Asset allocation under Solvency II - Adjusting investments for capital efficiency

ERIK HELLGREN FREDRIK UGGLA

Master of Science Thesis Stockholm, Sweden 2015

Tillgångsallokering i Solvens II - Inkludering av kapitalkrav vid investeringar

ERIK HELLGREN FREDRIK UGGLA

Examensarbete Stockholm, Sverige 2015

Tillgångsallokering i Solvens II -Inkludering av kapitalkrav vid investeringar

av

Erik Hellgren Fredrik Uggla

Examensarbete INDEK 2015:86 KTH Industriell teknik och management

Industriell ekonomi och organisation SE-100 44 STOCKHOLM

Asset allocation under Solvency II -Adjusting investments for capital efficiency

Erik Hellgren Fredrik Uggla

Master of Science Thesis INDEK 2015:86 KTH Industrial Engineering and Management

Industrial Management SE-100 44 STOCKHOLM

Examensarbete INDEK 2015:86

Tillgångsallokering i Solvens II – Inkludering av kapitalkrav vid investeringar

Erik Hellgren

Fredrik Uggla

Godkänt

2015-06-03

Examinator

Tomas Sörensson

Handledare

Gustav Martinsson Uppdragsgivare

SEB Kontaktperson

Marja Carlsson

Sammanfattning

Solvens II är ett nytt regelverk för försäkringsbolag inom EU som ska träda i kraft 2016.

Tidigare forskning har diskuterat effekterna av det nya regelverket och förutspår att det

kommer att påverka försäkringsbolagens tillgångsallokering. Syftet med denna studie är

att studera optimala tillgångsallokeringar för livbolag, både med avseende på interna

krav på risk och avkastning och externa kapitalkrav i Solvens II. En fallstudie utförs på

ett svenskt livbolag för att ta fram en modell för optimala tillgångsallokeringar, som även

tar hänsyn till livbolagets framtida utbetalningar. En optimal allokering tas fram med

hjälp av kvadratisk optimering på risk och kapitalkrav givet en viss förväntad avkastning

och den nuvarande allokeringen jämförs med olika optimala portföljer. Resultaten visar

att det är möjligt att optimera allokeringen både ur ett risk- och avkastningsperspektiv

samt kapitalkravsperspektiv, men att de optimala tillgångsportföljerna skiljer sig åt

markant. Detta arbete påvisar att det finns en betydande skillnad på risk, mätt genom

antingen historisk volatilitet eller kapitalkrav. Ett exempel är tillgångsklassen

hedgefonder som har en låg historisk volatilitet men har ett högt kapitalkrav i Solvens II.

Denna studie bidrar till befintlig forskning genom att utveckla ett ramverk för

investeringar för ett livbolag i Solvens II som tar hänsyn till kapitalkrav för olika

tillgångar.

Nyckelord: Solvens II, kapitalkrav, livförsäkring, marknadsrisk, portföljoptimering

Master of Science Thesis INDEK 2015:86

Asset allocation under Solvency II – Adjusting investments for capital efficiency

Erik Hellgren

Fredrik Uggla

Approved

2015-06-03 Examiner

Tomas Sörensson Supervisor

Gustav Martinsson Commissioner

SEB Contact person

Marja Carlsson

Abstract

Solvency II is a new regulatory framework concerning insurance companies in the

European Union, to be introduced in 2016. The effects of the regulation have been

discussed and previous literature believes it will have a significant effect on insurance

companies’ asset allocation. The aim of this thesis is to investigate the optimal asset

allocation for a life insurer with respect to internal risk-return requirements and external

capital requirements imposed by Solvency II. The thesis performs a case study on a

Swedish life insurer for the purpose of developing and evaluating an asset allocation

model which incorporates future liabilities of the life insurer. Through quadratic

optimization, the asset allocation is optimized for portfolios associated with a certain

expected return and the current allocation is compared to optimal portfolios. The results

show that it is possible to optimize the asset allocation from both a risk-return and

capital requirement perspective. However, they are subject to large shifts in asset

allocation. The thesis also shows that there is a large discrepancy of risk from a

standard deviation standpoint and regulatory capital charges. One example are hedge

funds which have shown a low historical volatility but are classified as an asset with high

risk in Solvency II. This study contributes to theory by providing an investment decision

framework for life insurers that includes capital charges for asset allocation.

