Modern multi factor analysis of bond portfolios critical implications for hedging and investing

Modern Multi-Factor Analysis of Bond Portfolios

DOI: 10.1057/9781137564863.0001


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DOI: 10.1057/9781137564863.0001


Modern Multi-Factor
Analysis of Bond
Portfolios: Critical
Implications for
Hedging and Investing
Edited by

Giovanni Barone Adesi
Professor, Università della Svizzera Italiana, Switzerland
and

Nicola Carcano
Lecturer, Faculty of Economics, Università della Svizzera
Italiana, Switzerland

DOI: 10.1057/9781137564863.0001


Selection and editorial content © Giovanni Barone Adesi and
Nicola Carcano 2016
Individual chapters © the contributors 2016
Softcover

f
reprint off the hardcover 1st edition 2016 978-1-137-56485-6
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permitting limited copying issued by the Copyright Licensing Agency,
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Any person who does any unauthorized act in relation to this publication
may be liable to criminal prosecution and civil claims for damages.
The authors have asserted their rights to be identified as the authors of this work
in accordance with the Copyright, Designs and Patents Act 1988.
First published 2016 by
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Palgrave® and Macmillan® are registered trademarks in the United States,
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ISBN: 978-1-137-56486-3 PDF
ISBN: 978-1-349-85024-2
A catalogue record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Barone Adesi, Giovanni, 1951– editor. | Carcano, Nicola, 1964– editor.
Title: Modern multi-factor analysis of bond portfolios : critical implications for

hedging and investing / [edited by] Giovanni Barone Adesi, Professor, Università
della Svizzera Italiana, Switzerland, Nicola Carcano, Lecturer, Faculty of
Economics, Università della Svizzera Italiana, Switzerland.
Description: New York : Palgrave Macmillan, 2015.
Identifiers: LCCN 2015037662
Subjects: LCSH: Bonds. | Bond market. | Investments. | Hedge funds. | Porfolio
management.
Classification: LCC HG4651 .M5963 2015 | DDC 332.63/23015195—dc23
LC record available at />www.palgrave.com/pivot
doi: 10.1057/9781137564863


Contents
List of Chart & Exhibits
List of Figures

vi
viii

Notes on Contributors

x

1

Introduction
Giovanni Barone Adesi and Nicola Carcano

1

2

Adjusting Principal Component Analysis
for Model Errors
Nicola Carcano

3

4

5

6

Alternative Models for Hedging Yield
Curve Risk: An Empirical Comparison
Nicola Carcano and Hakim Dall’O
Applying Error-Adjusted Hedging to
Corporate Bond Portfolios
Giovanni Barone Adesi, Nicola Carcano
and Hakim Dall’O
Credit Risk Premium: Measurement,
Interpretation and Portfolio Allocation
Radu C. Gabudean, Kwok Yuen Ng and
Bruce D. Phelps
Overall Conclusion
Giovanni Barone Adesi and
Nicola Carcano

6

21

47

78

111

References

115

Index

121

DOI: 10.1057/9781137564863.0001

v


List of Chart & Exhibits

Chart
1

Assessing the distance between the yields of the
2-year, 5-year, 10-year and 30-year treasury
bonds and the future notional coupon

35

Exhibits
1
2
3

4
5
6
7
8

vi

Testing alternative PCA-based strategies on US
treasury bonds: hedging quality indicators
Testing alternative PCA-based strategies on US
treasury bonds: average transaction fees
Testing alternative PCA-based strategies
including USD interest rate swaps: hedging
quality indicators
Testing the most common hedging techniques in
their traditional form
Testing the most common hedging techniques in
their error-adjusted form
Calculating the performance of hedging models
based on the initial cheapest-to-deliver bonds
Alternative hedging models based on bond

futures: sub-sample analysis
Sensitivity of PCA hedging models to small
changes in the coefficients

