portfolio choice - au purepure.au.dk/portal/files/32826640/pjessenthesis_dec2010.pdf · 2011. 1....

Portfolio Choice− and −

Structured Financial Products

Pernille Jessen

Finance Research Group

Department of Business Studies

Aarhus School of Business

Aarhus University

PhD Thesis

October 2010

Advisor: Peter Løchte Jørgensen

Members of the assessment committee

Professor Carsten Sørensen

Department of Finance

Copenhagen Business School

Professor Hans Bystrom

Department of Economics

Lund University

Associate Professor Stefan Hirth


Aarhus University

Date of the public defense

January 21, 2011

Contents

Acknowledgments iii

Summary v

Dansk Resume ix

Introduction xiii

1 Optimal Investment in Structured Bonds 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Investment Opportunities . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Asset Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 The Structured Bond . . . . . . . . . . . . . . . . . . . . . 9

1.3 Optimal Investment . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 The Investment Problem . . . . . . . . . . . . . . . . . . . 13

1.3.2 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.3 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . 19

1.3.4 Numerical Method and Benchmark Parameters . . . . . . 20

1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.1 The Benchmark Case . . . . . . . . . . . . . . . . . . . . . 22

1.4.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 25

1.4.3 Certainty Equivalent, Risk Premium, and Optimal Expected

Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 Optimal Responsible Investment 43

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

ii CONTENTS

2.2 Portfolio Responsibility . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2.1 ESG Ratings . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2.2 Investor Priorities . . . . . . . . . . . . . . . . . . . . . . . 51

2.2.3 Portfolio Responsibility . . . . . . . . . . . . . . . . . . . . 52

2.3 Investor Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3.1 Utility Function . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4 Mean Variance Analysis . . . . . . . . . . . . . . . . . . . . . . . 59

2.4.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.4.2 Responsible Investment . . . . . . . . . . . . . . . . . . . . 60

2.4.3 Sensitivity to Responsibility . . . . . . . . . . . . . . . . . 62

2.5 Example:

Investment in a Sustainability Index . . . . . . . . . . . . . . . . 63

2.5.1 DJSI World . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.5.2 Optimal Responsible Investment . . . . . . . . . . . . . . . 66

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 An Asset Protection Scheme for Banks

Exposed to Troubled Loan Portfolios 75

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.2 Asset Protection Scheme (APS) . . . . . . . . . . . . . . . . . . . 82

3.2.1 Construction and Motivation . . . . . . . . . . . . . . . . 82

3.2.2 The APS and Financial Stability . . . . . . . . . . . . . . 84

3.3 Credit Risk and APS Valuation . . . . . . . . . . . . . . . . . . . 85

3.3.1 Gaussian Innovations in the Asset Returns . . . . . . . . . 86

3.3.2 Normal Inverse Gaussian Innovations in the Asset Returns 86

3.3.3 Pricing of an APS . . . . . . . . . . . . . . . . . . . . . . . 89

3.4 Estimation and Valuation . . . . . . . . . . . . . . . . . . . . . . 90

3.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.4.2 Estimation of Parameters in the Asset Value Dynamics . . 92

3.4.3 Valuation of the APS . . . . . . . . . . . . . . . . . . . . . 93

3.5 Capital Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.5.1 Economic Capital . . . . . . . . . . . . . . . . . . . . . . . 94

3.5.2 Minimum Regulatory Capital . . . . . . . . . . . . . . . . 97

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Acknowledgments

I would like to thank my thesis advisor Peter Løchte Jørgensen and Michael

Christensen, head of the Department of Business Studies at Aarhus School of

Business, Aarhus University, for giving me this excellent opportunity to study

for a PhD degree. I am grateful to Peter Løchte Jørgensen for his outstanding

supervision and academic enthusiasm; without his encouragement and guidance,

this thesis could not have been conducted.

It has been a true pleasure to be part of the Finance Research Group at Aarhus

School of Business, Aarhus University. I would like to warmly thank all the

members of this group for academic discussions, collegial companionship, and for

generously sharing their knowledge with me when needed. In particular, I would

like to thank Anders Grosen, Thomas Kokholm, and Peter Løchte Jørgensen,

who I have had the privilege to do research collaborations with, and Stefan Hirth,

who has supported me in many academic activities. Moreover, I would like to

thank the remaining staff members and PhD students at the Department of

Business Studies for being great colleagues and for constituting a very pleasant

work environment.

During my PhD study, I spent nine months as a visiting scholar at University

of British Columbia in Vancouver, Canada, and I am very grateful to Peter Løchte

Jørgensen, Peter Ove Christensen, and Ron Giammarino for making this visit

possible. I would also like to thank Robert L. Heinkel from University of British

Columbia for taking the time to discuss my research ideas with me. Last but not

least, I am very thankful to the Tuborg Foundation for financially supporting my

stay in Canada and giving me the wonderful opportunity to attend a number of

overseas conferences.

Finally, I would like to express my deep gratitude to my loving family for their

care and encouragement throughout my study. I am especially grateful to my

iv Acknowledgments

sister and colleague, Cathrine, who has supported me greatly during my entire

academic life and who has not only provided inspiration for improvement of my

research but who has also been an invaluable friend along the way. My final

thanks go to Jacob for his patience, positive mind, and love.

Pernille Jessen

Aarhus, September 2010

Summary

The thesis consists of three self contained papers within the topics; portfolio

choice and structured financial products. The papers are briefly described here-

under.

Optimal Investment in Structured Bonds

Structured investment products are popular among small retail investors. How-

ever, these products regularly receive much criticism from academics and financial

experts who argue that the products are highly complex and too expensive. This

paper considers a generic structured product: the principal protected index-linked

note, which we denote a structured bond. The purpose of the paper is to suggest

possible explanations for the puzzle of why small retail investors hold structured

bonds.

We study optimal portfolio allocation in an investment universe consisting of

a structured bond, a stock index, and a bank account. The structured bond con-

tains a zero coupon bond and an option on an index which−as a benchmark−the

investor cannot trade. We find the optimal investment using expected utility

maximization and consider different utility functions and trading strategies.

Our results show that investors with a medium level of risk aversion give the

structured bond the highest portfolio weight. The structured bond holding is cost

sensitive, but it provides diversification that may compensate for this cost. The

correlation between the stock index and the index underlying the option of the

structured bond similarly affects the optimal investment due to diversification.

Finally, we show that the investor could obtain a higher expected utility by

directly investing in the index underlying the option instead of the structured

bond.

vi Summary

Optimal Responsible Investment

Responsible investment comprises investment analysis and decision-making pro-

cesses that incorporate environmental, social, and corporate governance (ESG)

criteria. Although this definition of responsible investment can be considered

ambiguous, the market for responsible investment has grown to a substantial size

during the past five years.

This paper studies the portfolio choice of responsible investors. The model

defines an investor specific measure of portfolio responsibility in terms of ESG

criteria, and this measure includes investor priorities regarding the different cri-

teria. When the model incorporates this measure in the portfolio choice, it aims

at addressing the ambiguity of the definition of responsible investment. This

investment model does not require investors to be responsible, however, it only

allows them to choose responsible investments.

The study incorporates the measure of portfolio responsibility into two dif-

ferent investment approaches from conventional investment theory. The first

approach is investor utility theory which describes preferences for portfolio re-

sponsibility with an additional argument in the utility function. The second

approach is mean variance analysis which captures portfolio responsibility with

an additional restriction to the investment problem.

The investor utility approach is intuitive, but it is difficult to implement. In

the mean variance approach, on the other hand, the optimal portfolio problem

has a simple, analytical solution and can be applied by all types of investors.

An example of investment in the Dow Jones Sustainability World Index (DJSI)

illustrates the latter approach.

An Asset Protection Scheme for Banks

Exposed to Troubled Loan Portfolios

In the aftermath of the financial crisis, many banks are highly exposed to the

credit risk of certain distressed sectors and fear severe losses which may threaten

their survival. To cope with this increased uncertainty, the banks need more

buffer capital, but the informational asymmetries inherent in banking relation-

ships can make urgent recapitalization prohibitively costly or impossible.

In this paper we examine a specific portfolio credit derivative, an Asset Pro-

tection Scheme (APS), and its applicability as a tool to restore financial stability

vii

and mitigate asymmetric information. As opposed to most governmental bailout

packages implemented across the world recently, the APS can be a fair valued

contract with an appropriate structure of incentives.

Within the structural credit risk modeling framework, we apply two alterna-

tive multivariate default risk models. The first model, included as a benchmark,

is the classical Gaussian Merton model based on Gaussian innovations in the log

asset returns. The second model seeks to capture extreme market risk; it allows

for jumps and the log asset returns have Normal Inverse Gaussian innovations.

Exchanging the normal factors in the Gaussian model with NIG factors adds

more flexibility to the distribution of asset returns while retaining a convenient

correlation structure.

Using a unique data set on annual, farm level data from 1996 to 2009, we

consider the Danish agricultural sector as a case study and price an APS on an

agricultural loan portfolio. We estimate a joint default risk model and price an

APS based on this loan portfolio via simulation using the two alternative credit

risk models. Moreover, we compute the economic capital for this loan portfolio

with and without an APS.

viii Summary

Dansk Resume

Afhandlingen bestar af tre selvstændige artikler indenfor emnerne; porteføljevalg

og strukturerede finansielle produkter. Artiklerne er kort beskrevet herunder.

Optimal Investering i Strukturerede Obligationer

Strukturerede investeringsprodukter er populære blandt sma private investorer.

Disse produkter er dog jævnligt genstand for megen kritik fra den akademiske

verden savel som fra finansielle eksperter, der pointerer at produkterne er meget

komplekse og alt for dyre. Denne artikel behandler et ofte forekommende struk-

tureret produkt, en indekseret obligation med hovedstolsgaranti, som vi betegner

en struktureret obligation. Formalet med artiklen er, at give mulige forklaringer

pa, hvorfor sma private investorer investerer i strukturerede obligationer.

Vi undersøger den optimale investeringsallokation i et investeringsunivers, der

bestar af en struktureret obligation, et aktieindeks og en bankkonto. Den struk-

turerede obligation indeholder en nulkuponobligation og en option pa et indeks,

som investoren som udgangspunkt ikke kan investere i. Vi finder den optimale

investering ved brug af forventet nyttemaksimering og benytter os af forskellige

nyttefunktioner og handelsstrategier.

Vores resultater viser, at investorer med medium risikoaversion tillægger den

strukturerede obligation den største porteføljevægt. Beholdningen af den struk-

turerede obligation er følsom overfor ændringer i omkostninger men giver diver-

sifikation, som muligvis kan kompensere for denne omkostning. Korrelations-

strukturen mellem aktieindekset og det indeks, der underligger optionen i den

strukturerede obligation, pavirker ligeledes den optimale investering grundet den

ønskede diversifikation. Afslutningsvist viser vi, at investoren ville kunne opna

en højere forventet nytte ved at investere direkte i det indeks, der underligger

optionen, i stedet for den strukturerede obligation.

x Dansk Resume

Optimal Ansvarlig Investering

Ansvarlige investeringer omfatter investeringsanalyser og beslutningsprocessor,

der inkorporerer miljømæssige, sociale og ledelsesmæssige (ESG) kriterier. Selvom

definitionen af ansvarlig investering kan anses for at være uklar, er markedet for

ansvarlige investeringer vokset til en anseelig størrelse igennem de seneste fem ar.

Artiklen studerer porteføljevalg indenfor ansvarlig investering og definerer et

investorspecifikt mal for portefølje-ansvarlighed med hensyn til ESG kriterier.

Malet beskriver investorens prioriteringer af de forskellige kriterier, og modellen

sigter mod at afhjælpe uklarheden i definitionen af ansvarlige investeringer ved

at indbefatte dette mal i porteføljevalget. Investeringsmodellen kræver ikke, at

investorer er ansvarlige, men den tillader, at de kan vælge at være det.

Investeringsmodellen inkorporerer portefølje-ansvarlighed i to forskellige til-

gange til investering kendt fra konventionel investeringsteori. I den første til-

gang beskriver nytteteori investorens præferencer for portefølje-ansvarlighed ved

hjælp af et nyt argument i investorens nyttefunktion. I den anden tilgang bruges

middelværdi-varians analyse til at indbefatte portefølje-ansvarlighed ved at tilføje

en ny restriktion til investeringsproblemet.

Nytteteorien er intuitiv, men den er vanskelig at implementere. I middelværdi-

varians analysen derimod har det optimale investeringsproblem en analytisk løs-

ning og kan bruges af alle typer af investorer. Et eksempel vedrørende indeksinve-

stering i Dow Jones Sustainability World Index illustrerer resultaterne for den

sidstnævnte metode.

En Forsikring af Bankers Udsatte Laneporteføljer

Under eftervirkningerne af den finansielle krise er mange banker i høj grad eks-

poneret overfor kreditrisiko i forbindelse med særligt økonomisk trængte sektorer

og frygter derfor svære tab, som kan true deres overlevelse. I handteringen af

denne tiltagende usikkerhed har bankerne brug for at tilføre ny buffer-kapital,

men den asymmetriske information, der er kendetegnet ved bankforhold, kan

gøre en hurtig rekapitalisering prohibitivt dyrt eller umuligt.

I denne artikel undersøger vi et specifikt kreditderivat, en Asset Protection

Scheme (APS), og dets anvendelighed som værktøj til at genoprette finansiel sta-

bilitet og reducere asymmetrisk information. I modsætning til de fleste statslige

genopretningspakker, der er blevet implementeret verden over, kan en APS være

xi

en fair kontrakt med en hensigtsmæssig incitamentsstruktur.

Vi anvender to forskellige multivariate kreditrisikomodeller indenfor struk-

turel kreditmodellering. Den første model, der er inkluderet som et sammen-

ligningsgrundlag, er den klassiske Gaussiske Merton model baseret pa Gaussiske

innovationer i log aktivafkast. Den anden model er i stand til at beskrive ekstrem

markedsrisiko; den tillader spring og log aktivafkast er udstyret med Normal In-

vers Gaussiske (NIG) innovationer. Ved at udskifte de normalfordelte faktorer

i den Gaussiske model med NIG fordelte faktorer opnas yderligere fleksibilitet i

fordelingen af aktivafkast mens en gunstig korrelationsstruktur bibeholdes.

Ved brug af et unikt datasæt med arlige, individuelle data fra 1996 til 2009

studerer vi den danske landbrugssektor som et case study og prisfastsætter en

APS pa en portefølje af landbrugslan. For hver af de to kreditrisikomodeller

estimerer vi modellen og prisfastsætter en APS baseret pa landbrugslanene via

simulation. Derudover beregner vi den økonomiske kapital for laneporteføljen

med og uden en APS.

xii Dansk Resume

Introduction

The thesis consists of three self contained papers within the topics; portfolio

choice and structured financial products. The field of research is financial eco-

nomics, and although the articles included in the thesis are diverse in their focus,

the main theme is the application and analysis of financial innovations.

Structured financial products have been subject to much criticism−only am-

plified by the financial crisis−but due to the persistent sophistication and com-

plexity of the financial market, financial innovations are inevitable. However, as

this thesis demonstrates, structured financial products have the potential to be

useful tools in both investment and risk mitigation. Still, in order to determine

how to use new, complex structured products, agents in the financial market need

correspondingly better understanding of the product properties.

In the following this introduction briefly motivates each of the three papers

constituting the thesis.

The first paper considers a structured investment product which is popular

among small retail investors. This product is the so-called structured bond that

contains a capital guarantee and a potential for an enhanced return that depends

on an index which is not part of the contract. Financial experts as well as

previous literature criticize structured bonds and claim that the products are

very expensive and that small retail investors cannot understand the properties

of the products they purchase.

Inspired by this discussion and from a theoretical point of view, the first paper

considers the role of the structured bond in the optimal portfolio of the small retail

investor. It uses conventional finance theory as well as behavioral elements in the

methodology to address the question of whether structured bonds provide small

retail investors with an enhanced expected investor utility. Since the structured

bond is a typical structured investment product, this paper begins the thesis at

xiv Introduction

the heart of the research topic: it studies a portfolio choice model considering

a structured product that has already existed for a number of years but is still

subject to much debate.

The second paper investigates the optimal portfolio choice of a socially re-

sponsible investor. Responsible investment is a type of investment where the

decision process includes considerations of environmental, social, and corporate

governance (ESG) aspects. The optimal allocation of responsible investments is

relevant to both institutional and retail investors due to new aspects of risk man-

agement and an increased interest in the non-financial properties of investment

portfolios.

This paper investigates how to incorporate ESG criteria into common, conven-

tional investment models. The study applies financial innovation to meet market

sentiments and new requirements to the investment decision process. Respon-

sible investment combines several investor objectives and produces investment

allocations which simultaneously regard the financial properties of an investment

and the required ESG portfolio profile. This study thus explores a current “hot”

topic−an area of development within investment management−and therefore it

appropriately constitutes the second paper of the thesis.

The third paper examines the potential of a particular portfolio credit deriva-

tive, the asset protection scheme (APS), as a tool for mitigating asymmetric

information and reestablishing financial stability related to a particular troubled

sector. We regard the APS as an insurance contract that can be established be-

tween, e.g., a government and one or several banks who own troubled loan port-

folios. As opposed to most governmental bailout packages implemented across

the world recently, the APS can be a fair valued contract with an appropriate

structure of incentives.

The APS is in essence a structured financial product established to manage

and transfer risk among major agents in the financial market. Since this paper

anchors in current problems in the financial market and describes anticipated

favorable perspectives in the application of the APS, this paper suitably ends the

thesis with a future outlook of how structured financial products can be utilized,

e.g., by financial authorities to strengthen financial stability.

Chapter 1

Optimal Investment in

Structured Bonds

2 Optimal Investment in Structured Bonds

3

Optimal Investment in Structured Bonds

Pernille Jessen Peter Løchte Jørgensen




Aarhus University

Abstract

Structured investment products are popular among small retail investors.

However, these products regularly receive much criticism from academics

and financial experts who argue that the products are highly complex and

too expensive. This paper considers a generic structured product: the

principal protected index-linked note, which we denote a structured bond.

The purpose of the paper is to suggest possible explanations for the puzzle

of why small retail investors hold structured bonds.

We study optimal portfolio allocation in an investment universe consisting

of a structured bond, a stock index, and a bank account. The structured

bond contains a zero coupon bond and an option on an index which−as

a benchmark−the investor cannot trade. We find the optimal investment

using expected utility maximization and consider different utility func-

tions and trading strategies. Our main results show that investors with a

medium level of risk aversion give the structured bond the highest portfo-

lio weight. The structured bond holding is cost sensitive, but it provides

diversification that may compensate for its cost. The correlation in the

market affects the optimal investment accordingly. Finally, the investor

could obtain a higher expected utility with direct investment in the index

underlying the option instead of the structured bond.

Key words: Portfolio choice, Structured bonds, Expected utility maximiza-

tion, Behavioral finance, Structured products

JEL Classification: G11


1.1 Introduction

Small retail investors face a set of investment opportunities that includes struc-

tured investment products. A common structured product is the principal pro-

tected index-linked note comprising a zero coupon bond and an option. For

simplicity, this paper denotes this product a structured bond. The properties of

the structured bond have to compensate for its cost, but academics and financial

experts argue that small retail investors cannot realistically evaluate these prop-

erties, stating that the structured bonds are highly complex and too expensive.

In Norway, for example, the government has set up financial regulation to prevent

retail investors from purchasing structured investment products. It thus seems

puzzling why small retail investors hold structured bonds in spite of warnings

and criticism.

This paper studies optimal investment in structured bonds in order to suggest

possible explanations for the puzzle of why small retail investors hold structured

bonds. The investment universe consists of a structured bond, a stock index, and

a bank account, and as a benchmark we assume that the small retail investor can-

not trade the index underlying the option of the structured bond. The investor

chooses the optimal portfolio within the investment universe using expected util-

ity maximization, and the analysis applies a set of utility functions from both the

conventional finance literature and behavioral finance theory. The trading strat-

egy is either a simple Buy-and-Hold strategy or a Fixed-Mix strategy allowing

annual rebalancing between the stock index and the bank account.