Key-words: Solvency II, capital requirements, life insurance, market risk, portfolio

optimization

Acknowledgements

Firstly, we would like to thank our supervisor at KTH, Gustav Martinsson, for hishelpful guidance and support throughout the work of this thesis. Furthermore, wewould like to thank Marja Carlsson and Mikael Anveden at SEB Global FinancialSolutions for giving us the opportunity for this thesis and invaluable input through-out the process. Finally, we would like to thank the anonymous life insurer that hasprovided us with data crucial for the completion of the thesis and shared importantinsights into the life insurance industry.

Stockholm, May 2015Erik Hellgren & Fredrik Uggla

Table of contents1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Purpose and research question . . . . . . . . . . . . . . . . . . . . . . 41.4 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Disposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Institutional setting 62.1 The insurance industry . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Life insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Regulatory framework . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Pillar I — Quantitative risk-based capital requirements . . . . 8

3 Literature review 103.1 Solvency II and implications on asset allocation . . . . . . . . . . . . 103.2 Standard and partial models’ implications on SCR . . . . . . . . . . . 123.3 Asset allocation optimization with regulatory constraints . . . . . . . 13

4 Theoretical framework 144.1 Quadratic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Mean-variance optimization . . . . . . . . . . . . . . . . . . . . . . . 14

4.2.1 E�cient frontier . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Interest rate swaps and zero rates . . . . . . . . . . . . . . . . . . . . 164.4 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 Market risk module and SCR . . . . . . . . . . . . . . . . . . . . . . 17

4.5.1 Interest rate risk . . . . . . . . . . . . . . . . . . . . . . . . . 184.5.2 Equity risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5.3 Property risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5.4 Currency risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5.5 Spread risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5.6 Concentration risk . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Methodology 255.1 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Reformulation of SCR formula . . . . . . . . . . . . . . . . . . . . . . 265.3 Discounting of expected cash flows in Solvency II . . . . . . . . . . . 28

5.4 Calculation of SCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 Mean-variance optimization . . . . . . . . . . . . . . . . . . . . . . . 30

5.5.1 Optimization problem . . . . . . . . . . . . . . . . . . . . . . 305.5.2 Estimating covariances . . . . . . . . . . . . . . . . . . . . . . 31

5.6 Minimization of SCR . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.7 Combined optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 325.8 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.8.1 Case study data . . . . . . . . . . . . . . . . . . . . . . . . . . 335.8.2 Index data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.8.3 Expected index returns . . . . . . . . . . . . . . . . . . . . . . 365.8.4 Interest rate swap data . . . . . . . . . . . . . . . . . . . . . . 37

5.9 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.10 Reliability and validity . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Results and analysis 396.1 Current allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.1.1 Allocation per asset class . . . . . . . . . . . . . . . . . . . . . 396.1.2 SCR contribution . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.2 Optimization without investment constraints . . . . . . . . . . . . . . 426.2.1 Mean-variance optimization . . . . . . . . . . . . . . . . . . . 426.2.2 Optimization with respect to SCR . . . . . . . . . . . . . . . . 456.2.3 Combined optimization . . . . . . . . . . . . . . . . . . . . . . 476.2.4 Results overview . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.3 Optimization with investment constraints . . . . . . . . . . . . . . . . 536.3.1 Investment constraints overview . . . . . . . . . . . . . . . . . 536.3.2 Mean-variance optimization . . . . . . . . . . . . . . . . . . . 536.3.3 Optimization with respect to SCR . . . . . . . . . . . . . . . . 566.3.4 Combined optimization . . . . . . . . . . . . . . . . . . . . . . 586.3.5 Results overview . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.4.1 Mean-variance optimization . . . . . . . . . . . . . . . . . . . 636.4.2 Optimization with respect to SCR . . . . . . . . . . . . . . . . 646.4.3 Combined optimization . . . . . . . . . . . . . . . . . . . . . . 656.4.4 Choice of c . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.4.5 Investment constraints . . . . . . . . . . . . . . . . . . . . . . 666.4.6 Sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Conclusion 68

7.1 Summary of findings . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.2 Contribution and further research . . . . . . . . . . . . . . . . . . . . 69

