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

List of Chart & Exhibits vi Giovanni Barone Adesi and Nicola Carcano 2 Adjusting Principal Component Analysis Nicola Carcano 3 Alternative Models for Hedging Yield Curve Risk: An Empi

Modern Multi-Factor Analysis of Bond Portfolios

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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

Selection and editorial content © Giovanni Barone Adesi and

Nicola Carcano 2016

Individual chapters © the contributors 2016

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, Saff ron 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.

Th e authors have asserted their rights to be identifi ed 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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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.

List of Chart & Exhibits vi

Giovanni Barone Adesi and Nicola Carcano

2 Adjusting Principal Component Analysis

Nicola Carcano

3 Alternative Models for Hedging Yield

Curve Risk: An Empirical Comparison 21

Nicola Carcano and Hakim Dall’O

4 Applying Error-Adjusted Hedging to

Giovanni Barone Adesi, Nicola Carcano and Hakim Dall’O

5 Credit Risk Premium: Measurement,

Interpretation and Portfolio Allocation 78

Radu C Gabudean, Kwok Yuen Ng and Bruce D Phelps

List of Chart & Exhibits

Chart

Exhibits

including USD interest rate swaps: hedging

List of Chart & Exhibits

List of Figures

for the NC IG corp index and their difference,

two different analytical duration measures,

conditional on the change in Treasury yields,

with Treasury returns, by sub-period, March

for the NC IG corp index, July 1989–

List of Figures

NC IG corp index conditional on the change in

of Treasuries with various returns of the non-call DGT IG

Treasury portfolios, as  of total net allocation,

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 contribu-tions 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 presi-dent of quantitative strategies with Barclays Risk Analytics and Index Solutions (BRAIS), where he designed and

Notes on Contributors

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 spend-ing 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

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

 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 instru-ments 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 expres-

sion 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 inter-est 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

dura-tion – introduced by Macaulay (1938) – for implementing immunizadura-tion

techniques Duration represents the first derivative of the price-yield relationship of a bond and was shown to lead to adequate immunization

The assumption of parallel yield curve shifts could be released thanks

to the concept of convexity which was initially related to the second

deriv-ative 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

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

orthogo-nal 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

 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 ing 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

prevail-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 Collin-Dufresne 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 rate 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 struc-ture 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

corpo-The fifth chapter of this book shifts the focus from pure hedging egies to optimal portfolio construction For many investors, analytical excess returns conform to their macro views: they wish to be exposed

strat-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

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 retical as well as practical conclusions and our key recommendations to practitioners actually engaged in interest rate risk management

theo-The book follows a stepwise construction approach We start from the simplest models in Chapter 2 and gradually move towards more sophis-ticated 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

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

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 proc-ess 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

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 ing based on PCA does not control the level of exposure to the model errors

hedg- 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

the cash flows of this portfolio in m time buckets The present value of

of these time buckets, basis risk comes from unexpected shifts in the

duration and maturity of the time bucket

For the sake of simplicity, we will assume that all rates are gales In other words, no interest rate changes are expected, so that:

martin-E[dR(t, D k )] = 0 for every k and every t Extending our framework to

account for expected rate changes is relatively simple Moreover, the impact

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

t represents the change in the l-th principal component between

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

Adjusting Principal Component Analysis

Usually (see, for example, Martellini and Priaulet (2001)), hedging

strategies assume the so-called self-fi nancing constraint:

hedg-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

fulfi ll the self-fi nancing constraint

In essence, the generalized version of PCA-hedging we intend to

consid-ered within the minimization of the expected immunization error, while these terms are ignored by simple PCA-hedging We show in

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):

1 1

1

2 2 2

M

k t k

­

¹¹

­ºº

­­

»

­ºº

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

 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

time

Let us analyze the set of hedging Equations (3.) more carefully It is

standard set of equations for PCA-hedging:

Accordingly, the only difference between our generalization and

is provided within the last term of the variance of the unexpected return (Equation (10.) in the Appendix):



