Credit risk quantification with the use of

2014 International Conference on Management, Education, Business, and Information Science(ICMEBIS 2014)
Credit risk quantification with the use of CreditRisk+
Boris Kollár1, a, Štefan Cisko2, b
1
Faculty of Operation and Economics of Transport and Communications, Department of Economics,
University of Žilina, Univerzitná 8251/1, 010 26 Žilina, Slovakia
2
Faculty of Operation and Economics of Transport and Communications, Department of Economics,
University of Žilina, Univerzitná 8251/1, 010 26 Žilina, Slovakia
a
email: [email protected], bemail: [email protected]
Keywords: Credit Risk; Default; Model
Abstract. The subject of this article is to present one of the credit risk quantification options. It
introduces the basic concepts related to CreditRisk+ model. Regular analysis and subsequent early
identification of required reserves amount should serve as protection against default and undesirable
economic consequences arising from it. In Slovakia, just like in whole global economy, there has
been significant increase of credit risk in business sector. This article presents basic approach to
CreditRisk+ with use of Poison distribution of defaults. The output of this model is loss probability
distribution with the use of individual group distribution.
Introduction
From mid-seventies of twentieth century the companies began to change defensive understanding
of risk to active one [1]. The current risk management is typified by the fact that risk should be treated
actively and risk should be respected in all decisions-making. The process of risk management is
therefore becoming an essential part of management, business culture and practice and is adapted to
particularities of business activities and processes. Risk management requires and must take into
account the nature and specificities of companies and activities, companies has to deal with. The main
objective of risk management is to limit the impact of various expected and unexpected effects and
events, but also to take opportunities they bring together with them and thereby contribute to
improvement of business performance. Today, more than ever is growing the importance of modern
enterprise management and application of innovative analytical programs and early warning systems.
That's why accurate and timely quantification of credit risk becomes important, as the scope and
volume of loans is constantly increasing in the business environment. It is also the case of our Central
European region which is closely connected to global economy due to our orientation on export [2].
One of the possibilities, that helps to manage risk is CreditRisk+ method.
CreditRisk+ method was developed by Credit Suisse Financial Products and belongs to category
of "default-mode" models. In contrast to known KMV model, default risk is not associated with
capital structure of enterprise. CreditRisk+ model does not give any assumptions about default
causes. At the end of risk period, each debtor may be situated only in two possible categories:
failure and non-failure. Among basic model inputs are exposures of financial institution to
individual borrowers and probabilities of borrower’s defaults. The rate of return is exogenous
constant parameter independent of market risk and downgrade risk in CreditRisk+ model [3].
Basic model quantification
It is assumed that:
• for a credit, probability of default in a given time horizon is the same for discretionary
equally long time horizon.
• for a large number of borrowers, the probability of individual borrower default is low and
the number of defaults, which will appear in given time period is independent of the amount
of the past horizon’s defaults.
978-0-9917647-3-0/EDUGait Press, Canada
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Under these assumptions, the probability distribution of default, during given time period is well
represented by Poisson distribution with parameter μ given.
for n = 0,1,2,...,
(1)
where n is number of defaults, m is average number of defaults during one year period,
,
(2)
represents borrower A default probability.
where
It is obvious, that there is usually finite amount of bonds in the portfolio, therefore Poisson
distribution, which specifies the probability of default for infinite amount of bonds, is only
approximation of default’s number distribution. However, if the number of borrowers is sufficiently
large, the probability, that default’s number exceeds the borrower’s number is negligible [4]. By
assuming Poisson distribution of default’s number, we expect that the standard deviation of default
rate will be approximated by square root of the default rate mean. In fact, we can observe, that
Poisson distribution underestimates the probability of default for all rating grades. This is caused by
variability of default intensity at time, that is modeled as a function of changing selected risk factors.
If the average number of defaults has stochastic nature and has Gamma distribution with parameters
μ and
default process can be represented by Poisson distribution.
Portfolio loss distribution
Derivation of single asset risk is based on expected loss calculation. To derive losses probability
of well-diversified portfolio, losses are divided into groups according to their size. Each group
contains borrowers with the same credit risk and is considered to be independent portfolio of bonds
with following marking [5]:
A ... borrower
LA ... possible loss
PA ... default probability of borrower A
εA ... expected loss
v j ... possible loss in group j
ε j ... expected loss in group j
μ j ... expected number of defaults in group j
Bonds portfolio of each borrowers group may be handled as an unchanging portfolio. Possible loss
of each borrower in group j can be obtained as:
vj = j × L.
(3)
By definition, then we get:
ε j= v j ×μ j
(4)
εA = LA × PA
(5)
Mark εA expected loss of borrower A, i.e.
, than εA is expected loss in one year horizon
in group j and is expressed by expected losses sum - εA
of all borrowers in group j:
(6)
Expected number of defaults in group j within one year period is then:
(7)
Derivation of losses probability distribution for whole portfolio consists of several
steps [2]:
1. Derivation of probability generating function for each group j:
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(8)
Because we assume that number of defaults is shall be governed by Poisson distribution,
we can derive:
(9)
2. Derivation of probability generating function for whole portfolio:
With assumption of individual groups’ independence, probability generating function for whole
portfolio is:
,
(10)
where
is expected number of defaults for whole bonds portfolio.
3. Probability distribution of losses for whole portfolio
From the probability generating function for whole portfolio, we can derive probable distribution
of losses as:
(11)
Model assessment
CreditRisk+ model is simple and easily implementable model for calculation of expected loss in
default state. CreditRisk + is appropriate model for credit risk calculation of homogeneous portfolio
consisting of large number of borrowers with low probability of default. It is based on Poisson
approximation of individual defaults. Certain disadvantage of the model comes from the fact is, that it
does not include downgrade risk [6].Model, in contrast to CreditMetrics method, aims at determining
volume of risk capital assets, estimates the distribution of expected losses and values at risk. Unlike
KMV model, this method does not focus on relationship between default risk and capital structure of
enterprise. Model does not use Monte Carlo simulation, therefore outputs are fully conditioned by
input data. Portfolio losses probability distribution may be derived from probabilistic generating
function with numerically stable algorithm. Calculation of CreditRisk+ model is calculated by
analytical method and is much faster than CreditMetrics model created by simulation method [7].
Another difference is also different approach to default rate. Swiss model considers it to be coherent
variable as can be seen at figure 1.
Figure (1) - Coherent default rate – CreditRisk+[8]
Conclusion
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Current world financial crisis increases the need for early and accurate credit risk estimation, this
need grows for the business, banking as well as for the government sector [9]. CreditRisk+ is
appropriate method for default estimation. In the past, there have been many comparison tests with
different results. In general, bigger differences between the default estimating models are in the case
of low credit price bonds portfolio than for a portfolio composed from higher credit score bonds.
Within the method CreditRisk + , credit event is determined by the default current condition, as it is
"default-mode" model type. At the same time, however, changes in default rate may signal
degradation in the credit quality of borrower [10]. According to us the biggest disadvantage of this
model comes from Poisson distribution, because it underestimates the probability of default for all
rating grades. On the contrary main advantage comes from easier way of calculation.
Acknowledgement
The contribution is an output of the science project VEGA 1/0656/14- Research of Possibilities
of Credit Default Models Application in Conditions of the SR as a Tool for Objective Quantification
of Businesses Credit Risks.
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