The questions arise as to where these mysterious default probabilities can be determined. One obvious source is historical information, but
PJ Po Pc o,c Po(1 Po) Pc(1 Pc).
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.
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
Another major source of default information is the capital mar- kets themselves. Indeed some would argue that this is the only viable route to establish a derivatives price. Table 3.11 illustrates that each company has a unique default premium driven in part by its rating and further by the sector. This spread depends on the maturity of the issue.
Yet market spread data is a classic example of the joint observation problem; credit spreads imply loss severity given a default but this can only be determined if you make an assumption about what this simultaneously states about recovery.
When all is said and done, most institutions price risk directly from the market price of traded securities. Roughly translated this means that the security issued by the credit will trade at a different price to a similar government security. The difference in price represents an implied default on the part of the borrower. Within the fixed income market, practitioners do not tend to discuss prices but rather yields.
This means the implied default is translated into a spread.
In Figure 3.22 we explore these points quantitatively using a zero coupon bond, the price will be given by
but also the bond can be priced by discounting using the yield spread, which can typically be observed in the secondary bond market, this
PV r1 P P RV
(1 ) (1 )100
risk free
d d
[
]
Table 3.11 The default swap premia for various names.
Default swap premium for protection on various credits
Credit 6 months 1 year 2 year 3 year
EDF 100 100 100 100
DT Lufthansa 100 100 100 100
Source:Bloomberg LP.
PV
100
RV Default
No default
Figure 3.22 The default process.
represents the excess over the risk free rate that compensates the investor for bearing the credit risk:
For example, if the recovery value is 88 and the spread is 50 bps then
The other major input required for derivative pricing is the default cor- relation. There are a number of choices relevant to its derivation. Direct default data can be used, and then the following formula to establish the correlation is used:
This can be written after replacing the variance (this can be found in Section 6.1):
Assuming we have the individual default probabilities implied from the derivative market, the remaining quantity is the joint default prob- ability which is very difficult to obtain. Most practitioners rely on a Merton like approach, because this is based on a very liquid marketplace (the equity, typically). Then we use
where Nis the cumulative normal distribution; BNis the cumulative bivariate normal. The threshold value is the return on the equity below which the company would default.
Correlation
(threshold , threshold ) (threshold ) (threshold ) (threshold ) [1 (threshold )]
(threshold ) [1 (threshold )]
1 2
1 2
1 1
2 2
BN
N N
N N
N N
È
ÎÍ ˘
˚˙ , Correlation
(1 ) (1 )1,2 1 2 .
1 1 2 2
P P P
P P P P
Correlation
joint default probality
company average company average company variance 1 company variance2 .
1 2
(
¥)
È
ÎÍ ˘
˚˙
¥ PD 0.005
1 0.88 4.2%.
P r
RV
D spread
(1 /100).
PV r 100 r (1 riskfree)(1 spread) ,