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1
A Multi-Objective Decision Framework for Credit Portfolio
Management
Juan C. Moreno-Paredes / School of Management University of Southampton. Highfield Campus
Southampton United Kingdom SO17 1BJ. E-mail: J.C.Moreno-Paredes@soton.ac.uk
Christophe Mues / School of Management University of Southampton. Highfield Campus
Southampton United Kingdom SO17 1BJ. E-mail: C.Mues@soton.ac.uk
Lyn C. Thomas / School of Management University of Southampton. Highfield Campus
Southampton United Kingdom SO17 1BJ. E-mail: L.Thomas@soton.ac.uk
Abstract
In this paper a framework is proposed which is meant to support the portfolio
optimisation problem, i.e. how to allocate bank’s resources across various sectors of
loans in order to maximise profitability and minimise risk. The framework enables
analysts to select an appropriate combination of performance measures and use a
multiple-objective optimisation technique to increase portfolio performance. A small
case study where the framework is applied shows how the risk and return can be
improved whilst simultaneously diversifying the credit portfolio.
Keywords: Credit Risk Management framework, Credit Portfolio Optimisation, Multi-
Objective Optimisation, Credit VaR, Credit Expected Shortfall.
1. Introduction
Since the Basel I and II accords (BCBS, 2005) established guidance for internal
models, the use of a variety of mathematical models and techniques in the financial
sector has become standard practice. Data driven models are widely used in the
financial sector to estimate the different levels of credit risk associated with obligors
involved in a credit transaction.
There is significant literature about techniques and methods to estimate credit risk at
the individual loan level (e.g. Thomas (2009); Thomas, Edelman and Crook (2004);
Thomas, Edelman and Crook (2002)).
On the other hand, further research is emerging to assess the overall risk of credit
portfolios (i.e. groups of loans) as interaction among the assets arises when they are
combined in the portfolio. For example, Cespedes (2002) shows how the correlation of
defaults can produce bigger losses in a credit portfolio. Additionally, phenomena like
2
concentration, correlation and contagion make the measurement of the risk of losses in
a credit portfolio more complex (Herbertsson (2011); Lütkebohmert (2009)).
An added complication with the credit portfolios of commercial and retail banks in
particular is that they are built of non-liquid assets like private loans. For that reason,
the quantitative analysis of such portfolios is challenging. For instance, for corporate
loans and bonds, risk metrics such as probability of default and loss given default are
constantly updated by the rating agencies, in contrast with private loans for whom no
such public information is found. Therefore, strategies aiming to improve the
performance of a credit portfolio are often made based on expert judgement. (Thomas
(2009); Lütkebohmert (2009); Schlottmann et al. (2010)).
A survey of approaches to optimise portfolios has to start with the Novel laureate
Markowitz (1952) who established the theoretical basis for portfolio analysis. Markowitz’
approach however cannot be applied directly to credit portfolios as the distribution of
the risk of losses does not follow a normal distribution (Cespedes, 2002). Also, the risk
in a credit portfolio is not adequately represented by one single risk measure. For that
reason, Zopounidis and Doumpos (2002) suggested a multi-objective approach to
integrate different risk metrics; however, they do not present details of how these
different metrics have to be calculated. Schlottmann and Seese (2004) present a hybrid
version of multi-objective evolutionary algorithms to optimise credit portfolios in
combination with CreditRisk+®, an actuarial model developed by Credit Suisse
Financial Products (CSFP, 1997) to calculate the risk of losses. They use a binary
encoding for modelling the decision variables which could be inefficient though to
optimise large portfolios. Additionally, this encoding is focused at the individual loan
level, whereas usually the main decisions in financial institutions are taken at a more
aggregate sector/segment level. Also, risk metrics such as concentration and
correlation among the sectors, are not yet taken into account as the methodology is
illustrated using a portfolio of one sector only.
Therefore, one of the main contributions in this paper is to propose a practical
framework where the risk metrics are explicitly disclosed and integrated using a Multi-
Objective Algorithm to support the credit portfolio optimisation process. The framework
integrates the profitability and risk perspectives; whereas the first perspective focuses
on finding opportunities to increase the portfolio’s return, the second perspective aims
to reduce the expected and unexpected losses and concentration risk of the portfolio.
3
We illustrate the suggested approach for a credit portfolio belonging to a financial
institution, thereby considering multi-sector and sector correlations. We also specify the
kind of data that would be needed for financial institutions to implement the approach.
This paper is organised as follows: In section 2, the profitability and risk perspective of
credit risk management are presented. Section 3 describes the process of modelling
the credit portfolio; in this section, the notation to represent the credit portfolio problem
and the measures associated with credit portfolio performance are established. Section
4 discusses the methods that are used to compute the solutions of the credit portfolio
optimisation problem and outlines the data required to perform this process. In section
5, the proposed framework is presented. In section 6, a practical case is presented in
order to illustrate how the framework operates. Finally, in section 7, conclusions are
drawn.