13
14

16
37
38
39
41
42

DOI: 10.1057/9781137564863.0002


List of Chart & Exhibits

Summary statistics on spreads related to
BBB-rated bonds
10 Variance reduction obtained by alternative hedging
strategies
11 Predictability of the hedging errors produced by alternative
hedging strategies

vii

9

DOI: 10.1057/9781137564863.0002

63
64
66


List of Figures
1
2

3

4

5

6

7

8

9

10

viii

Historical reported IG corporate index excess

returns
Analytical durations (DurOAD & DurDefAdj)
for the NC IG corp index and their difference,
July 1989–November 2012
Treasury yields and the difference between
DurOAD and DurDefAdj, for the NC IG corp
index, July 1989–November 2012
Comparison of OAS and the difference between
DurOAD and DurDefAdj for the NC IG corp
index, July 1989–November 2012
Statistics of various NC IG corp indices using
two different analytical duration measures,
July 1989–November 2012
Average ExRet (/mo) for NC IG corp index
conditional on the change in Treasury yields,
July 1989–November 2012
Correlations of various ExRetanalyt measures
with Treasury returns, by sub-period, March
2004–November 2012
Evolution of various empirical duration betas
for the NC IG corp index, July 1989–
November 2012
Rolling correlations of various ExRetemp with
Treasury returns, trailing 24 months, May
1991–November 2012
Statistics of various NC IG corp indices,
July 1989–November 2012

79

83

84

84

85

86

86

89

91
92

DOI: 10.1057/9781137564863.0003


List of Figures

Average ExRetanalyt and ExRetemp dyn (/mo) for
NC IG corp index conditional on the change in
Treasury yields, July 1989–November 2012
12 Cumulative NC IG corporate index ExRet performance
for various duration measures, July 1989–November 2012
13 Duration ratios (betas) for the IG corp index &
matched-DurOAD Treasury yields, January 1973–June 1989
14 Relation between IG corp index spreads & matched-DurOAD

Treasury yields, January 1973–June 1989
15 Correlation between IG corp spreads and matched-DurOAD
Treasury yields & level of matched-DurOAD Treasury yields,
January 1973–November 2012
16 Correlation of major assets’ performance with
macroeconomic variables, 1953–2011
17 Relationship of asset class performance with real GDP
growth (/y), 1953–2011
18 Relationship of asset class performance with CPI inflation
(/y), 1953–2011
19 Correlation of asset class returns with macroeconomic
variables, 1953–2011
20 Smoothed, de-meaned macroeconomic variables,
GDP growth & CPI inflation, Q1/1963–Q3/2012
21 Return statistics for various returns of the IG corp index,
January 1978–September 1981
22 Return statistics for mean-variance-optimal portfolios
of Treasuries with various returns of the non-call DGT IG
index, July 1989–November 2012
23 Net weight to Treasuries (scaled) for various corp/
Treasury portfolios, as  of total net allocation,
July 1989–November 2012
24 Cumulative performance of various ExRet measures
for the IG corporate index, January 1973–November 2012
25 Return statistics of various returns of the IG corporate
index, January 1973–November 2012

ix

11

DOI: 10.1057/9781137564863.0003

92
93
95
95

96
97
98
98
99
100
101

105

106
108
108


Notes on Contributors
Giovanni Barone Adesi is Professor of Finance Theory
at the Swiss Finance Institute, University of Lugano,
Switzerland. A graduate from the University of Chicago,
he has taught at the University of Alberta, University of
Texas, City University and the University of Pennsylvania.
His main research interests are derivative securities and