We solve the problem with a numerical procedure, and our results demon-

strate the properties of the structured bond. A benchmark case shows that the

model specification greatly affects the optimal portfolio weights. The choice of

utility function causes the major effect while the different trading strategies yield

almost identical results. Sensitivity analysis of the optimal weight in the struc-

tured bond shows that investors with a medium level of risk aversion give the

structured bond the highest portfolio weight. The structured bond holding is very

cost sensitive but provides diversification that may compensate for this cost. The

correlation in the market affects the optimal investment accordingly. Next, com-

paring certainty equivalents, risk premiums, and optimal expected utility levels

for each utility function indicates that the type of option in the structured bond

and the trading strategy only have little effect on the expected utility level. Fi-

1.1 Introduction 5

nally, we show that the investor could obtain a higher expected utility with direct

investment in the index underlying the option instead of investment in the struc-

tured bond. Our findings thus suggest that investors should include structured

bonds in their optimal portfolio only if they cannot access the index underlying

the option directly and only if the products then provide sufficient diversification

to compensate for their costs.

Much literature on investment in structured products refers to behavioral fi-

nance theory and makes use of, e.g., the pioneer study by Tversky and Kahneman

(1992) who introduce Cumulative Prospect Theory. Generally, behavioral finance

theory describes investors deviating from the rational behavior assumed by tra-

ditional expected utility theory. Since the area of retail structured investment

products may be subject to irrational investor behavior, as the following litera-

ture review argues, our paper considers a behavioral approach in addition to the

conventional expected utility setup.

In an oligopoly setup, Carlin (2006) studies strategic price complexity of struc-

tured products. He argues that investors deviate from rational behavior when

facing complex investment products, and therefore sellers of the products may

choose to enhance product complexity to encourage irrational investor behavior.

Moreover, when investors find it difficult to track down the best offer on a prod-

uct, they are more likely to choose to be uninformed about prices across the

industry. Carlin (2006) concludes that this synergy may lead to price dispersion

and lack of competition.

Bernard and Boyle (2008) agree that sellers of retail structured products ap-

ply designs giving the products a strong investor appeal. Structured products

can, for example, combine upside appreciation in bull markets with downside

protection in bear markets. If investors overweight the probability of a favorable

market outcome, they often prefer different types of structured products than

rational expectations predict. Bernard and Boyle (2008) also provide evidence

that financial advisors sometimes encourage this type of probability overweight-

ing in their product prospectus or written investment guides. Henderson and

Pearson (2009) contribute to this discussion by stating that financial institutions

are able to exploit investor cognitive biases when designing structured investment

products according to common market misinterpretations. The authors use this

argument to explain why some investors are willing to pay more for structured


products than the actual product value.

With respect to empirical evidence, Fischer (2007) investigates the motiva-

tion to invest in structured products. From a survey among German investors,

he finds mostly rational investment motives such as diversification and cost man-

agement but also irrational motives such as betting or inconsistent motives that

lead to contradictory strategies. Our paper basically studies the same question as

Fischer (2007), but we consider the motivation to invest in structured bonds in a

theoretical optimal investment model using expected utility maximization. Hens

and Rieger (2009) add to the empirical evidence using data from Germany and

Switzerland and show that most successful structured products are not optimal

for rational investors. To affect investment allocations, sellers of popular prod-

ucts use behavioral factors such as loss aversion, probability misestimation, and

framing, which is the particular way in which the seller presents the investment

decision.

Doebeli and Vanini (2010) study both the claimed and the actual preferences

of small retail investors with respect to structured products. Based on a field

experiment, their findings show that product descriptions with clear and simple

words motivate first-time investors in structured products and that individuals

behave in a consistent manner within a behavioral finance model. The results

support the hypothesis from the previous literature that sellers of structured

products concentrate the design of the products on investor motives, but the

results seem to contradict the claim that product complexity is a useful tool

for sellers of structured products. Finally, Doebeli and Vanini (2010) confirm

the empirical findings of Fischer (2007) because a considerable percentage of

the investors in their survey do not behave in accordance with what rational

expectations predict.

The structured bond provides a capital guarantee since the zero coupon bond

has a risk free, fixed payment at maturity. In the context of life and pension

research, Doskeland and Nordahl (2008) study guaranteed investments and find

that Constant Relative Risk Aversion (CRRA) utility functions cannot explain

the existence of any form of guarantees. However, they consider guarantees that

are effective on a yearly basis, and this turns out to be the major source of welfare

loss as opposed to guarantees that are only effective at maturity. The authors

generally explain the demand for products including guarantees using behavioral

1.2 Investment Opportunities 7

finance theory.

With respect to optimal portfolio choice and behavioral finance theory, lit-

erature is scarce. An important contribution is Bernard and Ghossoub (2010)

who consider optimal portfolio allocation for an investor behaving according to

Cumulative Prospect Theory (Tversky and Kahneman (1992)). They study a

one-period economy with one risk free asset and one risky asset and find that op-

timal investment is a function of a generalized Omega measure of the distribution

of the excess return on the risky asset over the risk free asset. The authors also

show that the investment is very sensitive to the skewness of the excess return

on the risky asset. Our paper addresses a similar problem, but we consider an

investment universe including also a structured bond and use conventional utility

maximization though additionally adopting a behavioral approach.

This paper contributes to the literature on retail structured products with a

theoretical investigation of optimal investment in structured bonds. Our analysis

adds to the discussion of the circumstances−with respect to both the investment

universe and the structured product−under which investors optimally include

structured bonds in their investment portfolio. Our assumption that small retail

investors face a limited investment universe leads to similar results as in previous

literature using behavioral finance theory, namely, that investors are likely to

include structured products in their optimal portfolios.

The paper is organized as follows. Section 2 presents the investment oppor-

tunities and the construction of the structured bond. Section 3 introduces the

optimal portfolio allocation problem, utility functions, and trading strategies.

Section 4 gives the numerical results, and Section 5 concludes.

1.2 Investment Opportunities

This section introduces the investment universe of the model. The three basic

elements are two stock indexes and a risk free bank account. The investor has

three investment opportunities: the first stock index, the bank account, and a

structured bond containing an option on the second stock index. We assume

that as a benchmark the small retail investor cannot invest directly in the second

stock index, so in order to get exposure to a different risk profile than the one of

the first stock index, the investor must purchase the structured bond.


In a Black-Scholes framework, this section states the asset dynamics character-

izing the basic elements of the investment universe. It then gives the construction

of a typical structured bond and specifies how costs enter into the contract in our

model.

1.2.1 Asset Dynamics

In the following we define all processes and random variables on a filtered proba-

bility space (Ω,F , (F)t≥0,P). The filtration (F)t≥0 satisfies the usual conditions,

see, e.g., Protter (1990), and P and Q are the actual and the risk neutral proba-

bility measure respectively.

The process B denotes the value of a risk free asset which we refer to as a

bank account. With the continuously compounded, constant interest rate r, the

dynamics of the bank account is

dB(t) = rB(t) dt, B(0) = B,

where B is the initial value of the account.

The process I1 denotes the value of a risky reference fund which we refer to

as a stock index, Index 1. The index trades freely in continuous time and has no

dividend payments, or alternatively the index immediately reinvests dividends.

The dynamics of I1 is

dI1(t) = µ1I1(t) dt+ σ1I1(t) dZ1(t), I1(0) = I1,

where µ1 is the constant expected rate of return, and σ1 is the constant volatility.

The standard Brownian motion Z1(t) drives the development of I1, and I1 is the

initial value of the index.

The process I2 denotes the value of an alternative risky reference fund which

we refer to as another stock index, Index 2. We consider options on this index

in relation to the structured bond, but as the point of departure we assume that

the small retail investor cannot invest directly in the index. Contrary to I1, the

model allows I2 to have dividend payments in order to show how the structured

bond participates in the option payment dependent on this factor. The dynamics


Option payment

100 Construction cost

Option ZCB payment /capital guarantee

Zero coupon bond

Initial investment Payment at maturityt=0 t=T

Figure 1.1: The figure shows the initial investment in a structured bond at t = 0 andthe payment to the investor at maturity t = T .

of I2 is

dI2(t) = (µ2 − δ)I2(t) dt+ σ2I2(t) dZ2(t), I2(0) = I2,

where µ2 is the constant expected rate of return, σ2 is the constant volatility,

and δ is the constant dividend yield. The standard Brownian motion Z2(t) drives

the development of I2, and I2 is the initial value of the index. The correlation

between Index 1 and Index 2 is constant such that

dZ1(t) · dZ2(t) = ρ dt,

where ρ is the correlation coefficient.

1.2.2 The Structured Bond

The basic elements of a structured bond are a zero coupon bond and an option.

The zero coupon bond repays notional at maturity, and hence the structured bond

has a capital guarantee. The option gives the structured bond the potential to pay

out more than the notional at maturity, but this depends on the development in

the value of the index underlying the option. The seller of the contract charges a

fee−a construction cost−in exchange for the construction of the structured bond,

and in our model the investor pays this cost at the initial investment. Figure 1.1

gives an overview of the structured bond payments.

The option gives the structured bond its particular profile and payment po-


tential. The option type can, for example, be Asian, basket, barrier, or vanilla,

and the option can be written on a stock index, currency exchange rate, basket

of assets, corporate bonds, etc. Although the general structure is uniform, in

practice structured bonds are very different from each other, and it is even rare

that a particular seller constructs similar issues. The products typically have

different option types, assets underlying the options, times to maturity, and cap-

ital guarantees. Consequently, comparison across sellers and products is very

difficult.1

The following states the cash flows and different option types of the structured

bond in this paper. The initial investment in the structured bond is G(0), the

payment to the investor at maturity is G(T ), and the capital guarantee is G.

If the seller of the structured bond wants to magnify the option feature of the

contract, the seller can choose G < G(0) or prolong the time to maturity T . In

our model the option embedded in the structured bond is basically Asian. The

option payment ΠA depends on a number of observations N of the development

in the underlying index with the time intervals ∆t = TN

. If N = 1, the option

is simply a plain vanilla option. In the numerical analysis, we consider both the

Asian and the vanilla option to compare a structured bond with a complex option

to one with a simple option.

A normalized Asian option written on I2 with a strike price I2(0) has the

payment

ΠA(T ) =

[I2(T )− I2(0)

I2(0)

]+

, (1.1)

at maturity t = T where I2(T ) is the arithmetic average 2

I2(T ) =1

N

N∑i=1

I2(i∆t). (1.2)

1See, for example, Jørgensen et al. (2009) where an overview of the Danish market forstructured bonds is given.

2If the option is Asian tail, however, the averaging period begins at t ∈ ]0, T [. Often t isclose to T and choosing the Asian tail option over a vanilla option can be considered a matterof avoiding an unfavorable market outlier observation of I2(T ). If the option is Asian tail, the

arithmetic average is I2(T ) = 1N

∑Ni=1 I2(t+ i∆t) for ∆t = (T − t)/N .


The payment G(T ) to the investor at maturity is

G(T ) = G+ θ ΠA(T )G, (1.3)

where θ is the participation rate of the structured bond in the option. If the seller

purchases both the zero coupon bond and the option in the financial market, the

seller may use this participation rate to scale the two elements appropriately.

In our model the seller derives θ at time t = 0 from the equation

e−rT G+ θ e−rTEQ0

[ΠA(T )

]G = G(0)(1− π), (1.4)

where π is the construction cost charged as a percentage of the initial investment.

Equation (1.4) relates the market value of the two elements in the structured

bond, the zero coupon bond and the option, and the price of the structured bond

less its cost. The first term on the left side of (1.4) is the price of a zero coupon

bond that pays G at maturity T when the continuously compounded interest rate

is r. The second term is the product of the participation rate θ and the option

price, i.e., it is the price of a θ-part of the option payment at maturity. The term

on the right side of (1.4) is the initial investment in the structured bond less the

construction cost π. In our model the investor pays this cost up-front at time

t = 0, but in practice the cost is often quoted in annual terms.

Equation (1.4) yields the participation rate

θ =G(0)(1− π)− e−rT Ge−rTEQ

0 [ΠA(T )] G, (1.5)

which shows that the participation rate is inversely proportional to the option

price. This relationship implies that if a seller exchanges an option in a given

structured bond with a cheaper option, the participation rate of the structured

bond increases.

The total cost of the structured bond π may be considered as consisting of two

elements. The prospectus of the structured bond quotes the disclosed cost which

we denote πd. In this model the investor has no access to derivatives trading,

and therefore the seller of the structured bond may charge a cost which is higher

than the disclosed cost. If there is a difference between the total cost π and the

disclosed cost πd, this difference can be regarded as a hidden cost which the seller


adds as a fee for constructing the product. We denote such a hidden cost πh, and

the total construction cost π is then

π = πd + πh, (1.6)

where πd > 0 and πh ≥ 0.

Jørgensen et al. (2010) consider the hidden cost of structured products for

retail investors. They define the total cost in percent as the relative difference

between the actual sales price G(0) and the fair value of the product Gfair. They

estimate the fair valueGfair as the sum of the fair values of the two elements which

the structured bond includes. The hidden cost is then the difference between the

total cost and the disclosed cost. If

G(0) > Gfair +G(0)πd, (1.7)

then the structured bond has a hidden cost πh > 0, and this is

πh =G(0)−Gfair −G(0)πd

G(0)= π − πd,

when πh is a percentage of the sales price G(0). This paper mainly considers the

total cost, but we keep in mind that in the Danish market for structured bonds,

for example, the hidden cost is on average positive πh > 0 according to Jørgensen

et al. (2010). See Figure 1.2 for a stylized illustration of the product costs.

To end this section, we show that the structured bond corresponds to a port-

folio insurance contract. In order to see this, we recall the put-call parity

Ccall − Cput = S(0)− e−rTK, (1.8)

where Ccall and Cput are the prices of a plain vanilla European call and put option

respectively. The options are written on a stock with the price process S, have a

strike price K, and run over the period [0, T ]. If we rearrange the terms in (1.8),

we get

e−rTK + Ccall = S(0) + Cput. (1.9)

On the left side of this equation are the elements of a structured bond including a

1.3 Optimal Investment 13

G(0)Disclosed cost Disclosed cost

G(0)πd G(0)πd( ) ( )Hidden cost

G(0)πh

G(0)(1-πd)

Gfair

Figure 1.2: The figure shows the cost elements of the structured bond. On the leftside are the elements which the small retail investor sees, and on the right side are theelements which the structured bond may actually cover when the fair value Gfair isestimated.

zero coupon bond and a vanilla call option on S. On the right side of (1.9) are the

elements of a portfolio insurance contract since the portfolio that consists of S is

guaranteed the sales price K at time t = T due to the vanilla put option. Because

of this correspondence between the structured bond and portfolio insurance, this

paper also indirectly includes results for structured products resembling portfolio

insurance contracts.

1.3 Optimal Investment

This section specifies the optimal investment problem. After an introduction to

the general investment problem, we give the set of utility functions and trad-

ing strategies which the problem takes into account. The section then states

the numerical method, which we use to solve the problem, and defines a set of

benchmark parameters.

1.3.1 The Investment Problem

We set up the general investment problem as follows. Investors are rational

maximizers of expected investor utility at maturity, and they do not withdraw

their investments throughout the time period [0, T ]. The initial wealth of the

investor isW (0), and at maturity the payment to the investor from the investment


portfolio is W (T ). The investor utility function U : R → R is well behaved; it is

monotonic, concave, continuous, and non-satiated. The problem of the investor

is then

maxw

EP0 [U(W (T ))] , (1.10)

for the given amount of initial wealth W (0) ≥ 0, i.e., at time t = 0 the investor

maximizes the expected utility of wealth at maturity. The weights w ≡ (wI , wG)

describe the relative portfolio weights in the stock index and in the structured

bond respectively, and they are the variables that the investor chooses in order

to maximize the expected utility.

The problem in (1.10) requires an expression for the wealth at maturity. We

assume that wI + wG ≤ 1 and that both wI ≥ 0 and wG ≥ 0 such that the

model does not allow short sales.3 The weight in the bank account is then the

remaining part (1− wI − wG). If the investor holds the portfolio until maturity

T , the payment to the investor is

W (T ) = wIW (0)I1(T )

I1(0)+ wGW (0)

G(T )

G(0)+ (1− wI − wG)W (0)

B(T )

B(0).

Since Index 1 and Index 2 follow geometric Brownian motions, the log-returns of

each stock index are normally distributed N(µ1T, σ21T ) and N(µ2T, σ

22T ) respec-

tively over the period [0, T ]. The payment at maturity is

W (T ) = wIW (0)e(µ1− 12σ21)T+σ1Z1(T )

+wGW (0)G+ θΠA(T )G

G(0)

+(1− wI − wG)W (0)erT . (1.11)

This term is a crucial part of the objective function in the maximization in

(1.10), but before we can solve the problem, we need to specify the possible

utility functions and trading strategies.

3This assumption reflects the market observation that small retail investors usually do notshort sell assets. In particular, short selling the structured bond is practically impossible for thesmall retail investor since it would mean that the investor should “issue” the product. However,as an experiment it might be interesting to consider relaxing the assumption prohibiting shortsales. In case the small retail investor optimally short sells the structured bond, we may arguethat the product is indeed too expensive.


1.3.2 Utility Functions

The paper considers four utility functions: three basic representatives from the

general class of hyperbolic absolute risk aversion (HARA) utility functions and

one function which reflects behavioral finance theory. The HARA utility functions

are well behaved, i.e., they are monotonic, concave, continuous, and globally

non-satiated. The study by Tversky and Kahneman (1992) inspires the fourth

function which has an S-shape around a point of reference. Hence, we denote

this function the S-shaped utility function.4 This section specifies these utility

functions.

HARA Utility

The class of HARA utility functions represented by the function U : R → Rhas the Absolute Risk Aversion (ARA)

ARA(x) = − U ′′(x)

U ′(x)=

1

αx+ β, (1.12)

which−as the name indicates−is hyperbolic in x.5 The HARA utility function

has an affine risk tolerance since the Absolute Risk Tolerance (ART) is a linear

function of x, i.e.,

ART (x) =1

ARA(x)= αx+ β,

and the risk caution is ART ′(x) = α. In the following we specify representative

examples of the HARA utility function for the three cases;

α 6= 0, β = 0,α = 0, β 6= 0, and

α 6= 0, β 6= 0.

Constant Relative Risk Aversion (CRRA)

The CRRA utility function is frequently applied in the finance literature, and

it is a representative of the HARA utility function class with α 6= 0 and β = 0

4Tversky and Kahneman (1992) use the term “S-shaped value function”, but the term“S-shaped utility function” appears often in the literature following their work.

5See, e.g., Luenberger (1998) or Munk (2007) for a review of the class of HARA utilityfunctions.


in (1.12). With support form empirical evidence, the classical studies by Friend

and Blume (1975) and Pindyck (1988) argue that individuals have approximately

constant relative risk aversions. For x ∈ R+, the CRRA utility function has the

definition

U1(x) =x1−γ

1− γ, γ ≥ 0 ∧ γ 6= 1, (1.13)

with the risk aversion coefficient γ which corresponds to α = 1γ

and β = 0

in (1.12). Since the Relative Risk Aversion (RRA) is xARA(x), equation (1.12)

yields that the relative risk aversion is constant−hence the name−with parameter

γ. The CRRA utility function has the convenient property that an optimal

solution to a maximization of the function has a strictly positive level of wealth

at maturity with probability one since limx→0 U′1(x) =∞ . Figure 1.3 illustrates

the CRRA utility function; it is the graph at the top left.

Exponential

The exponential utility function is a representative of the HARA utility function

class with α = 0 and β 6= 0. For x ∈ R, the function has the definition

U2(x) = −e−κx, (1.14)

with the constant absolute risk aversion coefficient κ which corresponds to α = 0

and β = 1κ

in (1.12). A constant absolute risk aversion is simplistic−and perhaps

not very realistic−but the exponential utility function is common in the litera-

ture, see, e.g., Luenberger (1998). Figure 1.3 illustrates the exponential utility

function; it is the graph at the top right.

