The probability of surviving, denoted by Q(t), up to a horizon period t.
This can be related to the default intensity by
The survival probability is conditional on surviving previous periods. We
gave a derivation of this in Section 1.11. The marginal for any period is
the difference of the survival probabilities neighbouring the period.
6.3 A portfolio as a set of standalones
Who really cares about individual loans and their ﬁnancial character-
istics? Business is in the context of a collection of credits. The port-
folio, unfortunately from the perspective of default, is a very complicated
beast. The industry approaches it in a sequence of stages, each stage
adds in another layer of reality. Or we could say complexity. It is
rather like the application of a physical science to explain experimen-
tal evidence, you are never quite sure how relevant the model actually
is, but provided it has some explanatory power there is a danger that
the science is substituted for reality when it comes to thinking about
the subject. Portfolio modelling is probably at this stage.
Qt u u
() ( ) . exp d
The fundamentals of credit 237
Figure 6.3 The number of outcomes for a default process.
As a portfolio manager the biggest risk is one of concentration. This
describes the case of too much exposure either to a name, sector or
economy. Any model worth its salt should identify concentration risk.
However one reservation over default modelling is that it has only
recently been able to address such issues in a remotely reasonable
What then is the source of difﬁculty? Well let us go to the top of the
mountain ﬁrst in order to gain an overall perspective, since we know
the nature of what we are trying to describe (which is a big step in
itself). The shortcoming is that we do not quite have the tool-set as yet
to adequately address the reality in a practical sense. The reality is
that a portfolio consists of many loans. Each counterparty has a
unique chance of defaulting; this default rate is a variable rate which
changes intraday. There will be some correlation between the borrow-
ers. But do not get too comfortable with correlation between two
issuers only because there is a high possibility of many joint default
events occurring. This is not adequately captured using just a trad-
itional correlation approach. Furthermore the volatility of the port-
folio losses cannot be captured using just a mean variance approach.
This is because the volatility only really tells you the average thickness
of the losses, we introduce Figure 6.4 which perfectly depicts this,
where we have two loss distributions which have the same volatility
but any portfolio manager would agree the losses are completely
Credit is the classic case of the community having a pre-conception
and then applying it blindly to another subject matter believing this
prescription is the correct one. The majority of institutions still believe
the risk in a credit portfolio can be adequacy addressed using an
approach based on mean variance volatility. Carrying the implication
238 Credit risk: from transaction to portfolio management
Figure 6.4 Two distributions having the same volatility.