Benchmarking Deutsche Bundesbank
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Benchmarking Deutsche Bundesbank's Default Risk Model, the KMV 

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  • exposé - matière potentielle : 2 years of fin
  • exposé - matière potentielle : plus market trend information
  • exposé
  • exposé - matière potentielle : information
  • fiche de synthèse - matière potentielle : the model dimensions
- 1 - Benchmarking Deutsche Bundesbank's Default Risk Model, the KMV Private Firm Model and Common Financial Ratios for German Corporations Stefan Blochwitz1 Thilo Liebig1 Mikael Nyberg Deutsche Bundesbank Deutsche Bundesbank KMV, LLC November 2000 Abstract: By comparing Gini curves and Gini coefficients that are determined on the same underlying dataset, we assess the discriminative power of Deutsche Bundesbank's Default Risk Model, KMV's Private Firm Model and common financial ratios for German corporations.
  • discriminative power
  • credit score
  • random gini curve
  • deutsche bundesbank
  • quantitative properties for further analysis
  • rating methodology
  • probabilities of default
  • minimum
  • model

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MA441: Algebraic Structures I
Lecture 2
8 September 2003
1Review:
A group G is a set with a binary operation that
satisfies four properties:
Closure
Associativity
Identity
Inverses
2Note:
The associativity property lets us write a com-
position without parentheses:
abc= a(bc)=(ab)c.
nFor a positive integer n, we write a for the
product of a taken n times.
1 nWhen n is negative, we mean (a ) .
0We take a = e.
3(From Chapter 0, page 5)
Division Algorithm
Leta,bbeintegerswithb >0. Thenthereexist
unique integers q,r with the property that
a= qb+r,
where 0 r < b.
Example:
Let a = 17 and b = 5. Then a = 3b + 2
(q =3,r =2).
4Definition:
Thegreatestcommondivisoroftwononzero
integers a and b is the largest of all common
divisors of a and b. We denote this integer by
gcd(a,b).
When gcd(a,b) = 1, we say that a and b are
relatively prime.
Fact: GCD is a linear combination
For any nonzero integers a,b, there exist in-
tegers s and t such that gcd(a,b) = as+ bt.
Moreover, gcd(a,b) is the smallest positive in-
teger of the form as+bt.
5By repeatedly applying the division algorithm
to two nonzero integers a and b, we can com-
pute gcd(a,b) and the linear combination
gcd(a,b)= as+bt.
Example:
a=17,b=5
17 = 35+2
5 = 22+1.
We can work backwards to write
1 = 5 22
2 = 17 35
1 = 5 2(17 35)
= 75 217.
6Note:
Let a,b be two relatively prime integers.
We can find s,t such that as+bt=1.
Then as 1 (mod b) and we say that a has a
multiplicative inverse modulo b.
Likewise, bt1 (mod a) and we say that b has
a multiplicative inverse modulo a.
7(From Chapter 1, page 33)
Definition:
Let G be a group of n elements.
ACayleytable(oroperationtable)isatable
with n rows, indexed by the elements of G,
and n columns, also indexed by G, such that
the table entry corresponding to (a,b) is the
product (or composition) ab in G.
Example
The dihedral group of an equilateral triangle,
D , has 6 elements corresponding to rotation3
by0,120,and240degreesandreflectionabout
an axis going through each vertex.
8(Chapter 2, page 49)
Definition:
Let S be a subset of a group G. We say that
S generates G if every element of G can be
written as a product of elements of S or their
inverses.
In other words, for any g in G, there are xi
1(i = 1...n) such that either x or x is in Si i
and
g = x x x .n1 2
We say that S is a set of generators for G.
9Example:
The dihedral group D is generated by a rota-4
tion and a flip. For example, let R = R and90
F = V be the flip about the vertical axis.
2 3 4Compute R,R ,R ,R = e. Then apply F to
2 3these four elements to get RF,R F,R F,F.
10