I. INTRODUCCIÓN
1.3. Teorías relacionadas al tema
1.3.2. Apoyo Social
Proof of Lemma 3.4 For the proof of the first two statements see Huschens and Vogl (2002, p. 287). The proof of the third state-ment follows from McNeil et al (2005, pp. 352–3) and the fact that P(Di= 1 | Z = z) = pi(z) and P(Di= 0 | Z = z) = 1 − pi(z).
Proof of Lemma 3.7
(i) For the proof of this statement see Schönbucher (2003, pp. 307–8), where a general default barrier is used instead of Φ−1(π ).
(ii) The proof of this statement follows from Schönbucher (2003, p. 308) and the fact that
Pr(nVn= k) = Pr
Vn= k
n
for k= 0, 1, . . . , n
(iii) Given that Z= z, the default variables Diare independent and identically Bernoulli distributed with parameter E[Di | Z = z] = p(z)∈ [0, 1].
it follows that Vnconverges in quadratic mean to p(Z), ie
n→∞limE[(Vn− p(Z))2]= 0
For each fixed value z∈R, the strong law of large numbers implies that Vn converges to p(z) with probability 1. As a result of this, convergence with probability 1 to p(Z) also holds unconditionally (Bluhm et al 2003, p. 89).
(iv) This statement follows from Bluhm et al (2003, Proposition 2.5.8) and the fact that−Φ−1(α)= Φ−1(1− α) for 0 < α < 1.
Proof of Lemma 3.9
(i) The stochastic independence of Z and Yiimplies that the random variables√
iZ and
1− iYi are also stochastically independent.
Since the distribution functions of these random variables are FZ
the distribution function of Aican be expressed as the convolution given in Equation 3.12.
(ii) Assumption 3.8 implies that E[Ai]=E[ (iii) This statement follows immediately from Equation 3.12.
Proof of Theorem 3.11
(i) From Equation 3.13, it follows that Diis Bernoulli distributed with default probability
Pr(Di= 1) = Pr(Ai F−Ai(pi))= FAi(F−Ai(pi))= pi
for i= 1, . . . , n, where the last equality follows from Equation 3.15.
(ii) The probabilities Pr(Di= 1 | Z = z) are defined uniquely for all z∈ ΩZ⊆R with Pr(Z ∈ ΩZ)= 1. Then
holds, where the last equality follows from Assumption 3.8(i).
Proof of Theorem 3.13
(i) This statement follows from Lemma 3.9 and the homogeneous correlation parameters of Assumption 3.5.
(ii) This statement follows from Theorem 3.11(i) and the homoge-neous default probabilities of Assumption 3.5.
(iii) This statement follows from Theorem 3.11(ii) and the homoge-neous default probabilities and correlation parameters of Assump-tion 3.5.
(iv) Equation 3.16 follows from Pr
and the fact that for given Z= z the default variables are independ-ent and idindepend-entically Bernoulli distributed with parameter p(z; FZ, FY) so that nVn| Z = z ∼ Bin(n, p(z; FZ, FY)).
(v) Given Z = z, the default variables Di are independent and identically distributed withE[Di| Z = z] = p(z; FZ, FY)∈ [0, 1].
Analogously to the proof of Lemma 3.7(iii), it follows from E[Vn| Z = z] = p(z; FZ, FY),
V[Vn| Z = z] =p(z; FZ, FY)(1− p(z; FZ, FY)) n
and
0 E[(Vn− p(Z; FZ, FY))2]
=E
p(Z; FZ, FY)(1− p(Z; FZ, FY)) n
1 4n that
n→∞limE[(Vn− p(Z; FZ, FY))2]= 0 ie, Vnconverges in quadratic mean to p(Z; FZ, FY).
For each fixed value z in a set ΩZ⊆R with Pr(Z ∈ ΩZ)= 1, the strong law of large numbers implies that Vnconverges to p(z; FZ, FY) with probability 1. Consequently, convergence with probability 1 to p(Z; FZ, FY)also holds unconditionally (Bluhm et al 2003, p. 89).
This follows from the existence of a regular conditional probability, which is guaranteed for real-valued random variables (Shorack 2000, pp. 168ff).
Proof of Lemma 3.15 The one-sided inequality of Cantelli for each random variable X with distribution function FX∈F0,1is (DasGupta 2008, p. 634)
FX(t) 1
1+ t2, t < 0
Let W be a random variable with distribution function
FW(t)=
⎧⎪
⎪⎨
⎪⎪
⎩ 1
1+ t2 for t < 0
1 otherwise
Then FX(t) FW(t) for all t∈R and therefore FW−(α) FX−(α)for all 0 < α < 1 and especially F−W(θ) F−X(θ)with
FW−(θ):= min{t | FW(t) θ} = min t < 0
1
1+ t2 θ
= aθ
This proves the first inequality in Equation 3.20. The second inequal-ity in Equation 3.20 follows from the definition of F−(θ)and F+(θ).