2. Perspectives of the credit portfolio management problem.
Credit portfolios from retail and commercial banks are characterised by a collection of
loans which are usually not liquid and therefore they are not easily sold to a third party
in the marketplace, unless they are properly packaged into “special purpose vehicles”
(SPV) (Thomas (2009)).
Each loan is a contract between a financial institution (FI) and individuals (called
obligors) where the FI lends an amount of money to the obligor who agrees to repay
later on. In some cases the obligors have to put some assets up as a guarantee
(collateral). In case the obligor defaults on its payments, the FI is able to repossess this
collateral in order to recover at least part of the lent money.
Credit portfolio management implies finding effective and efficient solutions as to how
to reorganise the portfolio in order to improve its profitability and reduce its total risk.
We refer to these (often competing) criteria as the profitability and risk perspective of
credit portfolio management.
Under the profitability perspective, the main objective of the portfolio managers is to
spot investment opportunities. Therefore elements such as return, profitability, pricing
and planning are key factors.
On the other hand, under the risk perspective, the portfolio managers have to identify
and anticipate real and potential losses as well as assess and quantify them; setting
4
exposure limits and capital reserves to mitigate and cover those potential losses are the
main elements to be considered (Thomas (2009); Van Gestel and Baesens (2009)).
In Figure 1, a graphical representation of these two perspectives is presented.
Figure 1 Perspectives and interactions in a credit portfolio.
Summing up, in order to further develop the risk and profitability perspectives it is
necessary to address the following questions:
a) What are the metrics associated with the performance of a credit portfolio?
b) What data should be used to calculate these metrics?
c) How is it possible to improve the performance of a credit portfolio reflecting
effectiveness and efficiency, i.e. higher returns and lower risk?
In the following sections, a framework is developed to address these questions.
3. Credit portfolio modelling
In this section, the notation to represent the credit portfolio optimisation problem and
the measures associated with credit portfolio performance are presented.
3.1. Credit portfolio definition
Following the definition of Markowitz (1952) according to which a portfolio can be
represented as a collection of assets that an investor can purchase using a predefined
amount of money, we define a credit portfolio as a set of credit operations (Assets). A
Soundness system of acredit portfolio risk management
(Risk perspective)
Soundness system of acredit portfolio risk management
(Profitability perspective)
S
E
G
M
E
N
T
S
/
S
E
C
T
O
R
S
Commercial
Manufacture
Corporate
Retail
Unsecure Loans
C
O
L
L
A
T
E
R
A
L
S
Land / Real Estate
Inventories and
Receivables
Cash
Marketable financial
instruments
Other Guaranties
and
Collaterals
Exposures Collateral
Loss Given Default (LGD)
Exposure Risk (EAD)
DefaultRisk (PD)
PortfolioStructure
(x1,…,xn)% of loans in n
sectors / segments
ConcentrationRisk
Stress testing& Limits
Scenario planning
Pricing
PotentialLosses
ExpectedReturns
Provisions and reserves
Regulatorycapital
RAROC &Profitability
5
credit operation is established when a FI (Investor) lends a predefined amount of
money (Exposure) to a counterparty (Obligor) that is willing to repay over a certain
period of time and in most of the cases paying an interest rate.
The FIs make strategic planning on their portfolios analysing not each individual credit
operation, but sets of them, grouped by segments or sectors such as real estate,
manufacturing, credit cards, agriculture and car loans.
Stratifying the portfolio into sectors makes it possible to analyse the risk due to
common factors, called systematic factors. The systematic factors are the underlying
common elements that could affect one significant part of the portfolio. For example, if
the real estate sector has performed badly in the economy recently, all the investments
in this sector could also be affected.
On the other hand, some portfolio metrics, as shown below, are specific to each loan in
the portfolio. For that reason it is important to consider two “levels” when a credit
portfolio is studied, i.e. the loan level and the sector level. The following notation is
used to represent these two levels:
Let be the proportion of money invested in loan in sector . The proportion
invested in sector is given by:
∑
Eq. 1
where : the number of loans in sector .
Definition 3.1: A credit portfolio can be defined as a vector , where is
the proportion of the portfolio invested in the sector ; ; ∑ and is the
number of sectors.
3.2. Credit portfolio performance metrics
The performance of a credit portfolio can be summarised by two main elements: the
returns produced by the repayments of the performing loans and the losses caused by
the default of loans. Hence, the default event is the main trigger that could produce
losses in the credit portfolio.
6
Definition 3.2: Let the random variable represent the default of a loan; hence
implies loan in sector defaults, or otherwise. The probability of default is
the associated probability of this event.
Definition 3.3: The risk of exposure or exposure at default ( ) is the book value of
the loans associated to loan in sector when the loan defaults. The EAD could be a
stochastic value, particularly when loans are credit lines or credit cards. BCBS (2005
parg. 311-315 and parg. 474 - 478) establishes the conditions that FIs should follow to
estimate EAD values. In this paper, the EADs are treated as deterministic values.
Definition 3.4: The severity of the loss ( ) is the portion of the value of the loans
that is lost after the loan defaults. The severity of the loss is also stochastic and its
expected value is the loss given default ( ) (Bluhm, Overbeck and Wagner, 2003).