risk management. Especially well-known are his contributions to the pricing of American commodity options and
the measurement of market risk.
Nicola Carcano holds a degree in Economics from the
LUISS University in Rome, an MBA from the New York
University, and a PhD in Financial Markets Theory from
the University of St Gallen. He teaches Structured Products
at the University of Lugano, Switzerland. After working as
a consultant and institutional portfolio manager, he is now
the Chief Executive Officer of Heron Asset Management.
His research focuses on fixed-income finance.
Hakim Dall’O received his PhD in Finance at the Swiss
Finance Institute in 2011. He has been working in both the
banking and the insurance industries as a quantitative risk
analyst for more than five years. Currently, he is working
in the security lending market as senior credit analyst.
Radu C. Gabudean co-manages American Century
Investments’ asset allocation strategies and conducts
related research. Prior to ACI, Gabudean was vice president of quantitative strategies with Barclays Risk Analytics
and Index Solutions (BRAIS), where he designed and

x

DOI: 10.1057/9781137564863.0004


Notes on Contributors

xi

implemented asset allocation strategies. Previously, he was a quantitative

portfolio modeler at Lehman Brothers and Barclays Capital. Gabudean
holds a BA from York University and a PhD (Finance) from New York
University.
Kwok Yuen Ng is a director in the Quantitative Portfolio Strategy group
at Barclays Capital. Ng is responsible for conducting studies on portfolio
strategies and index replication. Ng joined Barclays in 2008 after spending 20 years at Lehman Brothers, where he held a similar position. Prior
to that, he was a consultant at The Davidson Group and Software AG. Ng
holds an MS (Computer Science) from New York University.
Bruce D. Phelps is a managing director in global research at Barclays
Capital where he evaluates investment strategies on behalf of institutional
investors. Phelps joined Barclays in 2008 from Lehman Brothers where
he was managing director in research for eight years. Prior to that, he
was an institutional portfolio manager, a designer of electronic trading
systems and a forex trader. Phelps graduated with an AB from Stanford
and a PhD (Economics) from Yale.

DOI: 10.1057/9781137564863.0004



1

Introduction
Giovanni Barone Adesi and Nicola Carcano
Abstract: This chapter summarizes the motivation for
managing the risks related to interest rates changes and
the interest rate risk management techniques actually used
by most institutions and private investors: duration vector
(DV) models, principal component analysis (PCA) and
key rate duration (KRD). We highlight how a number of

studies conducting empirical tests of these models reported
puzzling results: models capable to better capture the
dynamics of the yield curve were not always shown to
lead to better hedging. In this chapter, we summarize the
contribution of each of the following chapters in explaining
these results and proposing alternative models capable of
adding value over the abovementioned traditional models
both for hedging and portfolio management.
Barone Adesi, Giovanni and Nicola Carcano, eds.
Modern Multi-Factor Analysis of Bond Portfolios:
Critical Implications for Hedging and Investing.
Basingstoke: Palgrave Macmillan, 2016.
doi: 10.1057/9781137564863.0005.

DOI: 10.1057/9781137564863.0005






Giovanni Barone Adesi and Nicola Carcano

Managing the risks related to interest rates changes is a highly relevant
issue for most institutional and private investors. In a broad sense, it
could even be argued that interest rate risk management is the single
most important global financial issue, at least in term of the involved
assets, since both institutions and private individuals invest on average
the majority of their assets in money-market and fixed-income instruments. Accordingly, these investors must face the issue of managing the
absolute volatility of these assets. In addition, many of these investors

