gaussians and the central limit theorem

The ability of CDOs to ameliorate effects of asymmetric information relies on the law of large numbers, which informally speaking states that the total payoff of a bundle of mortgages is close to the mean (or expected) value. More precisely, we have the central limit theorem: The sum of sufficiently many bounded and independent random variables will be approximately distributed like a Gaussian with same mean and variance. The Gaussian is a good approximation even when the number of variables is only in the hundreds. Therefore, if there are 100 mortgages, each paying 1 with probability 1/2 and paying 0 otherwise, the distribution of the total payoff is like a Gaussian with mean 50 and standard deviation 5. The probability that the payoff is outside [ 35, 65] (three standard deviations) is less than 0.2%.

Histogram of ProportionOfHeads

500

400

Frequency

300

200

100

0

0.40

0.45 0.50

ProportionOfHeads 0.55 0.60

0.65

In addition to their value as limiting distributions for the sum of independent random variables, Gaussians arise in one other way in finance: often the payoffs of assets themselves are assumed to be Gaussians. The joint distribution of these Gaussian valued assets is the well-known Gaussian copula:

Gaussian copula
Density of Gaussian copula
1 0.8 0.6 0.4 0.201 0.5

1. 8 1. 6 1. 4 1. 2 1 0.8 0.6 0.4 0.2 0

1 0.5

0 0 . 2 0.4 0.6 0.8

1

0.2 0.4 0.6 0.8 1

Although in the illustrative example we assumed binary payoffs for a single asset, similar results hold for asset yields that form a Gaussian copula with the same mean, variance, and covariance.

why introducing computational complexity into the picture greatly complicates the picture, and even takes away some of the theoretical benefits of CDOs. The full analysis appears in our longer paper.

The important twist we introduce in the above scenario is that a large bank is selling many CDOs and not just a single one. DeMarzo’s theorem does not generalize to this case, and indeed we will show that whether or not the CDOs ameliorate the lemons problem depends upon the computational ability of the buyer. The lemons problem gets ameliorated only if the buyer is capable of solving the densest subgraph problem, which is currently believed to be computationally difficult.

We will use the assumption—consistent with practice— that mortgages are grouped into classes depending upon factors such as the borrower’s credit score, geographic location, etc., and that default rates within a class are the same. Consider for simplicity a bank with N asset classes, each of which contains C assets. Some asset classes are “lemons”: assets in these classes will always default and have payoff 0. All other asset classes are good: assets in these classes pay 1/C with probability 1/2 and default (i.e., have payoff 0) with probability 1/2. The yields of assets from different asset classes are independent.

The buyer’s prior is that the number of lemon classes is uniformly distributed in [0, 2n] for some n< N/2, and the set of lemon classes is uniformly picked among all classes. However, the bank has additional information: it knows precisely which classes are lemons (this implies that it knows the number of lemons as well). This is the asymmetric information.

Since the expected number of lemon classes is n, each with payoff 0 and the remaining N–n good classes have payoff 1/2, a buyer purchasing the entire portfolio would be willing to pay the expected yield, which is (N–n)/2. Thus a wedge à la Akerlof arises for banks who discover that the number of lemons is lower than the expectation, and they would either exit the market, or would need to prefer cash by an amount that overcomes the wedge.

Of course, DeMarzo’s theorem allows this lemons prob-
lem to be ameliorated, via securitization of the entire
portfolio into a single CDO. As already mentioned, we are
interested in the case where the number of assets held by
the bank is large, and so, rather then using a single CDO, the
bank partitions them into multiple CDOs. Now how does the
bank’s extra information affect the sale? Clearly, it has new
cherry-picking possibilities, involving which asset to pack-
age into which CDO. We will assume that all transactions
are public and visible to all buyers, which means that seller
must do any such cherry picking in full public view.e
Now let us show that in principle derivatives should still
allow buyers to rule out any significant cherry picking, thus
ameliorating the lemon wedge. Consider the following:
the seller creates M new financial products, each of them
e This assumption of transparency only makes our negative results stron-
ger. It also may be a reasonable approximation if buyers are well-informed,
and recent financial regulation has mandated more transparency into the
market.