# Optimal Bespoke CDO Design via NSGA-II

Authors: Diresh Jewan; Renkuan Guo; Gareth Witten
Journal: Journal of Applied Mathematics and Decision Sciences
 (2009)

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## Abstract

This research work investigates the theoretical foundations and computational aspects of constructing optimal bespoke CDO structures. Due to the evolutionary nature of the CDO design process, stochastic search methods that
mimic the metaphor of natural biological evolution are applied. For efficient searching the optimal solution, the nondominating
sort genetic algorithm (NSGA-II) is used, which places emphasis on moving towards the true Paretooptimal
region. This is an essential part of real-world credit structuring problems. The algorithm further demonstrates
attractive constraint handling features among others, which is suitable for successfully solving the constrained portfolio
optimisation problem. Numerical analysis is conducted on a bespoke CDO collateral portfolio constructed from
constituents of the iTraxx Europe IG S5 CDS index. For comparative purposes, the default dependence structure is
modelled via Gaussian and Clayton copula assumptions. This research concludes that CDO tranche returns at all
levels of risk under the Clayton copula assumption performed better than the sub-optimal Gaussian assumption. It is
evident that our research has provided meaningful guidance to CDO traders, for seeking significant improvement of
returns over standardised CDOs tranches of similar rating.


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## Body

## 1. Introduction

Bespoke CDOs
provides tailored credit solutions to market participants. They provide both
long-term strategic and tactical investors with the ability to capitalise on
views at the market, sector and name levels. Investors can use these structures
in various investment strategies to target the risk/return profile or hedging
needs. These strategies can vary from leverage and correlation strategies to
macro and relative value plays [1].Understanding the risk/return trade-off dynamics
underlying the bespoke CDO collateral portfolios is crucial when maximising the
utility provided by these instruments. The single-tranche deal can be put
together in a relatively short period of time. This is aided by the development
of numerous advance pricing, risk management and portfolio optimisation
techniques.The most crucial tasks in putting together the bespoke
CDO is choosing the underlying credits that will be included in the portfolio.
Investors often express preferences on individual names, and there is likely to
be credit rating constraint and industry concentration limits imposed by the
investors and rating agencies [2].Given these various investor defined requirements, the
structurer is required to optimise the portfolio to achieve the best possible
tranche spreads for investors. This was a complicated task, however, with the
advent of faster computational pricing and portfolio optimisation algorithms,
aid structurers in presenting bespoke CDO which conform to the investment
parameters.The proper implementation of the decision steps lies
in the solution of the multiobjective, multiconstrained optimisation problem,
where investors can choose an optimal structure that matches their risk/return
profile. Optimal structures are defined by portfolios that lie on the Pareto
frontier on the CDO tranche yield/portfolio risk plane.Davidson [2] provides an interesting analogy between CDO
portfolio optimisation processes and evolutionary cycles espoused by Charles
Darwin. In the natural world, life adapts to suit the particulars of its
environment. To adapt to a specific environment, a simple but extraordinarily
powerful set of evolutionary techniques are employed—reproduction, mutation
and survival of the fittest. In this way, nature explores the full range of
possible structures to hone in on those that are most perfectly suited to the
surrounding environment.The creation of the CDO collateral portfolio can
broadly be seen in similar ways. Given a certain set of investor and/or market
constraints, such as the number of underlying credits, the notional for the
credits, concentration limits and the weighted average rating factor, credit
structurers need to be able to construct a portfolio that is best suited to the
market environment. If the portfolio does not suit the conditions, it evolves
so that only those that are “fittest,” defined by having the best CDO
tranche spread given the constraints, will survive. Many of the same techniques
used in the natural world can be applied to this constrained portfolio
optimisation problem. Evolutionary algorithms have received a lot of attention
regarding their potential for solving these types of problems. They possess
several characteristics that are desirable to solve real world optimisation
problems up to a required level of satisfaction.Our previous research work focused on developing a
methodology to optimise credit portfolios. The Copula Marginal Expected Tail
Loss (CMETL) model proposed by Jewan et  al. [3], is one that minimises credit portfolio
ETL subject to a constraint of achieving expected portfolio returns at least as
large as an investor defined level, along with other typical constraints on
weights, where both quantities are evaluated in the CMETL framework. Jewan et  al.
[3] have
shown that ETL optimal portfolio techniques, combined with copula marginal
(factor copula) distribution modelling of the portfolio risk factors can lead
to significant improvements in risk-adjusted returns.Our research work now investigates a new approach to
asset allocation in credit portfolios for the determination of optimal
investments in bespoke CDO tranches. Due to the complexity of the problem,
advance algorithms are applied solve the constrained multiobjective
optimisation problem. The nondominating sort genetic algorithm (NSGA-II)
proposed by Deb et  al. [4] is applied.NSGA-II is a popular second generation multiobjective
evolutionary algorithm. This algorithm places emphasis on moving towards the
true Pareto-optimal region, which is essential in real world credit structuring
problems. The main features of these algorithms are the implementation of a
fast nondominated sorting procedure and its ability to handle constraints
without the use of penalty functions. The latter feature is essential for
solving the multiobjective CDO optimisation problem.The study uses both Gaussian and Clayton copula models
to investigate the effects of different default dependence assumptions on the
Pareto frontier. Various real world cases are considered, these include the
constrained long-only credits and concentrated credit cases. Two objectives are
used to define the CDO optimisation problem. The first is related to the
portfolio risk, which is measured by the Expected-tail-loss (ETL). ETL is a
convex risk measure and has attractive properties for asset allocation
problems. The second objective is the CDO tranche return. This objective
requires a CDO valuation model. We apply an extension of the Implied Factor
model proposed by Rosen and Saunders
[5]. The extension is a result of the application of the
Clayton copula assumption. Rosen and Saunders [5] restricts their analysis to the Gaussian case,
but use a multifactor framework.The breakdown of the paper is as follows. The next
section briefly discusses the mechanics of bespoke CDOs. It outlines the three
improtant decision making steps involved in the structuring process. In section
four, a robust and practical CDO valuation framework based on the application
of the single-factor copula models given in Section3, is presented. This is in
conjunction with weighted Monte Carlo techniques used in options pricing. The
results of the study on the impact of the different copula assumption on the
market implied loss distribution is then presented. Finally the analysis on the
implied credit tail characteristics under the various copula assumptions is
given. Section 5 defines convex credit risk measures in a self contained
manner. Sections 4 and 5 establish the theory behind the objective
functions used in the CDO optimisation model. In Section 6 the generic model
for multiobjective bespoke CDO optimisation is presented. The components of
the NSGA-II are discussed and the algorithm outlined. This then paves the way
to perform a prototype experiment on a bespoke CDO portfolio constructed from
constituents of the iTraxx Europe IG S5 index. The final section highlights the
improtant research findings and discusses several areas of future study.


## 2. Bespoke CDO Mechanics

A bespoke CDO
is a popular second-generation credit product. This standalone single-tranche
transaction is referred to as a bespoke because it allows the investor to
customise the various deal characteristics such as the collateral composition,
level of subordination, tranche thickness, and credit rating. Other features,
such as substitution rights, may also play an improtant role [1]. In these
transactions, only a specific protion of the credit risk is transferred, unlike
the entire capital structure as in the case of standardised synthetic CDOs.
Most of these transactions involve 100–200 liquid corporate CDS.While the bespoke CDO provides great flexibility in
the transaction parameters, it is crucial that investors understand the
mechanics of the deal. A key feature of these transactions is the greater
dialogue that exists between the parties during the structuring process,
avoiding the “moral hazard” problem that existed in earlier CDO deals
[6].In a typical bespoke CDO transaction, there are three
main decision steps for potential investors:(1)
Credit Selection for the Reference portfolio: The first step in structuring a bespoke CDO is the selection of the credits for
the collateral portfolio. Investors can choose a portfolio of credits different
from their current positions. They can also sell protection on the subset of
names in their current portfolio in case they have overweight views on
particular credit and/or sectors.(2)
Defining the Subordination Level and Tranche
Size: Once a credit portfolio has been selected, investors must choose a
subordination level and tranche size. These variables determine the degree of
leverage and the required protection premium. Investors that are primarily
concerned about the credit rating of the CDO tranche, rated by major rating
agencies such as Moody's, S&P and Fitch, could choose the tranche size and
subordination level so as to maximise the premium for a selected rating [1].
Usually investors choose a subordination level that will provide a total spread
equal to some desired return target. This “tranching” of credit
portfolio risk can provide any desired risk/return profile. By choosing the
position of a tranche on a capital structure, investors can decouple their
views on default risk from their views on market risk. (See Rajan et  al. [1, page 203–220]
for trading strategies involving bespoke CDO transactions.)(3)
Substitution of Credits in the Reference
portfolio: The third distinctive feature of bespoke CDOs is the investors'
ability to dynamically manage their investment profit and loss by substituting
credits in the portfolio.
An optimal transaction
would be based on a collateral portfolio that lies on the efficient frontier on
the tranche return/risk plan, solution of a multiobjective optimisation
problem that satisfies both trade and regulatory constraints.A typical placement of a bespoke CDO is outlined in
Figure1.Figure 1
Placement of mezzanine tranche
in a typical bespoke CDO transaction.In the above schematic, the investor goes long the
credit risk in a mezzanine tranche. The challenge of issuing bespoke CDOs is
the ongoing need and expense of the risk management of the tranche position.
The common method of hedging these transactions is to manage the risk like an
options book. Greeks similar to those related to options can be defined for CDO
tranches. The distribution of risk in these transactions is not perfect,
leaving dealers exposed to various first and second order risks [7].


## 3. Factor Copula Models for Default Dependence Modelling

Copula-based
credit risk models were developed to extend the univariate credit-risk models
for the individual obligors to a multivariate setting which keeps all salient
features of the individual credit-risk models while incorporating a realistic
dependency structure between the obligor defaults.Copulas were first introduced by Sklar [8], in the
context of probabilistic metric spaces, however their applications to finance
have been very recent. The idea was first invoked within finance by Embrechths et  al. [9], in
connection with the inadequacies of linear correlation as a measure of dependence
or association.In credit portfolio modelling the copula approach and
factor models have become an industry-wide standard to describe the asset
correlation structure. Construction of mutual asset correlations between
entities of the portfolio on common external factors represents a very flexible
and powerful modelling tool. This modelling approach can be understood as a
combination of the copula and firm-value approach. This factor approach is
quite standard in credit risk modelling [10–14].
These models have been widely applied to credit derivatives modelling for
essentially two reasons:(i)
factor models
represent an intuitive framework and allow fast calculation of the loss
distribution function; and(ii)
the full
correlation matrix, which represents a challenging issue in large credit
portfolios, need not be fully estimated.
Factor models
can then be used to describe the dependency structure amongst credits using a
so called “credit-versus-common factors” analysis rather than a pairwise analysis.Credit risk models can be divided into two
mainstreams, structural models and reduced form models. In the reduced-form
methodology, the default event in these models is treated exogenously. The
central idea is to model the default counting processNc. This is a stochastic process which assumes only
integer values. It literally counts default events, with Nc(t) denoting the
number of events that have occurred up to time t. In the case of the mth-to-default,
the default time is given by the following:(3.1)τm=min{t∈[0,T]∣Nc(t)=m}. Using the standard settings for financial product
pricing, we fix a complete probability space (Ω,ℱ,ℚ) where the model
lives. Let ℚ denote the risk
neutral probability measure. All subsequently introduced filtrations are
subsets of the filtration ℱ and augmented
by the zero-sets of ℱ [15].We will consider a latent factorV such that
conditionally on V, the default times are independent. The factor
approach makes it simple to deal with a large number of names and leads to very
tractable results. We will denote by pti∣V=ℚ(τ(i)≤t∣ℋ^tvσ(Vt)), the conditional default probability of the name i, and by qti∣V=ℚ(τ(i)>t∣ℋ^tvσ(Vt)), the corresponding conditional survival probability.
Conditionally on V, the joint survival function is given by the
following:(3.2)ℚ(τ(1)≥t,τ(2)≥t,…,τ(m)≥t∣ℋ^tvσ(Vt))=∏i=1n𝔼[Nc(i)(t)∣ℋ^tvσ(Vt)]for m reference
entities. The filtration, ℋt represents all
available information in the economy at time t. The filtration ℋti represents all
available information restricted to obligor i and the
background process. These filtrations enable us to model the intensity of the
default process of an obligor independent of the information about the
remaining obligors.The next two subsections provide a description of the
two copula models used in the study. These two models will define the default
dependence structure, which is used in a Monte Carlo simulation, to derive the
credit portfolio loss distribution. These distributions are then used in
pricing, portfolio and risk management.


### 3.1. Gaussian Copula

A convenient
way to take into account the default dependence structure is through the
Gaussian copula model. This has become the market standard in pricing
multiname credit products. In the firm-value approach a company will default
when its “default-like” stochastic process,X, falls below a barrier. Define the stochastic
process,(3.3)Xi=ρiV+1−ρi2ϵi,where V and ϵi are independent
standard Gaussian variants, with Covar(ϵi,ϵj)≠0 for all i≠j. When ρ=0, this corresponds to independent default times while ρ=1 is associated
with the comonotonic case. Xi can be
interpreted as the value of the assets of the company, and V, the general state of the economy. The default
dependences come from the factor V. Unconditionally, the stochastic processes are
correlated but conditionally they are independent.The default probability of an entityi denoted by Fi can be observed
from market prices of credit default swaps. Under the copula model, each Xi is mapped to τ(i) using a
percentile-to-percentile transformation. In a Gaussian copula model the point Xi=x is transformed
into τ(i)=t with τ(i)=F−1(Φ(x)). It follows from (3.3) that the conditional
default probability is given by:(3.4)ℚ(Xi≤x∣V)=Φ(x−ρiV1−ρi2),where x=Φ−1(Fi(t)), and ℚ(X≤x)=ℚ(τ≤t),(3.5)ℚ(τ(i)≤t∣V)=Φ(Φ−1(Fi(t))−ρiV)1−ρi2).We also have that:(3.6)τ(i)=F−1(Φ(ρiV+1−ρi2ϵi)),this is used when we want to
simulate the default times. The Gaussian copula has no upper or lower tail
dependence. Embrechths et  al. [9] proves this result.