BCBS (2005 parag. 286 - 307) details the conditions to work out the LGDs in a credit
portfolio.
Using these previous metrics, the return for a particular loan can be modelled:
( ) ( )
Eq. 2
where is the annual interest rate associated with a particular loan, net of cost of
funding and expenses.
Given that is the probability of default of a particular loan and its loss given
default, the expected return can be expressed as follows:
( ) ( )
Eq. 3
It should be noted that Eq. 3 assumes independency between the LGDs and PDs.
Comments about the implications of this assumption are made in Bluhm, Overbeck and
Wagner (2003 pg 28); Allen and Saunders (2004); Miu and Ozdemir (2006).
Then the expected return of a credit portfolio is represented by:
( ) ∑∑ ( )
Eq. 4
7
On the other hand, the associated loss of a particular loan is given by:
( )
Eq. 5
Thus, the expected loss is:
( )
Eq. 6
Similarly, the expected loss (EL) in a credit portfolio is given by:
( ) ∑∑
Eq. 7
EL only captures potential losses based on the historical default experience of each FI.
Hence, holding reserves to absorb only expected losses are not enough for FIs (Bluhm,
Overbeck and Wagner, 2003). Therefore, metrics that estimate unexpected losses help
FIs set aside enough capital to cover additional losses.
Vasicek (2002) proposes Value at Risk (VaR) as a measure for unexpected losses in a
portfolio. VaR is a one-factor extension of the Merton model and is used in BCBS
(2005) to set the minimum capital requirement. This is the value so that the chance of
getting a greater loss is no more than for a prespecified ( ).
Let ( ) be the loss associated with the portfolio . For a fixed ( the credit
portfolio VaR at level can be defined as follows:
( ) ( ( )) Eq. 8
where is the supremum function.
Artzner et al. (1999) propose a framework with the properties that a risk measure
should have in order to be mathematically coherent. Even though VaR is widely used in
the financial sector and it is supported by BCBS (2005), this metric is not a coherent
risk measure as it fails in the subadditivity property and it is law invariant (Tasche,
2002).
8
The lack of subadditivity makes it complex to analyse the risk contribution of a portion
of the portfolio (Tasche (2002)). And because VaR is law invariant it can produce the
same result for portfolios with light and fat tails (see e.g. Embrechts, Klüppelberg and
Mikosch (1997)). For that reason a more robust metric should be considered.
In order to tackle the problems of the VaR measure, Acerbi and Tasche (2002) propose
the Expected Shortfall (also known as Conditional VaR or CVaR) as a risk metric to
estimate the unexpected losses in the portfolio. The definition of CVaR is presented as
follows:
( )
( )∫ ( )
Eq. 9
Definition 3.5. Economic Capital (EC) is the amount of capital that each FI estimates
by itself in order to cover its potential losses. In this paper the EC is given by:
( ) ( ) ( ) Eq. 10
Where ( ) is the unexpected losses, calculated using one of the mentioned
methods, e.g. VaR or CVaR, at level, of the portfolio .
Definition 3.7. The Risk-adjusted Return on Capital is a metric that combines the
portfolio’s return and its EC, as represented in Eq. 11.
( ) ( ) ( ) Eq. 11
The ( ) is economic capital at level of the portfolio .
Another element that should be taken into consideration when a credit portfolio is
analysed is the concentration risk. Concentration risk arises due to the uneven
distribution of the loans in the different sectors. Lütkebohmert (2009) makes a survey of
the different methods to calculate concentration risk, classifying them into two groups:
ad hoc methods and model-based methods. She also presents several properties that
a concentration index for credit portfolios should have in order to be mathematically
consistent.
9
One of the ad hoc methods presented by Lütkebohmert (2009) is the Herfindahl-
Hirschman (HHI index). Becker and Düllmann (2004) prove that HHI fulfils the
concentration index properties.
In this paper, the HHI is adapted to measure concentration in the sectors as follows:
( ) ∑( )
Eq. 12
Hence, portfolios that are highly concentrated in a few sectors will produce a HHI
nearer to 1, whereas more balanced portfolios will produce a HHI nearer to 0.
3.3. Integrating risk metrics to optimise the credit portfolio.
For improving the performance of a credit portfolio in terms of effectiveness and
efficiency, i.e. in order to find higher returns and lower risk, it is necessary to integrate
the different risk metrics and represent the problem in the form of an optimisation
program.
RAROC in particular combines the expected return and the EC. However as it is
argued in previous work (Moreno-Paredes (2009)), optimising the credit portfolio using
RAROC as the objective function can be misleading as it cannot guarantee effective
and efficient solutions; i.e. some solutions obtained using RAROC as the sole goal
metric may consist of portfolios with higher risk than the original ones.
Similarly, there are important efforts in the literature to develop mathematical models
capable of integrating concentration metrics into the EC calculations. Particularly,
Lütkebohmert (2009 p 72-73) argues that ad hoc methods like the HHI could be
unsuitable as the outcomes they produce cannot be related directly to the EC
estimation. For that reason, she suggests using model-based methods such as
Granularity Adjustment, Semi Asymptotic approach and methods based on Saddle-
Point Approximation. Details of these methodologies can be found in Gordy (2004);
Emmer and Tasche (2004) and Gordy (2002), respectively.