2 1

Adjusting Principal Component Analysis

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

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 interest to check if this approach and our generalized PCA-hedging lead to similar results

Our tests have been based on six portfolios of liabilities constructed

by varying the weights invested in the seven bonds The first three

port-folios have been identified as bullet portport-folios 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

 Nicola Carcano

on Equation (3.)), and a 2-component PCA based on minimizing the sum

Also in this case, we used real coupon bonds with gradually ing maturity from the CRSP database (which did not coincide with the bonds used for the liability portfolios)

lengthen-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 Fama-Bliss 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 pected return of portfolio H The quality of a hedging strategy has been

unex-measured by the Standard Error of Immunization (SEI), that is, the

Given the dependency between the results of different hedging gies on the same case and time period, we estimated statistical signifi-cance 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

strate-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

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

Adjusting Principal Component Analysis

for volatile portfolios of liabilities, which often impose a complete restructuring of the hedging portfolio In this case, the unit

transaction costs have been multiplied by the absolute values of the

that must be traded)

The results on the quality of the three hedging strategies are reported

in Exhibit 1 The error-adjusted 3-component PCA outperforms the traditional 3-component PCA on every single test case, and the differ-ence between the two strategies is mostly highly statistically significant

On average, the improvement in terms of SEI reaches 35 The simpler 2-component PCA based on minimizing the sum of the squared weights obtains similar results in terms of SEI as the error-adjusted 3-component PCA However, this simpler strategy displays on every single test case a

exhibit 1 Testing alternative PCA-based strategies on US treasury bonds:

hedging quality indicators (1992–2008; 204 monthly observations)

Short Bullet . . . . * . . *** . Medium Bullet . . . . . . *** . Long Bullet . . . . *** . . ** . Ladder . . . . *** . . *** . Barbell . . . . *** . . *** . Butterfly . . . . *** . . *** .

Notes: 1 Volatility indicates the standard deviation of the returns of the liability portfolio;

2 SEI (Standard Error of Immunization) indicates the average absolute value of the

hedging error The hedging error is the difference between the unexpected return of the liability portfolio and the unexpected return of the asset portfolio; 3Kurt indicates

the index of excess kurtosis of the hedging errors; 4 Statistical significance is related

to the average difference between the absolute value of the hedging errors for the

tested strategy and the traditional three-component (3-C) PCA: “*” indicates 10

significance, “**” indicates 5 significance, and “***” indicates 1 significance.

 Nicola Carcano

significantly higher level of kurtosis This highlights that – even though

on average it performs quite well – this strategy is subject to few very large mistakes This is due to its exposure to curvature changes as well as

to its higher exposure to model errors compared with the error-adjusted 3-component PCA (suggested by its consistently higher squared weights reported in Exhibit 2) Accordingly, our expectation that the error-ad-justed 3-component PCA should come out as the best hedging strategy

is confirmed by Exhibit 1

Transaction fees are reported in Exhibit 2 As expected, the traditional PCA strategy leads to high transaction fees The level of rebalance fees is particularly impressive, which also suggests that the weights implied by this strategy vary significantly over time Interestingly enough, the error-adjusted 3-component PCA displays consistently lower squared weights than the 2-component PCA based on minimizing the sum of the squared weights This highlights that the need to exactly match the sensitivity of

the two portfolios V and H to the dynamics of the first two principal

components limits the potential reduction in the squared weights for the latter strategy As a result, the error-adjusted 3-component PCA

is the preferable strategy also in terms of transaction fees The impact

of transaction fees should not be underestimated: hedging errors lead sometimes to losses and sometimes to profits, whereas transaction fees always lead to losses for the investors Accordingly, the longer the time

exhibit 2 Testing alternative PCA-based strategies on US treasury bonds:

average transaction fees (1992–2008; 204 monthly observations)

Traditional -C PCA Error-adjusted -C PCA

Squared weights minimizing -C PCA

up fees Rebalance fees Squared weights

Set-Set- up fees Rebalance fees

Note: 1 Squared Weights indicates the square root of the average sum of the squared

weights ϕ y expressed as a multiple of the average value of the liability portfolio.