### 3.2. Clayton Coupla

Most of the
existing copula models involve using the Gaussian copula which has symmetric
upper and lower tails, without any tail dependence. This symmetry fails to
capture the fact that firm failures occur in cascade in difficult times but not
in better times. That is, the correlation between defaults increases in
difficult times. The Clayton copula encapsulates this idea.This “class” of Archimedean copulas was
first introduced by Clayton [16] from his studies on the epidemiological chronic
diseases. Clayton [16] only developed a general form (without imposing any
parametric constraints) while Oakes [17] refined the copula in term of its
parameterisation. Friend and Rogge [18], Laurent and Gregory
[19], Schloegl and O’Kane [20], Schönbucher and Schubert [15], and Schönbucher [21] have been considering this
model in a credit risk context, primarily due to the lower tail dependence
which is ideal for credit risk applications.The asset value process in a single factor Clayton
copula is given by the following:(3.7)Xi=(1−log(ϵ)V)−1/θC,where V is the
systematic risk factor, a positive random variable, following a standard Gamma
distribution with shape parameter 1/θC with θC>0, and scale parameter equal to unity.Using the definition of default times and the Marshall and Olkin [22]
sampling algorithm, the conditional survival probability given by the
following:(3.8)qti|V=exp(V(1−Fi(t)−θC)).The default times are given
by,(3.9)τ(i)=F−1((1−log(ϵ)V)−1/θC).From expression (3.9), it is clear
that the stochastic intensities are proprotional to V. Thus the
latent variable acts as a multiplicative effect on stochastic intensities. High
levels of the latent variable are associated with shorter survival default
times. For this reason, V is called a
“frailty.”WhenθC=0, we obtain the product copula and this implies the
default times are independent. When θC→∞, the Clayton copula turns out to be the upper Frëchet
bound, corresponding to the case where default times are comonotonic [19]. As
the parameter θC increases, the
Clayton copula increases with respect to the supermodular order. This implies
an increasing dependence in default times, and hence has some direct
consequences for the pricing of CDO tranches as shown by Laurent and Gregory [19].


## 3.1. Gaussian Copula

A convenient
way to take into account the default dependence structure is through the
Gaussian copula model. This has become the market standard in pricing
multiname credit products. In the firm-value approach a company will default
when its “default-like” stochastic process,X, falls below a barrier. Define the stochastic
process,(3.3)Xi=ρiV+1−ρi2ϵi,where V and ϵi are independent
standard Gaussian variants, with Covar(ϵi,ϵj)≠0 for all i≠j. When ρ=0, this corresponds to independent default times while ρ=1 is associated
with the comonotonic case. Xi can be
interpreted as the value of the assets of the company, and V, the general state of the economy. The default
dependences come from the factor V. Unconditionally, the stochastic processes are
correlated but conditionally they are independent.The default probability of an entityi denoted by Fi can be observed
from market prices of credit default swaps. Under the copula model, each Xi is mapped to τ(i) using a
percentile-to-percentile transformation. In a Gaussian copula model the point Xi=x is transformed
into τ(i)=t with τ(i)=F−1(Φ(x)). It follows from (3.3) that the conditional
default probability is given by:(3.4)ℚ(Xi≤x∣V)=Φ(x−ρiV1−ρi2),where x=Φ−1(Fi(t)), and ℚ(X≤x)=ℚ(τ≤t),(3.5)ℚ(τ(i)≤t∣V)=Φ(Φ−1(Fi(t))−ρiV)1−ρi2).We also have that:(3.6)τ(i)=F−1(Φ(ρiV+1−ρi2ϵi)),this is used when we want to
simulate the default times. The Gaussian copula has no upper or lower tail
dependence. Embrechths et  al. [9] proves this result.


## 3.2. Clayton Coupla

Most of the
existing copula models involve using the Gaussian copula which has symmetric
upper and lower tails, without any tail dependence. This symmetry fails to
capture the fact that firm failures occur in cascade in difficult times but not
in better times. That is, the correlation between defaults increases in
difficult times. The Clayton copula encapsulates this idea.This “class” of Archimedean copulas was
first introduced by Clayton [16] from his studies on the epidemiological chronic
diseases. Clayton [16] only developed a general form (without imposing any
parametric constraints) while Oakes [17] refined the copula in term of its
parameterisation. Friend and Rogge [18], Laurent and Gregory
[19], Schloegl and O’Kane [20], Schönbucher and Schubert [15], and Schönbucher [21] have been considering this
model in a credit risk context, primarily due to the lower tail dependence
which is ideal for credit risk applications.The asset value process in a single factor Clayton
copula is given by the following:(3.7)Xi=(1−log(ϵ)V)−1/θC,where V is the
systematic risk factor, a positive random variable, following a standard Gamma
distribution with shape parameter 1/θC with θC>0, and scale parameter equal to unity.Using the definition of default times and the Marshall and Olkin [22]
sampling algorithm, the conditional survival probability given by the
following:(3.8)qti|V=exp(V(1−Fi(t)−θC)).The default times are given
by,(3.9)τ(i)=F−1((1−log(ϵ)V)−1/θC).From expression (3.9), it is clear
that the stochastic intensities are proprotional to V. Thus the
latent variable acts as a multiplicative effect on stochastic intensities. High
levels of the latent variable are associated with shorter survival default
times. For this reason, V is called a
“frailty.”WhenθC=0, we obtain the product copula and this implies the
default times are independent. When θC→∞, the Clayton copula turns out to be the upper Frëchet
bound, corresponding to the case where default times are comonotonic [19]. As
the parameter θC increases, the
Clayton copula increases with respect to the supermodular order. This implies
an increasing dependence in default times, and hence has some direct
consequences for the pricing of CDO tranches as shown by Laurent and Gregory [19].


## 4. Implied Factor Copula Models for CDO
Valuation

The current
market standard for pricing synthetic CDOs is the single-factor Gaussian copula
model introduced by Li
[23]. Numerous research has shown that a single parameter
model is unable to match the price of all quoted standardised CDO tranches
[24–32].It is common practice to quote an implied “correlation
skew”—a different correlation which matches the price of each tranche. This
assumption is analogous to the Black-Scholes implied volatility in the options
market. Implied tranche correlations not only suffer from interpretation
problems, but they might not be unique as in the case of mezzanine tranches,
and cannot be interpolated to price bespoke tranches [33].We define bespoke tranches in the following cases:(i)
the underlying
portfolio and maturity is the same as the reference portfolio, but the tranche
attachment and/or detachment points are different,(ii)
the underlying
portfolio is the same as the reference portfolio but the maturities are
different, or(iii)
the underlying
portfolio differs from the reference portfolio.
The following
subsection presents a robust and practical CDO valuation framework based on the
application of the single-factor copula models presented in the previous
section. The method to recover the credit loss distributions from the factor
copula structure is then presented. The implied factor model is then derived.
This development is in conjunction with weighted Monte Carlo techniques used in
options pricing. The Gaussian model presented here is a special case of the
multifactor Gaussian copula model proposed by Rosen and Saunders [5]. The application of the
Clayton copula model is an extension of Rosen and Saunders [5] work.The impact of the different copula assumptions on the
loss distribution is also investigated. The credit tail characteristics are
analysed. This is imperative to ensure that the copula model used has the
ability to capture the default dependence between the underlying credits, and
not severely underestimate the potential for extreme losses. The loss analysis
is performed on a homogeneous portfolio consisting of 106 constituents of the
iTraxx European IG S5 CDS index.


### 4.1. Synthetic CDO Pricing

The key idea
behind CDOs, is the tranching of the credit risk of the underlying portfolio. A
given tranchentr is defined by
its attachment and detachment points untrlower and untrupper respectively.
The tranche notional is given by: sntr=Nprot(untrupper−untrlower), where Nprot denotes the
total portfolio value.LetLtotal(t) be the
percentage cumulative loss in the portfolio value at time t. The total cumulative loss at time t is then Ltotal(t)Nprot. The loss suffered by the holders of tranche ntr from
origination to time t is a percentage Lntr(t) of the
portfolio notional value Nprot:(4.1)Lntr(t)=min{max{Ltotal(t)−Lntr,0},untrupper−untrlower}.We consider a transaction
initiated at time 0, with maturity T. In a CDO contract, the tranche investor adopts a
position as a protection seller. Similar to the assumption under the CDS
pricing, we assume that defaults occur at the midpoints between coupon payment
dates. The value of the protection leg is then given by the
following:(4.2)𝔼[PVprot(ntr)]=∑j=1JD(0,tj−tj−12)(𝔼[Lntr(tj)]−𝔼[Lntr(tj−1)]),where 𝔼[Lntr(⋅)] is the
expectation with respect to the risk neutral measure ℚ and is
calculated by a simulation model. The tranche loss profiles under each scenario
are calculated and stored. The expectation is found by calculating the weighted
average over all scenarios. This only applies if a weighted Monte Carlo scheme
is used.The tranche investors need to be compensated for
bearing the default risk in the underlying credits. The holders of tranchen receive a
periodic coupon payment. Let the coupon payment dates be denoted as 0=t0≤t1≤t2⋯≤tJ=T. The predetermined frequency of the coupon payment
dates is usually on a quarterly basis. The spread paid for protection on a
given tranche does not vary during the life of the contract. and is usually
quoted in basis points per annum. However, the tranche notional decays through
the life of the contract. The outstanding tranche notional at time t is given by the
following:(4.3)Noutntr(t)=(untrupper−untrlower−𝔼[Lntr(t)])Nprot.The expected outstanding tranche
notional since the last coupon date must be considered at the coupon payment
dates. This amount between coupon payment dates tj and tj−1 is simply the
average of Noutntr(tj) and Noutntr(tj−1). We assume again that defaults can only occur at the
midpoint between arbitrary coupon dates. The expected outstanding tranche
notional is denoted by:(4.4)𝔼[Noutntr(tj,tj−1)]=(untrupper−untrlower−𝔼[Lntr(tj)]+𝔼[Lntr(tj)]−𝔼[Lntr(tj−1)]2).Using this equation we can
compute the expected present value of the coupon payments:(4.5)𝔼[PVpremntr]=∑j=1Jsntr(tj−tj−1)D(0,tj)𝔼[Noutntr(tj,tj−1)].This fair spread of a tranche
can be computed by equating the expected present value of the protection and
premium legs. The market quotation for the equity tranche is to have a fixed
500 bps spread, and the protection buyer make an upfront payment of a fixed
percentage of the tranche notional. The protection seller receives the upfront
fee expressed as a percentage f of the tranche
notional, so that equity investors purchase the note at a discount f(ueq)Nprot. Only the premium leg is different for equity tranche
investors. This is given by the following:(4.6)𝔼[PVprem(eq)]=f(ueq)Nprot+∑j=1Jseq(tj−tj−1)D(0,tj)𝔼[Nout(eq)(tj,tj−1)].


### 4.2. Weighted Monte Carlo Techniques

Monte Carlo
algorithms can be divided (somewhat arbitrarily) into two categories: uniformly
weighted and nonuniformly weighted algorithms. nonuniform weights are a
mechanism for improving simulation accuracy. Consider a set ofM paths,
generated by a simulation procedure. A nonuniformly weighted simulation is one
in which the probabilities are not necessarily equal. Suppose that we assign,
respectively, probabilities p1,p2,…,pM, to the different paths. The value of the security
according to the nonuniform weights is,(4.7)Ωh=∑m=1MpmΛm, where Λ is the payoff
function.Two features of credit risk modelling which pose a
particular challenge under simulation based procedures, namely:(1)
it requires
accurate estimation of low-probability events of large credit losses; and(2)
default
dependence mechanisms described in the previous chapter do not immediately lend
themselves to rare-event simulation techniques used in other settings.
It is for these
reasons that we implement a weighted Monte Carlo simulation procedure to put
emphasis on determining directly the risk neutral probabilities of the future
states of the market, which will allow the accurate determination of the credit
loss distribution. This is in contrast to calibration of pricing models through
traditional methods.In what follows, we introduce the general modelling
framework, present the algorithm and discuss some technical implementation
details. (We use a similar methodology to Rosen and Saunders [5].) This generalised framework can be
applied to credit risk models that have a factor structure.A weighted Monte Carlo method can be used to find an
implied risk-neutral distribution of the systematic factors, assuming the
specification of a credit risk model with a specified set of parameter values.
According to Rosen and
Saunders [5] the justification for this approach follows working within a
conditional independence framework, where obligor defaults are independent,
conditional on a given systematic risk factor.The methodology to obtain the implied credit loss
distribution is summarised by the following steps.(i)
Latent factor scenarios. Define M scenarios on the systematic factor V.(ii)
Conditional portfolio loss distribution.
Calculate the conditional portfolio loss profile for scenario m.(iii)
Conditional tranche values: Infer the
tranche loss distributions conditional on scenario m.(iv)
Implied scenario probabilities (optimisation
problem). Given a set of scenario probabilities pm; tranche values are given as expectations over all
scenarios of the conditional values. We solve the resulting constrained inverse
problem to find a set of implied scenario probabilities pm.(v)
Implied credit loss distribution-Calculate the aggregate credit loss at each time step given the implied
scenario probabilities.
The first three
steps above have been discussed in detail in the previous sections of this
chapter. The focus now is placed on the optimisation problem for implying the
scenario probabilities.


#### 4.2.1. Mathematical Formulation

Leth:ℝM→ℝ∪{+∞} be a strictly
convex function and let wi,M denote the
weight of the ith path of M paths. Now
consider the optimisation problem:(4.8)minw1,M,w2,M,…,wM,M∑i=1Mh(wi,M)subjectto1M∑i=1Mwi,M=1,1M∑i=1Mwi,MGi=cG, for some fixed cG∈ℝN. The objective function is strictly convex and the
constraints are linear, so if a feasible solution exist, with a finite
objective function value, then there is a unique optimal solution w1,M,w2,M,…,wM,M. This optimum defines the weighted Monte Carlo
estimator,(4.9)Ψ^LCV=∑i=1Mwi,MΨi.The weights derived from
(4.8) can be made more explicit by introducing the Lagragian. The strict
convexity of the objective function is not a sufficient condition to guarantee
a unique solution.The classical approach to solving constrained
optimisation problems is the method of Lagrange multipliers. This approach
transforms the constrained optimisation problem into an unconstrained one, thereby
allowing the use of the unconstrained optimisation techniques.


#### 4.2.2. Objective Functions

Taking the
objective to be a symmetric separable convex function gives the optimal
probabilitiesp. This is an interpreted as the most uniform
probabilities satisfying the constraints, though different objective functions
imply different measures of uniformity. A common choice for the fitness measure
is entropy. This is a
particularity interesting and in some respects a convenient objective. The
principle of maximum entropy gives a method of generating a probability
distribution from a limited amount of information. It is a relatively well used
principle for the construction of probabilistic models.In this setting,(4.10)h=∑w*=1Mpw*logpw*.This case is put forward in Avellaneda et  al. [34], where a Bayesian
interpretation is given. The usual convention of 0log0 is followed. We
now provide a problem specific description of the optimisation problem using
the principle of maximum entropy.


#### 4.2.3. Constraint Description

For the bespoke
CDO pricing problem, the implied scenario probabilities must satisfy the
following constraints:(i)
the sum of all
scenario probabilities must equal to one;(ii)
the
probabilities should be positive:pi≥0foralli∈{0,1,…,M}; and bounded by 1;(iii)
we match the
individual CDS spreads (i.e., marginal default probabilities for each
name)(4.11)∑w*=1Mpw*pz*t∣Vw*=pz*t∣V∀z*∈{1,2,…,N};(iv)
the current
market prices of standard CDO tranches are matched given by(4.12)∑w*=1Mpw*PVprot(ntr)=∑w*=1Mpw*PVprem(ntr)∀ntr∈{1,2,…}tranches.
Because the
number of controls in the problem is typically smaller than the number of
replications, these constraints do not determine the probabilities [35].
We choose a particular set of probabilities by selecting the maximum entropy
measure as the objective. The problem specific optimisation problem is defined
by,(4.13)maxp∈ℝM∑w*=1Mpw*logpw*subjectto∑w*=1Mpw*pz*t∣Vw*=pz*t∣V∀z*∈{1,2,…,N},pw*≥0∀w*∈{0,1,…,M},∑w*=1Mpw*=1.The CDO tranche prices
constraint will only hold if the bespoke and index collateral portfolios have
the same underlying, while the constraint on default probabilities will always
hold.