Decreasing Relative Risk Aversion (DRRA)

The DRRA utility function is a representative of the HARA utility function class

with α 6= 0 and β 6= 0. The literature sometimes refers to this function as a

subsistence HARA utility function, see Munk (2007). Ogaki and Zhang (2001)

argue that a DRRA utility function is superior to the CRRA utility with respect

to household consumption, and Friend and Blume (1975) also recognize its po-

tential even though they favor the CRRA utility function. For x ∈ R, the DRRA


utility function has the definition

U3(x) =(x− x)1−ν

1− ν, ν ≥ 0 ∧ ν 6= 1, (1.15)

where ν is the risk aversion coefficient, and x is the subsistence level. This

specification corresponds to α = 1ν

and β = − xν

in (1.12).

The DRRA utility function has the relative risk aversion RRA(x) = νxx−x .

This expression implies that if the subsistence level is x = 0, U3 corresponds to

the CRRA utility function. For positive subsistence levels x > 0, the RRA is a

decreasing function of x whereas for negative subsistence levels x < 0, the RRA

is an increasing function of x. Therefore, the DRRA utility function requires

x > 0. Figure 1.3 illustrates the DRRA utility function; it is the graph at the

bottom left.

S-shaped Utility

A number of studies indicate that behavioral factors such as loss aversion af-

fect the optimal investment decision of the small retail investor. The article by

Tversky and Kahneman (1992) is the original study beginning this line of litera-

ture, but more recently empirical studies such as Fischer (2007) and Doebeli and

Vanini (2010) agree that the behavioral approach is useful in modeling investor

motives with respect to structured products. Furthermore, Doskeland and Nor-

dahl (2008) apply behavioral finance theory to explain the existence of capital

guarantees within insurance and pension contracts. Motivated by the findings

of Fischer (2007) and Doebeli and Vanini (2010) and since the structured bond

likewise includes a capital guarantee, this paper applies an alternative, behavioral

investor utility function with an S-shape.

The S-shaped utility function concentrates on the loss aversion of the investor.

The theory assumes that the investor does not focus on absolute levels of final

wealth but measures gains and losses in relative terms and evaluates the utility

from gains and losses separately. In the region of gains, the S-shaped utility

function is a concave utility function. In the region of losses, however, the investor

is very sensitive to loss when the wealth is just below the initial investment but

less sensitive when the loss occurs at a wealth level much smaller than the initial

investment. In other words, investors are risk averse over gains and risk seeking


0.5 1.0 1.5 2.0

-2.0

-1.5

-1.0

-0.5

CRRA

x

Util

ity

0.5 1.0 1.5 2.0

-0.4

-0.3

-0.2

-0.1

Exponential

x

Util

ity

0.5 1.0 1.5 2.0

-10

-50

DRRA

x

Util

ity

0.5 1.0 1.5 2.0

-2-1

01

23

S-shape

x

Util

ity

Figure 1.3: The figure shows the four utility functions considering initial investmentsof one unit, W (0) = 1. At the top left is the CRRA utility function with γ = 3, atthe top right is the exponential utility function with κ = 1, at the bottom left is theDRRA utility function with ν = 1.5 and subsistence level x = 0.5, and at the bottomright is the S-shaped utility function with ζ = 0.65.

over losses.

For x ∈ R, we define the S-shaped utility function as

U4(x) =

(x−1)1−ζ

1−ζ if x > W (0)

−(−x+1)1−ζ

1−ζ if x ≤ W (0),

where ζ ∈ (0, 1) is a risk aversion parameter, and W (0) is the initial investment.

Figure 1.3 illustrates the S-shaped utility function; it is the graph at the bottom

right.


1.3.3 Trading Strategies

We assume that the structured bond is illiquid and cannot be traded during the

investment period [0, T ]. This assumption reflects the market fact that structured

bonds are typically subject to very high bid-ask spreads, and consequently, trad-

ing is prohibitively expensive in the time period between the initial investment

and maturity. Index 1 and the bank account are nevertheless liquid assets, and

we assume that trading these has no transaction costs.

The model considers two different trading strategies. The first is a Buy-and-

Hold trading strategy where the investor simply keeps the initial allocation in

the stock index, bank account, and structured bond until maturity. The second

strategy is a Fixed-Mix trading strategy where the structured bond is still a

Buy-and-Hold asset, but now the allocation between the stock index and the

bank account rebalances to their initial relationship periodically. An attractive

feature of the Fixed-Mix trading strategy may be that it tries to “sell high” and

“buy low”. If an asset increases in value, the Fixed-Mix trading strategy sells part

of this asset holding at the next reallocation due to the required fixed relative

relationship between the assets in the portfolio. Likewise, if an asset decreases in

value, the Fixed-Mix trading strategy buys more of this asset to compensate for

the lower relative weight and to maintain the fixed relationship.

We regard the small retail investor as an unsophisticated investor who wishes

to limit the amount of attention that the investment portfolio requires during

the period [0, T ]. The Buy-and-Hold trading strategy is obviously without main-

tenance while the Fixed-Mix trading strategy requires maintenance when the

portfolio rebalances the stock index and the bank account. In order to keep the

Fixed-Mix trading strategy simple, we assume that the portfolio reallocates In-

dex 1 and the bank account annually. Theoretically, this rebalancing could also

happen continuously, but we consider this scenario unrealistic for small retail

investors, and in practice it would also be prohibitively expensive.

The Fixed-Mix trading strategy rebalances the stock index and bank account

to their initial relationship on an annual basis. This relationship is

λ =wIW (0)

wIW (0) + (1− wI − wG)W (0)=

wI1− wG

. (1.16)

The portfolio holdings of the stock index and the bank account are Ipf1 and Bpf


respectively. The time indicators t = t− and t = t+ denote the time prior to and

after the portfolio adjustment. The adjusted portfolio holdings at time t are then

Ipf1 (t+) = Ipf1 (t−) + ∆

Bpf (t+) = Bpf (t−)−∆,

where the adjustment ∆ rebalances the relationship of the index and bank account

back to the initial relationship λ in (1.16), i.e.,

λ =Ipf1 (t−) + ∆

Ipf1 (t−) +Bpf (t−).

The adjustment ∆ is therefore

∆ = λ(Ipf1 (t−) +Bpf (t−))− Ipf1 (t−).

The Fixed-Mix trading strategy does not require stochastic programming since

the only point in time when the investor chooses asset weights is at the ini-

tial investment. When the investor chooses (wI , wG), it corresponds to choosing

(λ,wG).

1.3.4 Numerical Method and Benchmark Parameters

We are now able to solve the problem

maxwI ,wG

EP0 [U(W (T ))]

s.t. wI , wG ∈ [0, 1] , wI + wG ≤ 1 , (1.17)

for the given amount of initial wealth W (0) ≥ 0, and we do so by the application

of a numerical optimization procedure using Monte Carlo simulations because an

analytical expression for the expected utility at maturity does not exist. We apply

the statistical software R and the optimization procedure constrOptim which al-

lows for linear inequality constraints.6 The optimization uses a simplex algorithm

based on the Nelder-Mead method performing a multi-dimensional, nonlinear op-

6See www.r-project.org to download R or the R manual which describes the optimizationprocedure constrOptim.


timization under linear inequality constraints. Nelder and Mead (1965) present

this algorithm, and in spite of the slow execution, the solutions are robust. The

optimization procedure considers 100,000 market outcomes of the two stock in-

dexes and computes the return for each index as well as for the bank account.

The Asian option considers annual observations in the arithmetic average used

to calculate the pay-off ΠA in (1.1). We price the Asian option using the ap-

proximative method by Turnbull and Wakeman (1991), and in case of the vanilla

option, we use the formula by Black and Scholes (1973) to price the option. Based

on the simulated market outcomes, the expected investor utility at maturity is a

function of w = (wI , wG), and the procedure constrOptim optimizes this function

to find the optimal portfolio allocation w.

We choose a set of model benchmark parameters for both the financial market

and for the structured bond. The set of market parameters Φm is

Φm =

r = 2% δ = 0

ρ = 30% W (0) = 100

µ1 = 6% I1(0) = 100

µ2 = 8% I2(0) = 100

σ1 = 15% B(0) = 100

σ2 = 22.5%

, (1.18)

where the market with two risky assets fulfills the no-arbitrage condition thatµ1−rσ1

= µ2−rσ2

. The set of parameters Φg that relates to the structured bond is

Φg =

G(0) = 105

G = 100

T = 5 yrs.

π = 3%

for Option ∈

Asian

Vanilla

. (1.19)

Finally, the benchmark for the utility functions are

Φu =

γ = 3

κ = 1

ν = 1.5

x = 12W (0)

ζ = 0.65

for U ∈

CRRA

Exp

DRRA

S-shaped

. (1.20)


1.4 Results

This section presents the results of the numerical analysis. We divide the results

into three parts. The first part shows the benchmark case of optimal investment

in the stock index, structured bond, and bank account. The second part considers

sensitivity analysis of the optimal weight in the structured bond to the factors:

risk aversion of the investor, cost of the structured bond, dividend on I2, and

correlation between the indexes. Finally, the third part presents certainty equiv-

alents, risk premiums, and optimal expected utilities. The results also include a

comparison of the optimal investment in the structured bond to an alternative

case where the investor can invest directly in Index 2 instead of the structured

bond.

1.4.1 The Benchmark Case

The benchmark case solves the optimal investment problem for each of the four

utility functions, two trading strategies, and two option types. The solution

includes an alternative case where the investor can invest directly in Index 2

instead of the structured bond.

Figure 1.4 presents the results and gives the optimal relative portfolio weights

for each combination of inputs. A common observation for all the utility func-

tions, trading strategies, and option types is that the optimal portfolio includes

the structured bond. The optimal weight in the structured bond ranges from 3%

in the case of the exponential utility function, Fixed-Mix trading strategy, and

Vanilla option to 45% in the case of the DRRA utility function, Fixed-Mix trading

strategy, and Asian option. The presence of the structured bond in the optimal

portfolio indicates that the structured bond provides the investment portfolio

with sufficient diversification to compensate for the construction cost−at least to

some extent.

Figure 1.4 also illustrates the differences in the utility functions. The CRRA

and DRRA utility functions yield the most similar results and the weight in

Index 1 is indeed very close to identical for the different trading strategies and

option types. The CRRA and DRRA utility functions have quite similar struc-

tures, but since the CRRA has a constant relative risk aversion, it gives more

weight to the bank account than the DRRA utility function with its decreasing

1.4 Results 23

B&H-A B&H-V FM-A FM-V B&H

Opt

. pf.

wei

ghts

0.0

0.2

0.4

0.6

0.8

1.0

CRRA

SB SB SB SB Index 2

Index 1 Index 1 Index 1 Index 1 Index 1

Bank Bank Bank Bank Bank


Opt

. pf.

wei

ghts

0.0

0.2

0.4

0.6

0.8

1.0

Exp.

SB SB

Index 2



Opt

. pf.

wei

ghts

0.0

0.2

0.4

0.6

0.8

1.0

DRRA

SB SB SB SB Index 2


Bank


Opt

. pf.

wei

ghts

0.0

0.2

0.4

0.6

0.8

1.0

S-shape

SB SB SB SB Index 2


Bank Bank Bank Bank Bank

Figure 1.4: The figure shows the results of the benchmark case for each combinationof utility function, trading strategy (Buy-and-Hold (B&H) and Fixed-Mix (FM)), andoption type (Asian (A) and vanilla (V)). For each utility function, the plot also includesthe case with direct investment in I2 instead of the structured bond.


relative risk aversion. The DRRA utility function inherently exhibits less risk

aversion for higher levels of wealth which implies that the combination of the

capital guarantee and the potential to participate in Index 2 fits the DRRA pref-

erence structure well. Thus, when the capital guarantee is in place, the upward

potential is more appreciated by the investor with the DRRA utility function

than with the CRRA utility.

Furthermore, Figure 1.4 shows that the exponential utility function represents

the least risk averse investor. This preference structure indicates that the investor

is only to a very limited extent prepared to pay a fee to get diversification since the

investor is willing to take on a large exposure to the risk of Index 1. The S-shaped

utility function, on the other hand, represents the most risk averse investor since

this preference structure is extremely sensitive to loss around the initial wealth,

i.e., when W (T ) ≈ W (0). This sensitivity entails that the investor prefers to

hold more than half the initial investment in the bank account to ensure a small

positive return. Moreover, the investor probably includes the structured bond in

the optimal portfolio since the capital guarantee facilitates this sensitivity.

The choice of trading strategy has very little effect on the results which means

that the investor does not gain much from reallocating Index 1 and the bank ac-

count annually−even with no transaction costs. As a consequence the investor

might as well choose the more simple Buy-and-Hold trading strategy. Addition-

ally, we observe minor differences in the optimal portfolio holdings for each option

type. In all cases investors give the structured bond more weight when the prod-

uct includes the Asian option type. This finding has an intuitive explanation

since the Asian option provides more diversification than the vanilla option.

The typical outcome of the alternative example allowing direct investment

in Index 2 instead of the structured bond is as follows: the weight in Index 2

is slightly less than the corresponding weight in the structured bond, and the

optimal portfolio places the remaining funds in the bank account. This finding is

intuitive since the risk averse investor replicates (part of) the capital guarantee

from the structured bond by investing in the bank account while at the same

time getting diversification from investing in Index 2.7

7This alternative example thus considers an investment universe consisting of Index 1, thebank account, and Index 2. A different way to study this point would be to additionally includethe structured bond in this example such that the investor could choose both the structuredbond and Index 2 in the optimal portfolio. In case the investor excludes the structured bond in

1.4 Results 25

1.4.2 Sensitivity Analysis

This section presents sensitivity analysis of the optimal weight in the structured

bond to the risk aversion of the investor, construction cost, dividend on I2, and

correlation between I1 and I2.

Risk Aversion

Figure 1.5 gives the sensitivity analysis of the optimal weight in the structured

bond to the risk aversion of the investor. Each of the four graphs represents a

utility function and indicates the region of risk aversion for which the optimal

portfolio has the largest amount of the structured bond for each trading strategy

and option type. In the top left figure, the CRRA utility function shows that

this region is γ ∈ [2, 4]. In the top right figure, the exponential utility function

illustrates that this region is κ ∈ [1, 3]. In the bottom left figure, the DRRA

utility function indicates that this region is ν ∈ [1, 2]. Finally, in the bottom

right figure, the S-shaped utility function peaks at approximately ζ = 0.5. For

low values of ζ, the investor is close to risk neutral while for slightly higher levels

than ζ = 0.5, the investor becomes very risk averse.8

Common to all utility functions are three observations. First, the case of the

structured bond with the Asian option has a higher weight than the vanilla case.

Second, the choice of the trading strategy does not affect the optimal investment

much. Third, for very low risk aversions, the structured bond does not enter

the optimal portfolio except for the DRRA case where the structured bond has

a low weight at low risk aversions. Furthermore, for higher risk aversions, the

structured bond only gradually gets a smaller weight except for the DRRA case

where the reduction is steep but levels off at β ∈ [4, 5].

Figure 1.6 gives an alternative representation of the sensitivity analysis of

the optimal weight in the structured bond to the risk aversion of the investor.

For each utility function, the graphs consider the Buy-and-Hold trading strategy

and the Asian option type and plot the changes in the optimal portfolio as the

risk aversion rises. The figure thus shows the changes in the optimal holding of

Index 1 and the bank account as well as the structured bond. This figure provides

the optimal portfolio, it is evident that the product is not preferable when the index underlyingthe option of the structured bond is an available investment opportunity.

8Appendix A illustrates the S-shaped utility function for different values of ζ.


0 1 2 3 4

0.0

0.1

0.2

0.3

0.4

Risk aversion - CRRA

Risk Aversion

Opt

. w

eigh

t in

SB

B&H-AsianB&H-VanillaFM-AsianFM-Vanilla

0 1 2 3 4

0.0

0.1

0.2

0.3

0.4

Risk aversion - Exp

Risk aversion

Opt

. w

eigh

t in

SB


0 1 2 3 4

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Risk aversion - DRRA

Risk aversion

Opt

. w

eigh

t in

SB


0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Risk aversion - S-shape

Risk Aversion

Opt

. w

eigh

t in

SB


Figure 1.5: The graphs show the sensitivity analysis of the optimal weight in thestructured bond to the risk aversion of the investor for each utility function, tradingstrategy, and option type.

1.4 Results 27

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

Risk Aversion

Opt

. pf.

Index 1

SB

Bank

B&H-A CRRA

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Risk Aversion

Opt

. pf.

Index 1

SB

Bank

B&H-A Exp

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

Risk Aversion

Opt

. pf.

Index 1

SB

Bank

B&H-A DRRA

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Risk Coefficient

Opt

. pf.

Index 1

SB

Bank

B&H-A S-shape

Figure 1.6: The graphs give an alternative representation of the sensitivity analysis ofthe optimal weight in the structured bond to the risk aversion of the investor. For eachutility function, the graphs consider the Buy-and-Hold trading strategy and the Asianoption type and plot the changes in the optimal portfolio as the risk aversion rises.


an indication of how the optimal portfolio shifts to a less risky position as the

risk aversion rises. The optimal weight in Index 1 diminishes as the risk aversion

increases while the bank account holding rises. The structured bond fills out a

“band” in between the risky and the risk free position which is an intuitive finding

since the product holds both a capital guarantee and an upward potential.

We conclude from the results in Figure 1.5-1.6 that the structured bond is

present in the optimal portfolio for many risk profiles, but generally it has the

highest weight for investors with a medium level of risk aversion.

Construction Cost


bond to the total cost of the structured bond π. Each graph indicates the region of

cost for which the optimal portfolio eliminates the structured bond. In the cases

of the CRRA, DRRA, and S-shaped utility functions, this region is approximately

π ∈ [0.06, 0.07]. In the case of the exponential utility function, with which the

investor is relatively prone to accept risk from Index 1 (as Figure 1.4 indicates),

this region of cost is π ∈ [0.04, 0.05].

Common to all utility functions is the finding that when the construction is

free, i.e., when π = 0, the structured bond has a high weight in the optimal

portfolio due to diversification and its combination of a risky and a risk free

investment. For each utility function, the case of the Asian option results in the

highest weight in the structured bond, regardless of trading strategy. The reason

for this finding is also diversification since the return on the structured bond with

the Asian option is less correlated with the return on Index 1 than the return on

the structured bond with the vanilla option.

Figure 1.8 gives an alternative presentation of the sensitivity analysis of the

optimal weight in the structured bond to the construction cost of the structured

bond π. For each utility function, the graphs consider the Buy-and-Hold trading

strategy and the Asian option type and plot the changes in the optimal portfolio

as the construction cost rises. The figure thus shows the changes in the optimal

holding of Index 1 and the bank account as well as the structured bond. We

observe that as the construction cost rises, the optimal portfolio holds larger

positions in both Index 1 and the bank account for all cases of utility functions

except for the exponential utility where the investor replaces the structured bond

1.4 Results 29

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0

0.1

0.2

0.3

0.4

0.5

Costs - CRRA

T

Opt

. w

eigh

t in

SB


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Costs - Exp

Cost

Opt

. w

eigh

t in

SB


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0

0.1

0.2

0.3

0.4

0.5

Costs - DRRA

Cost

Opt

. w

eigh

t in

SB


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0

00

.05

0.1

00

.15

0.2

00

.25

0.3

00

.35

Costs - S-shape

Cost

Op

t. w

eig

ht

in S

B

B&H-Asian

B&H-Vanilla

FM-Asian

FM-Vanilla

Figure 1.7: The graphs show the sensitivity analysis of the optimal weight in thestructured bond to the construction cost of the structured bond π.


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0

0.2

0.4

0.6

0.8

1.0

Cost

Opt

. pf.

Index 1

SB

Bank

B&H-A CRRA

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0

0.2

0.4

0.6

0.8

1.0

Cost

Opt

. pf.

Index 1

SB

B&H-A Exp

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0

0.2

0.4

0.6

0.8

1.0

Cost

Opt

. pf.

Index 1

SB

Bank

B&H-A DRRA

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.0

0.2

0.4

0.6

0.8

1.0

Cost

Opt

. pf.

Index 1

SB

Bank

B&H-A S-shape

Figure 1.8: The graphs give an alternative representation of the sensitivity analysis ofthe optimal weight in the structured bond to the construction cost of the structuredbond π. For each utility function, the graphs consider the Buy-and-Hold trading strat-egy and the Asian option type and plot the changes in the optimal portfolio as theconstruction cost rises.