Appendix A Solvency II regulation 74

Appendix B Additional results 75

Appendix C Asset class mapping 77

List of Figures1.1 Problem illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 SCR modules with sub-modules . . . . . . . . . . . . . . . . . . . . . 95.1 Liability cash flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.1 SCR contribution by risk factor . . . . . . . . . . . . . . . . . . . . . 416.2 Mean-variance optimization: expected return and standard deviation 436.3 Mean-variance optimization: asset allocation . . . . . . . . . . . . . . 446.4 SCR optimization: expected return and SCR . . . . . . . . . . . . . . 456.5 SCR optimization: asset allocation . . . . . . . . . . . . . . . . . . . 466.6 Combined optimization: expected return and standard deviation . . . 486.7 Combined optimization: expected return and SCR . . . . . . . . . . . 486.8 Combined optimization: asset allocation . . . . . . . . . . . . . . . . 496.9 Selected portfolios: expected return and standard deviation . . . . . . 516.10 Selected portfolios: expected return and SCR . . . . . . . . . . . . . 516.11 Mean-variance optimization with investment constraints: expected

return and standard deviation . . . . . . . . . . . . . . . . . . . . . . 546.12 Mean-variance optimization with investment constraints: asset allo-

cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.13 SCR optimization with investment constraints: expected return and

SCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.14 SCR optimization with investment constraints: asset allocation . . . . 576.15 Combined optimization with investment constraints: expected return

and standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . 596.16 Combined optimization with investment constraints: expected return

and SCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.17 Combined optimization with investment constraints: asset allocation 606.18 Selected portfolios with investment constraints: expected return and

standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.19 Selected portfolios with investment constraints: expected return and

SCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A.1 F up credit spread factors for corporate bonds . . . . . . . . . . . . . . 74A.2 F up credit spread factors for covered bonds . . . . . . . . . . . . . . . 75B.1 SCR implied correlation matrix between asset classes . . . . . . . . . 76B.2 Estimated correlation matrix between asset classes . . . . . . . . . . . 76

List of Tables4.1 Market risk correlation matrix . . . . . . . . . . . . . . . . . . . . . . 184.2 Interest rate stress factors . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Equity shocks by equity type . . . . . . . . . . . . . . . . . . . . . . . 214.4 Equity shock correlation matrix . . . . . . . . . . . . . . . . . . . . . 214.5 Credit quality step by rating . . . . . . . . . . . . . . . . . . . . . . . 235.1 Index mapping of asset classes . . . . . . . . . . . . . . . . . . . . . . 355.2 Asset classes: expected return and standard deviation . . . . . . . . . 376.1 Current asset allocation . . . . . . . . . . . . . . . . . . . . . . . . . 406.2 Asset classes: expected return and SCR contribution . . . . . . . . . 426.3 Mean-variance optimization: asset class frequency . . . . . . . . . . . 446.4 SCR optimization: asset class frequency . . . . . . . . . . . . . . . . 466.5 Selected portfolios: definition . . . . . . . . . . . . . . . . . . . . . . 506.6 Selected portfolios: results . . . . . . . . . . . . . . . . . . . . . . . . 526.7 Investment constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 536.8 Mean-variance optimization with investment constraints: asset class

frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.9 SCR optimization with investment constraints: asset class frequency . 576.10 Selected portfolios with investment constraints: results . . . . . . . . 62B.1 Risk factor sensitivity per asset class . . . . . . . . . . . . . . . . . . 75C.1 Asset classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Definitions

ALM — Asset liability managementBOF — Basic Own FundsEEA — European Economic AreaEIOPA — European Insurance and Occupational Pensions AuthorityEM credits — Emerging Market creditsEU — European UnionEUR — EuroHY credits — High Yield creditsIG credits — Investment Grade creditsMCR — Minimum Capital RequirementOECD — Organisation for Economic Co-operation and DevelopmentS&P — Standard & Poor’sSCR — Solvency Capital RequirementSD — Standard deviationSEK — Svensk kronaSTIBOR — Stockholm Interbank O�ered RateT-bills — Treasury billsUFR — Ultimate Forward RateVaR — Value-at-Risk

1 Introduction

This chapter will present the topic of the thesis by describing its background and givethe reader a short introduction to the insurance industry regulation. Relevant prob-lems are highlighted and discussed, after which the purpose and research question ofthe thesis are presented. This is followed by a discussion regarding the delimitationsof the thesis. Finally, a disposition for the remaining chapters is presented.

