1. COLLATERALIZED FUND
OBLIGATIONS
by
Nicola T.G. Padovani
This dissertation is submitted as part of the requirements for the award of the
degree in
Masters of Science in Mathematical Trading and Finance
Supervisor: Dr John Hatgioannides
Cass Business School
City University, London
November 2006

2. ABSTRACT
In this paper we aim to contribute to the limited literature available on Collateralized
Fund Obligations (CFO), structured finance products that issue securitized tranches
backed by pools of hedge fund vehicles. We compare different avenues of investment
in hedge funds either directly or via structures and investigate the reasons for the
limited number of CFO brought to market. We suggest that there is very limited publicly
available research in modeling fully the distributional characteristics of individual hedge
funds and what is their dependence structure when they are aggregated in pools. We
provide a framework for modeling their joint behavior and taking into account the
possibility of joint extreme negative returns by utilizing the tools of multivariate one
parameter Archimdean copula distributions. We calibrate the best fitting copula to a
variety of pools of HFRI strategy indices exhibiting various levels of dependence and
find the Clayton copula to be often the best fitting. We also provide a valuation
methodology by simulation, utilizing the identified copula function on the various
constructed pools and calculating pool volatility, excess return and breakeven spreads
for rated and unrated tranches. We find wide ranges in breakeven spreads depending
on the collateral’s dependence structure.

3. EXECUTIVE SUMMARY
In this dissertation we aim to analyze the products that take the name of Collateralized
Fund Obligations (CFO). While this denomination has also been applied to
securitizations of private equity investments, we will concentrate on deals where the
collateral pool is represented by hedge funds vehicles. Despite the expectation,
issuance in this new type of product has been in fact limited to a small number of
publicly rated transactions and a similarly limited number of privately placed deals.
We aim to offer several avenues of interpretation to try and explain the limited success
of these products and in particular we remark that published research in the
dependence structure of structured finance products backed by pools of hedge fund
collateral has lagged behind compared to “mainstream” Collateralized Debt Obligations.
We start by highlighting that research literature has reached consensus on accepting
that the specific characteristics of good returns and moderate volatility in the distribution
of hedge fund returns are compensation for specific risks assumed and these often
entail negative skewness and higher probability of extreme returns. We have seen also
that these return characteristics clearly reject the notion of normally distributed hedge
funds returns and that research has also highlighted the notion that linear correlation is
not a stable measure of the dependence structure among hedge funds. We therefore
have identified more stable groups of similar trading strategies (“clusters”) using data
from monthly broad based Hedge Fund Research HFRI indices, while also examining
and correcting the nature and prevalence of measurement biases and serial correlation
among these indices. We find that most monthly HFRI do exhibit serial correlation and
rely on previous research to evaluate other biases and proceed to adjust the data.
We proceed to examine all currently available avenues of investment in hedge funds,
distinguishing among direct allocation of capital and investment via structures.
We find that, like in other asset classes, investors can today decide whether they seek
an “alpha” or outperformance based investment (with accompanying risks) via either
single manager single strategy funds or a broader “beta” type of investment via an
investable index. Fund of funds represent an intermediate direct investment decision,
with institutional investors appreciating the additional layer of portfolio diversification,
monitoring and management and, crucially, of due diligence and administration.
Fund linked structures have also become important avenues for investment, especially
by institutional investors and these investors have been requiring increased levels of
structuring sophistication in order to customize the risk return profile, achieve capital
preservation and a degree of leverage while also often seeking an investment grade
rating. In this context we investigate if CFOs can be considered a viable solution to the

4. needs of institutional investors as they offer a range of tranches with varying risk return
profiles and credit support, different alpha or beta exposure (by selecting appropriate
underlying vehicles) and increased transparency and due diligence by active monitoring
by a manager and a rating agency.
We suggest that there is very limited publicly available research in modeling fully the
distributional characteristics of individual hedge funds and what is their dependence
structure when they are aggregated in pools. We provide a framework for modeling
their joint behavior and taking into account the possibility of joint extreme negative
returns by utilizing the tools of multivariate one parameter Archimdean copula
distributions.
We calibrate a variety of different pools (based on the identified clusters) of HFRI
strategy indices (dimension up to six) to the Clayton, Gumbel and Frank copulas using
a two step Maximum Likelihood Estimator procedure. We find that for most pools
constructed either by mixing strategies from different clusters or by aggregating all the
strategies from a specific cluster, the best fitting copula function is the Clayton one
displaying significant lower tail (extreme negative returns) dependency as predicated by
the Clayton family.
We provide a valuation methodology to five hypothetical CFOs on the various
previously constructed pools in order to calculate breakeven spreads for each tranche
by sampling from the calibrated multivariate Archimedean copula distribution and
utilizing a standardized tranching structure.
We find that a crucial impediment to the widespread appeal of CFOs may come from
investors’ difficulty in evaluating relative value among the vast array of possible CFO
structures, each with widely different dependence structures, ranges of spreads on
rated notes and potential upside returns for the equity tranche.