However, integrated models like the ones listed above could be complex to implement
for credit portfolios of commercial and retail banks, as these credit portfolios are mainly
composed of low liquidity assets such as private loans; for that reason insufficient
information is available to fulfil the requirements of these models.
10
An integration approach whereby it is possible to preserve each metric is presented by
Zopounidis and Doumpos (2002). They suggest Multi-Objective Optimisation
programming whereby each metric is maximised or minimised simultaneously. A similar
approach is developed by Schlottmann, Mitschele and Seese (2005) when they are
attempting to combine different risk measures to model credit risk, market risk and
operational risk in FIs.
The main advantage of the Multi-Objective framework is the flexibility that it gives to
include different metrics in the modelling process. However, one needs to clearly define
the rules and interactions between the solutions, in particular the dominancy criteria.
These are defined in the following sections.
3.4. Model representation
Following the multi-objective framework, the credit portfolio optimisation problem
(CPOP) can be summarised as in Table 1:
Decision variable:
: The proportion invested in each sector
Objective functions:
Profitability perspective:
( )
Eq. 13
Risk perspective:
( )
Eq. 14
( )
Eq. 15
Restriction: The entire budget should be invested
Subject to:
∑
Eq. 16
Restriction: Market conditions in each sector:
Eq. 17
11
where represents the no short sell allowance restriction
and is the sector maximum saturation.
Table 1. The credit portfolio optimisation problem (CPOP).
The Market conditions in Eq 17 are modelling the minimum and maximum exposure in
each sector. These are related to elements such as regulatory restrictions1 or market
saturation. This information should be set up by the FI’s managerial team.
The credit portfolio optimisation problem has the following assumptions:
1. The loan’s participation , within the sector, remains constant in the short
term2.
2. The , and return rates remain constant during a period of time3.
3. Each loan can only belong to one sector4.
The Multi-Objective framework makes it possible to combine different metrics, without
the challenging effort of integrating all into a single optimisation criterion. Also its
flexibility allows including more or other metrics.
4. Methods to compute the solutions
Generating solutions that solve the CPOP (Table 1) can be a challenge as the
computation of the unexpected losses is not straightforward; this is because it is not
obvious how to model the distribution of the losses of a credit portfolio. Therefore
computational methods are required to tackle this problem.
4.1. Methods to compute Unexpected Losses.
In the case of credit portfolios, the assumption of normality in the distribution of losses
would be misleading as the correlation between defaults can produce higher losses.
These distributions are usually characterised by having big kurtosis (fat tails) (see
Cespedes (2002)).
1 Some FIs want to maintain a minimal investment in strategic sectors. Also, in some countries regulators can
request a minimum participation of the portfolio in a particular sector such as agriculture. 2 It can be assumed that these proportions do not change drastically in the short term.
3 Usually the FIs re-calculate these values periodically to keep the portfolio data up-to-date.
4 In this case the FIs associate a loan to its most related sector.
12
Jorion (2009) and also Thomas (2009) produce a survey of the most popular methods
in the banking sector to estimate the distribution of losses in a credit portfolio.
One such method is CreditRisk+® proposed by Credit Suisse Financial Products
(CSFP, 1997). This approach considers the distributions of defaults and of the severity
of the losses to build the distribution function of losses. The distribution of the number
of defaults is modelled using the Poisson distribution, based on the assumption that the
PDs are small enough and independent among the loans. CreditRisk+® splits the
portfolio into sectors where the loans may have some systematic risk factors in
common. Initially, in CSFP (1997), the sectors are considered independent. Bürgisser
et al. (1999), Han and Kang (2008) and Fisher and Dietz (2011) present improved
versions of CreditRisk+® where it is possible to use correlated sectors.
Some advantages that CreditRisk+® has when compared with other methods are: it
produces deterministic solutions; therefore, random variations in the solutions,
introduced by methods based on simulation, are avoided. Also, this method is well
documented in the literature. For those reasons, CreditRisk+® is used in this paper in
order to illustrate how the framework operates. However, it is important to note that the
proposed framework is able to use any alternative method to estimate the unexpected
losses.
4.2. Methods to solve the CPOP.
As it is mentioned above, the CPOP is modelled under the Multi-Objective Optimisation
Problems (MOOP) approach. In general, solutions of MOOPs are characterised by the
following properties. Firstly, objectives are usually in conflict with each other, i.e. better
returns often imply higher risk. Secondly, the dominancy of the solutions; a possible
solution is called dominated by , if is better in all objectives than . In contrast is
called a non-dominated solution when there is no solution such that dominates .
Thirdly, there is often more than one possible non-dominated optimal solution. Finally,
all optimal solutions are located in a set called the Pareto-Optimal Front (also referred
to in the literature as the Efficient Front; refers to Markowitz (1952)). Figure 2 illustrates
these characteristics. Here the grey area represents the space of valid solutions for the
CPOP and a, b and c are three specific solutions to the problem.