This thesis aims at investigating the optimal asset allocation for a life insurer withrespect to capital requirements imposed by the new regulatory framework SolvencyII in the European Union (EU). The new regulation is expected to have a significantimpact on asset allocation decisions for life insurers since there has not been a majorchange to the regulation in a number of years while models for risk measurementshave become more sophisticated. The topic concerning solvency of insurance com-panies has become frequently discussed since the financial crisis in 2008 when manyinsurance companies were in financial distress to their low solvency ratio. This thesiswill study a Swedish life insurer and develop a general investment decision modelwhich finds the optimal asset allocation with regards to return, risk and solvencycapital requirement (SCR). The results show that the current asset allocation ofthe life insurer can be optimized further using a combined optimization, reducingstandard deviation and SCR while keeping the expected return constant. However,this implies a large shift in asset allocation to assets which are coherent from botha risk-return and regulatory perspective.

1.1 Background

The role of the insurance industry in our society is about minimizing individualfinancial risk. By charging customers a yearly premium for covering their risks incase of certain events, the individual hedges himself against contingent, potentiallosses. The premium that the insurance company charges is set using risk modelsassessing the likelihood of certain risks in order to be able to pay all the customersthat are a�ected by the events and still make a profit. Following the collection, aninsurer is faced with the challenge of investing the premiums in financial assets togenerate returns.

To understand how Solvency II will a�ect a life insurer, it is important to understandthe business model of a life insurance company. A life insurer collects premiums fromtheir customers and in return, guarantees them a certain pension at a future point

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in time. In order to pay the specified pension at a certain time in the future, the lifeinsurer needs to invest the premiums paid to generate returns which involves takinginvestment risk. One critical scenario is when the investments are a�ected by a fallin value of assets, thus making the insurer unable to meet its payment obligations,resulting in insolvency.

The measurement of risk has attracted regulators to the industry, making sure thatthe insurance companies do not take excessive risks. The level of regulation hasincreased in recent years, following the financial crisis in 2007-2008. During thecrisis, the largest insurance company in the world, American International Group(AIG), went bankrupt following, in hindsight, risky investments. One step towardsa more unified regulation within the EU is the introduction of Solvency II which isa directive that regulates the solvency of insurance companies and is scheduled tocome into full e�ect on 1 January 2016. The directive is concerned with the stabilityof the insurance industry with two main purposes: to limit systemic risk and protectpolicyholders. The directive sets forth rules and guidelines on capital bu�er withregard to the assets that life insurers hold, the liabilities they have and other factors.The capital bu�er should protect the life insurers from financial distress in the caseof unlikely events.

1.2 Problem formulation

Optimizing asset allocation in order to generate high risk-adjusted returns on theinvestment portfolio is central to the business of a life insurer. Generating returnsis important since the policyholders are entitled to specified amount of pension atfuture points in time. At the same time, the risk of such investments must be ade-quately measured and kept at a reasonable level. The returns must be stable enoughto ensure that the value of assets do not decrease below the value of the liabilities.Following the introduction of Solvency II, finding the optimal asset allocation hasbecome more complex. The SCR puts another restriction on the investments of thelife insurer. The standard formula for computing the SCR under the market riskmodule includes varying capital charges for di�erent types of assets. Thus, di�er-ent asset classes may consume more or less regulatory capital, potentially makingcurrent asset allocations unable to meet all requirements simultaneously. The ef-fects of Solvency II has been discussed in previous literature. Braun et al. (2013)uses a quadratic optimization program to show that e�cient portfolios with respectto risk-return are inadmissible when taking SCR into account. The author arguesthat the new regulation will heavily restrict the choice of investment and that the

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contractual obligations of life insurers could be hard to fulfil. Fitch Ratings (2011)compare assets from risk-return parameters as well as capital charges and show thatthe asset allocation of life insurers could need a significant re-allocation with theintroduction of Solvency II, due to the regulatory ”expensiveness” of certain assets.They highlight Government bonds issued by members of the European EconomicArea (EEA) as an example of an attractive asset since they are free of spread risk. Inaddition, Schwarzbach et al. (2014) shows that investments in stock is an importantasset in the current low-yield interest environment but that it has high capital re-quirements. The author shows that German life insurers might not be able to meetits payment obligations due to the unfavorable capital requirements on investmentsin equity.