#### 4.2.4. Augmented Lagrangian Methods

The augmented
Lagragian method seeks the solution by replacing the original constrained
problem with a sequence of unconstrained subproblems in which the objective
function is formed by the original objective of the constrained optimisation
plus additional “penalty” terms. These terms are made up of constraint
functions multiplied by a positive coefficient.For the problem at hand, the augmented Lagrangian
function is given by,(4.14)LM*(x;s;λM;μM)=∑m=1Mpmlogpm+λM(1)(1−∑i=1Mpi,M)+λM(2)∑k=1N(pkt∣V−∑i=1Mpi,Mpkt∣Vm)+12μM(1−∑i=1Mpi,M)2+12μM+(∑k=1N(pkt∣V−∑i=1Mpi,Mpkt∣Vm))2.


#### 4.2.5. Implementation Issues

In practice, a
significant proprotion of computation time is not spent solving the
optimisation problem itself, but rather computing the coefficients in the
linear constraints. The marginal default probability constraints require the
evaluation of the conditional default probability given in the previous chapter
for each name under each scenario.One has the option to only match the cumulative
implied default probability to the end of the lifetime of the bespoke CDO or
perhaps at selected times only. The advantage of dropping constraints is
twofold. Firstly it reduces the computational burden by reducing the number of
coefficients that need to be computed, thus leading to a faster pricing
algorithm. Secondly, it loosens the conditions required of the probabilitiesp, thus resulting in factor implied distributions with
superior “quality,” as given by higher values for the fitness function [5].
Matlab is used for implementing the model.


### 4.3. Interpreting the Implied Distributions

The valuation
of CDOs depends on the portfolio loss distribution. For the pricing of a CDO orCDO2 it is
sufficient to know the portfolio loss distributions over different time
horizons. The implied credit loss distributions should be considered relative
to the prior models, before deriving efficient frontiers for the bespoke
portfolios. This is crucial as deviations from the model will result in a
sub-optimal asset allocation strategy.Figure2 shows the implied and model distribution of
default losses for the benchmark portfolio under the Gaussian copula
assumption. Kernel smoothing was applied to the density results. This approach
is a nonparametric way of estimating the probability density function of a
random variable. This density estimation technique makes it possible to
extrapolate the data to the entire population.The comparison of the implied loss density under Gaussian copula assumption: (a) shows the deviation of the model density from the implied density at the 5 year horizon, (b) displays the implied credit loss density surface up to the 5 year horizon.
(a)(b)The first thing to note is the typical shape of credit
loss distributions. Due to the common dependence on the factorV, defaults are correlated. This association gives rise
to a portfolio credit loss density that is right-skewed and has a right-hand
tail. The loss surface as a whole is widening and flattening, with an
increasing expected portfolio loss.The Clayton copula model displays similar deviations.
This is shown in Figure3. The Clayton assumption will still under-estimate
small losses, but this effect is less pronounced than the Gaussian case. The
maximum portfolio loss obtained over a 5 year horizon is 53.7% as compared to
the 44.3% in the Gaussian case. The loss surface as a whole is widening and
flattening with an increasing expected portfolio loss. The maximum loss
increases from 44.2% over a 6 month horizon to 53.7% over five years. The
Clayton copula model still does not perfectly capture the market dynamics, but
is an improvement over the Gaussian case.The
comparison of the implied loss density under Clayton copula assumption:  (a) shows the deviation of the model density from the implied density at
the 5 year horizon, (b) displays the implied credit loss density
surface up to the 5 year horizon.
(a)(b)Similar sentiments on the weakness of Gaussian copula
are shared by Li and Liang [36].Figure4 uses a logarithmic scale for the
probabilities to show the tail effects more clearly.Figure 4
The comparison of implied tail
probabilities under different copula assumptions.The probabilities decrease very quickly under the
Gaussian and Clayton copula assumptions. The effect of thicker tails under the
Clayton copula can easily be seen to dominate the Gaussian copula. In the pricing
of the super senior tranche, the Clayton model exhibits higher expected losses
due to the excess mass concentrated in the tails of the distribution, the
tranche spread will be higher under this model than the Gaussian case. These
deviations in the implied distribution under different distributional
assumptions will filter through to the resulting efficient frontiers. Due to
the higher tail probabilities the Clayton ETL efficient frontiers will be
significantly different from frontiers resulting from the Gaussian assumption.
This feature will be shown in the subsequent sections.The results so far also have a practical edge for
credit risk management. The likelihood of extreme credit losses is increased
under the Clayton copula assumption. This is due to the lower tail dependency
exhibited by this copula function.


## 4.1. Synthetic CDO Pricing

The key idea
behind CDOs, is the tranching of the credit risk of the underlying portfolio. A
given tranchentr is defined by
its attachment and detachment points untrlower and untrupper respectively.
The tranche notional is given by: sntr=Nprot(untrupper−untrlower), where Nprot denotes the
total portfolio value.LetLtotal(t) be the
percentage cumulative loss in the portfolio value at time t. The total cumulative loss at time t is then Ltotal(t)Nprot. The loss suffered by the holders of tranche ntr from
origination to time t is a percentage Lntr(t) of the
portfolio notional value Nprot:(4.1)Lntr(t)=min{max{Ltotal(t)−Lntr,0},untrupper−untrlower}.We consider a transaction
initiated at time 0, with maturity T. In a CDO contract, the tranche investor adopts a
position as a protection seller. Similar to the assumption under the CDS
pricing, we assume that defaults occur at the midpoints between coupon payment
dates. The value of the protection leg is then given by the
following:(4.2)𝔼[PVprot(ntr)]=∑j=1JD(0,tj−tj−12)(𝔼[Lntr(tj)]−𝔼[Lntr(tj−1)]),where 𝔼[Lntr(⋅)] is the
expectation with respect to the risk neutral measure ℚ and is
calculated by a simulation model. The tranche loss profiles under each scenario
are calculated and stored. The expectation is found by calculating the weighted
average over all scenarios. This only applies if a weighted Monte Carlo scheme
is used.The tranche investors need to be compensated for
bearing the default risk in the underlying credits. The holders of tranchen receive a
periodic coupon payment. Let the coupon payment dates be denoted as 0=t0≤t1≤t2⋯≤tJ=T. The predetermined frequency of the coupon payment
dates is usually on a quarterly basis. The spread paid for protection on a
given tranche does not vary during the life of the contract. and is usually
quoted in basis points per annum. However, the tranche notional decays through
the life of the contract. The outstanding tranche notional at time t is given by the
following:(4.3)Noutntr(t)=(untrupper−untrlower−𝔼[Lntr(t)])Nprot.The expected outstanding tranche
notional since the last coupon date must be considered at the coupon payment
dates. This amount between coupon payment dates tj and tj−1 is simply the
average of Noutntr(tj) and Noutntr(tj−1). We assume again that defaults can only occur at the
midpoint between arbitrary coupon dates. The expected outstanding tranche
notional is denoted by:(4.4)𝔼[Noutntr(tj,tj−1)]=(untrupper−untrlower−𝔼[Lntr(tj)]+𝔼[Lntr(tj)]−𝔼[Lntr(tj−1)]2).Using this equation we can
compute the expected present value of the coupon payments:(4.5)𝔼[PVpremntr]=∑j=1Jsntr(tj−tj−1)D(0,tj)𝔼[Noutntr(tj,tj−1)].This fair spread of a tranche
can be computed by equating the expected present value of the protection and
premium legs. The market quotation for the equity tranche is to have a fixed
500 bps spread, and the protection buyer make an upfront payment of a fixed
percentage of the tranche notional. The protection seller receives the upfront
fee expressed as a percentage f of the tranche
notional, so that equity investors purchase the note at a discount f(ueq)Nprot. Only the premium leg is different for equity tranche
investors. This is given by the following:(4.6)𝔼[PVprem(eq)]=f(ueq)Nprot+∑j=1Jseq(tj−tj−1)D(0,tj)𝔼[Nout(eq)(tj,tj−1)].


## 4.2. Weighted Monte Carlo Techniques

Monte Carlo
algorithms can be divided (somewhat arbitrarily) into two categories: uniformly
weighted and nonuniformly weighted algorithms. nonuniform weights are a
mechanism for improving simulation accuracy. Consider a set ofM paths,
generated by a simulation procedure. A nonuniformly weighted simulation is one
in which the probabilities are not necessarily equal. Suppose that we assign,
respectively, probabilities p1,p2,…,pM, to the different paths. The value of the security
according to the nonuniform weights is,(4.7)Ωh=∑m=1MpmΛm, where Λ is the payoff
function.Two features of credit risk modelling which pose a
particular challenge under simulation based procedures, namely:(1)
it requires
accurate estimation of low-probability events of large credit losses; and(2)
default
dependence mechanisms described in the previous chapter do not immediately lend
themselves to rare-event simulation techniques used in other settings.
It is for these
reasons that we implement a weighted Monte Carlo simulation procedure to put
emphasis on determining directly the risk neutral probabilities of the future
states of the market, which will allow the accurate determination of the credit
loss distribution. This is in contrast to calibration of pricing models through
traditional methods.In what follows, we introduce the general modelling
framework, present the algorithm and discuss some technical implementation
details. (We use a similar methodology to Rosen and Saunders [5].) This generalised framework can be
applied to credit risk models that have a factor structure.A weighted Monte Carlo method can be used to find an
implied risk-neutral distribution of the systematic factors, assuming the
specification of a credit risk model with a specified set of parameter values.
According to Rosen and
Saunders [5] the justification for this approach follows working within a
conditional independence framework, where obligor defaults are independent,
conditional on a given systematic risk factor.The methodology to obtain the implied credit loss
distribution is summarised by the following steps.(i)
Latent factor scenarios. Define M scenarios on the systematic factor V.(ii)
Conditional portfolio loss distribution.
Calculate the conditional portfolio loss profile for scenario m.(iii)
Conditional tranche values: Infer the
tranche loss distributions conditional on scenario m.(iv)
Implied scenario probabilities (optimisation
problem). Given a set of scenario probabilities pm; tranche values are given as expectations over all
scenarios of the conditional values. We solve the resulting constrained inverse
problem to find a set of implied scenario probabilities pm.(v)
Implied credit loss distribution-Calculate the aggregate credit loss at each time step given the implied
scenario probabilities.
The first three
steps above have been discussed in detail in the previous sections of this
chapter. The focus now is placed on the optimisation problem for implying the
scenario probabilities.


### 4.2.1. Mathematical Formulation

Leth:ℝM→ℝ∪{+∞} be a strictly
convex function and let wi,M denote the
weight of the ith path of M paths. Now
consider the optimisation problem:(4.8)minw1,M,w2,M,…,wM,M∑i=1Mh(wi,M)subjectto1M∑i=1Mwi,M=1,1M∑i=1Mwi,MGi=cG, for some fixed cG∈ℝN. The objective function is strictly convex and the
constraints are linear, so if a feasible solution exist, with a finite
objective function value, then there is a unique optimal solution w1,M,w2,M,…,wM,M. This optimum defines the weighted Monte Carlo
estimator,(4.9)Ψ^LCV=∑i=1Mwi,MΨi.The weights derived from
(4.8) can be made more explicit by introducing the Lagragian. The strict
convexity of the objective function is not a sufficient condition to guarantee
a unique solution.The classical approach to solving constrained
optimisation problems is the method of Lagrange multipliers. This approach
transforms the constrained optimisation problem into an unconstrained one, thereby
allowing the use of the unconstrained optimisation techniques.


### 4.2.2. Objective Functions

Taking the
objective to be a symmetric separable convex function gives the optimal
probabilitiesp. This is an interpreted as the most uniform
probabilities satisfying the constraints, though different objective functions
imply different measures of uniformity. A common choice for the fitness measure
is entropy. This is a
particularity interesting and in some respects a convenient objective. The
principle of maximum entropy gives a method of generating a probability
distribution from a limited amount of information. It is a relatively well used
principle for the construction of probabilistic models.In this setting,(4.10)h=∑w*=1Mpw*logpw*.This case is put forward in Avellaneda et  al. [34], where a Bayesian
interpretation is given. The usual convention of 0log0 is followed. We
now provide a problem specific description of the optimisation problem using
the principle of maximum entropy.


### 4.2.3. Constraint Description

For the bespoke
CDO pricing problem, the implied scenario probabilities must satisfy the
following constraints:(i)
the sum of all
scenario probabilities must equal to one;(ii)
the
probabilities should be positive:pi≥0foralli∈{0,1,…,M}; and bounded by 1;(iii)
we match the
individual CDS spreads (i.e., marginal default probabilities for each
name)(4.11)∑w*=1Mpw*pz*t∣Vw*=pz*t∣V∀z*∈{1,2,…,N};(iv)
the current
market prices of standard CDO tranches are matched given by(4.12)∑w*=1Mpw*PVprot(ntr)=∑w*=1Mpw*PVprem(ntr)∀ntr∈{1,2,…}tranches.
Because the
number of controls in the problem is typically smaller than the number of
replications, these constraints do not determine the probabilities [35].
We choose a particular set of probabilities by selecting the maximum entropy
measure as the objective. The problem specific optimisation problem is defined
by,(4.13)maxp∈ℝM∑w*=1Mpw*logpw*subjectto∑w*=1Mpw*pz*t∣Vw*=pz*t∣V∀z*∈{1,2,…,N},pw*≥0∀w*∈{0,1,…,M},∑w*=1Mpw*=1.The CDO tranche prices
constraint will only hold if the bespoke and index collateral portfolios have
the same underlying, while the constraint on default probabilities will always
hold.


### 4.2.4. Augmented Lagrangian Methods

The augmented
Lagragian method seeks the solution by replacing the original constrained
problem with a sequence of unconstrained subproblems in which the objective
function is formed by the original objective of the constrained optimisation
plus additional “penalty” terms. These terms are made up of constraint
functions multiplied by a positive coefficient.For the problem at hand, the augmented Lagrangian
function is given by,(4.14)LM*(x;s;λM;μM)=∑m=1Mpmlogpm+λM(1)(1−∑i=1Mpi,M)+λM(2)∑k=1N(pkt∣V−∑i=1Mpi,Mpkt∣Vm)+12μM(1−∑i=1Mpi,M)2+12μM+(∑k=1N(pkt∣V−∑i=1Mpi,Mpkt∣Vm))2.


### 4.2.5. Implementation Issues

In practice, a
significant proprotion of computation time is not spent solving the
optimisation problem itself, but rather computing the coefficients in the
linear constraints. The marginal default probability constraints require the
evaluation of the conditional default probability given in the previous chapter
for each name under each scenario.One has the option to only match the cumulative
implied default probability to the end of the lifetime of the bespoke CDO or
perhaps at selected times only. The advantage of dropping constraints is
twofold. Firstly it reduces the computational burden by reducing the number of
coefficients that need to be computed, thus leading to a faster pricing
algorithm. Secondly, it loosens the conditions required of the probabilitiesp, thus resulting in factor implied distributions with
superior “quality,” as given by higher values for the fitness function [5].
Matlab is used for implementing the model.


