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
0.45
1
0.4
0.9
Cumulative Probability
Probability Density
0.35
0.8
0.3
0.7
0.25
0.6
0.5
0.2
0.4
0.15
0.3
0.1
0.2
0.05
0.1
0
0
0 2 4 6 8 10 12 14 16 18 20
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.