1.4 Results 31

only with Index 1. This result is intuitive since the structured bond comprises

elements with risk profiles similar to the profiles of both Index 1 and the bank

account.

In practice the prospectus of the structured bonds typically presents the dis-

closed cost as an annual percentage fee. If there are no hidden cost, i.e., if

πh = 0%, the up-front cost π = πd = 3% in our setting corresponds to the total

annual cost of 0.6% for the time horizon T = 5.9 Jørgensen et al. (2009) present

empirical results on disclosed costs in the Danish market for structured bonds. In

this market, the average annual disclosed cost is approximately 1%, which corre-

sponds to an up-front cost of π = πd = 4.8% in our model if πh = 0%. However,

in case that πd = 4.8% and πh ≥ 0%, Figure 1.7 suggests that the hidden cost

πh ∈ [0.01, 0.02] causes the structured bond to exit the optimal portfolio for the

cases of the CRRA, DRRA, and S-shaped utility functions. For the exponential

utility, the structured bond is already eliminated from the optimal portfolio when

πh = 0.

The results in Figure 1.7-1.8 are sensitive to, for example, the choice of risk

aversion parameters in the utility functions. The specific result that the optimal

portfolio eliminates the structured bond in the particular region of total cost

π ∈ [0.06, 0.07] is therefore illustrative.

Dividends


bond to the dividend yield δ on I2 for the cases of the exponential and the S-

shaped utility functions. This analysis excludes the remaining two cases of utility

functions because they produce identical results. The graphs in Figure 1.9 show

that the weight in the structured bond in the optimal portfolio remains constant

when the dividend yield on I2 rises. The figure also illustrates the changes in the

participation rates related to the two option types in the structured bond. These

participation rates, however, go up as the dividend yield rises. In practice this

effect means that the structured bond appears more appealing to the investor if it

has a dividend-paying asset underlying the option due to the higher participation

rate.

9The up-front disclosed cost πd corresponds to the annual disclused cost z such thatπd = z + z/(1 + r) + z/(1 + r)2 + . . .+ z/(1 + r)T−1.


0.000 0.005 0.010 0.015

-0.4

-0.2

0.0

0.2

0.4

0.6

Dividend - Exp

Dividend

Opt

. wei

ght i

n S

B

B&H-AsianB&H-VanillaFM-AsianFM-VanillaPart.Rate-AsianPart.Rate-Vanilla

0.000 0.005 0.010 0.0150.

00.

51.

0

Dividend - S-shape

Dividend

Opt

. wei

ght i

n S

B

B&H-AsianB&H-VanillaFM-AsianFM-VanillaPart.Rate-AsianPart.Rate-Vanilla

Figure 1.9: The graphs show the sensitivity analysis of the optimal weight in thestructured bond to the dividend yield δ on I2 for the exponential and the S-shapedutility functions. The figure also illustrates the participation rates of the structuredbond for each option type.

Market Correlation


bond to the correlation ρ between Index 1 and Index 2 for the case of the CRRA

and the DRRA utility functions. The top graphs show that the structured bond

has the highest weight in the optimal portfolio when the correlation between

Index 1 and Index 2 is low or negative. The DRRA function is less sensitive to

ρ when ρ < 0.3 than the CRRA function but both functions agree on the region

where the structured bond exits the optimal portfolio. In both cases this region is

approximately ρ ∈ [0.65, 0.85], and if ρ goes above this level, the structured bond

does not provide enough diversification to compensate for its cost π. Furthermore,

for negative correlation coefficients the structured bond appears to have almost

constant, relatively high optimal weights.

From the bottom graphs in Figure 1.10, we observe that when the correlation

is negative, the structured bond gives the portfolio a type of hedge and since it

includes a capital guarantee, the bank account is eliminated. This means that

the optimal level of Index 1 and the structured bond is almost constant although

the correlation decreases.

1.4 Results 33

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8

0.0

0.1

0.2

0.3

0.4

Correlation - CRRA

Correlation

Opt

. w

eigh

t in

SB


-0.4 -0.2 0.0 0.2 0.4 0.6 0.8

0.0

0.1

0.2

0.3

0.4

Correlation - DRRA

Correlation

Opt

. w

eigh

t in

SB


-0.5 0.0 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Rho

Opt

. pf.

Index 1

SB

Bank

B&H-A CRRA

-0.5 0.0 0.5

0.0

0.2

0.4

0.6

0.8

1.0

Rho

Opt

. pf.

B&H-A DRRA

Index 1

SB

B&H-A DRRA

Figure 1.10: The graphs show the sensitivity analysis of the optimal weight in thestructured bond to the correlation ρ between Index 1 and Index 2 for the CRRAand DRRA utility functions. The top graphs show the different cases for the tradingstrategies and option types while the bottom figures concentrate on the Buy-and-Holdtrading strategy and the Asian option. The latter case also shows the optimal weightsin the remaining assets.


1.4.3 Certainty Equivalent, Risk Premium, and Optimal

Expected Utility

After a brief introduction to the certainty equivalent, this section compares the

optimal investments with respect to their certainty equivalents, risk premiums,

and optimal expected utilities.

The certainty equivalent x∗ for a utility function U is the amount of certain

wealth that has a utility level equivalent to the expected utility of the random

wealth x.10 The definition of x∗ is

U(x∗) = E[U(x)]. (1.21)

For a risk averse investor, the certainty equivalent is smaller than the expected

value of the random wealth. This relationship corresponds to the finding that

risk averse investors have concave utility functions. The S-shaped utility function

is not concave, however, and therefore this section only compares the certainty

equivalents of the CRRA, DRRA, and exponential utility functions.

Since the utility functions we consider are invertible, we find the certainty

equivalents x∗ from (1.21) as

x∗ = U−1(E[U(x)]), (1.22)

for each optimal solution (wI , wG)∗.

The left side of Figure 1.11 presents the certainty equivalent returns of the

normalized initial investments (W (0) = 1). The plot gives the cases of each of the

three utility functions combined with trading strategies and option types as well

as it illustrates the alternative case where investment in I2 replaces the structured

bond. The figure shows that the certainty equivalent returns are very similar for

each utility function except for the alternative case. In this alternative case,

investors require a higher certain amount of wealth at maturity to substitute the

random outcome of the investment strategy.

The risk premium Λ that corresponds to the certainty equivalent x∗ is

E[U(x)] = U(x∗) = U(E[x]− Λ).

10See, e.g., Luenberger (1998) for a presentation of the theory on certainty equivalents.

1.4 Results 35

CRRA Exp DRRA

B&H-AsianB&H-VanillaFM-AsianFM-VanillaB&H-No_SB

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Certainty Equivalent (return)

CRRA Exp DRRA

B&H-AsianB&H-VanillaFM-AsianFM-VanillaB&H-No_SB

0.00

0.05

0.10

0.15

Risk premiums

Figure 1.11: The left plot gives the certainty equivalents as returns of the normalizedinitial investments (W (0) = 1) and the right plot presents the risk premiums thatcorrespond to these certainty equivalents.

For a given utility function, the risk premium that corresponds to the certainty

equivalent x∗ is therefore

Λ = E[x]− x∗.

The right side of Figure 1.11 presents the risk premiums that correspond to the

certainty equivalents. The plot shows that the risk premiums are very similar

for each utility function except for the alternative case where investment in I2

replaces the structured bond−as expected from the certainty equivalents.

Next, we consider the optimal utility levels compared to the average of the

combinations of trading strategies and option types for each utility function. The

left side of Figure 1.12 provides the results for the four combinations of trading

strategies and option types. The plot shows that the utility gain which an investor

gets from choosing the best combination is small (≤ 2%). The biggest difference

is in the case of the DRRA function where the investor obtains approximately 2%

more utility than the average when choosing the Buy-and-Hold trading strategy

and the Asian option type. The right side of Figure 1.12 gives the corresponding

result for the case that includes direct investment in I2 instead of the structured

bond. From this figure, it is clear that the investor obtains a higher optimal

utility level with investment directly in I2. The investor can improve the optimal


CRRA Exp DRRA S-shape


-0.0

15-0

.010

-0.0

050.

000

0.00

50.

010

0.01

5

Optimal expected utility deviations from mean

CRRA Exp DRRA S-shape

B&H-AsianB&H-VanillaFM-AsianFM-VanillaB&H-NoSB

-0.0

50.

000.

050.

10

Optimal expected utility deviations from mean

Figure 1.12: The left side of the figure compares the optimal utility levels for eachcombination of trading strategy and option type to the average of these combinationsfor each utility function. The right side of the figure gives the corresponding result forthe case that includes direct investment in I2 instead of the structured bond.

expected utility from the mean of the specific group of utility functions with

between 2% in case of the S-shaped utility function and 10% in case of the

DRRA utility function.

1.5 Conclusion

This paper studies optimal investment in structured bonds in order to suggest

possible explanations for why small retail investors hold structured bonds. The

investment universe consists of a structured bond, a stock index, and a bank

account, and as a benchmark we assume that the small retail investor cannot

trade the index underlying the option of the structured bond. We find the opti-

mal investment using expected utility maximization and consider different utility

functions and trading strategies.

A numerical procedure solves the problem, and the benchmark result shows

that the model specification chosen affects the optimal portfolio weights sig-

nificantly. A sensitivity analysis of the optimal weight in the structured bond

demonstrates that investors with a medium level of risk aversion give the struc-

tured bond the highest portfolio weight. The optimal holding of the structured

1.5 Conclusion 37

bond is very cost sensitive but provides diversification that may compensate for

this cost. The correlation in the market affects the optimal investment accord-

ingly. We also show that the investor could obtain a higher expected utility with

direct investment in the index underlying the option instead of investment in the

structured bond.

From these findings, we conclude that investors should mainly include struc-

tured bonds in their investment portfolio if they cannot directly access the index

underlying the option of the structured bond. Furthermore, in that case the in-

vestor should only invest in the structured bond if the product provides sufficient

diversification to compensate for its cost. Since the structured bond is very cost

sensitive, we suggest that investors are wary of potential hidden costs. Finally,

we encourage investors to consider whether they are able to diversify their in-

vestment portfolio in different ways than by holding structured products with

construction costs.

Acknowledgements

We would like to thank the participants at the following conferences and work-

shops where the paper has been presented: the Danish Doctoral School of Finance

Annual PhD Workshop 2008 in Vejle, Denmark; the Nordic Finance Network An-

nual PhD Workshop 2008 in Bergen, Norway; the 12th Congress on Insurance:

Mathematics and Economics 2008 in Dalian, China; the Quantitative Methods

in Finance Conference 2008 in Sydney, Australia; and the Midwest Finance As-

sociation Annual Meeting 2009 in Chicago, USA. Furthermore, we are thankful

to the committee members of Pernilles Thesis Proposal, Claus Munk and Jens

Lund, for valuable comments.

Bibliography

Bernard, C., Boyle, P. P., 2008. Locally-Capped Investment Products and the

Retail Investor. Conference Proceedings, European Financial Management As-

sociation (EFMA), June 2009.

Bernard, C., Ghossoub, M., 2010. Static Portfolio Choice Under Cumulative

Prospect Theory. Mathematics and Financial Economics 2 (4).

Black, F., Scholes, M., 1973. The Pricing of Options and Corporate Liabilities.

Journal of Political Economy 81 (3), 637–654.

Carlin, B. I., 2006. Strategic Price Complexity in Retail Financial Markets. SSRN

eLibrary.

Doebeli, B., Vanini, P., 2010. Stated and Revealed Investment Decisions Concern-

ing Structured Products. Journal of Banking and Finance 34 (6), 1400–1411.

Doskeland, T., Nordahl, H., 2008. Optimal Pension Insurance Design. Journal of

Banking and Finance 32 (3), 382–392.

Fischer, R., 2007. Do Investors in Structured Products Act Rationally? SSRN

eLibrary.

Friend, I., Blume, M. E., 1975. The Demand for Risky Assets. American Economic

Review 65 (5), 900–922.

Henderson, B. J., Pearson, N. D., 2009. The Dark Side of Financial Innovation.

SSRN eLibrary.

Hens, T., Rieger, M. O., 2009. The Dark Side of the Moon: Structured Products

from the Customer’s Perspective. SSRN eLibrary.

40 BIBLIOGRAPHY

Jørgensen, P. L., Nørholm, H., Skovmand, D., 2010. The Hidden Costs of Struc-

tured Products for Retail Investors. Unpublished working paper.

Jørgensen, P. L., Nørholm, H., Skovmand, D., Rasmussen, A.-S. R., 2009. Struc-

tured Bonds for Private Investors. A Market Overview. (In Danish). Finans

Invest 4, 5–12.

Luenberger, D. G., 1998. Investment Science. Oxford University Press, New York.

Munk, C., 2007. Financial Asset Pricing Theory. Lecture notes/Forthcoming

book, Oxford University Press 2010-2011.

Nelder, J. A., Mead, R., 1965. A Simplex Method for Function Minimization.

Computer Journal 7, 308–313.

Ogaki, M., Zhang, Q., 2001. Decreasing Relative Risk Aversion and Tests of Risk

Sharing. Econometrica 69 (2), 515–526.

Pindyck, R. S., 1988. Risk Aversion and the Determinants of Stock Market Be-

havior. Review of Economic Studies 70 (2), 183–190.

Protter, P., 1990. Stochastic Integration and Differential Equations. Springer-

Verlag, Heidelberg.

Turnbull, S. M., Wakeman, L. M., 1991. A Quick Algorithm for Pricing European

Average Options. The Journal of Financial and Quantitative Analysis 26 (3),

377–389.

Tversky, A., Kahneman, D., 1992. Advances in Prospect Theory: Cumulative

Representation of Uncertainty. Journal of Risk and Uncertainty 5, 297–323.

Appendix

A. S-shaped Utility

0.5 1.0 1.5 2.0

-0.5

0.0

0.5

1.0

S-shape, 0.05

x

Util

ity

0.5 1.0 1.5 2.0

-0.5

0.0

0.5

1.0

S-shape, 0.25

x

Util

ity

0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

1.5

S-shape, 0.45

x

Util

ity

0.5 1.0 1.5 2.0

-2-1

01

23

S-shape, 0.65

x

Util

ity

0.5 1.0 1.5 2.0

-6-4

-20

24

6

S-shape, 0.85

x

Util

ity

0.5 1.0 1.5 2.0

-20

-10

010

20

S-shape, 0.95

x

Util

ity

Figure 1.13: The figure shows the S-shaped utility function for the parameter valuesζ ∈ 0.05, 0.25, 0.45, 0.65, 0.85, 0.95. The graphs show that investors with low ζ pa-rameters are close to risk neutral while investors with high ζ’s are extremely sensitiveto loss around the initial investment, i.e., when W (T ) = W (0).

Chapter 2

Optimal Responsible

Investment

44 Optimal Responsible Investment

45

Optimal Responsible Investment

Pernille Jessen




Aarhus University

Abstract

The paper studies responsible investment portfolio allocation. The model

defines an investor specific measure of portfolio responsibility and incor-

porates this measure into two different investment approaches from con-

ventional investment theory. In the first approach, investor utility theory

describes preferences for portfolio responsibility. The utility setup is in-

tuitive, but an implementation requires information on investor tradeoffs

between portfolio risk, expected return, and responsibility. In the second

approach, mean variance analysis captures portfolio responsibility with an

additional restriction to the investment problem. This approach provides

an analytical solution to the optimal responsible investment problem and

offers sensitivity analysis regarding the required portfolio responsibility.

An example of index investment illustrates the results.

Key words: Responsible investment, Portfolio choice, Mean variance anal-

ysis, Expected utility maximization

JEL Classification: G11


2.1 Introduction

Responsible investment comprises investment analysis and decision-making pro-

cesses that incorporate environmental, social, and corporate governance (ESG)

criteria. This definition is the first principle in the UN Principles for Responsible

Investment (PRI), a set of voluntary guidelines regarding responsible investment

for institutional investors.1 Responsible investment includes a range of categories

such as socially responsible, sustainable, ethical, and green investment. The

vague definition of responsible investment makes this form of investment am-

biguous and difficult to manage from a quantitative point of view. Nevertheless,

today investment analysis subject to at least some ESG criteria is important in

the financial market.

The size of the market demonstrates the popularity of responsible investment.

According to Eurosif (2008), the global market for responsible investment is ap-

proximately EUR 5 trillion. The European market alone recorded an increase

from EUR 1,033 trillion in 2005 to EUR 2,665 trillion in 2007, i.e., the market

more than doubled during this period. The study by the Social Investment Forum

(2007) estimates that in the US responsible investment is almost 10% of the total

investment market. Furthermore, with respect to investment conventions, the

PRI (2009) reports that the principles have 538 signatories with USD 18 trillion

assets under management worldwide. The number of signatories still increases

notably, and in September 2010 it is up 50% to 808.2

This paper studies responsible investment with respect to portfolio allocation.

Since responsible investment has a vague definition, how do the market partici-

pants and PRI signatories incorporate ESG criteria in the investment decision?

Literature on responsible investment allocation is scarce, and therefore this paper

has the goal to identify the responsible investor’s optimal portfolio choice.

The model defines an investor specific measure of portfolio performance of

the underlying assets in terms of ESG criteria. The paper refers to this measure

as portfolio responsibility, and it has three inputs. The first input is ESG rating

data on the performance of the companies in the investment universe in terms of

ESG criteria. The second input is the individual investor’s priorities regarding

the different ESG criteria. These priorities specify which criteria the investment

1www.unpri.org/principles/2www.unpri.org/signatories/

2.1 Introduction 47

procedure includes as well as the relative importance of the criteria. The final

input is the portfolio weights, the endogenous model argument which scales the

portfolio responsibility appropriately.

Within conventional investment theory, this study suggests that two com-

mon investment approaches can encompass responsible investment allocation.

The first approach uses investor utility theory and proposes a utility function

that considers both portfolio responsibility and the financial properties of the

investment. If such utility function exists, it is straightforward to maximize the

expected utility at maturity to find the optimal responsible investment. The

second approach considers mean-variance analysis and adds to the original setup

a restriction on portfolio responsibility. This approach results in an analytical

solution to the optimal responsible investment problem, and it also leads to sen-

sitivity analysis of the optimal portfolio to the required portfolio responsibility.

A numerical example illustrates the second approach and shows how optimal in-

vestment in a sustainability index varies according to investor priorities and asset

ratings.

Only few academic studies consider investment decisions that incorporate

types of non-financial criteria. Beltratti (2005) investigates the utility costs to the

investing individual in an equilibrium model when the individual discriminates

against unethical companies. The analysis measures the effect on the market

value of the unethical company and finds that responsible investment has a cost

which depends on market circumstances. The model applies a utility function

that recognizes only financial performance, and this preference structure disre-

gards the intention behind the discrimination such that the investing individual

acts irrationally.

Wisebrod (2007) proposes a portfolio selection model that only considers ESG

criteria in the investment decision if such considerations do not affect the risk-

return profile. The model chooses the so-called socially dominant portfolio from

a subset of equally preferred optimal portfolios. Theoretically, this method is

not attractive since in a complete financial market, the optimal investment prob-

lem has only one solution. Finally, Hallerbach et al. (2004) suggest an interactive

programming method to allocate, evaluate, and re-allocate retail investment port-

folios cooperatively between the investment advisor and investor. Their approach

seems useful in practice but involves substantial communication, and the method


does not seem applicable for institutional investors.

This paper contributes to the literature on responsible investment allocation

because the investment approaches considered rest on conventional investment

theory and incorporate portfolio responsibility into these. The paper takes a nu-

anced view on responsible investment since it includes investor priorities regard-

ing ESG criteria, and it allows investors to weigh portfolio responsibility as they

wish. The study thereby suggests new quantitative methods for implementing

responsible investment strategies. The paper also contributes to the literature

on conventional portfolio allocation theory because it includes the view of the

investor on non-financial ESG criteria in the investment decision.

Most literature on responsible investment concerns the performance of re-

sponsible investment compared to conventional investment. Renneboog et al.