## 4.2.1. Mathematical Formulation

Leth:ℝM→ℝ∪{+∞} be a strictly
convex function and let wi,M denote the
weight of the ith path of M paths. Now
consider the optimisation problem:(4.8)minw1,M,w2,M,…,wM,M∑i=1Mh(wi,M)subjectto1M∑i=1Mwi,M=1,1M∑i=1Mwi,MGi=cG, for some fixed cG∈ℝN. The objective function is strictly convex and the
constraints are linear, so if a feasible solution exist, with a finite
objective function value, then there is a unique optimal solution w1,M,w2,M,…,wM,M. This optimum defines the weighted Monte Carlo
estimator,(4.9)Ψ^LCV=∑i=1Mwi,MΨi.The weights derived from
(4.8) can be made more explicit by introducing the Lagragian. The strict
convexity of the objective function is not a sufficient condition to guarantee
a unique solution.The classical approach to solving constrained
optimisation problems is the method of Lagrange multipliers. This approach
transforms the constrained optimisation problem into an unconstrained one, thereby
allowing the use of the unconstrained optimisation techniques.


## 4.2.2. Objective Functions

Taking the
objective to be a symmetric separable convex function gives the optimal
probabilitiesp. This is an interpreted as the most uniform
probabilities satisfying the constraints, though different objective functions
imply different measures of uniformity. A common choice for the fitness measure
is entropy. This is a
particularity interesting and in some respects a convenient objective. The
principle of maximum entropy gives a method of generating a probability
distribution from a limited amount of information. It is a relatively well used
principle for the construction of probabilistic models.In this setting,(4.10)h=∑w*=1Mpw*logpw*.This case is put forward in Avellaneda et  al. [34], where a Bayesian
interpretation is given. The usual convention of 0log0 is followed. We
now provide a problem specific description of the optimisation problem using
the principle of maximum entropy.


## 4.2.3. Constraint Description

For the bespoke
CDO pricing problem, the implied scenario probabilities must satisfy the
following constraints:(i)
the sum of all
scenario probabilities must equal to one;(ii)
the
probabilities should be positive:pi≥0foralli∈{0,1,…,M}; and bounded by 1;(iii)
we match the
individual CDS spreads (i.e., marginal default probabilities for each
name)(4.11)∑w*=1Mpw*pz*t∣Vw*=pz*t∣V∀z*∈{1,2,…,N};(iv)
the current
market prices of standard CDO tranches are matched given by(4.12)∑w*=1Mpw*PVprot(ntr)=∑w*=1Mpw*PVprem(ntr)∀ntr∈{1,2,…}tranches.
Because the
number of controls in the problem is typically smaller than the number of
replications, these constraints do not determine the probabilities [35].
We choose a particular set of probabilities by selecting the maximum entropy
measure as the objective. The problem specific optimisation problem is defined
by,(4.13)maxp∈ℝM∑w*=1Mpw*logpw*subjectto∑w*=1Mpw*pz*t∣Vw*=pz*t∣V∀z*∈{1,2,…,N},pw*≥0∀w*∈{0,1,…,M},∑w*=1Mpw*=1.The CDO tranche prices
constraint will only hold if the bespoke and index collateral portfolios have
the same underlying, while the constraint on default probabilities will always
hold.


## 4.2.4. Augmented Lagrangian Methods

The augmented
Lagragian method seeks the solution by replacing the original constrained
problem with a sequence of unconstrained subproblems in which the objective
function is formed by the original objective of the constrained optimisation
plus additional “penalty” terms. These terms are made up of constraint
functions multiplied by a positive coefficient.For the problem at hand, the augmented Lagrangian
function is given by,(4.14)LM*(x;s;λM;μM)=∑m=1Mpmlogpm+λM(1)(1−∑i=1Mpi,M)+λM(2)∑k=1N(pkt∣V−∑i=1Mpi,Mpkt∣Vm)+12μM(1−∑i=1Mpi,M)2+12μM+(∑k=1N(pkt∣V−∑i=1Mpi,Mpkt∣Vm))2.


## 4.2.5. Implementation Issues

In practice, a
significant proprotion of computation time is not spent solving the
optimisation problem itself, but rather computing the coefficients in the
linear constraints. The marginal default probability constraints require the
evaluation of the conditional default probability given in the previous chapter
for each name under each scenario.One has the option to only match the cumulative
implied default probability to the end of the lifetime of the bespoke CDO or
perhaps at selected times only. The advantage of dropping constraints is
twofold. Firstly it reduces the computational burden by reducing the number of
coefficients that need to be computed, thus leading to a faster pricing
algorithm. Secondly, it loosens the conditions required of the probabilitiesp, thus resulting in factor implied distributions with
superior “quality,” as given by higher values for the fitness function [5].
Matlab is used for implementing the model.


## 4.3. Interpreting the Implied Distributions

The valuation
of CDOs depends on the portfolio loss distribution. For the pricing of a CDO orCDO2 it is
sufficient to know the portfolio loss distributions over different time
horizons. The implied credit loss distributions should be considered relative
to the prior models, before deriving efficient frontiers for the bespoke
portfolios. This is crucial as deviations from the model will result in a
sub-optimal asset allocation strategy.Figure2 shows the implied and model distribution of
default losses for the benchmark portfolio under the Gaussian copula
assumption. Kernel smoothing was applied to the density results. This approach
is a nonparametric way of estimating the probability density function of a
random variable. This density estimation technique makes it possible to
extrapolate the data to the entire population.The comparison of the implied loss density under Gaussian copula assumption: (a) shows the deviation of the model density from the implied density at the 5 year horizon, (b) displays the implied credit loss density surface up to the 5 year horizon.
(a)(b)The first thing to note is the typical shape of credit
loss distributions. Due to the common dependence on the factorV, defaults are correlated. This association gives rise
to a portfolio credit loss density that is right-skewed and has a right-hand
tail. The loss surface as a whole is widening and flattening, with an
increasing expected portfolio loss.The Clayton copula model displays similar deviations.
This is shown in Figure3. The Clayton assumption will still under-estimate
small losses, but this effect is less pronounced than the Gaussian case. The
maximum portfolio loss obtained over a 5 year horizon is 53.7% as compared to
the 44.3% in the Gaussian case. The loss surface as a whole is widening and
flattening with an increasing expected portfolio loss. The maximum loss
increases from 44.2% over a 6 month horizon to 53.7% over five years. The
Clayton copula model still does not perfectly capture the market dynamics, but
is an improvement over the Gaussian case.The
comparison of the implied loss density under Clayton copula assumption:  (a) shows the deviation of the model density from the implied density at
the 5 year horizon, (b) displays the implied credit loss density
surface up to the 5 year horizon.
(a)(b)Similar sentiments on the weakness of Gaussian copula
are shared by Li and Liang [36].Figure4 uses a logarithmic scale for the
probabilities to show the tail effects more clearly.Figure 4
The comparison of implied tail
probabilities under different copula assumptions.The probabilities decrease very quickly under the
Gaussian and Clayton copula assumptions. The effect of thicker tails under the
Clayton copula can easily be seen to dominate the Gaussian copula. In the pricing
of the super senior tranche, the Clayton model exhibits higher expected losses
due to the excess mass concentrated in the tails of the distribution, the
tranche spread will be higher under this model than the Gaussian case. These
deviations in the implied distribution under different distributional
assumptions will filter through to the resulting efficient frontiers. Due to
the higher tail probabilities the Clayton ETL efficient frontiers will be
significantly different from frontiers resulting from the Gaussian assumption.
This feature will be shown in the subsequent sections.The results so far also have a practical edge for
credit risk management. The likelihood of extreme credit losses is increased
under the Clayton copula assumption. This is due to the lower tail dependency
exhibited by this copula function.


## 5. Risk Characterisation of Credit portfolios

Comparison of
uncertainty in outcomes is central to investor preferences. If the outcomes
have a probabilistic description, a wealth of concepts and techniques from
probability theory can be applied. The main objective in the following section
is to present a review of the fundamental work by Artzner et  al. [37] and Föllmer and Schied [38] from an
optimisation point of view, since a credit risk measure will represent one of
the objectives in the CDO optimisation problem.LetR denote the set
of random variables defined on the probability space (Ωp,ℱp,ℚp). We define Ωp as a set of
finitely many possible scenarios for the portfolio p. Financial risks are represented by a convex cone ℳ⊆(Ωp,ℱp,ℚp) of random
variables. Any random variable Lc in this set
will be interpreted as a possible loss of some credit portfolio over a given
time horizon. The following provides a definition of a convex
cone.Definition 5.1 (convex cone).
ℳ is a convex cone if,
(1)
Lc(1)∈ℳ and Lc(2)∈ℳ implies that Lc(1)+Lc(2)∈ℳ; and(2)
λ*Lc∈ℳ for every λ*≥0.Definition 5.2 (measures of risk).
Given some convex cone,ℳ of random
variables, a measure of risk Θ with domain ℳ is a
mapping:(5.1)Θ:ℳ→ℝ.From an economic perspective, Θ(Lc), can be regarded as the capital buffer that should be
set aside for adverse market movements. In order to measure and control the
associated risk Artzner et  al. [37] introduced an axiomatic framework of coherent risk
measures which were recently “generalised” by Föllmer and Schied [38] to convex risk
measures.Definition 5.3 (Convex risk measures).
A mappingΘ:ℳ→ℝ is called
a convex risk measure, if and only if it is
(1)
convex for every Lc(1) and Lc(2)∈ℳ, one has Θ(ΛLc(1)+(1−Λ)Lc(2))≤ΛΘ(Lc(1))+(1−Λ)Θ(Lc(2)) for some Λ∈ℝ;(2)
monotone for every Lc(1) and Lc(2)∈ℳ with Lc(1)≤Lc(2), one has Θ(Lc(1))≤Θ(Lc(2)); and(3)
translation invariant if a* is a constant
then Θ(Lc+a*1)=−a*+Θ(Lc), where 1 denotes the
unit vector.By adding positive homogeneity to these properties, one
obtains the following definition.Definition 5.4 (coherent risk measures).
A convex risk
measureΘ is called coherent, if in addition it is,
Positively homogeneous if Λ≥0 then Θ(ΛLc)=ΛΘ(Lc) holds.Denote the credit loss distribution of Lc by FLc(lc)=ℚ(Lc≤lc). In the analysis we are concerned solely with two
risk measures which are based on the loss distribution FLc, namely VaR and ETL. We now recall the definition of
these risk measures.Definition 5.5 (value-at-risk (VaR)).
Given some
confidence levelβ*∈(0,1), the Value-at-Risk (VaR) of the credit
portfolio at the confidence level β* is given by the
smallest number lc such that the
probability that the loss Lc exceeds lc is no larger
than (1−β*). Formally,(5.2)VaR=inf(lc∈ℝ:ℚ(Lc>l)≤1−β*).This definition of VaR coincides with the definition of an β*-quantile of
the distribution of Lc in terms of a
generalised inverse of the distribution function FLc. We observe this coincidence by
noting,(5.3)VaR=inf(lc∈ℝ:1−FLc(lc)≤1−β*)=inf(lc∈ℝ:FLc(lc)≥β*).For a random variable Y we will denote
the β*-quantile of
the distribution by qβ*(FY), and write VaRβ*(Y) when we wish to
stress that the quantile should be interpreted as a VaR number. A simple
definition of ETL which suffices for continuous loss distributions is as follows.Definition 5.6 (expected-tail-loss (ETL)).
Consider a lossL with continuous FLdf satisfying(5.4)∫ℝ|l|dFLc(lc)<∞.Then the Expected-Tail-Loss at confidence level α∈(0,1), is defined to be,(5.5)ETLβ*=𝔼[Lc∣Lc≥VaRβ*]=𝔼[Lc;Lc≥VaRβ*(Lc)]ℚ[Lc≥VaRβ*(Lc)].


## 6. Optimal Bespoke CDO Design

Credit
portfolio optimisation plays a critical role in determining bespoke CDO
strategies for investors. The most crucial tasks in putting together bespoke
CDOs is choosing the underlying credits that will be included in the portfolio.
Usually investors often express preferences on individual names to which they
willing to have the exposure, while there are likely to be credit rating
constraints and industry/geographical concentration limits imposed by rating
agencies and/or investors.Given these various requirements, it is up to the
credit structurer to optimise the portfolio and achieve the best possible
tranche spreads for investors. In the following analysis, we focus on the asset
allocation rather than credit selection strategy, which remains a primary
modelling challenge for credit structurers.Davidson [2] provides an interesting analogy between
bespoke CDO optimisation and Darwin's evolutionary cycles. In the natural
world, life adapts to suit the particulars of its environment. The adaptation
to a specific environment is possible due to the application of powerful set
evolutionary techniques—reproduction, mutation and survival of the fittest.
Nature then explores the full range of possible structures to hone in on those
that are most perfectly suited to the surroundings.Creating a portfolio for a CDO can broadly be seen in
similar ways. Given a certain set of investor defined, constraintsstructurers
need to be able to construct a credit portfolio that is best suited to the
market environment. Added to these constraints are market constraints such as
trade lot restrictions and liquidity and availability of underlying credits. If
the portfolio does not suit these conditions, it evolves so that only those
with the best fit (highest tranche spreads) will survive. Many of the same
techniques used in the natural world can be applied to this structuring
process.Evolutionary computation methods are exploited to
allow for a generalisation of the underlying problem structure and to solve the
resulting optimisation problems, numerically in a systematic way. The next
section will briefly discuss some of the basic concepts of multiobjective
optimisation and outline the NSGA-II algorithm used for solving the challenging
CDO optimisation problem. The CDO optimisation model is then outlined before
conducting a prototype experiment on the test portfolio constructed from the
constituents of the iTraxx Europe IG S5 index.