also have to face the issue of how the value of the assets invested in
money-market and fixed-income instruments changes relatively to the
value of their liabilities, an issue we commonly refer to using the expression Asset and Liability Management (ALM).
When we consider the essence of the interest rate risk management
techniques actually used by most institutions and private investors,
we conclude that the key points of these techniques have been mostly
developed a few decades ago. Of course, this does not necessarily imply
that these techniques are bad or out-of-date. However, one could expect
more technological advances actually applied in the framework of such
a critical topic. Accordingly, the main goal of this book is to describe
the value potentially added by more recent techniques to manage interest rate risk relatively to traditional techniques and to present empirical
evidence of such an added value.
Managing interest rate risk implies hedging the two components
of bond yields: the risk-free term structure of interest rates and the
corporate bond spreads. Different techniques to hedge the risk-free term
structure of interest rates have been developed over the past 40 years.
Initially, academicians and practitioners focused on the concept of duration – introduced by Macaulay (1938) – for implementing immunization
techniques. Duration represents the first derivative of the price-yield
relationship of a bond and was shown to lead to adequate immunization
for parallel yield curve shifts.1
The assumption of parallel yield curve shifts could be released thanks
to the concept of convexity which was initially related to the second derivative of the price-yield relationship (Klotz (1985)). Bierwag et al. (1987)
and Hodges and Parekh (2006) show that the usefulness of convexity is
generally not related to better approximating the price-yield relationship,
but rather to the fact that hedging strategies relying on duration- and
convexity-matching are consistent with plausible two-factor processes
describing non-parallel yield curve shifts. Further extensions of these
DOI: 10.1057/9781137564863.0005

Introduction



concepts were based on M-square and M-vector models introduced by
Fong and Fabozzi (1985), Chambers et al. (1988), and Nawalkha and
Chambers (1997). Similarly as for convexity, most of these models relied
on the observation that further-order approximations of the price-yield
relationship lead to immunization strategies which are consistent with
multi-factor processes accurately describing actual yield curve shifts.
Nawalkha et al. (2003) reviewed these duration vector (DV) models and
developed a generalized duration vector (GDV).
A second class of hedging models relied on a statistical technique
known as principal component analysis (PCA) which identifies orthogonal factors explaining the largest possible proportion of the variance of
interest rate changes. Litterman and Scheinkman (1988) showed that
a 3-factor PCA allows capturing the most important characteristics
displayed by yield curve shapes: level, slope and curvature.
A third approach relied on the concept of key rate duration (KRD)
introduced by Ho (1992). According to this approach, changes in all rates
along the yield curve can be represented as linear interpolations of the
changes in a limited number of rates, the so-called key rates.
The interest rate risk management techniques most commonly used in
practice rely on one of the three abovementioned approaches. However,
a number of studies conducting empirical tests of these models reported
puzzling results: models capable to better capture the dynamics of the
yield curve were not always shown to lead to better hedging. This was
the case of the volatility- and covariance-adjusted models tested by
Carcano and Foresi (1997) and of the 2-factor PCA tested by Falkenstein
and Hanweck (1997) which was found to lead to better immunization
than the corresponding 3-factor PCA.

These puzzling results contributed to limit the actual use of more
sophisticated yield curve models by practitioners. The second chapter
of this book analyzes possible explanations for these puzzling results in
the context of principal component analysis of government bond yields,
whereas the third chapter extends this analysis also to duration vector
and key rate duration models. In general, we find that – once we adjust
the models in order to control the exposure to model errors – empirical
results from government bond portfolios become broadly consistent
with economic theory.
The second component of bond yields which needs to be addressed
by interest rate risk management techniques is represented by the
corporate bond spreads. Hedging corporate bond spreads requires an
DOI: 10.1057/9781137564863.0005




Giovanni Barone Adesi and Nicola Carcano

understanding of the key economic factors explaining their existence
and dynamics. These factors have been the focus of a substantial amount
of research efforts over the last decade. Before these efforts, the prevailing opinion was the one reported by Cumby and Evans (1995): this
spread is driven mainly by expected default loss and tax premium. Later
research found that these factors cannot explain the cross-sectional and
time series dynamics of the spread and questioned the relevance of the
tax premium. Most scholars relied either on liquidity premiums or on
time-varying market risk premiums to explain this credit spread puzzle.
The relevance of an aggregate – as opposed to firm-specific – liquidity
premium for corporate bond spreads has been suggested by CollinDufresne et al. (2001): they find that these spreads are explained for 25
by expected default and recovery rate with the remaining 75 explained