(2008) provide an overview of the empirical literature on responsible investment

performance, and they conclude that responsible mutual funds appear to perform

similarly to their conventional peers. The report by the UNEP Finance Initiative

(2007) compares a number of academic and broker studies on ESG factors in

investment analysis and also finds little effect of ESG criteria on financial perfor-

mance. On the contrary, mean variance analysis states that imposing constraints

to the investment universe limits portfolio diversification and obstructs risk shar-

ing. This line of theory suggests that responsible investment lowers the expected

portfolio return, or alternatively, it magnifies portfolio risk. Dam (2008), Heinkel

et al. (2001), and Galema et al. (2008) apply different supply-demand arguments

to try to explain the empirical findings mentioned above. Galema et al. (2008)

state that when some investors discriminate against irresponsible companies, it

leads to overpricing stocks issued by responsible companies. Such over-valuation

implies lower stock returns and lower book-to-market values. As a result, they

argue, the market sees no positive nor negative effects of responsible investment

on performance. However, the question of performance of responsible investment

is still widely discussed and no consensus seems to have emerged.

Many institutional investors face a legal requirement stating that they only

meet their fiduciary duty if they do not pursue investment objectives which may

lessen financial return.3 Considering the argument from mean variance analysis,

institutional investors thereby risk being accused of neglecting their fiduciary

3See, for example, the Danish Financial Business Act no. 158 at www.finanstilsynet.dk/en.

2.2 Portfolio Responsibility 49

duty if they include ESG criteria in their investment decision. However, if an

institution can sufficiently argue that managing particular ESG issues is, for

example, part of their financial risk management, then the institution meets the

fiduciary duty even though it may have a cost.

This observation supports the incentive of this paper: in order for institutional

investors to meet their fiduciary duty, they may need to capture specific ESG

criteria in the responsible investment decision and incorporate them only in the

magnitude that the strategy of the institution specifies. The set of ESG criteria

must have an economical explanation, for example, some criteria may either serve

as risk management, support public and investor relations, or anticipate future

governmental regulation.

The paper is organized as follows. Section 2 defines ESG ratings, investor

priorities, and portfolio responsibility. Section 3 introduces investor utility theory

subject to portfolio responsibility. Section 4 applies mean-variance analysis and

introduces a new constraint regarding portfolio responsibility. Section 5 presents

a numerical example of investment in a sustainability index, and the last section

concludes.

2.2 Portfolio Responsibility

The model for responsible investment requires an investor specific measure of

performance of the underlying assets in terms of ESG criteria. This section

introduces the measure denoted portfolio responsibility, defines its inputs, and

makes an appropriate assumption that allows these inputs to be combined into a

portfolio measure.

2.2.1 ESG Ratings

A number of investment research companies specialize in ESG analysis and of-

fer ESG ratings of the publicly traded companies worldwide. The ratings give

characteristics of the individual assets in terms of a number of ESG criteria.

The following defines ESG ratings as they appear in this paper and gives an

introduction to the spectrum of ESG criteria that the ratings cover.

The model considers an investment universe of N assets and an ESG rating

procedure that evaluates K ESG criteria for each asset. The ESG rating is thus


a matrix K ∈ M(N,K), and the model assumes that the elements kij lie in the

interval [−1, 1] for all i ∈ 1, .., N and j ∈ 1, .., K,

K =

k11 k12 . . . k1K

k21 k22 . . . k2K

......

. . ....

kN1 kN2 . . . kNK

. (2.1)

The entry kij describes how well asset i performs with respect to ESG criterion

j. The following explains the range of the ratings, [−1, 1].4

• If kij = −1, asset i directly or indirectly covers activities that have very

negative effects on ESG criterion j. For an environmental criterion, it

includes activities that cause hazardous waste, substantial emissions, or

ozone depleting chemicals.

• If kij = 0, asset i has no positive nor negative effect on ESG criterion j.

In case criterion j is not relevant with respect to asset i, this rating also

applies.

• If kij = 1, asset i directly or indirectly covers activities that have a very

positive effect on ESG criterion j. For a social criterion, it includes support

of education, charitable giving, or employment of disabled.

The K rating criteria can basically consist of any set of standards that an

ESG research company specifies. The following gives two examples of company

specific approaches to ESG ratings. First, KLD Research5 includes in its ESG

ratings a number of themes within each of the categories: environmental, social,

and governmental aspects. Each theme has a subdivision into a number of issues

of strengths and concerns. In their environmental rating, for example, there is a

theme on climate change that considers producing clean energy as a strength and

aggravating climate change as a concern. Second, Jantzi-Sustainalytics6 considers

more than 100 ESG indicators grouped into the following areas: community

4Different ESG research companies use different ranges of ratings. This model considers therange [−1, 1] as a type of normalized rating range since most rating systems can be convertedinto this form.

5www.kld.com6www.sustainalytics.com.

2.2 Portfolio Responsibility 51

and society, customers, corporate governance, employees, environment, human

rights, and controversial business activities. Each of the indicators cover a number

of micro-level factors, and consequently, the total Jantzi rating matrix is very

comprehensive.

2.2.2 Investor Priorities

The model considers investors who include an individual measure of portfolio

responsibility in their portfolio choice. Each investor individually defines what

a responsible investment covers, and later on in the investment model, each in-

vestor determines how much weight portfolio responsibility gets in the investment

decision.

A vector ψ describes the investor specific attitude toward ESG criteria as

relative ESG priorities. Investor priorities regarding K ESG criteria is the K-

dimensional vector,

ψ =

ψ1

ψ2

...

ψK

, (2.2)

where ψj ∈ [0, 1] for all j ∈ 1, .., K. The entry ψj indicates how an investor

relates to criterion j. The model assumes that the K ESG criteria in ψ correspond

to the K rating criteria in the previous section.

The priorities has the range [0, 1] where zero indicates that the particular

ESG issue is unimportant to the investor and one indicates that the issue is

highly important. Consider, for instance, an investor who emphasizes the social

rating criterion Human Rights (HR) as well as the environmental rating crite-

rion Climate Change (CC). Except from these criteria, the investor disregards

ESG issues in the investment decision. The investor priority vector can then for


example read

ψ∗ =

...

ψHR

ψCC...

=

0

0.40

0.85

0

, (2.3)

where 0 represents appropriate vectors of zeros. The priority vector ψ∗ states

that the concern for climate change is more important to the investor than human

rights and that the remaining (K − 2) ESG criteria are irrelevant.

This paper denotes an investor who wishes to disregard ESG criteria in the

investment decision a neutral investor. The neutral investor has the priorities

ψ = [0 0 . . . 0]> = 0, and this case corresponds to a traditional portfolio alloca-

tion problem. As a result, the model in this paper does not require that investors

consider ESG criteria in their investment decision, but it allows them to do so.

2.2.3 Portfolio Responsibility

Next, the model defines how the ESG ratings and the investor priorities summa-

rize to a measure of portfolio responsibility. An assumption enables the definition.

Assumption 1 Investor priorities ψ regarding ESG criteria in the investment

decision are additive across the K rating criteria.

Assumption 1 is equivalent to the model property that the dot-products of the

individual rows of K and the vector ψ are meaningful. The definition below

clarifies why the assumption is necessary, but first the model needs notation for

the portfolio weights.

A vector w defines an investment portfolio, and the entries wj for j ∈ 1, .., Nin w represent relative asset weights for the N assets,

w =

w1

w2

...

wN

,N∑j=1

wj = 1. (2.4)

2.3 Investor Utility Theory 53

Definition 2 (Portfolio Responsibility) Let Assumption 1 hold. For the ESG

rating K, investor priorities ψ, and investment portfolio w, the cumulative level

of portfolio responsibility is

s = (Kψ)>w. (2.5)

The N-dimensional vector Kψ gives the investor specific ESG ratings of the N

assets.

The range of s is [−1, 1] when the model does not allow short-sales, i.e., when

wj ∈ [0, 1] for all j ∈ 1, .., N. When the model allows short-sales, i.e., when

wj ∈ R for all j ∈ 1, .., N, the range of s is R.

Definition 2 implies that any portfolio w has an explicit amount of portfolio

responsibility, and in principle portfolio allocation could occur on the basis of

maximizing the portfolio responsibility. However, investors consider portfolio

risk and expected return in addition to portfolio responsibility, and therefore the

forthcoming sections integrate s in the investment decision.

2.3 Investor Utility Theory

This section considers the first approach of this paper to the optimal responsible

portfolio choice. The analysis applies von Neumann-Morgenstern investor utility

theory to incorporate investor priorities regarding responsibility in the investment

decision. After summarizing the general utility framework, the section defines a

responsible investor in terms of utility theory and gives examples of utility func-

tions that illustrate the setup. A brief discussion states advantages, challenges,

and the applicability of the procedure.

2.3.1 Utility Function

Investment takes place in a one-period setting where at time t = 0 the portfo-

lio allocation occurs and at time t = T the investor evaluates the investment

decision. An investor utility function of financial return x is well-behaved; it is

usually monotonic, quasi-concave, continuous, and globally non-satiated. These

properties mean that the investor always prefers more wealth to less but has


a diminishing marginal utility the higher the return. Moreover, the investor is

risk-averse.

A Neumann-Morgenstern investor utility function has the general form

u : M → R, (2.6)

where the real-valued function u has a domain that consists of all possible out-

comes M of the investment. A maximization of the expected value of u over

the possible portfolio weights then gives a method to evaluate an investment

allocation.

This section considers the special case of (2.6) where the domain of u contains

both any investment return x of the investment at maturity and any portfolio

responsibility s. The investor utility function then maps u : (x, s) 7→ R. When

the model allows short sales, the domain of u is M = R × R, and when the

model does not allow short sales, the domain of u is M = [−1,∞) × [−1, 1].

Subsequently, this section focuses on the case of no short sales.

This utility theory entails a definition of a responsible investor as follows.

Definition 3 (Responsible Investor) An investor is a responsible investor if

the following relationship holds. For an increasing utility function u and for any

x ∈ [−1,∞):

∀ s1, s2 ∈ [−1, 1]∣∣ s1 ≥ s2 : u(x, s1) ≥ u(x, s2). (2.7)

A (strictly) higher portfolio responsibility s1 gives a (strictly) higher investor util-

ity. It means that

u(x, s1) = u(x, s2) + ∆u, (2.8)

where ∆u > 0 when s1 > s2 or alternatively that

u(x, s1) = u(x+ ∆r, s2). (2.9)

The term ∆u denotes a utility benefit, and ∆r denotes the equivalent financial

return premium.

The utility benefit ∆u measures the additional utility that a specific investor


gains from holding a responsible investment. This benefit is only informative in

relation to the particular utility function considered. The financial return pre-

mium ∆r is the amount of return that an investor is willing to give up in order to

be a responsible investor with portfolio responsibility s2 instead of s1. Alterna-

tively, ∆r is the compensation that the responsible investor requires to hold a less

responsible investment. The latter interpretation inspires the term premium. As

opposed to ∆u, ∆r gives a meaningful quantification of the preference for respon-

sibility. Both the utility benefit and return premium are, however, functions of

(x, s1, s2), and this dependency makes it difficult to make any general statements

on the sizes of ∆u and ∆r as well as to find the sensitivity of these variables to

portfolio responsibility.

Definition 3 allows the investor to incorporate non-financial factors into the

investment decision, but it requires information on the form of u. If a model

gives the form of u, it is simple to use a numerical optimization program to

find the optimal investment by maximizing the expected utility at maturity of

the investment. If a model does not give the form of u, on the other hand, the

problem calls for an empirical estimation of u on the basis of utility data which

may be difficult to acquire or non-existing. Since this paper will not empirically

motivate a responsible investor utility function, the following exemplifies three

theoretical functional forms.

2.3.2 Examples

A simple, general composition of u is

u(x, s) = f1(u1(x)) + f2(u2(s)), (2.10)

where u1 and u2 fulfill the usual conditions for utility functions. For the case

without short sales, the function u1 : [−1,∞) → R describes utility from the fi-

nancial outcome of the investment, and u2 : [−1, 1]→ R describes utility from the

portfolio responsibility. An intuitive example of (2.10) is the affine combination

of u1 and u2. For α ∈ [0, 1] it is

u(x, s) = (1− α)u1(x) + αu2(s), (2.11)


Return, x

-0.50.00.5

1.01.5

Pf. resp., s

-1.0-0.50.0

0.51.0

u(x,s)

-1.0

-0.5

0.0

0.5

Return, x

-0.50.00.5

1.01.5

Pf. resp., s

-1.0-0.50.0

0.51.0

u(x,s)

-0.5

0.0

0.5

Figure 2.1: The figure shows the case where u1 and u2 are simple, logarithmic utilityfunctions. The left function weighs financial return and portfolio responsibility equally,i.e., α = 0.5. The right function gives more weight to financial return and appliesα = 0.2.

where the parameter α describes the weight that an investor gives utility from

portfolio responsibility relative to utility from the financial outcome. The affine

combination allows the investor to determine the utility from holding a respon-

sible investment u2 first and then determine how much weight u2 should have

in the total utility. The intuition for this procedure is that an investor might

have idealistic ideas about responsible investment a priory, but when the actual

investment decision occurs, the investor may wish to put less weight on portfolio

responsibility.

Example 1: In the first case, both u1 and u2 are simple, logarithmic utility

functions of x and s respectively. Both functions adjust the domain with one

unit because the logarithmic function only exists for input greater than zero.

The utility function is

u(x, s) = (1− α) ln(x+ 1) + α ln(s+ 1). (2.12)

Figure 2.1 illustrates the function for α ∈ 0.2, 0.5. The left figure is the case

where financial return and portfolio responsibility have equal weights, i.e., where

α = 0.5. The right figure is the−perhaps more realistic−function where portfolio

responsibility has the weight α = 0.2.

Example 2: In the second case, u1 is an exponential utility function, and u2 is a

proportional function of s. For the risk aversion parameter γ and proportionality


Return, x0.0

0.51.0

1.5

Pf. resp., s

-0.50.0

0.51.0

u(x,s)

-0.5

0.0

0.5

Return, x0.0

0.51.0

1.5

Pf. resp., s

-0.50.0

0.51.0

u(x,s)

-1.0

-0.5

0.0

0.5

1.0

Figure 2.2: The left figure shows the case where u1 is an exponential utility functionwith risk aversion parameter γ = 2.5, and u2 is a simple proportional function of s.The right figure shows the case where u1 is once again an exponential utility function,but now u2 has an s-shape. The parameters are λ = 0.5 and γ = 1.5. Both figureshave the affine portfolio responsibility weight α = 0.2.

factor one, it is

u(x, s) = (1− α)(1− e−γx) + αs. (2.13)

The left side of Figure 2.2 shows the function where γ = 2.5 and α = 0.2. The

investor has a constant marginal utility gain from portfolio responsibility, and

it means that the investor appreciates additional portfolio responsibility equally

independent of where on the spectrum of s the current investment is.

Example 3: In the final case, u1 is once again an exponential utility function,

but now u2 has an s-shape. Tversky and Kahneman (1992) introduce the s-

shaped utility function (of financial wealth) and argue that an investor is more

sensitive to return around a certain point of reference. This point of reference is

typically the initial investment. In a similar way, the function u2 in this example

has the point of reference s = 0, and u2 has an s-shape around this point. The

form of u2 is

u2(s) =

s1−λ/(1− λ) for s > 0

0 for s = 0

−(−s)1−λ/(1− λ) for s < 0,

λ 6= 1, (2.14)

and u1 remains as (2.13) specifies. The right graph in Figure 2.2 illustrates this


example for λ = 0.5 and γ = 1.5. The graph shows that there is a shift in the

level of utility from portfolio responsibility around the point of reference, s = 0.

The intuition behind this example is that some investors may wish to define

themselves as responsible investors, and therefore it is important to them to hold

a portfolio with s > 0. Nevertheless, it is perhaps less important exactly how

big s is. Moreover, it might be disadvantageously for the investor to hold an

investment with a low portfolio responsibility, s < 0, but if an investor does so

anyway, it is perhaps less important exactly how small s is.

Discussion

The investor utility approach has the strength that it is intuitive−as argued below

using investor incentives−but it also has the main challenge that the user of the

model has to specify u. Due to lack of data, the user cannot estimate u and must

assume the form of u. As a result the responsible investor utility theory is not

easy to apply in retail investment, but some institutional investors might be able

to define their own u and then use the method.

The incentives to hold responsible investments differ across market partici-

pants. Institutional investors typically engage in responsible investment for the

purpose of long-term risk management related to, e.g., environmental risk fac-

tors, reputational risk, or risk related to future governmental regulation. Retail

investors, on the other hand, primarily hold responsible investments to align their

personal values with investment activities. Some researchers indicate that retail

investors obtain a type of non-financial return from responsible investment, see for

example Bollen (2007), Benson and Humphrey (2008), Renneboog et al. (2008),

or Nilsson (2008). In a similar manner, institutional investors may also get some-

thing more than financial return from responsible investment, for example, in

terms of risk management.

This first approach of the paper to optimal responsible investment introduces

this non-financial return into investor utility theory. In this setting, responsible

investors get a utility benefit ∆u > 0 from holding a responsible investment, and

a corresponding financial return premium ∆r > 0 exists, which alternatively can

compensate for a less responsible investment. This approach thus exemplifies how

responsible investors (by definition) have additional objectives than only financial

return.

2.4 Mean Variance Analysis 59

2.4 Mean Variance Analysis

This section considers the second approach of this paper to the optimal respon-

sible portfolio choice. It extends traditional mean variance analysis with an ad-

ditional constraint that captures portfolio responsibility. After the model setup,

this section solves the responsible investment problem analytically and finds an

expression for the sensitivity of the optimal portfolio to the required portfolio re-

sponsibility. Finally, a brief discussion reflects on the applicability of the model.

2.4.1 Setup

The model considers an investment universe that consists of N risky assets. The

risky assets have an N -dimensional rate of return vector X over the investment

period [0, T ]. The vector X has the first and second moment

E[X] = µ and Var(X) = Σ, (2.15)

where Σ is the variance-covariance matrix, and µ is the expected rate of return

vector of the assets. The model assumes that Σ is positive definite and that the

entries of µ are not all equal.

Investing W0 at time t = 0 in a portfolio with the relative portfolio weights

w yields the payment WT at maturity. This payment has the first and second

moment

E[WT ] = W0(1 + w>µ) and Var(WT ) = W 20 w>Σw. (2.16)

Mean variance analysis typically comprises the problem of minimizing the vari-

ance of the payment WT at maturity subject to a given level of required expected

return µ. Markowitz (1952) introduces this problem, and it is

minw

1

2w>Σw s.t.

w>µ = µ

w>1 = 1. (2.17)

This paper refers to the solution of the problem in (2.17) as neutral investment

since it corresponds to the portfolio allocation of the neutral investor, ψ = 0.

The first constraint in (2.17) specifies the required level of expected return while


the second constraint makes sure that the portfolio weights are relative portfolio

weights. The scalar 12

is only technical and does not affect the minimization

result.

2.4.2 Responsible Investment

In order to incorporate portfolio responsibility in this setting, the model imposes

another constraint. The problem that describes the optimal responsible portfolio

choice is

minw

1

2w>Σw s.t.

w>µ = µ

w>Kψ = s

w>1 = 1

. (2.18)

This problem is similar to the one in (2.17), but the second restriction in (2.18)

now specifies the required level of portfolio responsibility s.