### 6.1. Multiobjective Optimisation

Many real-world
problems involve simultaneous optimisation of several incommensurable and often
competing objectives. In single-objective optimisation the solution is usually
clearly defined, this does not hold for multiobjective optimisation problems.
We define formally, the multiobjective optimisation problem to define other
improtant concepts used in this chapter. All problems are assumed to be
minimisation problems unless otherwise specified To avoid inserting the same
reference every few lines, note that the terms and definitions are taken from
Zitzler [39], and adapted to CDO optimisation problem.Definition 6.1 (multiobjective optimisation problem).
A generalmultiobjective optimisation Problem (MOP)
includes a set of n* parameters
(decision variables), a set of k1 objective
functions, and a set of k2 constraints.
Objective functions and constraints are functions of the decision variables.
The optimisation goal is to obtain,(6.1)miny=f(x)=(f1(x),f2(x),…,fk1(x))subjectto,e(x)=(e1(x),e2(x),…,ek2(x))≤0,x=(x1,x2,...,xn)∈X,y=(y1,y2,...,yk1)∈Y,where x and y are the
decision and objective vectors, respectively, whilst X and Y denote the
decision and objective spaces, respectively. The constraints e(x)≤0 determine the
set of feasible solutions.Without loss of generality, a minimisation problem is
assumed here. For maximization or mixed maximisation/minimisation problems the
definitions are similar.Definition 6.2 (vector relationship).
For any two
vectorsr and r*(6.2)r=r*iffri=ri*∀i∈{1,2,…,k1},r≰r*iffri≤ri*∀i∈{1,2,…,k1},r<r*iffr≤r*⋀r≤r*.The relations ≥ and > are
similarly defined.Although the concepts and terminology of Pareto optimality
are frequently invoked, most often they are erroneously used in literature. We
now define this set of concepts to ensure understanding and
consistency.Definition 6.3 (Pareto dominance).
For any two
vectorsr and r*(6.3)r⊀r*ifff(r)≤f(r*)thenrdominatesr*,r⪯̸r*ifff(r)≤f(r*)thenrweaklydominatesr*,r~r*ifff(r)≰f(r*)⋀f(r*)≰f(r)thenrisindifferenttor*.The definitions for a
maximisation problems (≻,⋡,~) are analogous.Definition 6.4 (Pareto optimality).
A decision
vectorx∈Xf is said to be
nondominated regarding a set K⊂Xf iff(6.4)∄k∈K:k⊀K.If it is clear within the
context which set K is meant, it is
simply left out. Moreover, x is said to be
Pareto optimal iff x is nondominated
regarding Xf.The entirety of all Pareto-optimal solutions is termed the Pareto-optimal set; the corresponding
objective vectors form the Pareto-optimal frontier or
surface.Definition 6.5 (nondominated sets and frontiers).
LetK⊂Xf. The function 𝒩d(K) gives the set
of nondominated decision vectors in K:(6.5)𝒩d(K)={k∈K∣kisnondominatingregardingK}.The set 𝒩d(K) is the
nondominated set regarding K, the
corresponding set of objective vectors f(𝒩d(K)) is the nondominated
front regarding K. Furthermore, the set X𝒩d=𝒩d(Xf) is called the Pareto-optimal set and the set Y𝒩d=f(Xf), is denoted as the Pareto-optimal frontier.


### 6.2. Nondominating Sort Genetic Algorithm (NSGA-II)

The second
generation NSGA-II is a fast and elitist multiobjective evolutionary
algorithm. The main features are
(http://www.kxcad.net/ESTECO/modeFRONTIER320/html/userman/ch07s01s10.html).(i)
Implementation of a fast nondominated sorting
procedure: Sorting of the individuals of a given population is according
to the level of nondomination. Generally, nondominated sorting algorithms are
computationally expensive for large population sizes however, the adopted
solution performs a clever sorting strategy.(ii)
Implementation of elitism for multiobjective
search: Using an elitism-preserving approach introduces storing all
nondominated solutions discovered so far, beginning from the initial
population. Elitism enhances the convergence properties towards the true
Pareto-optimal set.(iii)
Adopting a parameter-less diversity
preservation mechanism: Diversity and spread of solutions is guaranteed
without use of sharing parameters, since NSGA-II adopts a suitable
parameter-less niching approach. This niche is accomplished by the crowding
distance measure, which estimates the density of solutions in the objective
space, and the crowded comparison operator, which guides the selection process
towards a uniformly spread Pareto frontier.(iv)
Constraint handling method does not make use
of penalty parameters: The algorithm implements a modified definition of
dominance in order to solve constrained multiobjective problems efficiently.(v)
Real-coded and binary-coded design variables: A new feature is the application of the genetic algorithms in the field of
continuous variables.


#### 6.2.1. Representation of Individuals

The first stage
of building the EA is to link the “real world” to the “EA
world.” This linking involves setting up a bridge between the original
problem context and the problem-solving solving space, where evolution takes
place. The objects forming possible solutions within the original problem
context are referred to as aphenotype, while their encoding are called genotypes or more commonly chromosomes.The mapping from phenotype space to the genotype space
is termedencoding. The inverse
mapping is termed decoding.
Choosing an appropriate representation for the problem being solved is
improtant in the design of a successful EA. Often it comes down to good
knowledge of the application domain [40].Many different encoding methods have been proposed and
used in EA development. Few frequently applied representations are: binary,
integer and real valued representation. Real-valued or Floating-point
representation is often the most sensible way to represent a candidate solution
to of a problem. This approach is appropriate when the values that we want to
represent as genes, originate from a continuous distribution.The solutions to the proposed CDO optimisation models
are real-valued. This study opted to use the real-valued encoding, for the sake
of operational simplicity. The genes of a chromosome are real numbers between 0
and 1, which represents the weights invested in the different CDS contracts.
However, the summation of these weights might not be 1 in the initialisation
stage or after genetic operations. To overcome this problem, the weights are
normalised as follows:(6.6)xi′=xi∑i=1Nxi.


#### 6.2.2. Evaluation Function

The role of theevaluation function is to
represent the requirements to which the population should adapt. This role
forms the basis of selection. Technically, the function assigns a quality
measure to the genotypes. Typically the function is composed from a quality
measure in the phenotype space and the inverse representation [40]. In the
evolutionary context the function is usually referred to as the fitness
function.In the bespoke CDO optimisation problem, we introduce
two objective to the problem. These objectives are namely the CDO tranche
return, and the portfolio tail risk measured by ETL. Various constraints are
then introduced to study the dynamics of the Pareto frontier under various
conditions.


#### 6.2.3. Population Models and Diversity Preservation

Unlike
operators, which operate on individuals, selection operators like parent and
survivor selection work at population level. In most EA applications, the
population size is constant and does not change during the evolutionary search.
Thediversity of the population
is a measure of the number of different solutions present.In NSGA-I, the well-known sharing function approach
was used, which was found to maintain sustainable diversity in a population
[4]. There are however two difficulties with this sharing function.
According to Deb et  al. [4], the performance of the sharing function method in
maintaining a spread of solutions depends largely on the value of σs, and since each solution must be compared with all
other solutions in the population, the overall complexity of the sharing
function approach is O(Npop2).Deb et  al. [4] replace the sharing function approach with a
crowded-comparison approach in NSGA-II. This eliminates both the above
difficulties to some extent. This approach does not require any user-defined
parameter for maintaining diversity among population members, and also has a
better computational complexity.To describe this approach, we first define a metric
for density estimation and then present the description of the
crowded-comparison operator.


#### 6.2.4. Crowded Comparison Operator

The density of
solutions surrounding a particular solution in the population must firstly be
estimated by calculating the average distance of two points on either side of
this point, along each of the objectives.Once the nondominated sort is complete, the crowding
distance is assigned. Individuals are selected based on rank and crowding
distance. The crowding-distance computation requires sorting the population according
to each objective function value in ascending order of magnitude. The boundary
solutions (solutions with smallest and largest function values) for each
objective function are assigned an infinite distance value. All other
intermediate solutions are assigned a distance value equal to the absolute
normalised difference in the function values of two adjacent solutions. This
calculation is continued with other objective functions. The overall
crowding-distance value is calculated as the sum of individual distance values
corresponding to each objective. Each objective function is normalised before
calculating the crowding distance. (See Deb et  al. [4] for more details regarding the
algorithm.)The complexity of this procedure is governed by the
sorting algorithm and has computational complexity ofO(M*NpoplogNpop), where there are M* independent
sorting of at most Npop individuals,
when all population members are in one front ℐ.After all population members in the setℐ are assigned a
distance metric, we can compare two solutions for their extent of proximity
with other solutions. A solution with a smaller value of this distance measure
is, in some sense, more crowded by other solutions.


#### 6.2.5. Nondomination Sorting Approach

In order to
identify solutions in the NSGA of the first nondominated front, each solution
is compared with every other solution in the population to find if it is
dominated (We summarise the algorithm discussed in Deb et  al. [4].). This procedure
requires O(k1Npop) comparisons for
each solution, where k1 is the number
of objectives. This process is continued to find all members of the first
nondominated level in the population, the total complexity is O(k1Npop2) computations.
At this stage, all individuals in the first nondominated front are found. In
order to find the individuals in the next nondominated front, the solutions of
the first front are discounted temporarily and the above procedure is repeated.
In the worst case, the task of finding the second front also requires O(k1Npop2) computations,
particularly when number of solutions belong to the second and higher
nondominated levels. This argument is true for finding third and higher levels
of nondomination.The worst case is when there areNpop fronts and
there exists only one solution in each front. This requires an overall O(k1Npop3) computations.
For each solution we calculate two entities:(1)
the domination
countnq, the number of
solutions which dominate the solution; and(2)
the set of
solutionsHq that the
solution dominates. This requires O(k1Npop2) comparisons.
Solutions in
the first nondominated front will have their domination count as zero. For each
solution with nq=0, each member q∈Hq is visited and
reduce its domination count by one. By doing so, these members belong to the
second nondominated front. This process continues until all fronts are
identified. For each solution in the second or higher level of nondomination,
the domination count can be at most Npop−1. Since there are at most such Npop−1 solutions, the
total complexity of the procedure is O(Npop2). Thus, the overall complexity of the procedure is O(k1Npop3).In NSGA-II the complexity reduction is due to the
realisation that the body of the first inner loop (for eachpi*∈𝒥i) is executed
exactly Npop times as each
individual can be the member of at most one front and the second inner loop
(for each q∈Hp) can be
executed at maximum Npop−1 times for
results in the overall computations. It is improtant to note that although the
time complexity has reduced to O(k1Npop2), the storage requirement has increased to O(Npop2).


#### 6.2.6. Parent Selection Mechanism

The selection
operator determines, which individuals are chosen for mating and how many
offspring each selected individual produces. Once the individuals are sorted
based on nondomination with crowding distance assigned, the selection is
carried out using a crowded comparison operator described above. The comparison
is carried out as below based on(1)
nondomination
rankprank*, that is,
individuals in FRONTi will have their
rank prank*=i; and(2)
crowding
distanceFRONTi(dj).
The selection
of individuals is through a binary tournament selection with the
crowed-comparison operator. In tournament selection the individuals are
chosen randomly from the population and the best individual from this group is
selected as parent. This process is repeated as often as individuals must be
chosen. These selected parents produce uniform at random offspring. The
parameter for tournament selection is the tournament size. This takes values
ranging from 2 to the number
of individuals in population [41].


#### 6.2.7. Mutation Operation

Mutation causes
individuals to be randomly altered. These variations are mostly small. They
will be applied to the variables of the individuals with a low probability.
Offspring are mutated after being created by recombination. Mutation of real
variables means, that randomly created values are added to the variables with a
low mutation probability. The probability of mutating a variable is inversely
proprotional to the number of variables (dimensions). The more dimensions one
individual has, the smaller is the mutation probability. Different papers
reproted results for the optimal mutation rate [41].The mutation step size is usually difficult to choose.
The optimal step-size depends on the problem considered and may even vary
during the optimisation process. Small mutation steps are often successful,
especially when the individual is already well adapted. However, large mutation
steps can produce good results with a faster convergence rate. According Pohleim
[41] good mutation operators should often produce small step-sizes with a
high probability and large step-sizes with a low probability.In the NSGA-II a polynomial mutation operator is used.
This operator is defined by the following:(6.7)Ψioffspring=Ψiparent+(Ψiup+Ψilow)δi,where Ψioffspring is the child,
and Ψiparent is the parent,
with Ψiup and Ψilow being the upper
and lower bounds of the parent component. δi is a small
variation which is calculated from a polynomial distribution given
below.(6.8)δi={(2ri)1/(ηm+1)−1ri<0.51−(2(1−ri))1/(ηm+1)−1ri≥0.5,where ri~𝒰(0,1) and ηm is mutation
distribution index.


#### 6.2.8. Recombination Operation

The
recombination operator produces new individuals in combining the information
contained in two or more parents in the mating population. This mating is done
by combining the variable values of the parents. Depending on the
representation of the variables different methods must be used.Using real-value representation, theSimulated Binary Crossover operator is
used for recombination and polynomial
mutation is used for mutating the offspring population [42].Simulated binary crossover simulates the binary
crossover observed in nature and is give by the following.(6.9)Ψ1,ioffspring=12((1−βk)Ψ1,kParent+(1+βk)Ψ2,kParent),Ψ2,ioffspring=12((1+βk)Ψ1,kParent+(1−βk)Ψ2,kParent),where βk(≥0) is a sample
from a random number generated having the density(6.10)fc(β)={12(ηc+1)βηc0≤β≤1,12(ηc+1)1βηc+2β≥1.This distribution can be
obtained from a u~𝒰(0,1) source. ηc is the
distribution index for the recombination operator.


#### 6.2.9. Main Loop

First, a random
parent population is created. The population is then sorted based on the
nondomination outlined above. Each individual is assigned a fitness value (or
rank) equal to its nondomination level, with 1 representing the best level.
Binary tournament selection, recombination, and mutation operators are then
applied to create an offspring population of sizeNpop.Due to the introduction of elitism, the current
population is compared to previously found best nondominated solutions, the
procedure is different after the initial generation. Figure5 outlines the tth generation of
the NSGA-II procedure.Figure 5
An outline of the NSGA-II procedure.The first step is to combined parent populationQ_t with the
offspring population R_t. The population is now of size 2Npop. The resulting population is then sorted according to
nondomination. Since all previous and current population members are included
in N_t=Q_t∪R_t, elitism is ensured.Solutions belonging to the best nondominated setFRONT(1), are of best solutions in the combined population and
must be emphasied more than any other solution in the combined population. If FRONT(1) is smaller than Npop, then all individuals in this set are chosen for the
next generation, else a truncation operation will have to be applied. If the
first case holds, then the remaining members of the next generation are chosen
from subsequent nondominated fronts in the order of their ranking.In general, the count of solutions in all sets fromFRONT(1) to FRONT(l) would be larger
than the population size. To choose exactly Npop population
members, the solutions of the last frontier are sorted and then the
crowded-comparison operator is applied in descending order. The best solutions
are chosen to fill all population slots. This procedure is valid for
minimisation problems only.The requisite diversity among nondominated solutions
is introduced by using the crowded comparison procedure, which is used in the
tournament selection and during the population truncation phases [4].


#### 6.2.10. Initialisation and Termination Conditions

Initialisation
is kept simple in most EA applications, the first population is seeded by
randomly generated individuals. In principle, problem-specific heuristics can
be used in this step, to create an initial population with higher fitness
values [40].We can distinguish two cases for an appropriate
termination condition. If the problem has a known optimum fitness level, then
an acceptable error bound would be a suitable condition. However, in many
multiobjective problems optimum values are not known. In this case one needs
to extend this condition with one that certainly stops the algorithm. As noted
by Eiben and Smith [40], a common choice is to impose a limit on the total number of fitness
evaluations.


