Constant Proportion Debt Obligations, Zeno's Paradox, and the Spectacular Financial Crisis of 2008

We study a coin-tossing model used by a ratings agency to justify the sale of constant proportion debt obligations (CPDOs), and prove that it was impossible for CPDOs to achieve in a finite lifetime the Cash-In event of doubling its capital. In the best-case scenario of a two-headed coin, we show that the goal of attaining the Cash-In event in a finite lifetime is precisely the goal, described more than two thousand years ago in Zeno's Paradox of the Dichotomy, of obtaining the sum of an infinite geometric series with only a finite number of terms. In the worst-case scenario of a two-tailed coin, we prove that the Cash-Out event occurs in exactly ten tosses. If the coin is fair, we show that if a CPDO were allowed to toss the coin without regard for the Cash-Out rule then the CPDO eventually has a high probability of attaining large net capital levels; however, hundreds of thousands of tosses may be needed to do so. Moreover, if after many tosses the CPDO shows a loss then the probability is high that it will Cash-Out on the very next toss. If a CPDO experiences a tail on the first toss or on an early toss, we show that, with high probability, the CPDO will have capital losses thereafter for hundreds of tosses; moreover, its sequence of net capital levels is a martingale. When the Cash-Out rule holds, we modify the Cash-In rule to mean that the CPDO attains a profit of 90 percent on its capital; then we prove that the CPDO game, almost surely, will end in finitely many tosses and the probability of Cash-Out is at least 89 percent. In light of our results, our fears about the durability of worldwide financial crises are heightened by the existence of other financial derivatives more arcane than CPDOs. In particular, we view askance all later-generation CPDOs that depend mean-reversion assumptions or use a betting strategy similar to their first-generation counterparts.