5. 3
Contents
1 Introduction ..................................................................................................................1
2 The hedge fund sector: return, risk and investment vehicles...............................3
2.1 Definition................................................................................................................3
2.2 Hedge fund risks and return characteristics.....................................................3
2.2.1 Factor Analysis and descriptive statistics...................................................4
2.2.2 Time varying correlation, cluster analysis and data biases......................7
2.2.3 Hedge Fund Research HFRI adjusted performance data....................11
2.2.4 Fund of funds’ returns statistical properties ...........................................14
2.3 Other risks in hedge fund investment.............................................................15
2.4 Hedge fund investment vehicles ......................................................................15
2.4.1 Single manager-single strategy hedge funds............................................17
2.4.2 Fund of hedge funds ...................................................................................18
2.4.3 Hedge fund indices......................................................................................19
3 A taxonomy of hedge fund structures....................................................................23
3.1 Leveraged structures...........................................................................................24
3.2 Principal protected securities ............................................................................25
3.2.1 Static hedging (option based) structures..................................................26
3.2.2 Static and dynamic threshold structures..................................................28
3.3 Risks in fund linked structures: the co-movement of alternative assets...30
4 Collateralized Fund Obligations..............................................................................33
4.1 Structural features and agency rating...............................................................34
4.1.1 Credit support and over collateralization ................................................35
4.1.2 Collateral pool diversification covenants.................................................38
4.2 Historical tranche pricing: CDO market supply and demand....................40
4.3 Modeling pools of HFRI strategies: a multivariate copula approach........42
4.3.1 Individual distribution fitting of HFRI strategy indices .......................43
4.3.2 Pools of HFRI strategies in a multivariate Archimdean copula
dependence structure.....................................................................................44
4.3.3 Archimedean copula distributions............................................................46
4.3.4 Calibration of pools of HFRI Indices to Archimdean copulas...........47
4.4 Tranche valuation in a Archimedean copula framework.............................51
5 Conclusions .................................................................................................................55

6. 3
List of Tables
Table 1 Unadjusted Descriptive Statistics: HFRI Strategy Indices and Conventional
Assets (Monthly data, Jan. 1995-June 2006) ............................................................................6
Table 2 SAS Cluster Convergence on HFRI Strategies.......................................................................8
Table 3 SAS Clusters Identified................................................................................................................8
Table 4 HFRI Inter Cluster Correlations................................................................................................8
Table 5 Serial Correlation in HFRI Strategies (monthly Data, Jan 1995~June 2006)................. 13
Table 6 Descriptive statistics on unbiased and unsmoothed HFRI strategies
(monthly data, Jan 1995-June 2006) ....................................................................................... 14
Table 7 Overview of Major Investable Indices (Gehin and Vaissie 2004).................................... 21
Table 8 Hedge Fund Structures............................................................................................................. 23
Table 9 Volatility test levels and advance rates for some CFO deals............................................. 36
Table 10 Tranching, Credit Support and Leverage for selected CFO deals ................................. 37
Table 11 Liquidity Profile for selected CFO deals............................................................................. 38
Table 12 CFO Portfolio diversification covenants for selected deals............................................ 39
Table 13 Tranche Pricing for selected CFO deals ............................................................................. 40
Table 14 CDO Spreads to LIBOR at the time of issue of Coast 2005 (Jpmorgan
2005)............................................................................................................................................. 41
Table 15 Other CDO issued at the time of Coast (Jpmorgan 2005).............................................. 41
Table 16 Fitted HFRI Strategies distributions( Monthly data, Jan 1995- June 2006).................. 43
Table 17 Generator functions, Inverse Generators, Parameter Space and
Dependency measures for multivariate Archimedean copulas.......................................... 47
Table 18 Copula estimation and parameters on selected pools....................................................... 49
Table 19 Hypothetical CFO tranching structure for all fitted pools .............................................. 52
Table 20 Tranche breakeven spreads and CFO return for selected pools.................................... 53
Table 21 Advance Rates for various types of collateral in securitizations
(Standard&Poor’s 2005)............................................................................................................ 59
Table 22 Hedge Fund Collateral Advance Rates ranges (Mahadevan and Schwartz
(2002)............................................................................................................................................ 59
Table 23 CDO spreads at time of issue of Diversified Strategies 2002 (Jpmorgan
2002)............................................................................................................................................. 61
Table 24 Other CDO deals issued at time of Diversified Strategies 2002 (Jpmorgan
2002)............................................................................................................................................. 61