### 6.3. Optimisation Problem

The optimal
bespoke structure can be derived by solving the following
problem:(6.11)minx∈[0,1]NΘ(x),maxx∈[0,1]Nsntr(x)subjectto∑i=1Nxi=1,where in the above problem, we
minimise the portfolio ETL, and simultaneously maximise the ntrth tranche spread
denoted by sntr subject to the
portfolio weights summing up to unity. This minimality may typically be the
case if only long positions in the underlying credits are allowed.For practical purposes, it may be crucial to include
other real-world constraints. Let𝒢* denote the set
of organisational, regulatory and physical constraints. The list of
constraints, which are commonly included into 𝒢* are summarised below.(1)
Upper and lower limits on issuer
weights: especially due to regulatory issues, the portfolio fractional amount invested
into each asset may be limited, that is, there may be individual lower and upper
bounds on the portfolio weights.(6.12)lupper(i)≤xi≤lupper(i)∀i∈{1,2,3,…,N}.
When short selling is disallowed, it can be explicitly
modelled by settingli=0 in this
formulation.(2)
Regional/industry concentration
constraints: due to rating agency and regulatory restrictions, the portfolio may have
industry and regional concentration limits, then(6.13)∑i=1kxi≤U(Iind,Igeo),where0≤U(Iind,Igeo)≤1,where the coordinate (Iind,Igeo) refers to a
particular industry Iind in region Igeo.(3)
Cardinality constraints: these
constraints limit the number of assets in the portfolio. They can either be
strict(6.14)𝒩(xi>0)=K*,0<K*≤a,K*∈ℕ.or lower and upper bounds on the
cardinality can be defined,(6.15)Kl*≤𝒩(xi>0)≤Ku*,0<Kl*<Ku*≤a,Kl*,Ku*∈ℕ.(4)
Weighted average rating factor (WARF)
constraints: due to regulatory or organisational restrictions, the
portfolio WARF must be above some minimum value,(6.16)Rprot≥Rtarget,where in the case of S and P
ratings Rprot∈{AAA,AA+,AA,AA−,…}. (Due to the
nature of CDOs, ratings higher than the portfolio rating can be achieved. In
this case the constraint may be defined as Rnthtranche≥Rtarget.)
Rating agencies presuppose that there are two broad
credit considerations that determine the risk of a CDO portfolio: collateral
diversity by industry and by reference entity, and the credit quality of each
asset in the portfolio. With respect to the latter, rating agencies use the
concept of the WARF. Each security in the CDO portfolio has a rating (actual or
implied) and each rating has a corresponding rating factor. The lower a
security's rating, the higher its corresponding rating factor. In order to
calculate the weighted average debt rating for a pool of assets, take the par
amount of each performing asset and multiply it by its respective rating
factor. Then, sum the resulting amounts for all assets in the pool and divide
this number by the sum of the par values of the performing assets [1].Krahnen and Wilde [43] show that
the common presumption about macro factor tail risk (extreme systematic risk)
being largely held by senior tranches is false. While senior tranches are in
fact primarily exposed to tail risk, the reverse is not true, as the share of
tail risk borne by senior tranches is quite limited. They conclude that tail
risk in standard CDO transactions is held by all tranches. This argument
motivates the use of a tail-risk measure like ETL in the tranche return-risk
optimisation to derive optimal structures. The multiobjective optimisation
problem including the above constraints is now defined in detail.(6.17)minx∈𝒳ETLβ*(L(tj)),maxx∈𝒳sn=∑j=1JD(0,(tj−tj−1)/2)(𝔼[Lntr(tj)]−𝔼[Lntr(tj−1)])∑j=1J(tj−tj−1)D(0,tj)𝔼[Nout(ntr)(tj,tj−1)]subjecttolupper(i)≤xi≤lupper(i)∀i∈{1,2,3,…,N},Rprot≥Rmin,𝒩(xi>0)=106.We do not include the
industry/region concentration constraints since the numerical analysis is
conducted on the benchmark portfolio consisting of 106 constituents of the
iTraxx Europe IG S5 index. The regional constraint will fall away, however, the
industry constraint will still hold. The study will analyse the industry
distributions of the resulting collateral portfolio's, for optimal [6,9]% mezzanine
tranches that define the pareto frontier. This is an improtant visualisation
tool for bespoke CDO investors, which depicts the trade-off between tranche
return and risk, to allow for picking a structure given the respective risk
preference. A flat Libor curve of 4.6% is applied. The CDO structure is priced
as at 18-April-2008.The two cases investigated are as follows.(1)
First an
examination of the pareto frontiers for bespoke CDO structures under both
Gaussian and Clayton copula assumptions is conducted. Long-only positions in
the underlying CDS are allowed. The upper trading limit in any particular
reference entity is set to 2.5% of the portfolio, whilst the lower limit is set
to 0.5% to satisfy the cardinality constraint of having 106 credits in the
collateral portfolio.(2)
Then an
investigation into the behaviour of the pareto frontiers under increasing upper
trading limit is conducted. This consequently allows an investigation of
concentration risk and its effects on the producing optimal CDO structures. The
upper trading limit is increased to 5% in this case.
Table 1 reprots
the NSGA-II parameter setting used to solve the optimisation problem (6.17).Table 1
NSGA-II parameter settings.
Evolutionary parameterValuePopulation size100Number of generations500Crossover probability90%Distribution index for crossover0.01Distribution index for mutation0.01Pool size50Tour size2The parameters settings were found to be most
appropriate after several runs of the algorithm. The computational time played
a crucial role in the assignment of parameter values.


### 6.4. Prototype Experiment: Pareto Frontier Analysis

#### 6.4.1. Optimal Structures with Long Only Credit Portfolio

Bespoke CDOs
are commonly preferred among investors because they can be used to execute a
variety of customised investment objectives and strategies. Flexibility in
choosing the reference portfolio allows investors to structure the portfolio
investments such that it can be properly matched with their investment
portfolios, as a separate investment strategy or as a hedging position. Most
often investors choose credit portfolios such that diversification is only as a
result of buying protection on reference entities in the underlying portfolio.We derive the pareto frontiers solving problem (6.17)
under both the Gaussian and Clayton copula assumptions when investors are
restricted to the long-only condition.Figure6 displays the difference in pareto frontiers
for the two default dependence assumptions. The first observation is that the
Gaussian allocation is sub-optimal. For all levels of portfolio risk, measured
by ETL, the Clayton allocation will result in higher [6,9]% tranche spreads.
This deviation is an increasing function of portfolio tail risk and widens to a
maximum of 120 bps for an ETL of 30% of the portfolio losses.Figure 6
The comparison of bespoke
CDO pareto frontiers under different default dependence assumptions.The minimum tail risk portfolio can provide a[6,9]%
mezzanine tranche that pays 180 bps under the Clayton allocation, whilst the
Gaussian allocation provides spread of 198 bps. However, there exist a 3.3%
difference in ETL for these minimum risk portfolios. A similar spread under the
Clayton allocation can be achieved with a ETL of just 16.95%.We concluded that the asset allocation strategies
based on the Gaussian copula will results in sub-optimal bespoke CDO
structures.


#### 6.4.2. Issuer Concentrated Credit Structures

We now
investigate the effects of issuer concentration on the pareto frontier. As we
concluded in the previous chapter, that investors demand a higher premium for taking
on concentration risk, combining this with the leverage effects of tranche
technology, will result in higher tranche spreads than the well diversified
case. The resulting frontiers are shown in Figure7 for the [6,9]% mezzanine
tranche.Figure 7
The portfolio notional
distribution for the concentrated portfolio case, by issuer credit rating.Figure7 compares the resulting pareto-frontiers for
the well-diversified and issuer concentrated cases. The CDO spread of the
concentrated case over the well-diversified case is an increasing function of
portfolio risk. For a 29.64% level of risk, a spread of 92 bps over the
well-diversified case can be achieved.


## 6.1. Multiobjective Optimisation

Many real-world
problems involve simultaneous optimisation of several incommensurable and often
competing objectives. In single-objective optimisation the solution is usually
clearly defined, this does not hold for multiobjective optimisation problems.
We define formally, the multiobjective optimisation problem to define other
improtant concepts used in this chapter. All problems are assumed to be
minimisation problems unless otherwise specified To avoid inserting the same
reference every few lines, note that the terms and definitions are taken from
Zitzler [39], and adapted to CDO optimisation problem.Definition 6.1 (multiobjective optimisation problem).
A generalmultiobjective optimisation Problem (MOP)
includes a set of n* parameters
(decision variables), a set of k1 objective
functions, and a set of k2 constraints.
Objective functions and constraints are functions of the decision variables.
The optimisation goal is to obtain,(6.1)miny=f(x)=(f1(x),f2(x),…,fk1(x))subjectto,e(x)=(e1(x),e2(x),…,ek2(x))≤0,x=(x1,x2,...,xn)∈X,y=(y1,y2,...,yk1)∈Y,where x and y are the
decision and objective vectors, respectively, whilst X and Y denote the
decision and objective spaces, respectively. The constraints e(x)≤0 determine the
set of feasible solutions.Without loss of generality, a minimisation problem is
assumed here. For maximization or mixed maximisation/minimisation problems the
definitions are similar.Definition 6.2 (vector relationship).
For any two
vectorsr and r*(6.2)r=r*iffri=ri*∀i∈{1,2,…,k1},r≰r*iffri≤ri*∀i∈{1,2,…,k1},r<r*iffr≤r*⋀r≤r*.The relations ≥ and > are
similarly defined.Although the concepts and terminology of Pareto optimality
are frequently invoked, most often they are erroneously used in literature. We
now define this set of concepts to ensure understanding and
consistency.Definition 6.3 (Pareto dominance).
For any two
vectorsr and r*(6.3)r⊀r*ifff(r)≤f(r*)thenrdominatesr*,r⪯̸r*ifff(r)≤f(r*)thenrweaklydominatesr*,r~r*ifff(r)≰f(r*)⋀f(r*)≰f(r)thenrisindifferenttor*.The definitions for a
maximisation problems (≻,⋡,~) are analogous.Definition 6.4 (Pareto optimality).
A decision
vectorx∈Xf is said to be
nondominated regarding a set K⊂Xf iff(6.4)∄k∈K:k⊀K.If it is clear within the
context which set K is meant, it is
simply left out. Moreover, x is said to be
Pareto optimal iff x is nondominated
regarding Xf.The entirety of all Pareto-optimal solutions is termed the Pareto-optimal set; the corresponding
objective vectors form the Pareto-optimal frontier or
surface.Definition 6.5 (nondominated sets and frontiers).
LetK⊂Xf. The function 𝒩d(K) gives the set
of nondominated decision vectors in K:(6.5)𝒩d(K)={k∈K∣kisnondominatingregardingK}.The set 𝒩d(K) is the
nondominated set regarding K, the
corresponding set of objective vectors f(𝒩d(K)) is the nondominated
front regarding K. Furthermore, the set X𝒩d=𝒩d(Xf) is called the Pareto-optimal set and the set Y𝒩d=f(Xf), is denoted as the Pareto-optimal frontier.


## 6.2. Nondominating Sort Genetic Algorithm (NSGA-II)

The second
generation NSGA-II is a fast and elitist multiobjective evolutionary
algorithm. The main features are
(http://www.kxcad.net/ESTECO/modeFRONTIER320/html/userman/ch07s01s10.html).(i)
Implementation of a fast nondominated sorting
procedure: Sorting of the individuals of a given population is according
to the level of nondomination. Generally, nondominated sorting algorithms are
computationally expensive for large population sizes however, the adopted
solution performs a clever sorting strategy.(ii)
Implementation of elitism for multiobjective
search: Using an elitism-preserving approach introduces storing all
nondominated solutions discovered so far, beginning from the initial
population. Elitism enhances the convergence properties towards the true
Pareto-optimal set.(iii)
Adopting a parameter-less diversity
preservation mechanism: Diversity and spread of solutions is guaranteed
without use of sharing parameters, since NSGA-II adopts a suitable
parameter-less niching approach. This niche is accomplished by the crowding
distance measure, which estimates the density of solutions in the objective
space, and the crowded comparison operator, which guides the selection process
towards a uniformly spread Pareto frontier.(iv)
Constraint handling method does not make use
of penalty parameters: The algorithm implements a modified definition of
dominance in order to solve constrained multiobjective problems efficiently.(v)
Real-coded and binary-coded design variables: A new feature is the application of the genetic algorithms in the field of
continuous variables.


### 6.2.1. Representation of Individuals

The first stage
of building the EA is to link the “real world” to the “EA
world.” This linking involves setting up a bridge between the original
problem context and the problem-solving solving space, where evolution takes
place. The objects forming possible solutions within the original problem
context are referred to as aphenotype, while their encoding are called genotypes or more commonly chromosomes.The mapping from phenotype space to the genotype space
is termedencoding. The inverse
mapping is termed decoding.
Choosing an appropriate representation for the problem being solved is
improtant in the design of a successful EA. Often it comes down to good
knowledge of the application domain [40].Many different encoding methods have been proposed and
used in EA development. Few frequently applied representations are: binary,
integer and real valued representation. Real-valued or Floating-point
representation is often the most sensible way to represent a candidate solution
to of a problem. This approach is appropriate when the values that we want to
represent as genes, originate from a continuous distribution.The solutions to the proposed CDO optimisation models
are real-valued. This study opted to use the real-valued encoding, for the sake
of operational simplicity. The genes of a chromosome are real numbers between 0
and 1, which represents the weights invested in the different CDS contracts.
However, the summation of these weights might not be 1 in the initialisation
stage or after genetic operations. To overcome this problem, the weights are
normalised as follows:(6.6)xi′=xi∑i=1Nxi.


### 6.2.2. Evaluation Function

The role of theevaluation function is to
represent the requirements to which the population should adapt. This role
forms the basis of selection. Technically, the function assigns a quality
measure to the genotypes. Typically the function is composed from a quality
measure in the phenotype space and the inverse representation [40]. In the
evolutionary context the function is usually referred to as the fitness
function.In the bespoke CDO optimisation problem, we introduce
two objective to the problem. These objectives are namely the CDO tranche
return, and the portfolio tail risk measured by ETL. Various constraints are
then introduced to study the dynamics of the Pareto frontier under various
conditions.


### 6.2.3. Population Models and Diversity Preservation

Unlike
operators, which operate on individuals, selection operators like parent and
survivor selection work at population level. In most EA applications, the
population size is constant and does not change during the evolutionary search.
Thediversity of the population
is a measure of the number of different solutions present.In NSGA-I, the well-known sharing function approach
was used, which was found to maintain sustainable diversity in a population
[4]. There are however two difficulties with this sharing function.
According to Deb et  al. [4], the performance of the sharing function method in
maintaining a spread of solutions depends largely on the value of σs, and since each solution must be compared with all
other solutions in the population, the overall complexity of the sharing
function approach is O(Npop2).Deb et  al. [4] replace the sharing function approach with a
crowded-comparison approach in NSGA-II. This eliminates both the above
difficulties to some extent. This approach does not require any user-defined
parameter for maintaining diversity among population members, and also has a
better computational complexity.To describe this approach, we first define a metric
for density estimation and then present the description of the
crowded-comparison operator.


### 6.2.4. Crowded Comparison Operator

The density of
solutions surrounding a particular solution in the population must firstly be
estimated by calculating the average distance of two points on either side of
this point, along each of the objectives.Once the nondominated sort is complete, the crowding
distance is assigned. Individuals are selected based on rank and crowding
distance. The crowding-distance computation requires sorting the population according
to each objective function value in ascending order of magnitude. The boundary
solutions (solutions with smallest and largest function values) for each
objective function are assigned an infinite distance value. All other
intermediate solutions are assigned a distance value equal to the absolute
normalised difference in the function values of two adjacent solutions. This
calculation is continued with other objective functions. The overall
crowding-distance value is calculated as the sum of individual distance values
corresponding to each objective. Each objective function is normalised before
calculating the crowding distance. (See Deb et  al. [4] for more details regarding the
algorithm.)The complexity of this procedure is governed by the
sorting algorithm and has computational complexity ofO(M*NpoplogNpop), where there are M* independent
sorting of at most Npop individuals,
when all population members are in one front ℐ.After all population members in the setℐ are assigned a
distance metric, we can compare two solutions for their extent of proximity
with other solutions. A solution with a smaller value of this distance measure
is, in some sense, more crowded by other solutions.