13
Figure 2 Efficient Front in MOOP.
Note that solution c is not an optimal solution, because it is dominated by a and b.
Particularly, for the CPOP the dominancy criteria can be defined as follows (see Deb
(2008)):
Definition 4.1. Let be a solution of the CPOP. dominates when at least one of
these conditions are true:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ).
Deb (2008 pg 28) explains in detail how dominancy plays a major role in the multi-
objective optimisation.
Generic optimisation methods called meta heuristics can be suitable for solving the
CPOP as conventional methods could have difficulties coping with the complexity
involved in the calculus of the objective functions, especially for the EC criterion. In
Moreno-Paredes (2009) it is shown how a conventional optimisation method such as
the Conjugate Gradient algorithm fails to solve this class of problems.
A Multi-Objective Evolutionary Algorithm (MOEA) is one of the meta-heuristics that has
been used successfully to solve these problems (Schlottmann and Seese (2004);
Branke et al. (2009); Moreno-Paredes (2009)). Deb (2008) presents a comprehensive
survey of these methods.
14
4.3. Data availability
In order to implement this framework, the FIs, particularly commercial and retail banks,
must have the following set of data for each loan: EAD, PD, LGD and Sector, as
illustrated in Table 2. For the computation of PDs and LGDs, the FIs have to collect
information of the default and recovery rates over a period of time long enough to
reflect economic downturn in these values (BCBS, 2005).
Loan ID EAD PD LGD Sector
O0001 £ 5,000.00 0.02 50% A
O0002 £ 3,000.00 0.07 80% B
O0003 £ 2,500.00 0.01 30% A
O0004 £ 6,000.00 0.07 40% C
Table 2 EAD, PD, and LGD; example.
Also, they have to calculate the Pearson correlation of the default rates among the
sectors over a period of time (see Table 3 for an example table).
Sector A B C
A
1.00
0.30
0.20
B
0.30
1.00 -
0.75
C
0.20 -
0.75
1.00
Table 3 Sector correlations; example.
Furthermore, Market conditions for each sector must be specified (see Table 4).
Sector LB UB
A 0.01 0.10
B 0.02 0.30
C 0.06 0.20
Table 4 Market conditions; example.
The default volatilities of the sectors are defined as the standard deviation of the
number of defaults in each sector divided by the average of the number of defaults in
each sector. Additionally, the returns of the portfolio are given by the annual interest
15
rates, less the cost of funding annually, commissions and fees incurred. The default
volatilities and the annual net returns are illustrated in Table 5.
Sector Default-
Volatilities Returns
A 0.02 1%
B 0.01 2%
C 0.03 3.5%
Table 5 Sector conditions; example.
These data are the inputs required to calculate the unexpected losses via
CreditRisk+®. Further details are given in section 6.
5. Framework outline
In Figure 3, the framework proposed in this paper is summarised.
Figure 3 Framework outline.
Inputs: Portfolio Risk
Indicators:
• PD, EAD, LGD
• Correlation
• PD volatilities
• Returns
• Market conditions
Output:
Strategies to
improve the
portfolio
Calculate the Portfolio
Performance Measures
(Return, EL, EC, HHI, etc.)
Solve the CPOP
Using MOEA
Populate the
model for the
CPOP
16
6. Practical Application
In order to illustrate how the proposed framework can be used, in this section a
practical application is presented. The data belongs to a credit portfolio of a commercial
bank in the United States with presence in Florida, New York and Texas and clients in
the USA and Latin-America. A summary of this database is given in Table 6:
Item Description
Total Assets 2.8 Billion of US$
Total Loans 2557 exposures
Number of Sectors 16
Table 6 Portfolio description.
The assets of the credit portfolio are loans to individuals and companies. For each loan
in the portfolio, values of PD, LGD and EAD are available. The bank estimates these
parameters using internal models. The portfolio is categorised into 16 different sectors:
1) Real Estate, 2) Retail, 3) Manufacture, 4) Trading, 5) Finance, 6) Services, 7) Other,
8) Mining, 9) Entertainment, 10) Information, 11) Agriculture, 12) Venezuela, 13)
Mexico, 14) Brazil, 15) Peru and 16) Colombia. Figure 4 shows how the portfolio is
distributed among the sectors.
Figure 4 Distribution of the portfolio by sector.
142%
413%
212%
136%
55%
35%
144%
124%
Rest of Sectors9%
Portfolio Distribution by Sector
17
The sectors of the portfolio are not independent; the inter-sector correlation matrix
between the sectors is presented in Table 7.
