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

All rights reserved. No reproduction, copy or transmission of this

publication may be made without written permission.

No portion of this publication may be reproduced, copied or transmitted

save with written permission or in accordance with the provisions of the

Copyright, Designs and Patents Act 1988, or under the terms of any licence

permitting limited copying issued by the Copyright Licensing Agency,

Saffron House, 6–10 Kirby Street, London EC1N 8TS.

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

PALGRAVE MACMILLAN

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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 http://lccn.loc.gov/2015037662

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

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