With the introduction of Solvency II, the problem for the life insurer is that previousasset allocations might not be feasible. If a life insurer keeps the same portfolio aspre-Solvency II, the amount of capital that is needed to be held might increase sig-nificantly. Furthermore, the life insurer needs to consider and balance both externalregulatory constraints and internal risk-return requirements when allocating assetsof the portfolio, as depicted in Figure 1.1.

Figure 1.1: Problem illustration

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1.3 Purpose and research question

The purpose of this thesis is to provide an investment decision framework for aSwedish life insurer that includes solvency capital charges for asset allocation andinternal risk-return requirements. The asset allocation of the portfolio will be alteredand the resulting changes in SCR will be measured. The most suitable portfoliowill come from an analysis of the relationship between expected return, standarddeviation and SCR. This will in turn lead to the development of e�cient portfoliofrontiers.

The main research question has been formulated as follows:

• How can a life insurance company integrate Solvency II capital requirementsinto the asset allocation decision?

In order to answer the research question, the thesis will aim to answer three sub-research questions deemed important for the study:

i) How can a portfolio be optimized from a risk perspective?

ii) How can a portfolio be optimized from a capital requirement perspective?

iii) How can a portfolio be optimized to satisfy both risk and capital requirements?

1.4 Delimitations

The study will investigate the case of a Swedish life insurer since it is believed tobe more relevant to study a real example than a fictional portfolio or a portfolioof the average life insurer. The investment decision framework developed will berelated to the investment rules, current allocation and liability cash flows of the lifeinsurer, and thus might be limited to the current environment for that life insurer.Furthermore, the study will only focus on optimizing the asset allocation within themarket risk module of Solvency II. The study will not address the SCR under theother sub-modules since they are not a�ected by asset allocation.

1.5 Contribution

Previous research has analyzed Solvency II, its implications for both life and non-life insurers in Europe and certain aspects of the asset allocation. The e�ects ofSolvency II on certain asset classes, the asset allocation of insurers and di�erences

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between internal models and the standard model for calculating market risk hasbeen studied. Since many insurers will have to use the standard model, SCR has tobe integrated in allocation decisions. However, there exists a research gap regardinghow a life insurer should allocate assets with regards to the new regulatory measuresand the optimal portfolio from a risk-return perspective.

The results from this study will be valuable for the life insurer in question as wellas other life insurance companies a�ected by Solvency II. The results will help lifeinsurance companies better understand the implications of asset allocations andhow they can more e�ectively reach a certain return with regards to risk and capitalrequirements. The academic contribution will be to add an investment decisionframework for life insurers and their asset allocation that includes solvency capitalcharges.

1.6 Disposition

The remainder of the thesis is structured as follows: Section 2 presents the institu-tional setting of the thesis, including the regulatory framework of Solvency II andthe insurance industry. Section 3 presents previous research within the field of Sol-vency II including a discussion about models used with respect to the regulation andoptimization of asset allocation. Section 4 presents the theoretical body of knowl-edge that is relevant for the thesis, including how to calculate SCR in the marketrisk module. Section 5 presents the methodology and data used in the thesis, includ-ing our developed model and a discussion regarding the data. Section 6 presents,analyses and discusses the results. Section 7 concludes the thesis by summarizingthe findings and proposing areas for further research.

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2 Institutional setting

In this chapter, an overview of the institutional setting for the topic of the thesis ispresented. First, a general overview of the insurance industry is presented, followedby a description of the life insurance industry in particular. Thereafter, a sectioncontaining a review of the coming European regulation of Solvency II is presented.This is followed by descriptions of the three pillars that constitute the regulation,with focus on Pillar I containing the market risk module.

2.1 The insurance industry

The insurance industry plays an important role in the national economy. It enablesindividuals and companies to obtain financial protection against di�erent types ofrisks which they might not have been able to pay for if any of the risks were tooccur. The insurance industry generally consists of two types of insurers, non-lifeand life. Non-life insurance is the practice of paying premiums to transfer the riskof damages caused by certain events on property such as a house, car or equipment.Life insurance pays compensation when the person insured is injured or dies, orwhen the insured person reaches a pensionable age. (Svensk Försäkring, 2013).

