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Unformatted text preview: expected systemic losses in a given network of interconnections, and the expected systemic losses without such interconnections. As before, ﬁx a set of n nodes
n
N = {1, 2, ..., n}, a vector of outside assets c = (c1 , c2 , ..., cn ) ∈ R+ and a vector of outside liabilities
n
b = (b1 , b2 , ..., bn ) ∈ R+ . Assume the net worth wi of node i is nonnegative before a shock is realized. ¯
Interconnections are determined by the n × n liabilities matrix P = (¯ij ).
p
Let us compare this situation with the following: eliminate all connections between nodes, that is,
¯
let P o be the n × n matrix of zeroes. Each node i still has outside assets ci and outside liabilities bi .
To keep their net worths unchanged, we introduce “ﬁctitious” outside assets and liabilities to balance
the books. In particular if ci − bi < wi we give i a new class of outside assets in the amount ci =
wi − (ci − bi ). If ci − bi > wi we give i a new class of outside liabilities in the amount bi = wi − (ci − bi ).
We shall assume that these new assets are completely safe (they are not subject to shocks), and that
the new liabilities have the same priority as all other liabilities.
Let F (x1 , ..., xn ) be a joint shock distribution that is homogeneous in assets, that is, F (x1 , ..., xn ) =
G(x1 /c1 , ..., xn /cn ) where G is a symmetric c.d.f. (Unlike in the preceding results on contagion we do
not assume that the shocks are independent.) We say that F is IFR if its marginal distributions are
¯
IFR; this is equivalent to saying that the marginals of G are IFR. Given F , let L be the expected total
¯
losses in the original network and let Lo be the expected total losses when the connections are removed
as described above.
¯
Theorem 3. Let N (b, c, w, P ) be a ﬁnancial system and let N o be the analogous system with all the
connections removed. Assume that the shock distribution is homogeneous in assets and IFR. Let β + =
maxi βi < 1 and let δi = P (Xi ≥ wi ). The ratio of expected losses in the original network to the expected
losses in N o is at most ¯
δi ci
L
¯ ≤ 1 + (1 − β + ) ci .
Lo (54) Proof. By assumption the marginals of F are IFR distributed. A general property of IFR distributions
is that “new is better than used in expectation,” that is,
∀i ∀wi ≥ 0, E [Xi − wi Xi ≥ wi ] ≤ E [Xi ] 21 (55) (Barlow and Proschan 1975, p.159). It follows that
∀i ∀wi ≥ 0, E [(Xi − wi )+ ] ≤ P (Xi ≥ wi )E [Xi ] = δi E [Xi ]. (56) ¯
By (47) we know that the total expected losses L can be bounded as
¯
L≤ (Xi − wi )ui (X )]. E [Xi ∧ wi ] + E [ (57) i i From (53) we know that ui ≤ 1/(1 − β + ) for all i; furthermore we clearly have Xi − wi ≤ (Xi − wi )+
for all i. Therefore
¯
L≤ E [Xi ∧ wi ] + (1 − β + )−1
i E [(Xi − wi )+ ]. (58) i From this and (56) it follows that
¯
L≤ E [Xi ∧ wi ] + (1 − β + )−1 δi E [Xi ] i E [Xi ] + (1 − β + )−1 ≤ δi E [Xi ]. (59) i When the network connections are excised, the expected loss is simply the expected sum of the shocks,
¯
that is, L...
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This document was uploaded on 02/20/2014 for the course ECON 101 at Pontificia Universidad Católica de Chile.
 Spring '11
 Traferri

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