### 6.2.5. Nondomination Sorting Approach

In order to
identify solutions in the NSGA of the first nondominated front, each solution
is compared with every other solution in the population to find if it is
dominated (We summarise the algorithm discussed in Deb et  al. [4].). This procedure
requires O(k1Npop) comparisons for
each solution, where k1 is the number
of objectives. This process is continued to find all members of the first
nondominated level in the population, the total complexity is O(k1Npop2) computations.
At this stage, all individuals in the first nondominated front are found. In
order to find the individuals in the next nondominated front, the solutions of
the first front are discounted temporarily and the above procedure is repeated.
In the worst case, the task of finding the second front also requires O(k1Npop2) computations,
particularly when number of solutions belong to the second and higher
nondominated levels. This argument is true for finding third and higher levels
of nondomination.The worst case is when there areNpop fronts and
there exists only one solution in each front. This requires an overall O(k1Npop3) computations.
For each solution we calculate two entities:(1)
the domination
countnq, the number of
solutions which dominate the solution; and(2)
the set of
solutionsHq that the
solution dominates. This requires O(k1Npop2) comparisons.
Solutions in
the first nondominated front will have their domination count as zero. For each
solution with nq=0, each member q∈Hq is visited and
reduce its domination count by one. By doing so, these members belong to the
second nondominated front. This process continues until all fronts are
identified. For each solution in the second or higher level of nondomination,
the domination count can be at most Npop−1. Since there are at most such Npop−1 solutions, the
total complexity of the procedure is O(Npop2). Thus, the overall complexity of the procedure is O(k1Npop3).In NSGA-II the complexity reduction is due to the
realisation that the body of the first inner loop (for eachpi*∈𝒥i) is executed
exactly Npop times as each
individual can be the member of at most one front and the second inner loop
(for each q∈Hp) can be
executed at maximum Npop−1 times for
results in the overall computations. It is improtant to note that although the
time complexity has reduced to O(k1Npop2), the storage requirement has increased to O(Npop2).


### 6.2.6. Parent Selection Mechanism

The selection
operator determines, which individuals are chosen for mating and how many
offspring each selected individual produces. Once the individuals are sorted
based on nondomination with crowding distance assigned, the selection is
carried out using a crowded comparison operator described above. The comparison
is carried out as below based on(1)
nondomination
rankprank*, that is,
individuals in FRONTi will have their
rank prank*=i; and(2)
crowding
distanceFRONTi(dj).
The selection
of individuals is through a binary tournament selection with the
crowed-comparison operator. In tournament selection the individuals are
chosen randomly from the population and the best individual from this group is
selected as parent. This process is repeated as often as individuals must be
chosen. These selected parents produce uniform at random offspring. The
parameter for tournament selection is the tournament size. This takes values
ranging from 2 to the number
of individuals in population [41].


### 6.2.7. Mutation Operation

Mutation causes
individuals to be randomly altered. These variations are mostly small. They
will be applied to the variables of the individuals with a low probability.
Offspring are mutated after being created by recombination. Mutation of real
variables means, that randomly created values are added to the variables with a
low mutation probability. The probability of mutating a variable is inversely
proprotional to the number of variables (dimensions). The more dimensions one
individual has, the smaller is the mutation probability. Different papers
reproted results for the optimal mutation rate [41].The mutation step size is usually difficult to choose.
The optimal step-size depends on the problem considered and may even vary
during the optimisation process. Small mutation steps are often successful,
especially when the individual is already well adapted. However, large mutation
steps can produce good results with a faster convergence rate. According Pohleim
[41] good mutation operators should often produce small step-sizes with a
high probability and large step-sizes with a low probability.In the NSGA-II a polynomial mutation operator is used.
This operator is defined by the following:(6.7)Ψioffspring=Ψiparent+(Ψiup+Ψilow)δi,where Ψioffspring is the child,
and Ψiparent is the parent,
with Ψiup and Ψilow being the upper
and lower bounds of the parent component. δi is a small
variation which is calculated from a polynomial distribution given
below.(6.8)δi={(2ri)1/(ηm+1)−1ri<0.51−(2(1−ri))1/(ηm+1)−1ri≥0.5,where ri~𝒰(0,1) and ηm is mutation
distribution index.


### 6.2.8. Recombination Operation

The
recombination operator produces new individuals in combining the information
contained in two or more parents in the mating population. This mating is done
by combining the variable values of the parents. Depending on the
representation of the variables different methods must be used.Using real-value representation, theSimulated Binary Crossover operator is
used for recombination and polynomial
mutation is used for mutating the offspring population [42].Simulated binary crossover simulates the binary
crossover observed in nature and is give by the following.(6.9)Ψ1,ioffspring=12((1−βk)Ψ1,kParent+(1+βk)Ψ2,kParent),Ψ2,ioffspring=12((1+βk)Ψ1,kParent+(1−βk)Ψ2,kParent),where βk(≥0) is a sample
from a random number generated having the density(6.10)fc(β)={12(ηc+1)βηc0≤β≤1,12(ηc+1)1βηc+2β≥1.This distribution can be
obtained from a u~𝒰(0,1) source. ηc is the
distribution index for the recombination operator.


### 6.2.9. Main Loop

First, a random
parent population is created. The population is then sorted based on the
nondomination outlined above. Each individual is assigned a fitness value (or
rank) equal to its nondomination level, with 1 representing the best level.
Binary tournament selection, recombination, and mutation operators are then
applied to create an offspring population of sizeNpop.Due to the introduction of elitism, the current
population is compared to previously found best nondominated solutions, the
procedure is different after the initial generation. Figure5 outlines the tth generation of
the NSGA-II procedure.Figure 5
An outline of the NSGA-II procedure.The first step is to combined parent populationQ_t with the
offspring population R_t. The population is now of size 2Npop. The resulting population is then sorted according to
nondomination. Since all previous and current population members are included
in N_t=Q_t∪R_t, elitism is ensured.Solutions belonging to the best nondominated setFRONT(1), are of best solutions in the combined population and
must be emphasied more than any other solution in the combined population. If FRONT(1) is smaller than Npop, then all individuals in this set are chosen for the
next generation, else a truncation operation will have to be applied. If the
first case holds, then the remaining members of the next generation are chosen
from subsequent nondominated fronts in the order of their ranking.In general, the count of solutions in all sets fromFRONT(1) to FRONT(l) would be larger
than the population size. To choose exactly Npop population
members, the solutions of the last frontier are sorted and then the
crowded-comparison operator is applied in descending order. The best solutions
are chosen to fill all population slots. This procedure is valid for
minimisation problems only.The requisite diversity among nondominated solutions
is introduced by using the crowded comparison procedure, which is used in the
tournament selection and during the population truncation phases [4].


### 6.2.10. Initialisation and Termination Conditions

Initialisation
is kept simple in most EA applications, the first population is seeded by
randomly generated individuals. In principle, problem-specific heuristics can
be used in this step, to create an initial population with higher fitness
values [40].We can distinguish two cases for an appropriate
termination condition. If the problem has a known optimum fitness level, then
an acceptable error bound would be a suitable condition. However, in many
multiobjective problems optimum values are not known. In this case one needs
to extend this condition with one that certainly stops the algorithm. As noted
by Eiben and Smith [40], a common choice is to impose a limit on the total number of fitness
evaluations.


## 6.2.1. Representation of Individuals

The first stage
of building the EA is to link the “real world” to the “EA
world.” This linking involves setting up a bridge between the original
problem context and the problem-solving solving space, where evolution takes
place. The objects forming possible solutions within the original problem
context are referred to as aphenotype, while their encoding are called genotypes or more commonly chromosomes.The mapping from phenotype space to the genotype space
is termedencoding. The inverse
mapping is termed decoding.
Choosing an appropriate representation for the problem being solved is
improtant in the design of a successful EA. Often it comes down to good
knowledge of the application domain [40].Many different encoding methods have been proposed and
used in EA development. Few frequently applied representations are: binary,
integer and real valued representation. Real-valued or Floating-point
representation is often the most sensible way to represent a candidate solution
to of a problem. This approach is appropriate when the values that we want to
represent as genes, originate from a continuous distribution.The solutions to the proposed CDO optimisation models
are real-valued. This study opted to use the real-valued encoding, for the sake
of operational simplicity. The genes of a chromosome are real numbers between 0
and 1, which represents the weights invested in the different CDS contracts.
However, the summation of these weights might not be 1 in the initialisation
stage or after genetic operations. To overcome this problem, the weights are
normalised as follows:(6.6)xi′=xi∑i=1Nxi.


## 6.2.2. Evaluation Function

The role of theevaluation function is to
represent the requirements to which the population should adapt. This role
forms the basis of selection. Technically, the function assigns a quality
measure to the genotypes. Typically the function is composed from a quality
measure in the phenotype space and the inverse representation [40]. In the
evolutionary context the function is usually referred to as the fitness
function.In the bespoke CDO optimisation problem, we introduce
two objective to the problem. These objectives are namely the CDO tranche
return, and the portfolio tail risk measured by ETL. Various constraints are
then introduced to study the dynamics of the Pareto frontier under various
conditions.


## 6.2.3. Population Models and Diversity Preservation

Unlike
operators, which operate on individuals, selection operators like parent and
survivor selection work at population level. In most EA applications, the
population size is constant and does not change during the evolutionary search.
Thediversity of the population
is a measure of the number of different solutions present.In NSGA-I, the well-known sharing function approach
was used, which was found to maintain sustainable diversity in a population
[4]. There are however two difficulties with this sharing function.
According to Deb et  al. [4], the performance of the sharing function method in
maintaining a spread of solutions depends largely on the value of σs, and since each solution must be compared with all
other solutions in the population, the overall complexity of the sharing
function approach is O(Npop2).Deb et  al. [4] replace the sharing function approach with a
crowded-comparison approach in NSGA-II. This eliminates both the above
difficulties to some extent. This approach does not require any user-defined
parameter for maintaining diversity among population members, and also has a
better computational complexity.To describe this approach, we first define a metric
for density estimation and then present the description of the
crowded-comparison operator.


## 6.2.4. Crowded Comparison Operator

The density of
solutions surrounding a particular solution in the population must firstly be
estimated by calculating the average distance of two points on either side of
this point, along each of the objectives.Once the nondominated sort is complete, the crowding
distance is assigned. Individuals are selected based on rank and crowding
distance. The crowding-distance computation requires sorting the population according
to each objective function value in ascending order of magnitude. The boundary
solutions (solutions with smallest and largest function values) for each
objective function are assigned an infinite distance value. All other
intermediate solutions are assigned a distance value equal to the absolute
normalised difference in the function values of two adjacent solutions. This
calculation is continued with other objective functions. The overall
crowding-distance value is calculated as the sum of individual distance values
corresponding to each objective. Each objective function is normalised before
calculating the crowding distance. (See Deb et  al. [4] for more details regarding the
algorithm.)The complexity of this procedure is governed by the
sorting algorithm and has computational complexity ofO(M*NpoplogNpop), where there are M* independent
sorting of at most Npop individuals,
when all population members are in one front ℐ.After all population members in the setℐ are assigned a
distance metric, we can compare two solutions for their extent of proximity
with other solutions. A solution with a smaller value of this distance measure
is, in some sense, more crowded by other solutions.


## 6.2.5. Nondomination Sorting Approach

In order to
identify solutions in the NSGA of the first nondominated front, each solution
is compared with every other solution in the population to find if it is
dominated (We summarise the algorithm discussed in Deb et  al. [4].). This procedure
requires O(k1Npop) comparisons for
each solution, where k1 is the number
of objectives. This process is continued to find all members of the first
nondominated level in the population, the total complexity is O(k1Npop2) computations.
At this stage, all individuals in the first nondominated front are found. In
order to find the individuals in the next nondominated front, the solutions of
the first front are discounted temporarily and the above procedure is repeated.
In the worst case, the task of finding the second front also requires O(k1Npop2) computations,
particularly when number of solutions belong to the second and higher
nondominated levels. This argument is true for finding third and higher levels
of nondomination.The worst case is when there areNpop fronts and
there exists only one solution in each front. This requires an overall O(k1Npop3) computations.
For each solution we calculate two entities:(1)
the domination
countnq, the number of
solutions which dominate the solution; and(2)
the set of
solutionsHq that the
solution dominates. This requires O(k1Npop2) comparisons.
Solutions in
the first nondominated front will have their domination count as zero. For each
solution with nq=0, each member q∈Hq is visited and
reduce its domination count by one. By doing so, these members belong to the
second nondominated front. This process continues until all fronts are
identified. For each solution in the second or higher level of nondomination,
the domination count can be at most Npop−1. Since there are at most such Npop−1 solutions, the
total complexity of the procedure is O(Npop2). Thus, the overall complexity of the procedure is O(k1Npop3).In NSGA-II the complexity reduction is due to the
realisation that the body of the first inner loop (for eachpi*∈𝒥i) is executed
exactly Npop times as each
individual can be the member of at most one front and the second inner loop
(for each q∈Hp) can be
executed at maximum Npop−1 times for
results in the overall computations. It is improtant to note that although the
time complexity has reduced to O(k1Npop2), the storage requirement has increased to O(Npop2).


## 6.2.6. Parent Selection Mechanism

The selection
operator determines, which individuals are chosen for mating and how many
offspring each selected individual produces. Once the individuals are sorted
based on nondomination with crowding distance assigned, the selection is
carried out using a crowded comparison operator described above. The comparison
is carried out as below based on(1)
nondomination
rankprank*, that is,
individuals in FRONTi will have their
rank prank*=i; and(2)
crowding
distanceFRONTi(dj).
The selection
of individuals is through a binary tournament selection with the
crowed-comparison operator. In tournament selection the individuals are
chosen randomly from the population and the best individual from this group is
selected as parent. This process is repeated as often as individuals must be
chosen. These selected parents produce uniform at random offspring. The
parameter for tournament selection is the tournament size. This takes values
ranging from 2 to the number
of individuals in population [41].


## 6.2.7. Mutation Operation

Mutation causes
individuals to be randomly altered. These variations are mostly small. They
will be applied to the variables of the individuals with a low probability.
Offspring are mutated after being created by recombination. Mutation of real
variables means, that randomly created values are added to the variables with a
low mutation probability. The probability of mutating a variable is inversely
proprotional to the number of variables (dimensions). The more dimensions one
individual has, the smaller is the mutation probability. Different papers
reproted results for the optimal mutation rate [41].The mutation step size is usually difficult to choose.
The optimal step-size depends on the problem considered and may even vary
during the optimisation process. Small mutation steps are often successful,
especially when the individual is already well adapted. However, large mutation
steps can produce good results with a faster convergence rate. According Pohleim
[41] good mutation operators should often produce small step-sizes with a
high probability and large step-sizes with a low probability.In the NSGA-II a polynomial mutation operator is used.
This operator is defined by the following:(6.7)Ψioffspring=Ψiparent+(Ψiup+Ψilow)δi,where Ψioffspring is the child,
and Ψiparent is the parent,
with Ψiup and Ψilow being the upper
and lower bounds of the parent component. δi is a small
variation which is calculated from a polynomial distribution given
below.(6.8)δi={(2ri)1/(ηm+1)−1ri<0.51−(2(1−ri))1/(ηm+1)−1ri≥0.5,where ri~𝒰(0,1) and ηm is mutation
distribution index.