The Swedish insurance industry generated a premium income of around SEK 260 bnin 2013 and invested SEK 3,400 bn in the global economy (Svensk Försäkring, 2013).The total insurance industry had a balance sheet of SEK 3,400 bn of which SEK 2,900bn (85%) constitute life insurance companies (Statistiska centralbyrån, 2013). Theamount can be put in comparison to the Swedish gross domestic product, amountingto SEK 3 710 bn (World Bank, 2013), highlighting the specific importance of thelife insurance industry in the Swedish economy. In 2013, there were 404 registeredinsurance companies in Sweden of which most are small, local non-life insurancecompanies. The market is very concentrated to a few large companies and corporategroups. This is illustrated by the market shares of the five largest companies withinnon-life and life insurance of 83 percent and 62 percent respectively. The number ofinternational insurance companies present on the Swedish market has grown duringthe last couple of years and today amount to 41 companies.

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2.1.1 Life insurance

The focus of this thesis is life insurance products for occupational pensions. Theseare insurance products that contain insurance and savings components and are clas-sified as pension insurance. The life insurance companies have a ”reverse productioncycle”, since they charge their customers in advance and pay the cost of insuranceclaims in the future. This leads to a large sum of premiums that needs to be in-vested in a portfolio of assets each year. The assets shall cover the liabilities thatthe insurance company faces with high certainty, but at the same time yield return.A holistic perspective, frequently referred to as Asset Liability Management (ALM)therefore has to be adopted that considers risk and expected return of the assets inrelation to the value of the liabilities. A decline in the value of the invested assetsof a life insurer is mainly a risk for customers in the form of lower future pension(above the guaranteed pension). According to Finansinspektionen (2014), the lifeinsurance companies in Sweden are not generally posing a systemic risk, as the fourmajor banks do. This is due to short-term predictable payments on an aggregatedlevel, their size in relation to the total financial sector is small and that the insurersdo not have the same level of inter-linkage between them as the banks have. How-ever, on an international scale, the systemic risk of insurance companies has beensubject of much debate, notably during the financial crisis of 2007-08 when AIG wasbailed out by the U.S. government (The Economist, 2013).

2.2 Regulatory framework

The Solvency II initiative was launched in the year 2000 by the EU to create anew prudential supervisory system for risk assessment in the European insuranceindustry (Linder & Ronkainen, 2004). It will act as a framework for solvency rulesfor insurance companies that all EU member states should introduce in their re-spective insurance legislation in the coming years (Finansinspektionen, 2015). Thedevelopment of a new and EU-wide insurance regulation has proven to be a compli-cated and time consuming process. From the years 2001-2003, general specificationsof the new regulatory system were discussed and investigated (Eling et al., 2007).The European Commission requested several reports, of which one was the KPMG(2002) study that proposed a three pillar structure, similar to Basel II, for insur-ance solvency regulation. Since then, there has been an ongoing debate on how tooperationalize the overall purpose of Solvency II into specific rules and guidelines.The European Insurance and Occupational Pensions Authority (EIOPA) has been

7

advising the European Commission on the Solvency II project since 2004, includingfive quantitative impact studies and the development of technical standards. Therevision of the solvency framework aims to take current developments in insurance,risk management and finance techniques into account and streamline the way in-surance groups are supervised across the EU (EIOPA, 2014a; FSA, 2015). Sincethe regulation is introduced with the objective to cover unexpected losses, SolvencyII will also protect policyholders and reduce the risk for market disruption (FSA,2015). New freedom of investment for insurers will also be implemented throughthe ”prudent person principle”, meaning that the insurer is free to invest in anytype of asset as long as capital requirements are met (EIOPA, 2014a). Also, thenew regulation will take group diversification e�ects into account, contributing to amore e�cient capital allocation for shareholders.

Solvency II is organized under three pillars covering quantitative requirements (Pil-lar I), qualitative requirements (Pillar II) and enhanced disclosure and transparencyrequirements (Pillar III) (EIOPA, 2013). Pillar I regulates how technical provisionsand capital requirement ratios are to be calculated. The technical provision can beseen as a best estimate of the insurance and reinsurance obligations, also called liabil-ities (EIOPA, 2014a). The second pillar includes requirements for risk management,governance and specifications of supervisory processes with authorities to ensurethat the regulatory framework is combined with the insurers own risk-managementmodels and business decisions. Pillar III addresses transparency, reporting and dis-closure in order to enhance market discipline and increase comparability which inturn leads to a higher level of competition in the market.