Adjusting Principal Component Analysis

horizon of the hedging need, the greater the role of transaction fees in determining the optimal hedging strategy

Interest rate derivatives, like bond futures or interest rate swaps, are often used for hedging yield curve risk From a theoretical standpoint, there should be no significant drawbacks to use derivatives in the context

of the error-adjusted hedging model presented above One simply needs

to adapt the return Equation (8.) to the sensitivity of the derivatives to changes in the zero rates and to re-derive the hedging Equations (3.) as shown in the Appendix In practice, it should be of interest to check if the results reported for bond-based hedging also apply to derivative-based hedging

Therefore, we decided to apply interest rate swaps to the hedging problem described above More precisely, we replaced the three bonds

of shorter maturity within the hedging portfolio H with the three more

liquid swap contracts, namely the 2-year, the 5-year and the 10-year contracts We defined the sensitivity of swaps to rate changes following

extracted from the Datastream database On each month, we re-balanced the hedging strategy in order to use swaps of constant maturity

Our results for hedging strategies including swaps are reported in Exhibit 3 The accuracy of the hedging strategies – summarized by the SEI – is generally lower compared with Exhibit 1 This is not surprising, since swap rates include a spread over US Treasury rates; this spread varies over time and does not always move together with US Treasury rates Additionally, swaps are mainly traded over-the-counter and the quality of their prices tends to be lower compared with liquid US Treasury bonds Nevertheless, the superiority of the error-adjusted model over the tradi-tional 3-component PCA is fully confirmed for every single hedging case

by Exhibit 3 Interestingly enough, the simpler 2-component PCA based

on minimizing the sum of the squared weights often obtains significantly poorer results than the error-adjusted model, when swaps are used This confirms the exposure of the simpler strategy to large hedging errors, which we have already reported from Exhibit 1 When hedging errors are greater, the lower exposure to these errors ensured by the error-adjusted model provides more substantial advantages compared with simpler models Finally, also in this case the error-adjusted 3-component PCA displays consistently lower squared weights than the two alternative strate-gies, thus leading to lower transaction costs

 Nicola Carcano

2.3 Conclusions

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 claimed 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 Also, this adjustment leads more sophisticated 3-component PCA to be more reliable than 2-component PCA, as theory would suggest Therefore, our results suggest that the poor performance of the 3-component PCA tested by Falkenstein and Hanweck (1997) relatively to a 2-component PCA is likely to be explained by high exposure to model errors much more than by the instability of the third principal component

exhibit 3 Testing alternative PCA-based strategies including USD interest rate swaps:

hedging quality indicators (1998–2008; 127 monthly observations)

Short-Term Bullet . . . . *** . . . *** . . Medium-Term Bullet . . . . *** . . . *** . . Long-Term Bullet . . . . *** . . . *** . . Ladder . . . . *** . . . *** . . Barbell . . . . *** . . . * . . Butterfly . . . . *** . . . . .

Notes: 1 SEI (Standard Error of Immunization) indicates the average absolute value of the hedging error The hedging error is the difference between the unexpected return of the liability portfolio and the unexpected return of the asset portfolio; 2 Kurt indicates the index of excess kurtosis

of the hedging errors; 3 W 2 is related to the squared weights It indicates the square root of the

average sum of the squared weights ϕ y expressed as a multiple of the average value of the liability portfolio; 4 Statistical significance is related to the average difference between the absolute value

of the hedging errors for the tested strategy and the traditional three-component (3-C) PCA:

“*”indicates 10 significance, “**”indicates 5 significance and “***”indicates 1 significance.

