+is)(4.14)
1-Á"
As r is independent of n, the same shortfall risk hedging rate r must also be valid for a
limitless number of periods of loan term, in which the expectation value of the repayment of
capital accordingtoSection3.3.1is preciselyzero! This canbe verified, inthat equation(4.5)
is writtenfor a limitless geometric series witha limitless number of periods, thus:
Á Á · b(1+i)
1 =i · +Á" + (4.15)
1-Á (1+is) · (1-Á)
AsÇ 1alwaysapplies, Á 0alwaysapplies
d"
s
likewise. This assumption is permissible, as there is in practice no such thing as free credit .
This means that Á" is a null consequence and equation (4.15) turns into:
Á Á · b(1 + i)
1 = i · + (4.16)
1 - Á (1 + i) · (1 - Á)
Á · b · (1 + i)
(1 - Á) = i · Á + (4.17)
(1 + is)
Appendix 5: Chapter 4 Derivations 157
The comparison shows that equations (4.7) and (4.17) are identical and thus must lead to the
same conclusion. The value
Á" = Á(1 - b) (4.18)
is none other than the credit shortfall risk, which is dependent on the shortfall risk Á of the
borrower and on the probable breakdown distribution rate b.
SECTION 4.5 DERIVATION
The deviation L from the nominal amount is calculated as the difference of:
L = › - L (4.30)
This means:
L > 0 appreciation profit
L
From (4.29) and (4.30) there results:
l l
Çlj · i · L Çll · L Çlj-1 · Á · b · L · (1 + i)
L = › - L = + + - L
(1 + isl)j (1 + isl)l (1 + isl)j
j=1 j=1
(4.31)
Abbreviation using L results in:
l l
L Çlj · i Çll Çlj-1 · Ál · b · (1 + i)
= » = + + - 1 (4.32)
L (1 + isl)j (1 + isl)l (1 + isl)j
j=1 j=1
In the case of both summands, we are again concerned with geometric series. By using the
appropriate overall equations, there results:
Çll Çll
Çl
i · · 1 -
(1 + isl) (1 + isl)l Çll Ál · b · (1 + i) · 1 - (1 + isl )l
» = + + - 1 (4.33)
Çl
(1 + isl)l (1 + isl) · 1 - Çl
1 -
(1 + isl) (1 + isl)
Reformulation of the first and third summands gives:
Çll Çll
i · Çl · 1 -
(1 + isl)l Çll Ál · b · (1 + i) · 1 - (1 + isl)l
» = + + - 1 (4.34)
(1 + isl - Çl) (1 + isl)l (1 + isl - Çl)
Giving them a common denominator results in:
Çll
i · Çl · (1 + isl)l · 1 - + Çll · (1 + isl - Çl)
(1 + isl)l
» = (4.35)
(1 + isl - Çl) · (1 + isl)l
Çll
Ál · b · (1 + i) · (1 + isl)l · 1 - - (1 + isl - Çl) · (1 + isl)l
(1 + isl )l
+
(1 + isl - Çl) · (1 + isl)l
158 Risk-adjusted Lending Conditions
Multiplying them out in the numerator results in:
l
i · Çl · 1 + isl - Çll + Çll · (1 + isl - Çl)
» =
(1 + isl - Çl) · (1 + isl)l
l
Ál · b · (1 + i) · 1 + isl - Çll - (1 + isl - Çl) · (1 + isl)l
[ Pobierz całość w formacie PDF ]