Sect
or 1
Sect
or 2
Sect
or 3
Sect
or 4
Sect
or 5
Sect
or 6
Sect
or 7
Sect
or 8
Sect
or 9
Sect
or 1
0
Sect
or 1
1
Sect
or 1
2
Sect
or 1
3
Sect
or 1
4
Sect
or 1
5
Sect
or 1
6
Sector 1 1.00 0.41 0.54 0.84 -0.05 0.25 0.41 0.96 0.12 0.61 0.19 0.09 0.28 0.27 0.13 0.01
Sector 2 0.41 1.00 0.34 0.41 -0.16 0.10 0.99 0.39 0.82 0.00 0.07 0.09 -0.19 -0.04 0.12 0.30
Sector 3 0.54 0.34 1.00 0.42 0.10 0.53 0.34 0.60 -0.02 0.34 0.33 -0.05 0.34 0.41 0.42 0.16
Sector 4 0.84 0.41 0.42 1.00 -0.01 0.02 0.41 0.83 0.25 0.42 0.33 -0.01 0.45 -0.08 -0.12 -0.15
Sector 5 -0.05 -0.16 0.10 -0.01 1.00 0.40 -0.16 -0.05 -0.08 0.01 0.47 -0.16 0.45 -0.15 -0.25 -0.38
Sector 6 0.25 0.10 0.53 0.02 0.40 1.00 0.10 0.24 -0.06 0.35 0.31 -0.21 0.02 0.15 0.19 -0.04
Sector 7 0.41 0.99 0.34 0.41 -0.16 0.10 1.00 0.39 0.82 0.00 0.07 0.09 -0.19 -0.04 0.12 0.30
Sector 8 0.96 0.39 0.60 0.83 -0.05 0.24 0.39 1.00 0.02 0.51 0.18 -0.07 0.27 0.28 0.19 0.04
Sector 9 0.12 0.82 -0.02 0.25 -0.08 -0.06 0.82 0.02 1.00 -0.10 0.11 0.15 -0.17 -0.33 -0.17 -0.01
Sector 10 0.61 0.00 0.34 0.42 0.01 0.35 0.00 0.51 -0.10 1.00 -0.09 0.11 0.21 -0.04 -0.23 -0.45
Sector 11 0.19 0.07 0.33 0.33 0.47 0.31 0.07 0.18 0.11 -0.09 1.00 -0.19 0.67 -0.03 0.09 -0.10
Sector 12 0.09 0.09 -0.05 -0.01 -0.16 -0.21 0.09 -0.07 0.15 0.11 -0.19 1.00 0.16 0.26 0.03 0.40
Sector 13 0.28 -0.19 0.34 0.45 0.45 0.02 -0.19 0.27 -0.17 0.21 0.67 0.16 1.00 -0.06 -0.20 -0.21
Sector 14 0.27 -0.04 0.41 -0.08 -0.15 0.15 -0.04 0.28 -0.33 -0.04 -0.03 0.26 -0.06 1.00 0.79 0.60
Sector 15 0.13 0.12 0.42 -0.12 -0.25 0.19 0.12 0.19 -0.17 -0.23 0.09 0.03 -0.20 0.79 1.00 0.57
Sector 16 0.01 0.30 0.16 -0.15 -0.38 -0.04 0.30 0.04 -0.01 -0.45 -0.10 0.40 -0.21 0.60 0.57 1.00
Table 7 Correlation matrix among the sectors.
The volatilities and the annual net returns for each sector of the portfolio are given by
Table 8:
Sector Volatilities Returns
1 0.22 1.91%
2 0.17 0.41%
3 0.33 -0.72%
4 0.22 0.29%
5 0.06 -1.77%
6 0.15 0.15%
7 0.17 0.39%
8 0.35 0.06%
9 0.17 -0.83%
10 0.41 -1.46%
11 0.89 0.11%
12 0.73 0.87%
13 0.32 0.78%
14 0.21 -0.83%
15 0.36 0.12%
16 0.25 -0.34%
Table 8 Volatilities and annual returns by sector.
The Market Conditions (L and U) were calculated by dividing the minimum and
maximum exposure that the bank is willing to invest in this sector over the total
exposure of the credit portfolio, respectively. The resulting market conditions are
presented in Table 9.
18
Sector L U
1 0.000016819% 100%
2 0.000046288% 100%
3 0.000002066% 100%
4 0.000007322% 100%
5 0.000001269% 100%
6 0.000003842% 100%
7 0.000001414% 100%
8 0.000000036% 100%
9 0.000000580% 100%
10 0.000000217% 100%
11 0.000000072% 100%
12 0.000010729% 100%
13 0.000000761% 100%
14 0.000000399% 100%
15 0.000000399% 100%
16 0.000000471% 100%
Table 9 Market conditions.
In order to compute the VaR and CVaR, the version of CreditRisk+® by Haaf, Rieß and
Schoenmakers (2004) is used in combination with the Bürgisser et al. (1999) approach.
By integrating both versions it is possible to have a numerically stable version of
CreditRisk+® which also allows for calculating the CVaR and dealing with correlated
sectors.
To solve the associated CPOP (see Table 1), Non-Dominated Sorting Genetic
Algorithm type II (NSGA II), a type of MOEA developed by Deb and Goel (2001) is
used. The initial settings and stop conditions of this algorithm are the same as specified
by Moreno-Paredes (2009); however these parameters settings would have to be
adjusted for other credit portfolios. The problem of parameter selection goes beyond
the scope of this paper.