by a single factor which is not strongly related to variables traditionally
used as proxies for systematic risk and liquidity. They conclude that this
factor could be linked to more sophisticated proxies for liquidity.
Time-varying market risk premiums have been emphasized by Elton
et al. (2001). They find that, using traditional Fama-French factors, 85
of the spread that is not accounted for by taxes and expected default can
be explained as a reward for bearing systematic risk. Since the expected
default loss and tax premium are relatively static, this risk premium is
responsible for most of the dynamics of corporate bond spreads.
The fourth chapter of this book starts from the evidence reported by
the abovementioned studies on the dynamics of corporate bond spreads
in order to develop and test more advanced models for hedging corporate bond portfolios. We find that hedging strategies relying only on
T-bond futures provide results which can hardly be improved by equity
derivatives or Credit Default Swaps (CDS). These results may contradict
common practical beliefs. Nevertheless, they are consistent with previous
findings that stock market variables are less important than term structure variables to explain investment-grade bond returns and confirm
recent empirical evidence of a non-default component of corporate
spreads which becomes critical in times of unusual turbulences.
The fifth chapter of this book shifts the focus from pure hedging strategies to optimal portfolio construction. For many investors, analytical
excess returns conform to their macro views: they wish to be exposed
to any change in corporate default probabilities/recoveries, including
any change correlated with changes in Treasury yields. Other investors
want a corporate excess return uncluttered by the effects of correlated
DOI: 10.1057/9781137564863.0005


Introduction



movements in corporate spreads and Treasury yields. This chapter
focuses on presenting the techniques to implement the abovementioned
investment views and on back-testing their empirical results.
Finally, the sixth chapter of the book summarizes our overall theoretical as well as practical conclusions and our key recommendations to
practitioners actually engaged in interest rate risk management.
The book follows a stepwise construction approach. We start from the
simplest models in Chapter 2 and gradually move towards more sophisticated models in the following chapters. In each chapter, the additional
layers of complexity are firstly explained and motivated and secondly
tested relying on extensive sets of empirical data.

Note
 The original formulation of duration relied on flat yield curves, but this
restriction was overcome thanks to the formulation proposed by Fisher and
Weil (1971). For an extensive review of how the concept of duration was
developed during the last century, see Bierwag (1987).

DOI: 10.1057/9781137564863.0005


2

Adjusting Principal
Component Analysis
for Model Errors
Nicola Carcano
Abstract: Several papers which tested alternative ways
of hedging against yield curve risk reported that models
capturing the dynamics of the yield curve better do not
necessarily lead to better hedging. We claim that the main
reason for these counterintuitive observations could have

been the level of exposure to the model errors and tested
a generalized model of PCA-hedging which controls the
overall exposure to these errors. The results we obtained
both for bond-based and for swap-based hedging clearly
confirm our claim. Controlling the exposure to model
errors leads to an average reduction in the hedging errors
of 35. An additional, important advantage of controlling
the exposure to model errors is a substantial reduction in
the transaction fees implied by the hedging strategies.
Barone Adesi, Giovanni and Nicola Carcano, eds.
Modern Multi-Factor Analysis of Bond Portfolios:
Critical Implications for Hedging and Investing.
Basingstoke: Palgrave Macmillan, 2016.
doi: 10.1057/9781137564863.0006.



DOI: 10.1057/9781137564863.0006


Adjusting Principal Component Analysis



The level of interest in Liability Driven Investments (LDI) and, more
generally, in accurate techniques of asset and liability management
has grown up significantly over the last decade. This follows a process of de-risking which has been implemented worldwide by many
institutional investors. Accordingly, the approaches to effectively hedge
against interest rate risk have become significantly more sophisticated
than the initial models based on duration and convexity. The theories