Adjusting Principal Component Analysis

An additional, important advantage of controlling the exposure to model errors is a substantial reduction in the transaction fees implied

by the hedging strategies (since the exposure to model errors is tively correlated with the amounts of bonds to be traded) This factor

posi-is likely to play a very important role in bond markets with limited liquidity

Other approaches to hedge against yield curve risk – like those based

on the M-vector – are subject to model errors In theory, the principle described in this paper could also be applied to these approaches An empirical verification of the possible advantages of these applications is left to further research

2.4 Appendix

There is large empirical evidence that – for holding periods not longer than one month – the effect of rate changes on the return provided by a zero bond can be plausibly approximated by its duration For example, Hodges and Parekh (2006) highlighted that convexity is not important

as a parameter of the second-order approximation of the price-yield relationship, but rather as a proxy for the sensitivity to the second principal component (which is normally related to the slope of the term structure) Also the above mentioned M-square models, like the one in Fong and Fabozzi (1985), move from a similar outset

Accordingly, we can approximate the overall unexpected return ψ provided by the combination of the two portfolios V and H as follows:

 Nicola Carcano

Since the residuals of a PCA have mean equal to zero and are ent from the principal components, the expected squared value of the unexpected return is:

2 1

2

2 1

independency among the principal components as well as definition (4.), the last equation becomes:

(10.)

Our goal is to minimize the last equation subject to the self-financing constraint This implies setting the first partial derivatives of the follow-ing Lagrangian function equal to zero:

Adjusting Principal Component Analysis

In order to complete our proof, we now need to check that the order condition of the minimization is fulfilled We remind that the mathematical formulation of this condition is:

which fulfill the hedging Equations (3.) We introduce the following definitions:

M l

t l l

M l

t l l

2 2 2 2

The elements of the Hessian matrix can be derived from Equation (12.) and written as:

n

lk k j k t k

, , , ,

Accordingly, the Hessian matrix can be written as:

v´(DxL(φ t *, μ t*))v = v´[W tE[k t k t ´ + Θ t]W t ´]v > 0 (17.)and the second-order condition of the minimization is fulfilled.

Notes

This chapter reproduces the text of the paper “Yield Curve Risk Management: Adjusting Principal Component Analysis for Model Errors” by Nicola Carcano,

Journal of Risk, 12, 1, 3–16, 2009 It was republished with the kind permission of

Incisive Risk Information (IP) Limited.

 An extensive review of these approaches is available in Nawalkha et al (2005).

 Past studies reported that the overall effect on hedging errors from changing our

assumption of no expected rate changes into an assumption of equal expected holding period returns is very small See, for example, Carcano and Foresi (1997).

 A 3-component PCA based on minimizing the sum of the squared weights

ϕ y2 would request the use of five coupon bonds Accordingly, it would imply

a higher degree of freedom and could not be fairly compared with the other strategies.

 An alternative measure of the quality of a hedging strategy is represented by the standard deviation of the hedging errors Both measures have some pros and cons We analyzed the results of the tested strategies based on both quality measures and arrived at identical findings For the sake of brevity, we will report our results only in terms of SEI.

 Hedging strategies using derivatives can be implemented without the

self-financing constraint (2.) In these cases, M assets are sufficient to implement

the strategy, and the hedging Equations (3.) no longer include the Lagrange

multiplier μ t However, for complete hedging investors generally ensure that the underlying value of the hedging portfolio (considering the notional principal for derivatives) equals the value of the portfolio to be hedged This is guaranteed by the self-financing constraint (2.) and is the approach we followed.

Alternative Models for

Hedging Yield Curve Risk:

An Empirical Comparison

Nicola Carcano and Hakim Dall’O

Abstract: This chapter extends the results presented in

Chapter 2 to further interest rate hedging models We present the results of explicitly accounting for the variance

of the model errors displayed by each zero rate We find out that the reduction in both the hedging errors and the transaction costs is substantial: the errors are reduced on average by 17 for the PCA model, by 39 for the KRD model and by 53 for the DV model What is perhaps more important is that the error adjustment makes the optimal weights of the hedging strategies far more stable Also, we

do find that the error-adjusted PCA model systematically outperforms all alternative models Finally, this chapter shows that bond futures can effectively be used to hedge the yield curve risk of a bond portfolio.