Parameter Value
Pc: Probability of Crossover 0.9
Pm: Probability of mutation 0.9
Max number of generation 500
Initial Population Random
initialization
Number of max generation 500
Number of runs 50
Table 10 MOEA initialisation parameters.
19
The numerical representation and genetic procedures in the NSGA II algorithm were
implemented using the approaches in Moreno-Paredes (2009) and Dinovella and
Moreno-Paredes (2005).
A graphical representation of the solutions produced by the MOEA is shown in Figure
5. Here, the values of the portfolio metrics are plotted for each solution, the colours
representing the RAROC.
Figure 5 Solutions from the MOEA.
In Table 11, we decide to select the solution with the higher RAROC and label it as
“Best Solution”, but this selection is arbitrary and it is made just to show how the
optimisation process helps to improve the performance of the credit portfolio.
Initial
Portfolio
Best
Solution
Improvement
(%)
Average
Solution
Average
Improvement
(%)
Standard
deviation
Expected Return (MM US$) 22.23$ 27.81$ 25.1% 25$ 14.4% 1.51$
Economic Capital via CVaR
(MM US$) 26.28$ 21.70$ -17.4% 23$ -13.7% 0.96$
HHI 0.2202 0.2198 -0.2% 0.2010 -8.7% 0.0114
RAROC 0.8458 1.2818 52% 1.1231 33% 0.0878
Table 11 Summary of the results.
Initial Portfolio
20
The results suggest a potential increment of 25% in the return in this portfolio along
with a reduction of 17% in the Economic Capital, improving the RAROC up to 52%, with
a modest reduction in the concentration index. On average, solutions suggest
increments of 14% in the return, a reduction of 14% in the Economic Capital, 33%
increment in the RAROC and a reduction of 8.7% in the concentration index. It is also
important to highlight the value of 0.0114 of the standard deviation of the RAROC,
suggesting a low dispersion among the solutions coming from the MOEA.
Figure 6 shows a comparison between the distribution among the sectors of the initial
portfolio and the best portfolio solution identified by the CPOP. Specifically, this
distribution shows a reduction in sector 1 and an important increment in sector 13,
making the portfolio more diversified.
Figure 6 Comparison of portfolio distributions.
7. Conclusions
Credit portfolios of commercial and retail banks are majority made of non-liquid assets
like loans. Elements such as the lack of public data makes the assessment of these
portfolios challenging. In this paper, a framework is developed to assess and improve
these types of credit portfolios.
Its aim is to find effective and efficient strategies, i.e. strategies that not only contribute
to mitigating possible losses but also convey opportunities to increase the return of a
credit portfolio.
142%
413%
212%
136%
55%
35%
144%
124%
Rest of Sectors9%
Initial Portfolio by Sector
135%
411%2
11%
1324%
1211%
34%
Rest of Sectors4%
Best Solution by Sector
21
The framework proposed in this work can be summarised as having four major
components:
Credit portfolio measures: a set of mathematical formulas are proposed to
assess the performance of a credit portfolio.
Computational methods: several methodologies are suggested to compute the
magnitude of the unexpected losses that a credit portfolio could face.
Optimisation model: A multi-objective optimisation program is proposed to
integrate the different risk measures with the main objective to find strategies
that improve the credit portfolio performance.
Optimisation solver: A Multi-Objective Evolutionary Algorithm is proposed as an
alternative heuristic method to solve the optimisation model.
In the practical example application presented in this work, it is illustrated how the
framework can be used to identify and evaluate possible strategies for improving the
portfolio in terms of risk and returns. In this particular case, the framework shows how
to increase the return, reduce potential losses, whilst diversifying the credit portfolio.
An important characteristic of the proposed framework is its flexibility, as the multi-
objective approach allows risk analysts to add more objectives and constraints in a
straightforward manner. In that sense, a possible extension of this work is to
incorporate other risk measures such as contagion. However, it is not always obvious
how to measure contagion as interbank data is probably needed. Another possible
enhancement of this work is to model the correlation among the loans in each sector.
References
Acerbi, C. and Tasche, D. (2002), 'On the coherence of expected shortfall', Journal of Banking & Finance, Vol. 26, No. 7, pp. 1487-1503.
Allen, L. and Saunders, A. (2004), 'Incorporating systematic influences into risk measurements: A survey of the literature', Journal of Financial Services Research, Vol. 26, No. 2, pp. 161-192.
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999), 'Coherent measures of risk', Mathematical Finance, Vol. 9, No. 3, pp. 203-228.
BCBS (2005), 'International convergence of capital measurement and capital standards: A revised framework', Basel: Basel Committee on Banking Supervision – Bank for International Settlements [Online], Available: www.bis.org.
Becker, S. and Düllmann, K. (2004), 'Measurement of concentration risk - a theoretical comparison of selected concentration indices': Deutsche Bundesbank.
Bluhm, C., Overbeck, L. and Wagner, C. (2003), An introduction to credit risk modeling, London: Chapman & Hall/CRC.
