low/no documentation LTV >= 90 FICO > 620 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 2004 2005 2006
Figure 13: Subprime piggyback percentage
% of firsts with
piggyback
second
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Appendix 1: Derivation of Propositions 1 and 2
To verify Proposition 1, let F(x, a1, b1) and F(x, a2, b2) be two beta distributions with identical medians xm. Then F(x, a1, b1) and F(x, a2, b2) intersect at x=0, where F(x, a1, b1)=F(x, a2, b2)=0; at x=1, where F(x, a1, b1)=F(x, a2, b2)=1; and at x= xm, where F(x, a1, b1)=F(x, a2, b2)=1/2. Next, we show that these are the only points at which the two distributions will intersect.
The beta distribution is defined as F(x, a, b)=B(x, a, b)/B(0, a, b), where B(x, a, b) is the beta function.
Calculating the second derivative of the beta distribution, we obtain:
(A‐1) ∂2F/∂x2= [(a‐1)xa‐2(1‐x)b‐1 – (b‐1)xa‐1(1‐x)b‐2]/B(x, a, b)
It follows that:
(A‐2) ∂2F/∂x2 > (=) (<) 0 if and only if [(a‐1)(1‐x) – (b‐1)x] > (=) (<) 0.
Hence, there can be at most one x0 such that ∂2F/∂x2= 0; i.e., at most one inflection point of the
distribution, implying that any two beta distributions can intersect at most once in (0, 1). It follows that
if F(x, a1, b1) and F(x, a2, b2) intersect at a common median value xm, then either F(x, a1, b1) > F(x, a2, b2) for all x < xm, or vice versa. It then follows immediately that F(x, a1, b1) > F(x, a2, b2) for all x < xm if and only if F(x, a1, b1) has the wider interquartile range.
To verify Proposition 2, let F(x, a1, b1) and F(x, a2, b2) be two beta distributions. From (A‐2), if ai < 1 and bi ≥ 1 for i=1 and 2, then the distributions are strictly concave; if ai ≥ 1 and bi < 1 for i=1 and 2 they are strictly convex. In either case, the distributions will intersect only at 0 and 1, from which it follows
that F(x, a1, b1) > F(x, a2, b2) for all x in (0, 1) if and only if F(x, a1, b1) has the wider interquartile range.