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: 10.1057/9781137564863.0007

 Nicola Carcano and Hakim Dall’O

We define yield curve risk as the risk that the value of a financial asset might change due to shifts in one or more points of the relevant yield curve As such, it represents one of the most widely spread financial risks: each institution having to match future streams of assets and liabilities is exposed – up to a certain extent – to it

A simple, but effective way to cope with yield curve risk is to match positive with negative cash-flows Unfortunately, the dates and the amounts of future cash-flows are often subject to constraints, so that implementing an accurate matching might either not be possible or be

very expensive In these cases, immunization techniques are employed to

manage yield curve risk These techniques make the sensitivity of the assets and the liabilities to yield curve changes similar to each other,

so that the overall balance sheet will not be largely affected by these changes

Initially, academicians and practitioners focused on the concept of

duration – introduced by Macaulay (1938) – for implementing

immuniza-tion techniques Duraimmuniza-tion represents the first derivative of the price-yield relationship of a bond and was shown to lead to adequate immunization

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)) However, the impact of interest rate changes over a few weeks is normally well-approximated by duration 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

Extensions of these strategies 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 approxima-tions 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 immunization models relied on a statistical

tech-nique known as principal component analysis (PCA) which identifies

Alternative Models for Hedging Yield Curve Risk

orthogonal factors explaining the largest possible proportion of the ance 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 Accordingly, models matching the sensitivity of assets and liabilities to these three components should lead to high-quality hedging

vari-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 key rates A significant extension of this approach in the presence of restricting constraints was developed by Reitano (1996)

In practice, yield curve hedging techniques mostly rely on one of these three classes of models However, we are not aware of conclusive evidence

empirical tests of these models reported puzzling results: models ble 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

capa-Carcano (2009) tests a model of PCA-hedging which controls the exposure to model errors He finds that – by introducing this adjustment – 3-component PCA does lead to better hedging than 2-component PCA,

as theory would suggest On this basis, he claims that random changes in the exposure to model errors might have led previous empirical tests of alternative hedging models to inconclusive results

The goal of this chapter is to identify the model capable of ing yield curve risk based on a sound empirical evidence We expect the exposure to model errors to play a crucial role in determining the performance of alternative models Once we give appropriate consideration to this exposure, the success of the tested models should mainly depend on how well the underlying stochastic process catches the actual dynamics of the yield curve Accordingly, we extend all three mainstream immunization approaches in order to account for model errors and compare them among themselves and with their traditional implementations

minimiz- Nicola Carcano and Hakim Dall’O

We rely on previous evidence that three factors are sufficient to explain the vast majority of the yield curve dynamics and test only three-factor models Accordingly, we expect the quality of the resulting error-adjusted hedging strategies to be comparable By construction, the PCA model explains the largest possible part of the variance of yield curve shifts based on three orthogonal factors Accordingly, we suspect that – once we account for model error exposure – this model would slightly outperform the alternative models

We test the three models by hedging portfolios of US T-bonds and T-notes through T-bonds and T-notes futures The results confirmed our expectations: even though we could not clearly rank the models based on their traditional version, hedging based on PCA is consistently the best choice when the error-adjustment is introduced This adjustment also improves the performance of all three models and leads to substantially more stable hedging equations

The remainder of the paper is organized as follows: Section 3.1 presents the hedging models we are going to test and their theoretical justification Section 3.2 describes our dataset and testing approach Section 3.3 reports our results, both on the full sample as well as on three sub-samples, while Section 3.4 concludes and indicates some possible directions for future research

3.1 The hedging methodology

We consider the problem of immunizing a risk-free bond portfolio which

be invested in each of the four US T-note/T-bond futures (the 2-year, the 5-year, the 10-year and the 30-year contracts) We group the cash flows of the bond portfolio and of the cheapest-to-deliver (CTD) bonds underly-

ing the futures in n time buckets Following the most common approach

to this immunization problem as in Martellini and Priaulet (2001), we

impose the so-called self-financing constraint:

Alternative Models for Hedging Yield Curve Risk

CTD bonds of the hedging portfolio In practice, the weightings of the hedging portfolio are often constrained, even though the form of these constraints can differ from the last equation Accordingly, we feel that including a constraint would make our empirical tests more realistic

We decide to analyze the quality of alternative hedging models on a relatively short hedging horizon, which is set equal to one month This choice is motivated by the fact that many institutional investors and portfolio managers do have a time horizon of 1–3 months, when they set

up their hedging strategies After this period, they mostly reconsider the whole hedging problem and determine a new strategy

The market risk for the portfolio to be hedged comes from unexpected shifts in the corresponding continuously compounded zero-coupon

the corresponding time bucket We assume, for simplicity, that all rates

compute the unexpected return of a bond as the return in excess of what would have been obtained if the promised yield had remained constant

Approximating the dynamics of the term structure through a limited number of factors results in a difference between the modeled and actual

dynamics of interest rates, the model error For a generic three-factor

model of the term structure of interest rates, we can describe the

dR t D k c F lk t t D

l l

As reported in several papers, like Hodges and Parekh (2006), the impact of monthly rate changes on the price of a zero-coupon bond can

be well-approximated by its duration Accordingly, we will follow this simplifying approach

Estimating the sensitivity of future prices to changes in zero rates is more complex In the past, researchers implementing hedging strate-gies through note and bond futures attempted to simplify the problem One approach has been to calculate the sensitivity of futures through standard regression analysis (an example of this approach can be found

 Nicola Carcano and Hakim Dall’O

in Kuberek and Norman (1983)) Such an approach implicitly assumes that the sensitivity of the future price is constant over time, whereas practitioners know well that this sensitivity varies significantly with the underlying CTD bond Since we intend to test realistic hedging strate-gies, we decide not to follow this approach

The approach we decide to follow in the estimation of bond future sensitivity can be better described by splitting the market price of these futures in two components: the theoretical price excluding the value of the embedded options and the basis

The theoretical price of a bond future excluding the value of the

embedded options FP can be represented by the following expression:

,

the accrued interests of the cheapest-to-deliver bond on the expiration

date s of the future contract The only cash-flows of this bond which are

relevant for the valuation of the future contract are the ones maturing

after the expiration date s.

Approximating the effect of rate changes on the price of a zero bond

by its duration, the percentage sensitivity of the future price to these changes can be expressed as:

e CF

e CF

cf e

t

t

s

s t

related to the CTD cash-flow with maturity k and is defined based on the

Alternative Models for Hedging Yield Curve Risk

last two equations The sensitivity of the future price to changes in zero

for all k < s.

The second component of the market price of a future is represented by the basis Within the basis, we can identify three further components:The carry, which we estimate as the difference between the yield



of the CTD bond and the 1-month risk-free rate applied from the starting date of the hedging period to the expiration of the future

contract The basis net of carry – the so-called net basis – is the sum

of the components 2 and 3

A possible mispricing between the cash and the future market and/



or data quality issues, like the difference in the time at which spot and future prices are observed (5 pm for bonds, 2 pm for futures) and in their meaning (mid price for bonds, closing price for

For all components of the basis, we need to distinguish expected from unexpected changes If the latter changes display a dependency on yield curve shifts, this would represent a further source of future price sensi-tivity to such shifts and would influence the optimal hedging strategy.For the carry, given our focus on the next expiring future, actual changes during the hedging period are dominated by its time-decay Accordingly, we estimate this time-decay as a component of futures’ expected return and neglect unexpected changes due to modifications in the yield curve shape

For the net basis, we follow Grieves et al (2010) in the assumption that this value should be expected to be linearly amortized in order to get to zero by the contract expiration An analysis performed on the average absolute value of the net basis confirms that the hypothesis of a linear amortization is fully consistent with empirical evidence Accordingly,

we estimate the unexpected change as the difference between the actual value of the net basis at the end of the hedging period and its expected value In the expiring months of future contracts (therefore, in one-third

of our test sample), we impose this difference to be zero because of the