## 6.2.8. Recombination Operation

The
recombination operator produces new individuals in combining the information
contained in two or more parents in the mating population. This mating is done
by combining the variable values of the parents. Depending on the
representation of the variables different methods must be used.Using real-value representation, theSimulated Binary Crossover operator is
used for recombination and polynomial
mutation is used for mutating the offspring population [42].Simulated binary crossover simulates the binary
crossover observed in nature and is give by the following.(6.9)Ψ1,ioffspring=12((1−βk)Ψ1,kParent+(1+βk)Ψ2,kParent),Ψ2,ioffspring=12((1+βk)Ψ1,kParent+(1−βk)Ψ2,kParent),where βk(≥0) is a sample
from a random number generated having the density(6.10)fc(β)={12(ηc+1)βηc0≤β≤1,12(ηc+1)1βηc+2β≥1.This distribution can be
obtained from a u~𝒰(0,1) source. ηc is the
distribution index for the recombination operator.


## 6.2.9. Main Loop

First, a random
parent population is created. The population is then sorted based on the
nondomination outlined above. Each individual is assigned a fitness value (or
rank) equal to its nondomination level, with 1 representing the best level.
Binary tournament selection, recombination, and mutation operators are then
applied to create an offspring population of sizeNpop.Due to the introduction of elitism, the current
population is compared to previously found best nondominated solutions, the
procedure is different after the initial generation. Figure5 outlines the tth generation of
the NSGA-II procedure.Figure 5
An outline of the NSGA-II procedure.The first step is to combined parent populationQ_t with the
offspring population R_t. The population is now of size 2Npop. The resulting population is then sorted according to
nondomination. Since all previous and current population members are included
in N_t=Q_t∪R_t, elitism is ensured.Solutions belonging to the best nondominated setFRONT(1), are of best solutions in the combined population and
must be emphasied more than any other solution in the combined population. If FRONT(1) is smaller than Npop, then all individuals in this set are chosen for the
next generation, else a truncation operation will have to be applied. If the
first case holds, then the remaining members of the next generation are chosen
from subsequent nondominated fronts in the order of their ranking.In general, the count of solutions in all sets fromFRONT(1) to FRONT(l) would be larger
than the population size. To choose exactly Npop population
members, the solutions of the last frontier are sorted and then the
crowded-comparison operator is applied in descending order. The best solutions
are chosen to fill all population slots. This procedure is valid for
minimisation problems only.The requisite diversity among nondominated solutions
is introduced by using the crowded comparison procedure, which is used in the
tournament selection and during the population truncation phases [4].


## 6.2.10. Initialisation and Termination Conditions

Initialisation
is kept simple in most EA applications, the first population is seeded by
randomly generated individuals. In principle, problem-specific heuristics can
be used in this step, to create an initial population with higher fitness
values [40].We can distinguish two cases for an appropriate
termination condition. If the problem has a known optimum fitness level, then
an acceptable error bound would be a suitable condition. However, in many
multiobjective problems optimum values are not known. In this case one needs
to extend this condition with one that certainly stops the algorithm. As noted
by Eiben and Smith [40], a common choice is to impose a limit on the total number of fitness
evaluations.


## 6.3. Optimisation Problem

The optimal
bespoke structure can be derived by solving the following
problem:(6.11)minx∈[0,1]NΘ(x),maxx∈[0,1]Nsntr(x)subjectto∑i=1Nxi=1,where in the above problem, we
minimise the portfolio ETL, and simultaneously maximise the ntrth tranche spread
denoted by sntr subject to the
portfolio weights summing up to unity. This minimality may typically be the
case if only long positions in the underlying credits are allowed.For practical purposes, it may be crucial to include
other real-world constraints. Let𝒢* denote the set
of organisational, regulatory and physical constraints. The list of
constraints, which are commonly included into 𝒢* are summarised below.(1)
Upper and lower limits on issuer
weights: especially due to regulatory issues, the portfolio fractional amount invested
into each asset may be limited, that is, there may be individual lower and upper
bounds on the portfolio weights.(6.12)lupper(i)≤xi≤lupper(i)∀i∈{1,2,3,…,N}.
When short selling is disallowed, it can be explicitly
modelled by settingli=0 in this
formulation.(2)
Regional/industry concentration
constraints: due to rating agency and regulatory restrictions, the portfolio may have
industry and regional concentration limits, then(6.13)∑i=1kxi≤U(Iind,Igeo),where0≤U(Iind,Igeo)≤1,where the coordinate (Iind,Igeo) refers to a
particular industry Iind in region Igeo.(3)
Cardinality constraints: these
constraints limit the number of assets in the portfolio. They can either be
strict(6.14)𝒩(xi>0)=K*,0<K*≤a,K*∈ℕ.or lower and upper bounds on the
cardinality can be defined,(6.15)Kl*≤𝒩(xi>0)≤Ku*,0<Kl*<Ku*≤a,Kl*,Ku*∈ℕ.(4)
Weighted average rating factor (WARF)
constraints: due to regulatory or organisational restrictions, the
portfolio WARF must be above some minimum value,(6.16)Rprot≥Rtarget,where in the case of S and P
ratings Rprot∈{AAA,AA+,AA,AA−,…}. (Due to the
nature of CDOs, ratings higher than the portfolio rating can be achieved. In
this case the constraint may be defined as Rnthtranche≥Rtarget.)
Rating agencies presuppose that there are two broad
credit considerations that determine the risk of a CDO portfolio: collateral
diversity by industry and by reference entity, and the credit quality of each
asset in the portfolio. With respect to the latter, rating agencies use the
concept of the WARF. Each security in the CDO portfolio has a rating (actual or
implied) and each rating has a corresponding rating factor. The lower a
security's rating, the higher its corresponding rating factor. In order to
calculate the weighted average debt rating for a pool of assets, take the par
amount of each performing asset and multiply it by its respective rating
factor. Then, sum the resulting amounts for all assets in the pool and divide
this number by the sum of the par values of the performing assets [1].Krahnen and Wilde [43] show that
the common presumption about macro factor tail risk (extreme systematic risk)
being largely held by senior tranches is false. While senior tranches are in
fact primarily exposed to tail risk, the reverse is not true, as the share of
tail risk borne by senior tranches is quite limited. They conclude that tail
risk in standard CDO transactions is held by all tranches. This argument
motivates the use of a tail-risk measure like ETL in the tranche return-risk
optimisation to derive optimal structures. The multiobjective optimisation
problem including the above constraints is now defined in detail.(6.17)minx∈𝒳ETLβ*(L(tj)),maxx∈𝒳sn=∑j=1JD(0,(tj−tj−1)/2)(𝔼[Lntr(tj)]−𝔼[Lntr(tj−1)])∑j=1J(tj−tj−1)D(0,tj)𝔼[Nout(ntr)(tj,tj−1)]subjecttolupper(i)≤xi≤lupper(i)∀i∈{1,2,3,…,N},Rprot≥Rmin,𝒩(xi>0)=106.We do not include the
industry/region concentration constraints since the numerical analysis is
conducted on the benchmark portfolio consisting of 106 constituents of the
iTraxx Europe IG S5 index. The regional constraint will fall away, however, the
industry constraint will still hold. The study will analyse the industry
distributions of the resulting collateral portfolio's, for optimal [6,9]% mezzanine
tranches that define the pareto frontier. This is an improtant visualisation
tool for bespoke CDO investors, which depicts the trade-off between tranche
return and risk, to allow for picking a structure given the respective risk
preference. A flat Libor curve of 4.6% is applied. The CDO structure is priced
as at 18-April-2008.The two cases investigated are as follows.(1)
First an
examination of the pareto frontiers for bespoke CDO structures under both
Gaussian and Clayton copula assumptions is conducted. Long-only positions in
the underlying CDS are allowed. The upper trading limit in any particular
reference entity is set to 2.5% of the portfolio, whilst the lower limit is set
to 0.5% to satisfy the cardinality constraint of having 106 credits in the
collateral portfolio.(2)
Then an
investigation into the behaviour of the pareto frontiers under increasing upper
trading limit is conducted. This consequently allows an investigation of
concentration risk and its effects on the producing optimal CDO structures. The
upper trading limit is increased to 5% in this case.
Table 1 reprots
the NSGA-II parameter setting used to solve the optimisation problem (6.17).Table 1
NSGA-II parameter settings.
Evolutionary parameterValuePopulation size100Number of generations500Crossover probability90%Distribution index for crossover0.01Distribution index for mutation0.01Pool size50Tour size2The parameters settings were found to be most
appropriate after several runs of the algorithm. The computational time played
a crucial role in the assignment of parameter values.


## 6.4. Prototype Experiment: Pareto Frontier Analysis

### 6.4.1. Optimal Structures with Long Only Credit Portfolio

Bespoke CDOs
are commonly preferred among investors because they can be used to execute a
variety of customised investment objectives and strategies. Flexibility in
choosing the reference portfolio allows investors to structure the portfolio
investments such that it can be properly matched with their investment
portfolios, as a separate investment strategy or as a hedging position. Most
often investors choose credit portfolios such that diversification is only as a
result of buying protection on reference entities in the underlying portfolio.We derive the pareto frontiers solving problem (6.17)
under both the Gaussian and Clayton copula assumptions when investors are
restricted to the long-only condition.Figure6 displays the difference in pareto frontiers
for the two default dependence assumptions. The first observation is that the
Gaussian allocation is sub-optimal. For all levels of portfolio risk, measured
by ETL, the Clayton allocation will result in higher [6,9]% tranche spreads.
This deviation is an increasing function of portfolio tail risk and widens to a
maximum of 120 bps for an ETL of 30% of the portfolio losses.Figure 6
The comparison of bespoke
CDO pareto frontiers under different default dependence assumptions.The minimum tail risk portfolio can provide a[6,9]%
mezzanine tranche that pays 180 bps under the Clayton allocation, whilst the
Gaussian allocation provides spread of 198 bps. However, there exist a 3.3%
difference in ETL for these minimum risk portfolios. A similar spread under the
Clayton allocation can be achieved with a ETL of just 16.95%.We concluded that the asset allocation strategies
based on the Gaussian copula will results in sub-optimal bespoke CDO
structures.


### 6.4.2. Issuer Concentrated Credit Structures

We now
investigate the effects of issuer concentration on the pareto frontier. As we
concluded in the previous chapter, that investors demand a higher premium for taking
on concentration risk, combining this with the leverage effects of tranche
technology, will result in higher tranche spreads than the well diversified
case. The resulting frontiers are shown in Figure7 for the [6,9]% mezzanine
tranche.Figure 7
The portfolio notional
distribution for the concentrated portfolio case, by issuer credit rating.Figure7 compares the resulting pareto-frontiers for
the well-diversified and issuer concentrated cases. The CDO spread of the
concentrated case over the well-diversified case is an increasing function of
portfolio risk. For a 29.64% level of risk, a spread of 92 bps over the
well-diversified case can be achieved.


## 6.4.1. Optimal Structures with Long Only Credit Portfolio

Bespoke CDOs
are commonly preferred among investors because they can be used to execute a
variety of customised investment objectives and strategies. Flexibility in
choosing the reference portfolio allows investors to structure the portfolio
investments such that it can be properly matched with their investment
portfolios, as a separate investment strategy or as a hedging position. Most
often investors choose credit portfolios such that diversification is only as a
result of buying protection on reference entities in the underlying portfolio.We derive the pareto frontiers solving problem (6.17)
under both the Gaussian and Clayton copula assumptions when investors are
restricted to the long-only condition.Figure6 displays the difference in pareto frontiers
for the two default dependence assumptions. The first observation is that the
Gaussian allocation is sub-optimal. For all levels of portfolio risk, measured
by ETL, the Clayton allocation will result in higher [6,9]% tranche spreads.
This deviation is an increasing function of portfolio tail risk and widens to a
maximum of 120 bps for an ETL of 30% of the portfolio losses.Figure 6
The comparison of bespoke
CDO pareto frontiers under different default dependence assumptions.The minimum tail risk portfolio can provide a[6,9]%
mezzanine tranche that pays 180 bps under the Clayton allocation, whilst the
Gaussian allocation provides spread of 198 bps. However, there exist a 3.3%
difference in ETL for these minimum risk portfolios. A similar spread under the
Clayton allocation can be achieved with a ETL of just 16.95%.We concluded that the asset allocation strategies
based on the Gaussian copula will results in sub-optimal bespoke CDO
structures.


## 6.4.2. Issuer Concentrated Credit Structures

We now
investigate the effects of issuer concentration on the pareto frontier. As we
concluded in the previous chapter, that investors demand a higher premium for taking
on concentration risk, combining this with the leverage effects of tranche
technology, will result in higher tranche spreads than the well diversified
case. The resulting frontiers are shown in Figure7 for the [6,9]% mezzanine
tranche.Figure 7
The portfolio notional
distribution for the concentrated portfolio case, by issuer credit rating.Figure7 compares the resulting pareto-frontiers for
the well-diversified and issuer concentrated cases. The CDO spread of the
concentrated case over the well-diversified case is an increasing function of
portfolio risk. For a 29.64% level of risk, a spread of 92 bps over the
well-diversified case can be achieved.


## 7. Conclusions

The main
objective of this research has been to develop and illustrate a methodology to
derive optimal bespoke CDO structures using the NSGA-II technology. Until
recently, the derivation of such structures was a near to an impossible task.
However, with the advent of advance pricing and optimisation techniques applied
in this research, credit structurers can use these tools to provide investors
with optimal structures. Investors can use this type of methodology to compare
similar rated credit structures and make an informed investment decision,
especially under the current credit market conditions.The most improtant finding is that the Gaussian copula
allocation produces suboptimal CDO tranche investments. Better tranche returns
at all levels of risk are obtained under the Clayton assumption. In the
constrained long-only case, for all levels of portfolio risk, measured by ETL,
the Clayton allocation will result in higher[6,9]% tranche spread. This
deviation was shown to be an increasing function of portfolio tail risk and
widen to a maximum of 120 bps for an ETL of 30%.It was also demonstrated that significant improvement
of returns over standardised CDOs tranches of similar rating can be achieved
using the methodology presented in this paper. The findings also concluded that
the leverage effects should became more pronounced for tranches lower down the
capital structure. More concentrated bespoke CDO portfolios result in higher
tranche spreads. Investor demand higher premiums for taking on the
concentration risk. For a 29.64% level of portfolio tail risk, an excess return
of 92 bps over the well diversified case can be achieved.Since the current analysis is restricted to
single-factor copula models, one obvious extension for future studies should be
on the added benefits multifactor copula models may provide in producing
optimal CDO structures. On the use of evolutionary technology, a future study
should compare NSGA-II results to the rival, the second generation of the
strength pareto evolutionary algorithm (SPEA-2) proposed by the Zitzler et  al. [44]. Stability
of and convergence to the pareto frontier for optimal CDO structures should
also be investigated.



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*Source: 925169-2009-03-29.xml*