A Lagrangian approach solves the optimal responsible investment problem as

follows. The Lagrange-function with multipliers (λ1, λ2, λ3) is

L(w, λ1, λ2, λ3) =1

2w>Σw− λ1(w>µ− µ)

−λ2(w>Kψ − s)− λ3(w>1− 1). (2.19)

It has the first order conditions

∂L∂w

= Σw− λ1µ− λ2Kψ − λ31 = 0 (2.20)

w>µ− µ = 0 (2.21)

w>Kψ − s = 0 (2.22)

w>1− 1 = 0. (2.23)

Since Σ is invertible, (2.20) can be written as

w = Σ−1[µ Kψ 1]

λ1

λ2

λ3

. (2.24)

2.4 Mean Variance Analysis 61

The remaining first order conditions (2.21)-(2.23) are

[µ Kψ 1]>w =

µ

s

1

, (2.25)

and it means that when both sides of (2.24) are multiplied by [µ Kψ 1]>, it gives µ

s

1

= [µ Kψ 1]>w

= [µ Kψ 1]>Σ−1[µ Kψ 1]

λ1

λ2

λ3

. (2.26)

Define A = [µ Kψ 1]>Σ−1[µ Kψ 1]. If A is invertible, then (2.26) yields λ1

λ2

λ3

= A−1

µ

s

1

. (2.27)

From (2.24) and (2.27) the optimal responsible portfolio w∗ is then

w∗ = Σ−1[µ Kψ 1]A−1

µ

s

1

, (2.28)

and the variance of the optimal responsible portfolio is

(σ∗)2 = w∗>Σw∗

= [µ s 1]A−1

µ

s

1

. (2.29)


2.4.3 Sensitivity to Responsibility

The optimal portfolio depends on the required portfolio responsibility s. All else

being equal, it can be written as

w∗(s) = Σ−1[µ Kψ 1]A−1

µ

s

1

≡ B

µ

s

1

, (2.30)

where B is an (N, 3)-matrix of parameters. Each optimal asset weight w∗i for

i ∈ 1, .., N in the vector w∗ depends on (s, µ) since

w∗i = b1iµ+ b2is+ b3i, (2.31)

where b.. denotes entries in the matrix B. The weights are linear functions of

the required responsibility level s, and the sensitivity of asset weight w∗i to s is

therefore

∂w∗i∂s

= b2i. (2.32)

The optimal portfolio risk (σ∗) that results from solving the optimal respon-

sible investment also depends on the required portfolio responsibility. All else

being equal, it is

(σ∗)2(s) = [µ s 1]A−1

µ

s

1

= (µa11 + sa21 + a31)µ

+(µa12 + sa22 + a32)s

+(µa13 + sa23 + a33)1, (2.33)

where a.. denotes entries in the matrix A−1. The sensitivity of the optimal vari-

2.5 Example:Investment in a Sustainability Index 63

ance to the required portfolio responsibility is therefore

∂(σ∗)2

∂s= 2a22s+ µ(a21 + a12) + a32 + a23. (2.34)

The sensitivity of the variance is thus a linear function of s.

Discussion

The key advantages of mean variance approach are that the portfolio selection has

a simple, analytical solution and that the model can−in principle−be applied by

all types of investors. Furthermore, the requirement of a certain level of portfolio

responsibility is objective and intuitive.

A limitation of the approach is that institutional investors may have to apply

negative screening procedures to the investment universe in addition to the opti-

mal responsible investment model. If a very irresponsible company, for example,

has very advantageous financial properties, the investment model may include the

company in the portfolio in spite of the negative effect on portfolio responsibility.

This event depends on the investor priorities as well as the required portfolio re-

sponsibility. If an investor puts great emphasis on portfolio responsibility, i.e., has

high∑K

i=1 ψ and s, then including the irresponsible company is very unlikely. An

additional limitation, which is relevant to responsible investment companies, is

that the model cannot capture the effects of active ownership or other anticipated

changes in ESG ratings.

2.5 Example:

Investment in a Sustainability Index

This section illustrates the extended mean variance theory in Section 2.4 with a

numerical example. After the introduction of the investment opportunities, the

example shows how the optimal responsible investment differs according to the

portfolio responsibility factors.


2.5.1 DJSI World

The Dow Jones Sustainability World Index (DJSI) tracks the performance of

the 10% best performing companies in the Dow Jones Global Total Stock Market

Index in terms of corporate sustainability. The guide DJSI (2009) gives the details

on the index and explains the best-in-class method which the index applies in

the asset selection process.

This example considers an investment universe of N = 4 assets; it is the Dow

Jones Industrial Average (DJ), the Nasdaq Composite (NQ), the S&P500 Index

(SP), and the Dow Jones Sustainability World Index (DJSI). Data on DJSI is

available from September 1999 onwards from the homepage of the Dow Jones

Sustainability Indexes,7 and the remaining market data is available from Yahoo

Finance.8

The example considers monthly data from September 1999 to June 2010. The

average monthly log-return vector µ over the period for the four indexes is

µ =

µDJ

µNQ

µSP

µDJSI

=

−16.88 bps

−20.94 bps

−17.43 bps

−17.05 bps

. (2.35)

Because µ is negative, however, the example considers µ inappropriate as a fore-

cast for the future index values. The example therefore assumes the expected,

monthly log-return

µ =

µDJ

µNQ

µSP

µDJSI

=

120 bps

140 bps

115 bps

100 bps

. (2.36)

Concerning the correlation structure, on the other hand, the example applies

the historical variance-covariance matrix Σ of the monthly log-return during the

7www.sustainability-index.com8www.finance.yahoo.com


0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.00

80.

010

0.01

20.

014

0.01

6

Std. dev.

Req

uire

d re

turn

Short salesNo short sales

DJNQSPDJSI

Figure 2.3: The figure shows the efficient frontier for neutral investment with andwithout short sales, and it also plots the individual indexes. An investor who investsonly in the DJSI faces either a lower expected return or a higher portfolio risk comparedto portfolios on either of the efficient frontiers.

period. The variance-covariance matrix is

Σ =

0.00227 0.00256 0.00205 0.00055

0.00256 0.00649 0.00314 0.00071

0.00205 0.00314 0.00225 0.00066

0.00055 0.00071 0.00066 0.00283

. (2.37)

According to mean variance analysis, responsible investment limits investment

opportunities and therefore has a financial cost. Figure 2.3 gives the efficient

portfolio frontier for N = 4 for the neutral investment associated with the original

problem in (2.17) and indicates the individual assets. The figure shows that a

responsible investor who wishes to invest only in the DJSI accepts either a lower

expected return or a larger amount of portfolio risk−or a combination of both.

The horizontal and vertical distances from the DJSI asset to the efficient frontier

measure the sizes of the possible compromises.

In this example investing only in DJSI as opposed to the entire market of

N = 4 with short sales has a cost of a lower expected return of approximately


32 bps per month for the same level of standard deviation. Alternatively, this cost

corresponds to a 0.0142% higher standard deviation per month for the same level

of required expected return. These costs of responsible investment correspond

to a 24% lower expected return or a 38% higher risk than the efficient frontier

portfolios with similar risk and expected return respectively. In case the investor

only considers portfolios without short sales, however, the cost of responsible

investment decreases.

2.5.2 Optimal Responsible Investment

After general model assumptions, this section solves the optimal responsible in-

vestment problem, compares the solution to the corresponding neutral invest-

ment, and shows how the responsible investment changes when the sustainability

factors change.

Model Assumptions

The example makes assumptions on three model inputs which remain unchanged

during the rest of the example except when different premises clearly apply. First,

the ESG rating matrix K weighted with the investor priorities ψ for the four

indexes is

(Kψ) =

(Kψ)DJ

(Kψ)NQ

(Kψ)SP

(Kψ)DJSI

=

0.05

0.05

0.05

0.75

. (2.38)

The DJSI thus contributes to the portfolio responsibility 15 times as much as

the remaining individual assets. The numbers in (2.38) are assumptions because

this study has no access to rating data K and because Kψ is investor specific.

Secondly, the required monthly expected return is the average of the vector µ,

i.e., it is µ ≡ 14

∑4i=1 µi = 118.8 bps. Finally, the required level of portfolio re-

sponsibility is set to s = 0.35 for the responsible investor while s is irrelevant for

the neutral investor.

The mean variance analysis in Section 2.4 considers a large, general invest-

ment universe, but when the investment universe is small, the required level of


DJ NQ SP DJSI

Optimal neutral inv.

Por

tfolio

wei

ghts

0.0

0.2

0.4

0.6

0.8

DJ NQ SP DJSI

Optimal responsible inv.

Por

tfolio

wei

ghts

0.0

0.2

0.4

0.6

0.8

DJ NQ SP DJSI

Optimal neutral inv. (ss)

Por

tfolio

wei

ghts

-0.4

0.0

0.4

0.8

DJ NQ SP DJSI

Optimal responsible inv. (ss)

Por

tfolio

wei

ghts

-0.4

0.0

0.4

0.8

Figure 2.4: The figure compares the optimal responsible investment to the optimalneutral investment for the case with short sales (ss) and the case without short sales.

portfolio responsibility may cause unnecessary limitations to the optimal invest-

ment. To avoid this problem, this example assumes that the investor applies

the constraint w>Kψ ≥ s in the responsible investment problem (2.18) instead

of w>Kψ = s. This assumption means that a responsible investor focuses on

the required minimum level of portfolio responsibility and accepts−but do not

seek−additional portfolio responsibility.

The example applies a numerical procedure to solve the optimal investment

problem because of the relaxed assumption of s and because the example consid-

ers both the case with short sales and without short sales. The analytical solution

in Section 2.4 to the optimal investment problem applies to the case with short

sales where w>Kψ = s. The statistical software R therefore solves the general

quadratic programing problem numerically using the package quadprog.9

9www.r-project.org


0.0 0.1 0.2 0.3

-1.0

-0.5

0.0

0.5

1.0

Required responsibility (ss)

Required responsibility

Por

tfolio

wei

ghts

DJSINQSPDJOpt. pf. resp.

0.0 0.1 0.2 0.30.

00.

20.

40.

60.

81.

0



Por

tfolio

wei

ghts

DJSINQSPDJOpt. pf. resp.

Figure 2.5: The figure shows how the optimal investment changes when the level ofrequired responsibility s rises. The left figure demonstrates the case with short sales,and the right figure shows the case without short sales. The dotted line illustrates theamount of portfolio responsibility in the optimal portfolio.

Results

Initially, Figure 2.4 compares the optimal responsible investment to the neutral

investment for the case with short sales (ss) and the case without short sales. The

plots show that the optimal investment in the DJSI rises in case of responsible

investment as expected. The optimal investment in the remaining assets also

change to compensate for the additional weight in DJSI and to limit portfolio

risk. In the case with short sales, the optimal neutral investment and optimal

responsible investment are more similar than in the case without short sales. The

reason is that the problem with short sales meets the restriction smore easily since

the problem counterbalances the additional weight in DJSI with short selling.

The parameters that determine the portfolio responsibility, s and (Kψ)DJSI ,

affect the optimal investment. To demonstrate the effect of these parameters,

the example gives three illustrations as follows. First, Figure 2.5 shows how the

optimal investment changes when the level of required responsibility s rises. The

left figure demonstrates the case with short sales, and the right figure shows

the case without short sales. The dotted line illustrates the amount of portfolio

responsibility in the optimal portfolio. The graphs show that the restriction on


0.70 0.75 0.80 0.85 0.90 0.95 1.00

-1.0

-0.5

0.0

0.5

1.0

DJSI rating (ss)

DJSI rating

Por

tfolio

wei

ghts

DJSINQSPDJ

0.70 0.75 0.80 0.85 0.90 0.95 1.00

0.0

0.2

0.4

0.6

0.8

DJSI rating

DJSI rating

Por

tfolio

wei

ghts

DJSINQSPDJ

Figure 2.6: The figure plots the optimal investment when the DJSI priority weightedrating changes, i.e., when (Kψ)DJSI ∈ [0.70, 1.00]. The left figure gives the case withshort sales, and the right figure gives the case without short sales.

s becomes binding when s > 0.29 for the case with short sales and already when

s > 0.22 for the case without short sales. The weight changes are piecewise linear,

and in the case without short sales, the slopes are steeper because the investor

has fewer opportunities to compensate for the additional weight in DJSI.

Second, the example considers the effect of changing the priority weighted rat-

ing of the DJSI, i.e., changing the investor specific ESG rating. Figure 2.6 shows

how the optimal investment changes over the range (Kψ)DJSI ∈ [0.70, 1.00]. The

optimal weight in DJSI falls as the priority weighted rating rises because the

problem meets the requirement s = 0.35 with less weight in DJSI. The changes

are continuous but not linear, and once again the slopes are steeper for the case

without short sales because the problem has to compensate for diversification

effects.

As the third and final illustration, the example considers the change in optimal

portfolio standard deviation as the responsibility factors change the optimal in-

vestment. The right graph in Figure 2.7 shows how the standard deviation rises

when the required responsibility goes up. The left graph in Figure 2.7 shows

how the standard deviation falls when the priority weighted rating of DJSI goes

up. In both situations the effect is more severe for the case without short sales.


0.0 0.1 0.2 0.3

0.04

10.

042

0.04

30.

044

0.04

50.

046

0.04

7

Required responsibility and std. dev.


Std

.dev

.


0.70 0.75 0.80 0.85 0.90 0.95 1.00

0.04

10.

042

0.04

30.

044

0.04

50.

046

0.04

7

DJSI rating and std. dev.

DJSI rating

Std

.dev

.


Figure 2.7: The figure plots the standard deviation of the optimal investment asfunctions of two responsibility factors; the right figure considers the required portfolioresponsibility, and the left figure considers the (priority weighted) rating of DJSI. Thenumbers in the right figure relates to the solution in Figure 2.5, and the numbers inthe left figure relates to the solution in Figure 2.6.

These graphs illustrate the same point from two different angels: for the given

requirements s and µ, the standard deviation is the factor that compensates for

prioritizing portfolio responsibility in the investment decision.

Analytical Sensitivity to the Required Responsibility

Section 2.4.3 derives the analytical sensitivity of the optimal portfolio weights to

the required responsibility s. The matrix B in (2.30) gives the sensitivities in its

second column. In this example these sensitivities are

∂w∗

∂s=

∂wDJ/∂s

∂wNQ/∂s

∂wSP/∂s

∂wDJSI/∂s

=

b12

b22

b32

b42

=

0.57

0.74

−2.74

1.42

. (2.39)

When the restriction on the required portfolio responsibility is binding, the opti-

mal weights change with linear slopes corresponding to the entries of the vector

2.6 Conclusion 71

∂w∗

∂s. The left graph in Figure 2.5 confirms these sensitivities; when the required

sensitivity exceeds s ≈ 0.29, the restriction becomes binding and the optimal

portfolio weights change with the linear slopes as described in (2.39).

2.6 Conclusion

The paper studies responsible investment portfolio allocation. The model defines

an investor specific measure of portfolio responsibility in terms of ESG criteria.

This measure includes investor specific priorities regarding the different criteria,

and hence the model addresses the ambiguity of the definition of responsible

investment. The study incorporates portfolio responsibility into two different in-

vestment approaches from conventional investment theory. The first approach

is investor utility theory which describes preferences for portfolio responsibil-

ity with an additional argument in the utility function. The second approach

is mean variance analysis which captures portfolio responsibility with an addi-

tional restriction to the investment problem. An example of investment in the

Dow Jones Sustainability World Index (DJSI) illustrates the results of the mean

variance setup.

The paper gives two simple approaches to optimal responsible investment,

both with the advantage that they take a nuanced view on responsible invest-

ment since they incorporate ESG ratings and investor priorities. The investor

utility approach is intuitive, but it is difficult to implement because it requires an

assumption or estimation of the utility function u. The mean variance approach,

on the other hand, has the key advantage that the optimal portfolio problem

has a simple, analytical solution, and the model can be applied by all types of

investors.

The paper argues in favor of a nuanced view on responsible investment. The

reason is that investors may benefit from choosing the ESG factors that matter

in their investment decision in terms of total investment utility or in terms of,

for example, risk management. The last argument especially applies to investors

who need to present economic motivations to consider ESG factors in investment

decisions in order to fulfill their fiduciary duty.


Acknowledgements

I would like to thank Peter Løchte Jørensen for valuable comments and sugges-

tions. I would also like to thank the participants at the following conferences

where the paper has been presented: the Canadian Business Ethics Research

Network Conference 2009 in Ottawa, Canada; the Principles for Responsible In-

vestment Academic Conference 2009 in Ottawa, Canada; the Global Dialogue

Conference 2009 in Aarhus, Denmark; the 1st World Finance Conference 2010 in

Viana do Castelo, Portugal; and the European Financial Management Associa-

tion Annual Conference 2010 in Aarhus, Denmark.

Bibliography

Beltratti, A., 2005. Capital Market Equilibrium with Externalities, Production

and Heterogeneous Agents. Journal of Banking and Finance 29 (12), 3061–

3073.

Benson, K. L., Humphrey, J. E., 2008. Socially Responsible Investment Funds:

Investor Reaction to Current and Past Returns. Journal of Banking and Fi-

nance 32, 1850–1859.

Bollen, N., 2007. Mutual Fund Attributes and Investor Behaviour. Journal of

Financial and Quantitative Analysis 42, 689–708.

Dam, L., 2008. Corporate Social Responsibility and Financial Markets. PhD the-

sis, University of Groningen, The Netherlands.

DJSI, Sep. 2009. Dow Jones Sustainability World Index Guide Book, v. 11.1.

Available online from www.sustainability-index.com.

Galema, R., Plantinga, A., Scholtens, B., 2008. The Stocks at Stake: Return and

Risk in Socially Responsible Investment. Journal of Banking and Finance 32

(12), 2646–2654.

Hallerbach, W., Ning, H., Soppe, A., Spronk, J., 2004. A Framework for Man-

aging a Portfolio of Socially Responsible Investments. European Journal of

Operational Research 153 (2), 517–529.

Heinkel, R., Krause, A., Zechner, J., 2001. The Effect of Green Investment on Cor-

porate Behavior. Journal of Financial and Quantitative Analysis 35, 431449.

Markowitz, H. M., 1952. Portfolio Selection. Journal of Finance 7(1), 77–91.

Eurosif, 2008. European SRI Study 2008. Available online from www.eurosif.org.

74 BIBLIOGRAPHY

Social Investment Forum, 2007. Report on Socially Responsible Investing Trends

in the United States, 2007. Available online from www.socialinvest.org.

UNEP Finance Initiative, Oct. 2007. Demystifying Responsible Investment Per-

formance. Available online from www.unepfi.org.

Nilsson, J., 2008. Investment with a Conscience: Examining the Impact of Pro-

Social Attitudes and Perceived Financial Performance on Socially Responsible

Investment Behavior. Journal of Business Ethics 83, 307–325.

PRI, 2009. Report on progress 2009. UNEP Finance Initiative.

Renneboog, L., Ter Horst, J., Zhang, C., 2008. Socially Responsible Investments:

Institutional Aspects, Performance, and Investor Behavior. Journal of Banking

and Finance 32, 1723–1742.

Tversky, A., Kahneman, D., 1992. Advances in Prospect Theory: Cumulative

Representation of Uncertainty. Journal of Risk and Uncertainty 5, 297–323.

Wisebrod, J. M., 2007. Social Impact Ratings: How to make Responsible Invest-

ment Appealing. Finance and the Common Good/Bien Commun 27, 29–40.

Chapter 3

An Asset Protection Scheme

for Banks Exposed to

Troubled Loan Portfolios

76An Asset Protection Scheme for Banks


77

An Asset Protection Scheme for Banks


Anders Grosen Pernille Jessen Thomas Kokholm




Aarhus University

Abstract

We examine a specific portfolio credit derivative, an Asset Protection

Scheme (APS), and its applicability as a tool to restore financial stability

and reduce asymmetric information. As opposed to most governmental

bailout packages implemented across the world recently, the APS can be a

fair valued contract with an appropriate structure of incentives.

Within the structural credit risk modeling framework, we apply two al-

ternative multivariate default risk models; the classical Gaussian Merton

model and a model based on Normal Inverse Gaussian (NIG) processes.

Exchanging the normal factors in the Gaussian model with NIG factors

adds more flexibility to the distribution of asset returns while retaining a

convenient correlation structure.

Using a unique data set on annual, farm level data from 1996 to 2009, we

consider the Danish agricultural sector as a case study and price an APS on

an agricultural loan portfolio. Moreover, we compute the economic capital

for this loan portfolio with and without an APS.

Key words: Asset Protection Scheme, Bank regulation, Credit risk, Port-

folio credit derivatives, Normal Inverse Gaussian, Asymmetric information

JEL Classification: G13, G21, G32, G38



3.1 Introduction

In the aftermath of the financial crisis, many banks are highly exposed to the

credit risk of certain distressed sectors and fear severe losses which may threaten

their survival. To cope with this increased uncertainty, the banks need more

buffer capital. However, the informational asymmetries inherent in banking re-

lationships suggest that urgent recapitalization may be prohibitively costly or

impossible. Consequently, the market for borrowing and for raising new equity

capital for banks may collapse and lead to a credit crunch for both banks and

their borrowers.

