predictable yield, and hence more attractive to risk-averse lenders such as retirement funds.

Consider the following simplistic example: suppose a bank holds a portfolio of 100 mortgages, and that each mortgage yields $0 if the borrower defaults and $1 million otherwise. For now assume the probability of default is 10%, which implies that the expected yield (and hence fair price) for the entire portfolio is $90 million. The bank would like to get all or most of these mortgages off its books because this is favorable for regulatory reasons. Since each individual mortgage may be unacceptably risky for a risk-averse buyer, the bank holding the mortgages can do the following. Create two new assets by combining the above 100 mortgages. Set a threshold, say $80 million. The first asset, called the senior tranche, has claim to the first 80 million of the yield; and the second, called the junior tranche, has claim to the rest. These assets are known as collateralized debt obligations (CDOs). The bank offers the senior tranche to the risk-averse buyer and holds on to the junior tranche (Figure 1).

Now a risk-averse buyer may reason as follows: if the mortgage defaults are independent events (which is a big if, though in practice justified by pooling mortgages made to a geographically diverse group of homeowners, whose defaults are presumably independent) then the senior tranche is extremely unlikely to ever yield less than its maximum total value of $80 million. In fact for this to happen,

more than 20 mortgages need to default, which happens only with probability . Thus the senior tranche is a very safe asset with a highly predictable yield, and such derivatives were often rated by credit-rating agencies to be as safe as a U.S. treasury bond. In real-life CDOs the mortgage yields, payment streams, and tranching are all much more complex, but the basic idea is the same. the lemons problem enters: There is an obvious lemons problem in the above scenario because of asymmetric information: The bank that issued the mortgages has the most accurate

figure 1. aggregating many mortgages into a single asset makes the yield more predictable due to the law of large numbers (central limit theorem). this assumes that the yields of the different mortgages are independent random variables.

I want to get good but very safe returns.

I want to buy a house, but might default on my loan

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Aggregating mortgages gives an asset with predictable yields

estimate of the rate of default, whereas the buyer may have a less precise idea—say that the rate lies between 10% and 15%. Economic theory says that in addition to transforming the risk profile of the asset, the CDO also protects the investor from this lemons problem. The crucial observation is that even if the probability of default was 15%, the probability of the senior tranche yielding its maximum of $80 million would still be roughly 99.9%, and hence the tranching insulates the buyer from the information-sensitive part of the mortgage pool.

In fact, as was shown by DeMarzo, 11 the choice of the threshold can be used as a signaling mechanism that allows the bank to transmit in a trustworthy way the true default value. We now illustrate the idea behind this result. Consider the problem from the bank’s viewpoint. It is interested in getting rid of as many mortgages—specifically, the largest possible portion of the entire portfolio—from its book as possible, so it wants to set the threshold as high as possible. Suppose it knows the default rate is 15%. This is the highest possible default rate, so it can simply sell the whole bundle of mortgages (i.e. set the threshold to 100%). The buyers will use their most pessimistic evaluation (that the default rate is 15%) and pay the price of $85 million. Both the seller and the buyer are satisfied because the price is just equal to the estimated yield. Suppose on the other hand that the bank knows the default rate is actually only 10%, the lowest possible rate. Now setting the threshold to 100% is no longer its best strategy, since the buyers will just pay $85 million while the bundle is now worth $90 million. To signal its confidence in the quality of mortgages, the bank will tranch the pool, set the threshold to 80%,c and offer to hold the riskier part— the junior tranche. Knowing that the bank will not offer this lower threshold when the default rate is high (indeed, the best threshold for default rate 15% is 100%), the rational buyers should correctly interpret this as a signal to the true default rate and pay close to $80 million for the senior tranche. Again, both the seller and the buyer are satisfied because the price is almost equal to the estimated yield of the senior tranche.

Making the above intuitive argument precise in the usual rational expectations framework of economic theory takes some work, and was done in DeMarzo, 11 where it is shown that the CDO is the optimum solution to the lemons problem in a setting somewhat more general than the above simplistic one. Specifically, the CDO allows the lemon cost—i.e., the difference in valuation of the security by buyer and bank required for the sale to occur—to approach 0.d That is, the bank’s secret information does not lead to large market inefficiencies. Henceforth we will refer to this as DeMarzo’s Theorem.

4. Wh Y ComPLexIt Y matteRs

We presented above the traditional justification for CDOs from economic theory. Now we explain at an intuitive level c The exact threshold here depends on a number of factors, including default rate and discount factor. The discount factor shows how much the seller prefers cash to assets. The threshold can be computed exactly using methods in DeMarzo. 11 d A nonzero difference in valuation or wedge between the bank and buyer arises because the buyer holds cash and the bank holds the mortgages, and the bank prefers cash to mortgages because of regulatory or other reasons.