The third inequality in Equation 3.20 can be derived from the first inequality in Equation 3.20 as follows. If FX∈F0,1, then the distribu-tion funcdistribu-tion F−Xof−X is also contained inF0,1. The first inequality in Equation 3.20 gives
−
θ
1− θ F−−X(1− θ) and therefore
−F−−X(1− θ) bθ
Using F+X(θ)= −F−−X(1− θ) (Föllmer and Schied 2004, p. 177), the third inequality in Equation 3.20 follows.
Proof of Lemma 3.16
(i) If X has a dichotomous distribution, then Pr(X= a) = θ = 1 − Pr(X = b)
for a parameter 0 < θ < 1 and two real numbers a < b. FX ∈ F0,1
implies the two restrictions
E[X] = aθ + b(1 − θ) = 0 and E[X2]= a2θ+ b2(1− θ) = 1 for a and b. For fixed θ, these equations hold if and only if a = aθ
and b= bθ.
(ii) The distribution function in Equation 3.22 and the two quantile functions in Equations 3.23 and 3.24 follow immediately from the dichotomous distribution defined by Equation 3.21.
Proof of Theorem 3.17 The theorem follows immediately from Equa-tions 3.20 and 3.25.
Proof of Lemma 3.18
(i) Let Z and Y be stochastically independent random variables with FZ = FY = Gθ and 0 < θ < 1. The discrete random variable
A := √Z + In general, the support may consist of four different points
s1<min{s2, s3} max{s2, s3} < s4
for k= 0, 1, . . . , n with Pr(Z = aθ)= θ and Pr(Z = bθ)= 1−θ. From Theorem 3.13(iii), it follows for the two Bernoulli parameters in the mixture distribution of Vnthat
p(aθ; Gθ, Gθ)= Pr(D1= 1 | Z = aθ)
= Pr(A F−A(π )| Z = aθ)
= Pr(A max{s2, s3} | Z = aθ)
= Pr(A < s4| Z = aθ)
= Pr(
Z+
1− Y < s4| Z = aθ)
= Pr(
aθ+
1− Y < s4) (3.33)
= 1 and
p(bθ; Gθ, Gθ)= Pr(A F−A(π )| Z = bθ)
= Pr(
bθ+
1− Y < s4) (3.34)
= Pr(Y < bθ)
= θ
where Equations 3.33 and 3.34 follow from the independence of Z and Y. Therefore
Pr(Vn= 1) = p(aθ; Gθ, Gθ)nPr(Z= aθ)+ p(bθ; Gθ, Gθ)nPr(Z= bθ)
= θ + θn(1− θ) which implies Equation 3.26, if
α >1− (θ + θn(1− θ)) =√
1− π(1 − (1 −√
1− π)n) (ii) Theorem 3.13 implies
V∞= Pr(D1= 1 | Z) = Pr(A FA−(π )| Z) From the derivation above, it follows that
Pr(A F−A(π )| Z = aθ)= 1 and Pr(A F−A(π )| Z = bθ)= θ Therefore, the distribution of V∞is given by
Pr(V∞= 1) = Pr(Z = aθ)= θ Pr(V∞= θ) = Pr(Z = bθ)= 1 − θ
⎫⎬
⎭ (3.35)
which implies Equation 3.27, if α > 1− θ =√ 1− π.
Proof of Theorem 3.19
(i) Pr(Vn 1) = 1 implies VaRα[Vn] 1 and AVaRα[Vn] 1.
If α >√
1− π(1 − (1 −√
1− π)n) and if FZ = FY = Gθ with θ= 1 −√
1− π, then VaRα[Vn] = AVaRα[Vn] = 1, ie, the upper bounds are reached. Using Gθ∈G, the equations
max{VaRα[Vn]| FZ, FY∈G} = max{AVaRα[Vn]| FZ, FY∈G} = 1 result. Since the upper bound 1 is reached as maximum in the setG and sinceG ⊂ F0,1, Equations 3.28 follow.
Using Equation 3.2 with value-at-risk and average value-at-risk as risk measures and with Vn as the random variable, the worst-case loss distribution in Equation 3.29 follows from Equation 3.32 with Pr(Z = aθ)= θ, Pr(Z = bθ) = 1 − θ, p(aθ; Gθ, Gθ) = 1 and p(bθ; Gθ, Gθ)= θ, which are derived in the proof of Lemma 3.18.
(ii) The analogous reasoning for V∞with α >√
1− π leads to max{VaRα[V∞]| FZ, FY∈G} = max{AVaRα[V∞]| FZ, FY∈G}
= 1
Since the upper bound 1 is reached as maximum in the setG and sinceG ⊂ F0,1, Equations 3.30 follow.
For the random variable V∞ the worst-case loss distribution follows from Equation 3.35 with θ= 1 −√
1− π.
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