In such an unfavorable situation, the government may wish to interfere if

it considers the given sector “too big” or “too important” to fail. We suggest

that any interference should fulfil at least one of the two following objectives.

First, it should reestablish confidence in the banks’ balance sheets and decrease

asymmetric information, hereby easing the issuing of new equity capital. Second,

it should release bank capital which should enable lending to financially healthy

borrowers, helping these to invest and be innovative.

For this purpose, we analyze a particular portfolio credit derivative, the Asset

Protection Scheme (APS), and provide a practical application of this instrument

using the Danish agricultural sector as a case study. The APS was initially in-

troduced by the regulatory authorities in the UK as a response to the financial

crisis, and it is a type of credit derivative which intuitively can be considered

as an insurance contract covering part of the default risk in a loan portfolio. A

governmental insurance contract covering extreme market events can be a con-

venient political tool to meet the objectives stated above. An APS can be fairly

priced and non-subsidiary as opposed to most governmental bailout packages

implemented across the world recently.

More specifically, the APS has the cash flows as follows. If we consider an

insurer who sells an APS to a bank, then the bank pays a fixed fee to the insurer

on a regular basis in return for compensation in case the loan portfolio losses

exceed a predefined threshold. If this threshold is exceeded, a given fraction of

the subsequent losses remains the liability of the bank which thus retains financial

motivation to service the individual loans in the portfolio even after the insurance

contract has gone into effect.

The APS introduced above should be seen as a tool to overcome the prob-

3.1 Introduction 79

lems arising from the informational asymmetries in banking relationships after a

negative shock to the economy. These problems are serious as the core business

of banks is to mitigate issues of asymmetric information between lenders and

borrowers as pointed out by Diamond (1984) and others (see Bhattacharya and

Thakor (1993) for a review). The consequences can be dramatic as illustrated

by the collapse of the banking markets in the wake of the financial crisis. This

collapse can be explained by the fact that equilibria in markets with asymmetric

information are inherently unstable as initially described by Akerlof (1970). In

such markets, even small changes in market conditions can have the consequence

that the market disappears (or reappears). In this paper, the problems in raising

new equity capital for banks after a negative shock to the economy is addressed.

If banks cannot issue equity in economic downturns, it might lead to a severe

contraction in the supply of credit. This problem is−as demonstrated in Repullo

and Suarez (2008)−aggravated by the Basel II accord since capital requirements

prescribed for credit risk increase during downturns.

Within the structural credit risk modeling framework introduced by Merton

(1974), we consider two alternative multivariate default risk models. The first

model, included as a benchmark, is the classical approach based on Gaussian

innovations in the log asset returns. The second model seeks to capture extreme

market risk; it allows for jumps and the log asset returns have Normal Inverse

Gaussian innovations, see Barndorff-Nielsen (1997). Multivariate credit modeling

using the Gaussian model by Merton (1974)−or variations of it−already proved

itself insufficient in the early stages of the financial crisis since it failed to model

realistic loss distributions. Exchanging the normal factors in the Gaussian model

with NIG distributed factors adds more flexibility to the distributional proper-

ties of asset returns while retaining a convenient correlation structure. The NIG

process has been successfully applied in finance, see, e.g., Rydberg (1997) for

stock market modeling and Nicolato and Skovmand (2010) for interest rate mod-

eling. The NIG distribution has been used for multivariate credit modeling in

Kalemanova et al. (2007) who consider a one factor copula approach, but a mul-

tivariate structural NIG model has not previously been applied in the literature

on credit risk. However, related to the approach of this paper, a multivariate

model is proposed in a study by Luciano and Schoutens (2006) where the log

asset returns are Variance Gamma distributed.



Katchova and Barry (2005) study credit risk of agricultural lending within

the structural Merton model and estimate capital requirements for agricultural

lenders under the Basel II framework. They show that these capital requirements

vary substantially depending on the riskiness and homogeneity of the portfolio.

However, Pederson and Zech (2009) find that the Merton model is inappropriate

for the modeling of agricultural credit risk. The authors argue that the assump-

tion of asset dynamics following geometric Brownian motions is not realistic.

Stokes and Brinch (2001) let the probability of default depend on the value of

farmland. A significant part of farm assets consists of farmland: according to the

US Department of Agriculture (2009), the average farm in the US in 2007 had

approximately 90% of its assets tied up in farm real estate and machinery. In

comparison, according to Statistics Denmark,1 the average farm in Denmark in

2007 had roughly 82% of its assets in farm real estate. Therefore, if the market

for farm land experiences a sudden negative shock, we expect that this event

will significantly affect the value of farm assets. Yan and Barry (2006) suggest,

however, that the value of farmland is non-depreciable and that agricultural de-

faults may not necessarily lead to losses for the lender. Using data from the

US Department of Agriculture on farm land values for the period from 1950 to

2002, the authors show that when the loan collateral is farmland, it is expected

to cover the loan balance with relatively high probability in case of default. Even

though this finding is historically meaningful, our current economic situation is

very likely to lead to different results.

Figure 3.1 shows the development in land prices in the US, Denmark, and

a number of European countries. For the US, UK, Netherlands, and Denmark,

the graphs depict a steep increase in prices during the previous six years, and

in Denmark this development has been followed by a sudden decrease. The

figure indicates that some countries may currently experience an over-valuation

of farmland and therefore may face future decreasing prices.

Sherrick et al. (2000) consider the valuation of credit risk in agricultural loans

and estimate the cost of insuring against credit risk in pools of agricultural loans.

Using an actuarial approach and contingent claim analysis, they find that the

value of such insurance depends on the size of the pool, deductibles, timing,

premiums, adverse selection, and under-writing standards. Furthermore, they

1www.statistikbanken.dk

3.1 Introduction 81

2.000

2.500

Farm land value in the USaverage value per acre (USD)

250.000

300.000

Farm land value in Denmarkaverage value per hectare (DKK)

500

1.000

1.500

2.000

2.500


50.000

100.000

150.000

200.000

250.000

300.000


Source: USDA see http://usda.mannlib.cornell.edu Source: Statistics Denmark see www.statistikbanken.dk

0

500

1.000

1.500

2.000

2.500

1994 1996 1998 2000 2002 2004 2006 2008 2010


0

50.000

100.000

150.000

200.000

250.000

300.000

1992 1994 1996 1998 2000 2002 2004 2006 2008


Farm land value in Europe

0

500

1.000

1.500

2.000

2.500

1994 1996 1998 2000 2002 2004 2006 2008 2010


0

50.000

100.000

150.000

200.000

250.000

300.000

1992 1994 1996 1998 2000 2002 2004 2006 2008


30 000

35.000

40.000

45.000

Farm land value in Europeaverage value per hectare (EUR)

0

500

1.000

1.500

2.000

2.500

1994 1996 1998 2000 2002 2004 2006 2008 2010


0

50.000

100.000

150.000

200.000

250.000

300.000

1992 1994 1996 1998 2000 2002 2004 2006 2008


10.000

15.000

20.000

25.000

30.000

35.000

40.000

45.000


Spain

Netherlands

Sweden

United Kingdom

Source: EuroStat see http://epp.eurostat.ec.europa.eu

0

500

1.000

1.500

2.000

2.500

1994 1996 1998 2000 2002 2004 2006 2008 2010


0

50.000

100.000

150.000

200.000

250.000

300.000

1992 1994 1996 1998 2000 2002 2004 2006 2008


0

5.000

10.000

15.000

20.000

25.000

30.000

35.000

40.000

45.000

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008


Spain

Netherlands

Sweden

United Kingdom

Figure 3.1: The development in average farmland prices for the US, Denmark,and a number of European countries.

show that insurance costs are highly sensitive to these factors for small pools but

become relatively insensitive when the size of the pool grows.

Our numerical implementation of the APS valuation builds on a unique data

set on Danish agricultural panel data. The Danish Agricultural Advisory Service

(Landbrugsraadgivningen) has generously offered us annual, farm level data for

the period from 1996 to 2009. We estimate a joint default risk model and price

an APS based on a hypothetical loan portfolio via simulation using the two

alternative models. Moreover, we compute the economic capital for this loan

portfolio with and without an APS for the two models. Brief descriptive statistics

offer intuition on the development in the sector during the past 14 years, the

period of time where a price bubble seems to have emerged in farmland prices

(see Danmarks Nationalbank (2009)).

The contribution to the financial literature on credit risk is the empirical

analysis of a unique data set on agricultural loans. Moreover, we demonstrate how



100%

Covered by

Incentive for the insured

Portfolio to maintain the loan

Loss portfolio, α2

Covered by the insurer (APS seller)

First Loss, α1

0%

Covered by the insured (APS buyer)

Covered by the insurer (APS seller)

Figure 3.2: The figure shows how the APS covers extreme losses in a loan portfolio.The APS takes effect if the portfolio loss exceeds a given first loss threshold α1. Inthis case, the bank still covers a fraction α2 of the following losses and thus retains anincentive to maintain the portfolio.

to apply a multivariate structural NIG model to credit risk. The contribution to

the banking literature is the suggestion of how to use a specific credit derivative

with a desirable structure of incentives to cope with asymmetric information

problems in banking when market conditions are extreme. Finally, the paper

contributes by suggesting how to address the serious problems concerning the

financial side of the Danish agricultural sector−or any other particular, distressed

sector experiencing similar problems.

3.2 Asset Protection Scheme (APS)

This section describes the APS and explains how it can be used as a fair valued

tool by financial authorities.

3.2.1 Construction and Motivation

The APS seller provides the buyer with protection against credit losses incurred

on a given portfolio when the losses exceed a specified “first loss” threshold α1

in return for a fee. If the first loss level is passed, the protection seller cov-

3.2 Asset Protection Scheme (APS) 83

0 20 40 60 80 100

010

020

030

040

0

Tranche Loss

# Defaults

Tra

nche

Los

s

APS buyerAPS seller

Figure 3.3: The figure shows the tranche loss as a function of number of defaults in anunderlying portfolio of size EUR 1,000 with 100 equally weighted loans and a recoveryrate of 40%. The APS has the parameters α1 = α2 = 10%.

ers (1− α2) of the following portfolio loss while the buyer still bears the resid-

ual loss, α2. Figure 3.2 shows a stylized version of the APS structure. For

a description of the properties and terms of the APS contracts implemented

in the UK, see The Royal Bank of Scotland Group (RBS) (2009), HM Treasury

(2009a), and HM Treasury (2009b).

The covered assets remain on the balance sheet of the protection buyer who

economically and legally still owns the assets. The first loss threshold α1 serves

as a deductible, which is a common feature of insurance contracts. After the first

loss threshold is exceeded, the protection buyer still covers a fraction α2 of the

further losses and thus retains an appropriate structure of incentives to continue

the diligent management of the loan portfolio. Since the APS slices portfolio

credit risk into tranches similar to a Collateralized Debt Obligation (CDO), we

use the corresponding terminology. Figure 3.3 shows the tranche loss as a function

of the number of defaults in the underlying portfolio.

Generally, the covered assets can be of any asset class; residential mortgages,

consumer finance, bonds, loans, structured products, or derivatives. In case of a

governmentally issued APS, the underlying portfolio would typically represent a

certain distressed sector related to the current financial climate.



3.2.2 The APS and Financial Stability

In the following, we describe circumstances under which financial authorities can

apply the APS as a tool to enhance financial stability. In particular, we consider

two situations where the APS can be useful.

The first case was illustrated during the financial crisis by the collapse of

banks’ funding in the money market. After a major negative shock to the econ-

omy, the market for bank lending may experience an increase in asymmetric

information and face undesirable consequences. Banks with lower quality loan

portfolios are more interested in issuing equity than banks with higher quality

loan portfolios, and this may result in prices for new equity being prohibitively

high. Consequently, the market for bank equity capital may experience large

price discounts−or even a market collapse−when increases in banks’ buffer cap-

ital and access to the money market funding are most needed. An APS can help

free up liquidity in the secondary markets by substantially reducing the tail risk

of losses of the insured institution, hereby allowing it to stabilize its capital base

and in an orderly manner decrease leverage over time.

In this situation, the APS manages the risk of the taxpayers: if the troubled

institutions are considered “too important” or “too big” to fail, the government

may be forced to interfere with the industry when it is threatened by extreme

market events. However, when the government is the seller of the APS, the

government collects compensation for this (politically implicit) guarantee in the

form of fee payments. From the perspective of the taxpayer, the government

should prefer the formal insurance contract with this compensation instead of

the implicit guarantee.

The second case where the APS can be useful concerns the situation when a

governmental authority has taken over a troubled bank. Normally the author-

ity will try to sell the troubled loan portfolio to another, well-functioning bank,

however, due to the asymmetric information in banking relationships, it might be

difficult to find a bank willing to take over the troubled loan portfolio. In order

to ease this situation, the government can attach an APS as a state guarantee to

the troubled portfolio in order to mitigate the asymmetric information concerns.

Both the borrowers in the troubled loan portfolio and the government can benefit

from such contract. First, transferring the loan portfolio secures a banking rela-

tionship for the troubled debtors. Lack of a personal banking relationship may

3.3 Credit Risk and APS Valuation 85

have severe negative effects on already troubled loans. Second, the government

lacks banking expertise and operational systems, and therefore we expect that

the government has a strong interest in selling the activities of the troubled bank.

In Denmark, for example, the Financial Stability company2 is part of an agree-

ment between the Danish state and the Danish financial sector. The company

was established in October 2008 as a reaction to the financial crisis, and it han-

dles troubled banks and seeks to support the banking industry with various state

guarantees. However, an establishment like Financial Stability is a liquidation

company and does not necessarily have the banking expertise to maintain the

loan portfolio of a troubled bank.

In the following, we consider a practical application of the first case when we

perform a numerical example based on the Danish agricultural sector.

3.3 Credit Risk and APS Valuation

Consider a filtered probability space(Ω,F , Ftt≥0 ,P

)where the asset value pro-

cesses Ait for i = 1, ...,M are adapted to the filtration Ftt≥0. Let K1, . . . , KM

denote the total principal value of debt of company 1 to M at time zero. We

model the asset values as

Ait = Ai0 expX it

, (3.1)

where X it is a stochastic process.

For a given tenor structure

0 = T0 < T1 < . . . < Tn < . . . , (3.2)

we define the default time of company i as

τ i = inft ∈ T1, . . . , Tn, . . . | Ait < Ki

. (3.3)

In this framework the defaults are modeled via the log asset returns X its, and

we consider two alternative specifications: correlated Gaussian innovations and

correlated Normal Inverse Gaussian innovations.

2http://www.finansielstabilitet.dk/en/



3.3.1 Gaussian Innovations in the Asset Returns

Assume that the value of the assets of a company follows a geometric Brownian

motion under the physical measure. At time t the i’th company’s asset value is

Ait = Ai0 expµit+ σiW

it

, (3.4)

where µi is the average annual log return, σi is the instantaneous volatility,

and W it is a Wiener process driving the development. W i

t is allowed to corre-

late with the Wiener process W jt of company j with parameter ρi,j such that

dW jt dW

it = ρi,jdt.

The risk neutral dynamics of the asset values is found by setting the instan-

taneous rate of return equal to the risk free rate. For company i at time t, the

risk neutral dynamics is

Ait = Ai0 exp

(r − σ2i /2)t+ σiW

it

. (3.5)

In the estimation related to our case study, the parameters µi, σi, and ρi,j

for i, j = 1, . . . ,M are estimated from historical data on farm asset values, see

Section 3.4.2.

3.3.2 Normal Inverse Gaussian Innovations in the Asset

Returns

In this section, we allow for more flexibility in the log asset values by letting them

have normal inverse Gaussian dynamics. Correlated NIG processes are generated

by time-changing/subordinating correlated Brownian motions with an Inverse

Gaussian (IG) process

X it = µiIt + δiW

iIt , (3.6)

where the Brownian parts W it s are correlated but independent of the common IG

subordinator It. For normalization reasons, we consider an IG time-change with

E [It] = t, and it leaves one parameter κ for the variance. Since It is positive, we

describe it using the moment generating function

E[euIt]

= etl(u) ∀u ∈ R, (3.7)


where (see Cont and Tankov (2004))

l (u) =1

κ

(1−√

1− 2κu). (3.8)

From the moment generating function, the central moments of the time-change

can be found by differentiation, and they are reported in Table 3.1.

Table 3.1: Central moments of the IG process It with parameter κ.

EIt t

E (It − t)2 tκ

E (It − t)3 3tκ2

E (It − t)4 15tκ3 + 3t2κ2

The characteristic function of a Levy process Xt has the form

E[eiuXt

]= etΨ(u), (3.9)

where Ψ(u) is the characteristic exponent. For an NIG process, this is given by

Ψ (u) =1

κ

(1−

√1− 2iµuκ+ u2δ2κ

), (3.10)

which can be used to find the central moments of the NIG process. Table 3.2

reports the central moments of both the Gaussian and the NIG process.

Table 3.2: Central moments of the Gaussian process with parameters µ, σ andthe Normal Inverse Gaussian process with parameters µ, δ, κ.

Gaussian NIGEXt µt µt

E (Xt − µt)2 σ2t δ2t+ µ2κt

E (Xt − µt)3 0 3δ2µκt+ 3µ3κ2t

E (Xt − µt)4 3σ4t2 3δ4κt+ 15µ4κ3t+ 18δ2µ2κ2t+ 3 (δ2t+ µ2κt)2

Assume that under the physical measure the asset value follows an exponential

Levy process

Ait = Ai0 expX it

, (3.11)



where X it is an NIG process with parameters δi, µi, κ. In general, the process

defined in (3.11) will not be a martingale. We specify the dynamics of the asset

value under an equivalent martingale measure by introducing a mean correcting

drift mi. The asset value is then

Ait = Ai0 expmit+X i

t

, (3.12)

where the martingale condition imposes that (see Raible (2000))

mi = r −Ψi (−i) . (3.13)

Denote the correlation between the Brownian motions W it and W j

t by ρi,jBrown.

Then the correlation between log asset returns of company i and j is given by

(see Cont and Tankov (2004))

ρi,j =δiδjρ

i,jBrown + µiµjκ√

δ2i + µ2

iκ√δ2j + µ2

jκ. (3.14)

In the estimation related to our case study, the model parameters are esti-

mated by matching the model moments in Table 3.2 to historical moments of the

return data on individual farmers, see Section 3.4.2.

Simulation

Inspection of equation (3.6) reveals the two-step recipe for simulating correlated

NIG processes. First, we simulate IG distributed increments with mean tj − tj−1

and variance (tj − tj−1)κ of the common subordinator It and denote these incre-

ments by

∆Itj = Itj − Itj−1. (3.15)

Second, we simulate the log asset returns with

logAitj = logAitj−1+mi (tj − tj−1) + µi∆Itj + δi

√∆ItjZ

i, (3.16)

where the Zis are drawn from a multivariate standard normal distribution with

covariance matrix entries ρi,jBrown.


3.3.3 Pricing of an APS

Similar to a synthetic CDO tranche, an APS is a swap contract. It has the value

of the expected present value of its cash flows; the APS seller receives periodic

payments from the APS buyer (premium leg) and makes contingent payments

to the buyer if defaults exceed the first loss threshold (protection leg). From

the perspective of the buyer, the value of the APS is the difference between the

expected present value of the protection leg and the expected present value of

the premium leg.

Typically, the debt in company i consists of a mortgage and a loan in the

bank

Ki = Ki,Mort +Ki,Bank, (3.17)

where the mortgage lender has higher seniority. The bank thus holds the more

risky part of the debt, and the total principal value in the bank portfolio is

F =M∑i=1

Ki,Bank. (3.18)

Then the loss process in the bank portfolio is

Lt =M∑i=1

minKi − Aiτ i , Ki,Bank

1τ i≤t, (3.19)

since each defaulted company, indicated by 1τ i≤t, has the loss of either the full

bank loan Ki,Bank or the principal value of the debt less the asset value−whatever

amount is the smallest.