22
Branke, J., Scheckenbach, B., Stein, M., Deb, K. and Schmeck, H. (2009), 'Portfolio optimization with an envelope-based multi-objective evolutionary algorithm', European Journal of Operational Research, Vol. 199, No. 3, pp. 684-693.
Bürgisser, P., Kuth, A., Wagner, A. and Wolf, M. (1999), 'Integrating correlation', Risk, Vol. July, pp. 57-60.
Cespedes, J.C.G. (2002), 'Credit risk modelling and basel ii', ALGO Research Quarterly, Vol. 5, No. 1.
CSFP (1997), 'A credit risk management framework', Available: http://www.csfb.com/institutional/research/assets/creditrisk.pdf.
Deb, K. (2008), Multi-objective optimization using evolutionary algorithms, West Sussex: John Wiley & Sons LTD.
Deb, K. and Goel, T. (Year), 'Controlled elitist non-dominated sorting genetic algorithm for better convergence’', First International Conference on Evolutionary Multi-Criterion Optimization, 2001.
Dinovella, P. and Moreno-Paredes, J.C. (2005), 'A computational approach to improve the benders‟ decomposition method', Revista de la Facultad de Ingeniería de la U.C.V., Vol. 20, No. 2, pp. 5 - 14.
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997), Modelling extremal events, Berlin: Springer.
Fisher, M. and Dietz, C. (2011), 'Modelling sector correlations with creditrisk+', The Journal of Credit Risk, Vol. 7, No. 4, pp. 1-20.
Gordy, M.B. (2002), 'Saddlepoint approximation of creditrisk+', Journal of Banking and Finance, Vol. 7, No. 26, pp. 1335- 1353.
Gordy, M.B. (2004), 'Granularity adjustment in portfolio credit risk measurement', IN, Szego, G. (ed.) Risk measures for the 21st century.
Haaf, H., Rieß, O. and Schoenmakers, J. (2004), 'Numerical stable computation of creditrisk+', IN, Gundlach, M. and Lehrbass, F. (eds.), Creditrisk+ in the banking sector, Berlin: Springer.
Han, C. and Kang, J. (2008), 'An extended creditrisk+ framework for portfolio credit risk management', The Journal of Credit Risk, Vol. 4, No. 4, pp. 63-80.
Herbertsson, A. (2011), 'Modelling default contagion using multivariate phase-type distributions', Review of Derivatives Research, Vol. 14, No. 1, pp. 1-36.
Jorion, P. (2009), 'Financial risk manager handbook', 5th ed., Hoboken, NJ: Wiley. Lütkebohmert, E. (2009), Concentration risk in credit portfolios, Berlin: Springer-Verlag. Markowitz, H. (1952), 'Portfolio selection', Journal of Finance, Vol. 7, No. 1, pp. 77-91. Miu, P. and Ozdemir, B. (2006), 'Basel requirements of downturn loss given default:
Modeling and estimating probability of default and loss given default correlations', Journal Credit Risk, Vol. 2, No. 2, pp. 43-68.
Moreno-Paredes, J.C. (2009), An implementation of a multi-objective evolutionary approach for credit portfolio optimisation, MSc in Risk Management, University of Southampton.
Schlottmann, F., Mitschele, A. and Seese, D. (2005), 'A multi-objective approach to integrated risk management', IN, Coello, C.a.C., Aguirre, A.H. and Zitzler, E. (eds.), Evolutionary multi-criterion optimization, pp. 692-706.
Schlottmann, F. and Seese, D. (2004), 'A hybrid heuristic approach to discrete multi-objective optimization of credit portfolios', Computational Statistics & Data Analysis, Vol. 47, No. 2, pp. 373-399.
Schlottmann, F., Seese, D., Lesko, M. and Vorgrimler, S. (2010), 'Risk-return analysis of credit portfolios', IN, Gundlach, M. and Lehrbass, F. (eds.), Creditrisk+ in the banking industry, Berlin: Springer, pp. 259-278.
Tasche, D. (2002), 'Expected shortfall and beyond', Journal of Banking & Finance, Vol. 26, No. 7, pp. 1519-1533.
23
Thomas, L.C. (2009), Consumer credit models: Pricing, profit and portfolios, Oxford: Oxford University Press.
Thomas, L.C., Edelman, D.B. and Crook, J.N. (2002), Credit scoring and its applications, Philadelphia, PA: Society for Industrial and Applied Mathematics.
Thomas, L.C., Edelman, D.B. and Crook, J.N. (2004), Readings in credit scoring : Foundations, developments, and aims, Oxford ; New York: Oxford University Press.
Van Gestel, T. and Baesens, B. (2009), 'Credit risk management basic concepts: Financial risk components, rating analysis, models, economic and regulatory capital', Oxford: Oxford University Press.
Vasicek, O. (2002), 'Loan portfolio value', Risk, Vol. 15, No. 12. Zopounidis, C. and Doumpos, M. (2002), 'Multi-criteria decision aid in financial
decisionmaking: Methodologies and literature review', Journal of Multi-Criteria Decision Analysis, Vol. 11, pp. 167-186.
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