underpinning these approaches mostly rely on the concepts of key rate
duration introduced by Ho (1992), of duration vectors (like the M-square
model of Fong and Fabozzi (1985) and the M-vector models proposed
by Nawalkha and Chambers (1997) and Nawalkha et al. (2003)) or on
Principal Component Analysis (PCA).1
Hedging based on PCA is one of the most common techniques used by
institutional investors to minimize the basis risk from shifts in the yield
curve. In theory, accounting for the third principal component should
improve the quality of hedging, since it allows to hedge also against
changes in the curvature of the yield curve (this point was highlighted
by Litterman and Scheinkman (1988)).
However, Falkenstein and Hanweck (1997) presented empirical
evidence suggesting that hedging based on PCA should rely on two
principal components rather than on three. They attributed the poor
performance of three-component PCA-hedging to the instability of
the third component. Also other papers (like Carcano and Foresi
(1997)) reported that models which should – in theory – allow to better
capture the dynamics of the yield curve do not necessarily lead to better
hedging.
We believe that these observations deserve further analysis and claim
that they can be explained by the interaction of the two main factors
influencing the size of the hedging errors:
 The difference between the modeled and the actual dynamics of the
yield curve; we will call this difference model error.
 The level of exposure of the overall portfolio (represented by the
sum of the assets and the liabilities) to the model errors.
It is intuitive that a higher exposure to the model errors could outbalance
the positive effect of a more sophisticated yield curve model capable
of reducing the size of these errors. We remind that traditional hedging based on PCA does not control the level of exposure to the model
errors.

DOI: 10.1057/9781137564863.0006




Nicola Carcano

The main goal of this paper is to test a generalized model of hedging
based on PCA, which controls the overall exposure to the model errors,
and to compare it with the plain-vanilla model based on PCA. This
should allow us to understand how much results like the ones reported
by Falkenstein and Hanweck (1997) can be explained by the level of
exposure to the model errors.

2.1

The hedging models

Let us consider the problem of immunizing a given portfolio of liabilities
which at time t has a value of Vt. Let us assume that we have grouped
the cash flows of this portfolio in m time buckets. The present value of
the liabilities included in the i-th time bucket amounts to Ai. For each
of these time buckets, basis risk comes from unexpected shifts in the
corresponding zero-coupon risk-free rate R(t,Dk), where Dk indicates the
duration and maturity of the time bucket.
For the sake of simplicity, we will assume that all rates are martingales. In other words, no interest rate changes are expected, so that:
E[dR(t, Dk)] = 0 for every k and every t. Extending our framework to
account for expected rate changes is relatively simple. Moreover, the impact
of this simplification on our empirical results is likely to be negligible.2
Hedging interest rate risk relies on approximating the dynamics of

the term structure through a limited number of factors. This leads to
a difference between the modeled and the actual dynamics of interest
rates, the model error. In a PCA framework, the model error ε for the
zero-coupon rate of duration Dk can be defined as:
dR t , Dk x

M

£c

lk

Ctl  a t , Dk

(1.)

l 1

where Clt represents the change in the l-th principal component between
time t and t + 1 and clk represents the sensitivity of the zero-coupon rate
of maturity Dk to this change. M represents the number of considered
principal components.
Our problem consists in investing the assets in a hedging portfolio H
of coupon bonds which can minimize the overall basis risk from shifts
in the yield curve. The optimal amount to be invested in a specific
coupon bond y is indicated by: ϕy. The percentage of the present value
DOI: 10.1057/9781137564863.0006


Adjusting Principal Component Analysis

of bond y represented by the cash flow with maturity Dk is indicated
by: wy,k.
Usually (see, for example, Martellini and Priaulet (2001)), hedging
strategies assume the so-called self-financing constraint:
M 1

Êb

y ,t

 H t  Vt

(2.)

