OFRwp0009_GlassermanYoung_HowLikelyContagionFinancialNetworks

# From 37 we see that a simple sucient condition for

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Unformatted text preview: j &gt; c1 . Thus contagion is impossible if wj /w1 &gt; β1 (λ1 − 1). (32) 2≤j ≤m Let us therefore assume that w1 + (1/β1 ) 2≤j ≤m o wj ≤ c1 . Deﬁne the random variables Yi = Xi /ci . Weak contagion from 1 to D holds if wj ) ≤ P (Y1 &gt; w1 /c1 )P (Y2 &gt; w2 /c2 ) · · · P (Ym &gt; wm /cm ) P (Y1 &gt; w1 /c1 + (1/β1 c1 ) 2≤j ≤m = P (Y1 &gt; w1 /c1 )P (Y1 &gt; w2 /c2 ) · · · P (Y1 &gt; wm /cm ). (33) where the latter follows from the assumption that the Yi are i.i.d. By assumption Y1 is IFR, hence P (Y1 &gt; s + t|Y1 &gt; s) ≤ P (Y1 &gt; t) for all s, t ≥ 0. (See for example Barlow and Proschan 1975, p.159.) It follows that P (Y1 &gt; wk /ck ) 1≤k≤m = P (Y1 &gt; w1 /c1 )P (Y1 &gt; w1 /c1 + w2 /c2 |Y1 &gt; w1 /c1 ) · · · P (Y1 &gt; w1 /c1 + w2 /c2 + · · · + wm /cm |Y1 &gt; w1 /c1 + w2 /c2 + · · · + wm−1 /cm−1 ) ≤ P (Y1 &gt; w1 /c1 )P (Y1 &gt; w2 /c2 ) · · · P (Y1 &gt; wm /cm ) Together with (33) this shows that contagion from 1 to D is weak provided that P (Y1 ≥ w1 /c1 + (1/β1 c1 ) wj ) ≤ P (Y1 &gt; 2≤j ≤m wk /ck ). (34) 1≤k≤m This clearly holds if wj ≥ w1 /c1 + (1/β1 c1 ) 2≤j ≤m wk /ck , (35) 1≤k≤m which is equivalent to wj ≥ (1/β1 c1 ) 2≤j ≤m λ−1 . j wj /cj = 2≤j ≤m (36) 2≤j ≤m Since c1 = λ1 w1 , we can re-write (36) as λ−1 . j wj /w1 ≥ β1 λ1 2≤j ≤m (37) 2≤j ≤m We have therefore shown that if contagion from 1 to D = {2, 3, . . . , m} is possible at all, then (37) is a suﬃcient condition for weak contagion. From (37), we see that a simple suﬃcient condition for weak contagion is cj ≥ β1 c1 , j = 2, . . . , m, and the condition m j =2 wj &gt; β1 (c1 − w1 ) makes contagion impossible. 16 The assumption of independent shocks to diﬀerent nodes is conservative. If we assume that direct shocks to diﬀerent nodes are positively dependent, as one would expect in practice, the bounds in Theorems 1 and 2 will be lower and the relative likelihood of default through contagion even smaller. A particularly simple case arises when the primary shocks Xi are independent with a negative exponential distribution of form f (xi ) = µe−µxi , xi ≥ 0. (38) If we further assume that c1 = · · · = cm ≡ c and β1 = 1, then the two probabilities compared in (33) are equal, both evaluating to exp(−µ j ∈D wj /c). In this sense, the exponential distribution is a borderline case in which the probability of a set of defaults from a single shock is roughly equal to the probability from multiple independent shocks. We say “roughly” because the left side of (33) is an upper bound on the probability of default through contagion, and in practice the βi are substantially smaller than 1. In the example of Figure 2(a), we have seen that contagion from the central node requires a shock greater than 80, which has probability exp(−80µ) under an exponential distribution. For direct defaults, it suﬃces to have shocks greater than 5 at the peripher...
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