a CDO depending on a pool of D of the underlying assets. We assume MD = NC and that every asset occurs in exactly one pool. Clearly, assets in the same class should be distributed to different products in order to ensure the diversity of the products. Each CDO will have assets from distinct classes (and hence have stochastically independent yields) so that sum of their yields follows the central limit theorem. Suppose each one of the M products has the following design: it pays off N/(3M) units as long as the number of assets in its pool that defaulted is at most for some parameter t (set to be about ), and otherwise it pays 0. Henceforth we call such a product a “binary CDO”; it can be viewed as the senior tranche of a simple CDO.f Assets contained in the same product come from different classes, the yields of good assets are uniformly iid, so the expected number of defaults among D good assets is D/2 and the standard deviation is . Now the central limit theorem applies and the total number of defaults may be assumed to be distributed like a Gaussian. Thus so long as the fraction of lemon classes is much smaller than the safety margin of t standard deviations, the probability of default for an individual CDO is tiny. Thus, if V denotes the combined expected yield of these M products, then V ≈ M × N/3M = N/3. (The exact value of V is unimportant below.)

If the bank were to pick the pools truly randomly— i.e., the entire portfolio of assets is randomly partitioned into the M pools—then the portfolio’s expected yield is only mildly affected by the presence of lemons. Specifically, if V is the expected yield when there are no lemon classes, then it can be shown that the yield is still V – o(n) (i.e. larger than V – n for any  > 0) when the number of lemon classes is 2n, the maximum possible. In this sense derivatives can help significantly reduce the lemon wedge from n to o(n), thus performing their task of allowing a party to sell off the least information-sensitive portion of the risk.

However, the above description assumed that the seller
creates the pools disinterestedly using pure randomness. But
this may be against his self-interest given his secret informa-
tion! Given that the seller’s interest is to give out the minimum
yield possible, as long as this is undetected by the buyer, it
turns out that his optimum strategy is to pick some subset of
m of the financial products, and ensure that the lemon assets
are overrepresented in the pools of these m products—to an
extent about which is just enough to significantly skew
the probability of default. We call this subset of n CDOs the
“boobytrap.” Thus the CDOs in the boobytrap have a much
higher probability of default than buyers expect, causing the
expected yield of the entire portfolio of CDOs to be smaller by
an amount proportional to m (roughly mN/(3M) ).g
With some settings of m the tampered derivative can
have much smaller yield than the yield of V–o(n) obtained
f This is a so-called synthetic binary option. The more popular CDO derivative
described above behaves in a similar way, except that if there are defaults
above the threshold (in this case ) then the payoff is not 0 but
the defaults are just deducted from the total payoff. We call this a “tranched
CDO” to distinguish it from the binary CDO.
g The non-booby trapped CDOs will have a slightly smaller probability of
default than in the untampered (i.e., random) case, but a simple calculation
shows that this will only contribute a negligible amount to the yield.

by random pooling. The question is whether buyers can be fooled by this cherry picking. We have to consider two cases, based on the buyer’s computational powers. fully rational (computationally unbounded) buyer: He will not be fooled. Even though he does not know the set of lemon classes, he knows thanks to random graph theory (see the excellent references of Alon and Spencer2 and Bollobás7) that in a randomly chosen portfolio of CDOs the possibility of accidentally setting up such a boobytrap is vanishingly remote. Therefore it suffices for him to rule out the existence of any boobytrap in the presented portfolio: he enumerates over all possible 2n-sized subsets of the N classes and verifies that none of them are over-represented in any subset of m products. The same calculations as above guarantee him that in this case the yield of the derivative is at least V – o(n), even though he does not know the identity of the lemon classes. Thus a seller has no incentive to plant a boobytrap for a fully rational buyer, and the lemon wedge is indeed ameliorated greatly if buyers are fully rational.

real-life buyer, who is feasibly rational (computationally bounded): For him the above computation for detecting boobytraps is infeasible even for moderate parameter values.

To get an appreciation of the infeasible problem lurking here, it helps to take a graph-theoretic view of the problem. Recall that a bipartite graph consists of two disjoint sets of vertices A, B such that each edge has an endpoint in both A and B. We can use a bipartite graph to represent the portfolio of CDOs: A is the set of asset classes and B is the set of CDOs, and an edge (a, b) indicates that the CDO numbered b contains an asset from the asset class numbered a (see Figure 2).

Of course the buyer will also try other possible algorithms to detect the boobytrap. If the bank randomly throws assets into CDOs, then this graph that represents the portfolio is some kind of random graph. If the bank creates a boobytrap as described above, then the boobytrap corresponds to a dense subgraph in this bipartite graph: it is a subset of asset classes (the lemons) and a subset of CDOs (the boobytrapped ones) where the number of edges lying between them is substantially higher than it would be in a random graph.

The problem of detecting a boobytrap is equivalent to the so-called hidden dense subgraph problem, which is widely believed to be intractable. In fact the conjecture is that there is no efficient way to distinguish the truly random bipartite

figure 2. using a bipartite graph to represent asset classes and derivatives. there are M vertices on top corresponding to the derivatives and N vertices at the bottom corresponding to asset classes. each derivative references D assets in different classes.