y 1

Traditional hedging based on PCA implies setting the expected hedging error due to the modeled behavior of interest rates equal to zero.
Accordingly, the hedging portfolio H must be composed of M + 1 bonds
in order to match the dynamics of the M principal components and to
fulfill the self-financing constraint.
In essence, the generalized version of PCA-hedging we intend to
test implies that the error terms in Equation (1.) should be considered within the minimization of the expected immunization error,
while these terms are ignored by simple PCA-hedging. We show in
the Appendix that it is sufficient to assume independency among the
model errors in order to obtain the following set of hedging equations
which apply to every y (that is, to any of the M + 1 assets composing the
hedging portfolio H):

max
êĐ n
Ô M 1
ả ạ
ả Đ ; =
ư ă Ê clk Dk w y , k ,t ã ă Ê clk Dk Ê b j ,t w j , k ,t
Ak ,t ã  ư
M
Ư j 1
à ãá ư
2
á ăâ k 1
ư â k 1
2Ê E Đ Ctl ả ô
 *t
â
á
Đ M 1
ả
l 1
ư n 2 2 2
ư
ư Ê k k clk Dk w y k ,t ă Ê b j ,t w j , k ,t Ak ,t ã
ư
â j 1
á
ơ k 1
ằ



(3.)
where t is the Lagrange multiplier and k is defined as:
2

2

M

m a t , Dk x k 2k m C t , Dk  k 2k Ê clk2 E ĐâCtl ảá

2

(4.)

l 1

The set of hedging Equations (3.) is subject to the self-financing
constraint (2.).
In theory, the assumption that the model error for a given rate k
is independent from the model errors for all other rates could be
DOI: 10.1057/9781137564863.0006




Nicola Carcano

considered tautological: if we really believe that only three factors
explain the systematic dynamics of the yield curve, the dynamics which

are not explained by these factors are by definition unsystematic. And
unsystematic residuals are commonly considered completely random by
financial modelers. In practice, residuals of a PCA on the yield curve will
display a non-zero correlation. However, for sophisticated models like
3-factor PCA the correlation absolute value will tend to be smaller than
for less sophisticated models. Also, positive and negative correlations
will largely offset each other, so that their overall impact on the optimal
hedging strategy is likely to be limited. Checking that this assumption
does not prevent error-adjusted PCA from significantly improving the
hedging quality is one of the main goals of our empirical tests.
The motivation for definition (4.) is represented by the empirical
evidence that the volatility σε of the model errors is proportional to
the volatility σC of the modeled rate shifts. Both estimates of volatility
significantly vary over time, whereas their ratio θk displays a much lower
variability. For the sake of simplicity, we assumed θk to be constant over
time.
Let us analyze the set of hedging Equations (3.) more carefully. It is
immediate to see that if we set θk equal to zero for every k, we obtain the
standard set of equations for PCA-hedging:
M 1

n

£ b £c
y ,t

y 1

k 1

m
lk

Dk w y , k ,t  £ clk Dk , A Ak ,t

(5.)

k 1

which must be true for each principal component l. Equations (5.) ensure
that the sensitivities of the two portfolios V and H to the dynamics of
each principal component are equal. This highlights that traditional
PCA-hedging is a special case of the generalized hedging strategy we will
test. Namely, it is the case assuming no model errors.
Accordingly, the only difference between our generalization and
simple PCA-hedging consists in the term of Equations (3.) including θk.
Within this term, θk represents the size of the expected model errors for
rate R(t,Dk), whereas the exposure of the hedging strategy to these errors
is provided within the last term of the variance of the unexpected return
(Equation (10.) in the Appendix):
M 1
§ C l 2 ¶ c 2 D 2 §¨ A
b w ¶·
E
£
y ,t
y , k ,t
© t ¸ lk k © k ,t £
l 1
y 1

¸
M



2

(6.)

