combined exposure is generally much lower than the individual prob-
ability of default. The probability is called a ‘joint default’.
The joint probability of default is given by the equation below:
where P
J
is the joint probability of default; is the correlation between
the obligor and the counterparty; P
o
is the probability of default by the
obligor; P
c
is the probability of default by the counterparty.
The reader may be disappointed to learn that there is no commonly
accepted methodology of systematically pricing credit derivatives. This
unfortunate state of affairs exists because of the underlying nature of a
credit default and the difficulty of replicating a claim synthetically.
Given the behavior of credit, that is characterized by an occasional
large loss, it is very difficult to construct a replicating portfolio and hence
an arbitrage pricing schema. Another source of difficulty is the lack of
‘calibration’ information. To remain positive it can be argued that most
of the ingredients of a good pricing model have been recognized and we
reproduce these as follows:
Probability of default of the underlying credit.
Loss given a default of the underlying credit, that is projected recov-
ery rate.
Probability of default of the counterparty.
Correlation between the underlying and the counterparty.
Maturity of the deal.
The most fundamental uncertainty surrounds default, and then given
a default, the amount of capital that can be returned. A good deriva-
tive pricing model should predict the premium given the inputs of say,
a default probability of 1 per cent, and that the loss given default is
50 per cent. There are now a number of accepted models and we provide
an example of a common approach encountered within the market in
Section 3.14.
A number of secondary considerations come into play when pricing
CDSs including liquidity, regulatory capital requirements together with
market sentiment.
3.13 Source of pricing
The questions arise as to where these mysterious default probabilities
can be determined. One obvious source is historical information, but
PPP P P P P
Joco,co o c c
  ()().11
168 Credit risk: from transaction to portfolio management
there are a number of shortcomings with this approach. The first is the
relevance of history to the future; the second concerns the actual set
of historical data.
We familiarize the reader with both the various terminologies and the
data which will confront them in the market. Table 3.10 is represen-
tative of the type of default data available for homogenous pools as they
vary across time. The marginal rates are simply the probability of a
sample company defaulting in that year – it varies across time; be very
careful on how you combine these across different periods, to generate
a resulting figure known as the cumulative default probability. For
example if the annual default probability is 1 per cent then the cumu-
lative default over 2 years is not 2 per cent, but slightly less because it
must have survived in the first year to default in the second (it will be
99% 1% 1%).
The most comprehensive default analysis was performed on US
corporate bonds. The reader should be aware prior to an examina-
tion of the literature that there are two common definitions of default
employed; the first is the ratio of names that default divided by those
remaining.
The second is to take the ratio of the nominal of debt defaulting against
the nominal value outstanding. You will encounter the terms marginal
defaults and cumulative; the marginal default is simply the default rele-
vant to any 1-year and the cumulative is the sum of the marginals.
As stated, care is required in interpreting these results. Just adding
marginals in this way will produce a figure which will tend to overstate
the default probability on any one issuer, because the actual default
probability is conditional upon the issuer surviving up to the period of
interest. Strictly they should only be applied if starting with the same
notional on each period.
There are two basic methodologies based around these distinctions,
Altman and Kishore determine the default probability on a pool of rela-
tively homogeneous corporate debt, broken down by rating category.
Moody’s, Carty and Lieberman determine the issuer default on a pool
enlarged to encompass convertible and foreign bonds.
The details of this analysis can be seen in the excellent text by Altman.
Credit derivatives 169
Table 3.10 Marginal default rates.
Year 1(%) Year 2(%) Year 3(%) Year 4(%)
Pool A 4.3 4.2 5.6 7
Pool B 2.2 2.1 1.8 1.5
Pool C 7 7.2 7.3 7.5

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