We consider an APS with annual premium payments, first loss threshold α1,

and an incentive-percentage α2 (see Figure 3.2). The loss in the APS at time t

is given by

LAPSt = max Lt − α1F, 0 (1− α2) , (3.20)

since the APS only experiences loss if Lt > α1F . The present value of the

protection payments for an APS with maturity Tn is

πProt = E

[n∑j=1

B (0, Tj)(LAPSTj

− LAPSTj−1

)], (3.21)



where B (0, Tj) is the discount factor for the maturity Tj. The present value of

the premiums equals

πPrem = E

[n∑j=1

B (0, Tj) sAPS

(F (1− α1) (1− α2)− LAPSTj

)], (3.22)

where the total value covered by the APS is F (1− α1) (1− α2). The fair spread

sAPS that makes the initial value of the contract zero can be found by equating

(3.21) and (3.22), and it is given by

sAPS =πProt

E[∑n

j=1 B (0, Tj)(F (1− α1) (1− α2)− LAPSTj

)] . (3.23)

3.4 Estimation and Valuation

In this section, we use our data set and estimate the real world dynamics of the

farm asset values as described in (3.4) and (3.11) for our two model specifications.

From these dynamics, we derive the risk-neutral dynamics in (3.5) and (3.12)

and price an APS on our agricultural loan portfolio via simulation using the two

alternative models.

3.4.1 Data

Our numerical implementation of the APS valuation builds on a unique data

set on Danish agricultural panel data provided by the Danish Agricultural Ad-

visory Service (Landbrugsraadgivningen). The data set holds annual, farm level

observations for the period from 1996 to 2009 and has detailed information on

the value of assets of the individual farmer, amounts and types of debt, value of

farmland, and more. We let the agricultural loan portfolio consist of the farms

for which we have data for all 14 years. It amounts to 208 farms, and in 2009 the

portfolio has a total principal value of F = 702, 723, 413 DKK.

Figure 3.4 gives the development in average asset value, debt, and debt ratio

over the period 1996 to 2009. The top graph in the figure shows the development

in the average asset values and debt. We see that both quantities have increased

significantly during the period but that the mean asset values have decreased

since 2008 due to the decrease in farmland prices. The middle graph illustrates

3.4 Estimation and Valuation 91

1996 1998 2000 2002 2004 2006 2008 20100.4

0.5

0.6

0.7

0.8Mean debt ratio 1996-2009

Deb

t rat

io

1996 1998 2000 2002 2004 2006 2008 20100.18

0.2

0.22

0.24Std dev of debt ratio 1996-2009

Std

dev

1996 1998 2000 2002 2004 2006 2008 20100

1

2

3

4x 10

7 Mean asset and debt values 1996-2009

DK

K

Mean asset valuesMean total debt

Figure 3.4: The figure shows the development in average asset value, debt, anddebt ratio.

the average debt ratio defined as the total debt to asset values. We see that the

increasing asset values during 1996 to 2008 have caused a decreasing debt ratio,

but since 2008 the debt ratio has gone up as a consequence of the decreased asset

values. The bottom graph shows the standard deviation of the debt ratios for

each year across farmers. We observe that the homogeneity in terms of debt ratio

was high in 2006 but has decreased significantly since then. We interpret this

result as a consequence of the falling land prices. While many farmers are still

financially healthy, many farmers are now close to default, and thus the dispersion

of the rates of solvency of farmers has become larger.



3.4.2 Estimation of Parameters in the Asset Value Dy-

namics

Estimation of the parameters in the two models is performed by matching the

model moments to the empirical moments. Table 3.3 reports the average of

the estimated annual return, standard deviation, skewness, and excess kurtosis

defined by

µi = EX i (3.24)

σi =

√E (X i − µi)2 (3.25)

si =E (X i − µi)

3

σ3i

(3.26)

ki =E (X i − µi)

4

σ4i

− 3, (3.27)

where X i is the annual log asset return of farm i.

Table 3.3: The average across farms of the annual return, standard deviation,skewness, and excess kurtosis.

µ 0.1120σ 0.1681s 0.7173

k 1.6590

For the Gaussian model, the first two moments can be matched precisely but

skewness and excess kurtosis equal zero. Only the individual volatilities σis are

important for valuation, but for risk management purposes, the first moment also

matters. The pairwise correlations are all set equal, i.e., ρi,j = ρ for i, j = 1, ..,M

where i 6= j, and estimated as the average of the pairwise correlations, ρ = 0.2146.

In order to match higher moments of the log asset returns data, we consider

the NIG framework. In the multivariate NIG model, the two free parameters µi

and δi for each name and the common parameter κ are estimated by minimizing

the objective function

M∑i=1

w1 (µi − µi)2 + w2 (σi − σi)2 + w3 (si − si)2 + w4

(ki − ki

)2

, (3.28)

3.4 Estimation and Valuation 93

where w1, .., w4 are the weights given by (see Bartlett (1950))

wn = 1− n

5. (3.29)

The optimal value of the common parameter is κ = 0.4286 while the average

values of the two free parameters are µ = 0.0594 and δ = 0.1317. The correlation

between the Brownian components ρi,jBrown is found by inverting equation (3.14)

for a fixed pairwise correlation equal to ρ = 0.2146.

3.4.3 Valuation of the APS

We estimate the parameters in the two models from the data set and price a

hypothetical APS with α1 = 0.1, α2 = 0.1, maturity T = 5 years, and annual

premium payments via simulation of 1,000,000 sample paths of the two models.

The risk-free rate of return is set constant and equal to 3%. This assumption can

be relaxed but one has to bear in mind that the driving process for the interest

rate should be of the same type as the driving process in the firm assets, see, e.g.,

Cox et al. (1985) in the Gaussian case and Nicolato and Skovmand (2010) in the

NIG case.

Based on the data, we compute the spreads in the two models to

sAPSGaussian = 59 bp

sAPSNIG = 41 bp.

The price for the Gaussian model is higher than for the NIG case since the NIG

model is able to match the positive skewness of the empirical distribution of the

asset value returns. This skewness results in a lower individual probability of

default which infers a lower price of the APS.

If the value of farmland is experiencing a price bubble burst, it is relevant to

consider scenarios where farm asset values decrease. Therefore, we price the APS

on the agricultural loan portfolio where the asset values have decreased 10% and

30% respectively. Based on these decreases and all other things equal, the fair

spreads are reported in Table 3.4 and compared to the current situation (0%).

We observe in Table 3.4 that as expected the spreads are increasing for de-

creasing asset values. Moreover, this increase is not linear because−when using



Table 3.4: Based on a decrease in asset values of 10% to 30%, the table showsthe fair APS spreads in the two model specifications.

Model Gaussian NIG0% 59 bp 41 bp10% 144 bp 111 bp30% 770 bp 718 bp

the option terminology−the “option to default” becomes closer to being in the

money as the asset values fall. The prices for the Gaussian model are once again

higher than for the NIG case since the NIG model matches the positive skewness

of the realized asset value returns. However, this effect becomes relatively smaller

when the asset values decrease.

3.5 Capital Requirements

This section considers the capital requirements for a hypothetical bank holding

the given agricultural loan portfolio, before and after the bank enters the APS.

The capital requirements of a financial institution have great impact on its perfor-

mance since it is expensive to hold more capital than necessary. We consider the

economic capital of the bank and comment on the minimum regulatory capital

requirements in this context.

3.5.1 Economic Capital

Economic capital is the risk capital that ensures the technical solvency of a bank

with a given probability over a given time horizon. The bank computes the

economic capital under the physical measure typically with a Value at Risk (VaR)

method (see Jorion (1997)).

McNeil et al. (2005) describe the economic capital as follows. The bank is

conservatively capitalized over a given time period [t, t+ ∆] if it holds equity E

for which it with a “high” probability β holds that

P(∆Lt+∆ − E[∆Lt+∆] < E) ≥ β, (3.30)

3.5 Capital Requirements 95

where ∆Lt+∆ = Lt+∆ − Lt is the loss in the loan portfolio over the given period

of time. When the APS is included in the portfolio, we consider the premium

payments as “losses” and add them to the total loss. The premium payments are

computed from the spreads reported in the first row of Table 3.4.

The economic capital (EC) is defined as the amount of capital E that makes

the inequality in (3.30) binding. Hence

EC = V aRβ(∆Lt+∆)− E[∆Lt+∆]

= V aRβ(∆Lt+∆ − E[∆Lt+∆]), (3.31)

where V aRβ is the β-quantile of the loss distribution. The economic capital is

then considered sufficient to limit the probability of technical insolvency to (1−β)

over the given time period [t, t+ ∆].

Equation (3.31) computes the contribution to the total economic capital from

the loan portfolio. In general this will overestimate the required economic capital

since it does not take into account the diversification effects from adding the loan

portfolio to the bank assets. For a fully integrated model of economic capital see,

e.g., McNeil et al. (2008).

Based on 1,000,000 sample paths, Figure 3.5 shows the loss distributions on a

horizon of 5 years for both the original loan portfolio (top) and for the portfolio

with an APS (bottom). The principal value of the loan portfolio amounts to

F = 702, 723, 413 DKK. The left side of the figure refers to the case where

asset values follow a multivariate geometric Brownian motion, and the right side

shows the NIG case. For the portfolio without the APS, we observe that the loss

distribution has a much heavier right tail for the NIG model than the Gaussian

case, resulting in a much higher VaR and expected loss. For the portfolio with

the APS, we observe that the loss distributions have two peaks. The first peak

corresponds to the losses in the retained equity tranche, i.e., the portfolio loss

from zero to α1F . The second peak arises because the losses above the threshold

level are compressed due to the fraction α2 retained by the APS buyer.

In the figure, we indicate the expected losses and VaRs reported in Table 3.5

from which the economic capital numbers are computed. In the Gaussian model,

the economic capitals are almost identical with or without the APS. Ignoring the

premiums paid for entering the APS, the expected loss is lower when the APS

is included while the VaRs are identical because the probability of exceeding the



0 0.5 1 1.5 2

x 108

0

1

2

3

4

5x 10

−8

pdf

GaussianLoss distribution without the APS

Loss0 0.5 1 1.5 2

x 108

0

1

2

3

4

5x 10

−8

pdf

NIGLoss distribution without the APS

Loss

0 0.5 1 1.5 2

x 108

0

1

2

3

4

5x 10

−8

pdf

Loss distribution with the APS

Loss0 0.5 1 1.5 2

x 108

0

1

2

3

4

5x 10

−8pd

fLoss distribution with the APS

Loss

EL VaR EL VaR

VaR

EC

EC

EC

EL

ELVaR

EC

Figure 3.5: Estimated density functions for the losses on a horizon of 5 yearsfor the Gaussian and the NIG model with and without the APS. The VaRs arecomputed as the 97.5% quantiles of the loss distributions.

threshold is lower than 2.5%. Since the premiums only shift the loss distribution

to the right, we observe the counter-intuitive result that the economic capital

is slightly higher in this case, an increase of 201,000 DKK or 0.45%. In the

NIG model, on the other hand, the probability of exceeding the loss threshold

is greater than 2.5%, and therefore the economic capital is reduced significantly

when the portfolio includes the APS, a reduction of 31,531,000 DKK or 47.42%.

Inspection of Figure 3.5 indicates that the APS should be priced higher for

the NIG model contradicting the actual prices computed in Section 3.4.3. This

disparity occurs because the prices are computed under the risk-neutral measure

while the loss distributions used for risk management purposes are estimated

under the physical measure. As described in Section 3.4.2, the parameters are

3.5 Capital Requirements 97

Table 3.5: The Expected Loss (EL), Economic Capital (EC), and the97.5% quantile of the loss distributions. The face value of the portfolio isF = 702, 723, 413 DKK, and the values in the table are in DKK.

Gaussian NIGWithout APS

EL 18,047,000 41,509,000Quantile (0.975) 62,264,000 108,000,000EC 44,217,000 66,495,000EC/F 6.3% 9.5%With APS

EL 34,637,000 50,363,000Quantile (0.975) 79,055,000 85,327,000EC 44,418,000 34,964,000EC/F 6.3% 5.0%

estimated by minimizing the objective function in (3.28) which is a weighted sum

of the squared moment differences. As a result the annual expected log returns

µis turns out quite differently in the NIG case (average of 0.0594) than those

of the Gaussian model where the empirical log returns are matched precisely

(average of 0.1120). This difference in the expected return is apparent under the

physical measure while unimportant under the risk-neutral measure.

3.5.2 Minimum Regulatory Capital

The regulatory capital is the mandatory capital that the financial authorities

specify. The banks’ minimum regulatory capital requirements−Pillar I of the

Basel II framework−are calculated as 8% of the risk weighed assets based on

a formal set of detailed rules. The Basel II minimum regulatory capital will

typically differ from the banks economic capital due to differences in model,

data, and parameters although both measures of buffer capital aim at keeping the

bank solvent in a worst case scenario based on a VaR methodology. Nevertheless,

these differences between economic capital and minimum regulatory capital will

typically be minimized as banks’ internal economic capital figures are central

to supervisors’ assessment of regulatory capital adequacy under Pillar II of the

Basel II framework. It must also be expected that differences between economic

and regulatory capital will be further diminished as loopholes in Basel II will be



closed by the forthcoming Basel III regulatory capital requirements. Therefore,

we expect that the capital relief by introducing an APS can be based on the

changes in economic capital.

For a portfolio without the APS, the minimum regulatory capital requirement

(CR) is

CRNo APS = F ·RW1 · 0.08, (3.32)

where RW1 is the risk weight of the assets in the portfolio, and F is the principal

value. For a portfolio with an APS, this requirement is

CRAPS = α1 · F ·RW2 · 0.08 + (1− α1)α2 · F ·RW3 · 0.08, (3.33)

where RW2 and RW3 are risk weights for the equity tranche and for the retained

fraction α2 of loss exceeding the first loss respectively.

For the minimum regulatory requirements to be reasonable, it should hold

that CRNo APS > CRAPS. The risk weights RW1, RW2, and RW3, which are

defined by the regulatory authorizes, determine whether entering an APS gives

a minimum regulatory capital relief. Hence, we argue that these risk weights

related to the APS should be specified in a meaningful manner.

3.6 Conclusion

We examine a specific portfolio credit derivative, an Asset Protection Scheme,

and its applicability as a tool to restore financial stability and reduce asymmetric

information. We consider the Danish agricultural sector as a case study and price

an APS on an agricultural loan portfolio. Moreover, we compute the economic

capital for this loan portfolio with and without the APS. With respect to credit

modeling, we apply two alternative multivariate models within the structural

credit risk framework; the classical Gaussian Merton model and a model based

on Normal Inverse Gaussian processes.

We argue that that the APS can serve as a convenient tool to reduce asym-

metric information and thereby help banks in raising new equity. As the financial

crises demonstrated, especially in times of financial distress, it is crucial to be

able to raise new equity capital. We find that the price for the Gaussian model

is higher than for the NIG case due to the fact that the NIG model is able to

3.6 Conclusion 99

match the positive skewness of the realized asset value returns. Finally, we find

that the reduction in economic capital when introducing the APS is greater for

the NIG model than the Gaussian case.

Of these two models, we suggest that the NIG is used for both risk manage-

ment and pricing of the APS since this model is more flexible and able to describe

extreme market risk. Furthermore, the economic capital numbers computed in

our example indicate that the Gaussian model is not suitable since the economic

capital increases after entering the APS.

Acknowledgements

We thank the Danish Agricultural Advisory Service (Landbrugsraadgivningen)

for providing us the data set on Danish farms. We also thank Peter Løchte

Jørgensen, Elisa Nicolato and David Skovmand for valuable comments. Fur-

thermore, we are thankful to the participants at the Nordic Finance Network

Workshop 2010 and the 14th International Congress on Insurance: Mathematics

and Economics 2010 where the paper has been presented.

Bibliography

Akerlof, G., 1970. The Market for “Lemons”: Quality Uncertainty and the Market

Mechanism. The Quarterly Journal of Economics 84 (3), 488–500.

Barndorff-Nielsen, O., 1997. Normal Inverse Gaussian Distributions and Stochas-

tic Volatility Modelling. Scandinavian Journal of Statistics 24 (1), 1–13.

Bartlett, M., 1950. Periodogram Analysis and Continuous Spectra. Biometrika

37 (1-2), 1.

Bhattacharya, S., Thakor, A. V., 1993. Contemporary Banking Theory. Journal

of Financial Intermediation 3 (1), 2–50.

Cont, R., Tankov, P., 2004. Financial Modelling with Jump Processes. Chapman

& Hall/CRC.

Cox, J., Ingersoll Jr, J., Ross, S., 1985. A Theory of the Term Structure of Interest

Rates. Econometrica 53 (2), 385–407.

Diamond, D. W., 1984. Financial Intermediation and Delegated Monitoring. The

Review of Economic Studies 51 (3), 393–414.

Jorion, P., 1997. Value at Risk. Irwin Homewood, IL.

Kalemanova, A., Schmid, B., Werner, R., 2007. The Normal Inverse Gaussian

Distribution for Synthetic CDO pricing. The Journal of Derivatives 14 (3),

80–94.

Katchova, A. L., Barry, P. J., 2005. Credit Risk Models and Agricultural Lending.

American Journal of Agricultural Economics 87 (1), 194–205.

102 BIBLIOGRAPHY

Luciano, E., Schoutens, W., 2006. A Multivariate Jump-Driven Financial Asset

Model. Quantitative Finance 6 (5), 385–402.

Danmarks Nationalbank, Jun. 2009. Financial Stability 2009, 1st half. Available

from the Danmarks Nationalbank, www.nationalbanken.dk.

HM Treasury, Dec. 2009a. Royal Bank of Scotland: Details of Asset Protec-

tion Scheme and Launch of the Asset Protection Agency. Available from

www.hm-treasury.gov.uk.

HM Treasury, Feb. 2009b. UK Asset Protection Scheme. Available from

www.hm-treasury.gov.uk.

The Royal Bank of Scotland Group (RBS), Nov. 2009. The Royal Bank of Scot-

land Group plc (RBS) – Announcement on the APS and State Aid Discussions.

Available from www.rbs.com.

US Department of Agriculture, Dec. 2009. 2007 Census Publications. The Census

of Agriculture. Available from www.agcensus.usda.gov.

McNeil, A., Frey, R., Embrechts, P., 2005. Quantitative Risk Management: Con-

cepts, Techniques, and Tools. Princeton University Press.

McNeil, A. J., Kirchner, A., Kretzschmar, G. L., 2008. The Case for Fully Inte-

grated Models of Economic Capital. SSRN eLibrary.

Merton, R. C., 1974. On the Pricing of Corporate Debt: The Risk Structure of

Interest Rates. Journal of Finance 29 (2), 449–470.

Nicolato, E., Skovmand, D., 2010. Empirical Analysis of Affine LIBOR Market

Models. Working Paper.

Pederson, G. D., Zech, L., 2009. Assessing Credit Risk in an Agricultural Loan

Portfolio. Canadian Journal of Agricultural Economics 57 (2), 169–185.

Raible, S., 2000. Levy Processes in Finance: Theory, Numerics, and Empirical

Facts. University of Freiburg, Doctoral Thesis.

Repullo, R., Suarez, J., 2008. The Procyclical Effects of Basel II. SSRN eLibrary.

BIBLIOGRAPHY 103

Rydberg, T., 1997. The Normal Inverse Gaussian Levy Process: Simulation and

Approximation. Stochastic Models 13 (4), 887–910.

Sherrick, B. J., Barry, P. J., Ellinger, P. N., 2000. Valuation of Credit Risk in

Agricultural Mortgages. American Journal of Agricultural Economics 82 (1),

71–81.

Stokes, J. R., Brinch, B. M., 2001. Valuing Agricultural Mortage-Backed Securi-

ties. Journal of Agricultural and Applied Economics 33 (3), 493–511.

Yan, Y., Barry, P. J., May 2006. Loss-Given-Default on Farm Real Estate Loans:

Probability of Full Recovery. Agricultural Finance Review 66 (1), 47–59.

portfolio choice - au purepure.au.dk/portal/files/32826640/pjessenthesis_dec2010.pdf · 2011. 1....

Documents