DOI: 10.1057/9781137564863.0006


Adjusting Principal Component Analysis



The purpose of the term of Equations (3.) including θk is to introduce a
penalty for the exposure to model errors. In other words, our generalized
PCA-hedging implements a trade-off between the precision of matching
the sensitivity of portfolio V to each principal component (the exclusive
goal of simple PCA-hedging) and the level of exposure to model errors.
A slightly simpler way to limit the exposure of PCA-hedging to
model errors and to transaction costs could be to minimize the sum of
the squared weights ϕy2, while ensuring that the simple PCA-hedging
Equations (5.) are fulfilled. From a theoretical point of view, this approach
is difficult to justify: as highlighted by expression (6.), the exposure of the
hedging strategy to the model errors is more complex than the sheer sum
of the squared weights ϕy2. However, for the sake of simplicity, it could be
of interest to check if this approach and our generalized PCA-hedging
lead to similar results.

2.2

The results

We intended to base our tests on real bond prices reported by the CRSP
database. Accordingly, the portfolios of liabilities have been constructed
by using seven real coupon bonds with gradually lengthening maturity:
the maturity of the first bond varies between 2 months and 3 years,
whereas the maturity of the seventh bond varies between 23 and 26 years.
When the maturity of a bond has no longer fitted within the corresponding maturity bucket, the bond has been replaced by another bond of
appropriate maturity.
Our tests have been based on six portfolios of liabilities constructed
by varying the weights invested in the seven bonds. The first three portfolios have been identified as bullet portfolios because a large portion
of the liabilities matures on one date in the – respectively – short-term
(up to 5 years), medium-term (between 8 and 12 years), and long-term
(beyond 23 years) future. The second three portfolios replicate common
bond portfolio structures: ladders (evenly distributed liabilities), barbells
(most liabilities mature either in the short-term or in the long-term), and
butterflies (liabilities mature either in the short-term or in the long-term
and assets mature in the medium-term).
For each portfolio of liabilities, we built the hedging portfolio H in
three alternative ways: a traditional three-component PCA (based on
Equation (5.)), a generalized, error-adjusted 3-component PCA (based
DOI: 10.1057/9781137564863.0006


Nicola Carcano



on Equation (3.)), and a 2-component PCA based on minimizing the sum
of the squared weights ϕy2. All three tested models imply to construct the
hedging portfolio with four bonds, which makes the comparison fair.3
Also in this case, we used real coupon bonds with gradually lengthening maturity from the CRSP database (which did not coincide with the
bonds used for the liability portfolios).
The alternative hedging strategies have been tested on 204 monthly
holding periods from the January 1, 1992, to December 31, 2008. The PCA
parameters have been estimated on the monthly Unsmoothed FamaBliss zero-coupon rates between May 1975 and December 1991. The same
rates have been used for discounting the cash flows to present value. The
methodology followed for the calculation of these zero-coupon rates is
described in Bliss (1997).
The hedging equations have been solved at the beginning of each
month; the resulting weights have been applied for the following month.
For each monthly observation, we calculated the hedging error as the
difference between the unexpected return of portfolio V and the unexpected return of portfolio H. The quality of a hedging strategy has been
measured by the Standard Error of Immunization (SEI), that is, the average absolute value of the hedging error.4
Given the dependency between the results of different hedging strategies on the same case and time period, we estimated statistical significance following an approach of matched pairs experiment. In other
words, we calculated the difference between the absolute value of the
hedging errors generated by two strategies on the same case and holding
period. Our inference referred to the mean value of this difference.
Additionally to the SEI, we also reported the index of excess kurtosis
of the hedging errors. High positive values for this index indicate very fat
tails, which in this case implies higher probability of large hedging errors
(with negative or positive value). Most investors are adverse to fat tails.
Accordingly, for comparable levels of SEI, hedging strategies displaying
lower kurtosis should normally be preferred.
Finally, we estimated the average sum of the squared weights ϕy2 and
the transaction fees implied by the alternative hedging strategies. The
transaction costs for each bond unit have been estimated as one half of

the bid/ask spread reported for a certain bond at a certain date by the
CRSP database. Two types of transaction fees have been estimated:


Set-Up Fees, which represent the costs of implementing the
hedging strategy from scratch. These fees are particularly relevant
DOI: 10.1057/9781137564863.0006