7. 3
List of Figures
Figure 1 Summary Table of possible Risk Factors’ effect on different HF Strategies
(CIDSM 2006) ...............................................................................................................................5
Figure 2 Hedge Fund Databases Biases (MarHedge 2005).............................................................. 10
Figure 3 Assets under Management (CISDM 2006).......................................................................... 16
Figure 4 Fixed Threshold Structure (Mattoo 2005).......................................................................... 28
Figure 5 Variable Threshold Structure (Mattoo 2005)...................................................................... 29
Figure 6 A typical CFO Transaction Structure (Standard&Poor's2006) ....................................... 34
Figure 7 Frank Copula calibrated to Devonshire pool on 3 dimensions....................................... 50
Figure 8 Clayton Copula calibrated to Centrepoint pool on 3 dimensions................................... 50
Figure 9 Market Timing Strategy Fitting.............................................................................................. 62
Figure 10 Fixed Income Arbitrage Strategy Fitting ........................................................................... 62
Figure 11 Centrepoint CFO copula parameter MLE estimation in R ........................................... 63

8. 1 Introduction
Alternative Investments have become the latest asset class to be used as collateral in
structured finance transactions. Many commentators in the last years have been quick
to predict strong growth in a sector that seemed ideally placed to take advantage of the
increasing appetite of investors both for diversified “absolute” returns and for structured
tranched products utilizing credit enhancement techniques.
In this dissertation we aim to analyze from various points of view the products that take
the name of Collateralized Fund Obligations (CFO). While this denomination has also
been applied to securitizations of private equity investments, we will concentrate on
deals where the collateral pool is represented by one specific type of alternative
investments, hedge funds vehicles. Despite the expectation, issuance in this new type
of product has been in fact limited to a small number of publicly rated transactions and
a similarly limited number of privately placed deals.
In this paper we aim to offer several avenues of interpretation to try and explain the
limited success of these products and in particular we remark that published research in
structured finance products backed by hedge fund collateral has lagged behind
compared to “mainstream” Collateralized Debt Obligations. To this aim, we provide a
framework for modeling the joint behavior of hedge fund assets using multivariate
Archimedean copula distributions. The approach is then extended to calculate
breakeven spreads on the CFO tranches.
Chapter two starts with a review of the wide range of hedge fund vehicles currently
available with a view of highlighting benefits and pitfalls of their use as collateral to
tranched securities. We will concentrate on the statistical properties of hedge fund
returns, the existence and nature of biases in the available performance and dispersion
data and other issues concerning the reliability of co-movement indicators such as
linear correlation, within vehicles in this asset class. The HFRI family of hedge fund
strategy indices will be introduced and groups (or styles) of highly correlated hedge
fund indices will be constructed using clustering analysis; these groups will be used as
collateral in the structuring of hypothetical CFOs later on in chapter five.
Chapter three reviews structured products based on the hedge fund asset class and
places CFO within the broader family of Fund Linked structures and highlights the key
areas in which these competing structures differ from CFOs and analyses why other

9. 2
structures that offer capital protection and an investment grade by a rating agency
continue to enjoy widespread investor interest.
Chapter four analyses in detail how techniques typical of market value Collateralized
Debt Obligations have been applied to the structuring of CFOs, highlighting the
peculiarities of Hedge fund securitizations as regards credit enhancement and liquidity
and diversification covenants and how little literature is available on more advanced
methods of modeling the dependence of returns among hedge fund vehicles making up
the collateral pool. A modeling and pricing model is proposed utilizing simulation where
the joint distribution among collateral assets is catered for in a multivariate
Archimedean copula framework. Fair spreads and probability of loss are calculated for
a series of CFOs based around hypothetical pools of collateral created by aggregating
HFRI strategy indices. Some considerations are drawn on the suitability of the spread
offered on the rated and unrated tranches to compensate for the risks assumed.
Conclusions follow highlighting our findings and key avenues for further development in
research applied to the investigation of dynamic multivariate Archimdean copulas.