2.2.1 Pillar I — Quantitative risk-based capital requirements

Pillar I sets out quantitative requirements, including the rules to value assets andliabilities, calculate capital requirements and identify eligible own funds to coverthese requirements. Solvency II will have forward-looking and economic capitalrequirements, i.e. tailored specifically after risks to each insurer to allow for optimalallocation of capital across the EU. In order to allow for timely and proportionatesupervisory intervention, two quantitative risk-based capital requirements will bemeasured: SCR and the minimum capital requirement (MCR). SCR is defined asthe Value-at-Risk (VaR) of the Basic Own Funds (BOF) subject to a confidencelevel of 99.5% over a one-year period, corresponding to an annual default probabilitybelow 0.5% (Gatzert & Martin, 2012; Fischer & Schlütter, 2014).

8

BOF is defined as the current value of assets less liabilities and should be su�cientto absorb unexpected losses, i.e. it should be higher than the SCR. The Solvency IIdirective defines certain stress scenarios for di�erent asset classes which a�ects lifeinsurers’ BOF. The SCR is subsequently calculated from the changes in the BOF.Under the stress scenarios, the BOF should be positive in order for the insurancecompany to comply with the regulation. The SCR calculation is divided into di�er-ent modules and sub-modules, according to Figure 2.1, which are then aggregatedaccording to predefined formulas. The market risk module is a central part of Sol-vency II and it captures e�ects of changes in valuation of financial instruments thata�ect BOF. Each market sub-risk is associated with at least one stress scenario.For a life insurer, both assets and liabilities are usually sensitive to changes in e.g.interest rates, making the market risk module the major contributor to the totalSCR (EIOPA, 2011).

Figure 2.1: SCR modules with sub-modulesThe figure illustrates the modules and sub-modules used to calculate the SCR.(EIOPA, 2014b)

9

3 Literature review

This section reviews earlier findings that constitute the basis of this study. It con-tains a review of the current literature on Solvency II and its implications on assetallocation, the use of di�erent models when calculating capital requirements and op-timal asset allocation under regulatory constraints. The Solvency II framework hasnot yet been formally implemented, resulting in a limited amount of literature on thesubject.

3.1 Solvency II and implications on asset allocation

Solvency I, the current regulatory framework for life insurers in the EU, came intoe�ect in the 1970s and has been important for the quality of supervision. Today,insurance companies face a di�erent business environment and risk managementmethods have changed significantly, creating a need for new regulation (Linder &Ronkainen, 2004). In comparison, the banking sector has stricter and more up-dated regulation in the form of the current Basel II and the ongoing adoption ofthe forthcoming Basel III framework. Solvency II is an EU directive with the aimto update the regulatory framework for supervision of insurance companies. Thenew regulation takes an holistic approach toward capital standards with the mainconcern of the amount of capital an insurance company need to hold to reduce therisk of insolvency in the case of unlikely events. The introduction of the regulationwill likely require major changes for insurance companies (Eling et al., 2007). Theregulation itself has been reviewed, analyzed and discussed a number of times (Lin-der & Ronkainen, 2004; Eling et al., 2007; Do�, 2008). Floreani (2013) analyses anddiscusses the VaR-based capital requirement imposed by Solvency II using a theo-retical model which includes risk-return profile, default probability and systematicvs. diversifiable risk. The author argues that the choice of risk measure is the mostimportant issue of Solvency II and by using a VaR capital requirement measure,it exposes insurance companies to a potentially huge systemic e�ect and increasedfragility of the insurance system since the VaR measure is unable to distinguishbetween systematic and diversifiable risk.

In addition to reviewing Solvency II in itself, the implications of the regulation oncertain parts of the market risk module has also been investigated. van Bragt et al.(2010) investigate the impact of Solvency II guidelines on the risk-return trade-o� for life insurance companies using the Dutch (FTK) regulatory framework and

10

reach the conclusion that capital requirements would change drastically comparedto Solvency I. By reducing the short-term risk, the long-term expected returns maydecrease for life insurers. The authors instead suggest an approach where multi-period calculations for di�erent scenarios should be performed to find the true risk-return trade-o� and adjust the investment policies thereafter.

Fitch Ratings (2011) has looked on the potential e�ects on European insurers’ assetallocation with the introduction of Solvency II. By comparing the capital chargeson asset classes and comparing them to the expected return, they have analyzedthe risk-adjusted return and discuss eventual re-allocations. They expect that therelative regulatory ”expensiveness” of for example longer dated corporate bonds willreduce insurance companies’ holdings of them. Government bonds from membersof the European Economic Area (EEA), on the other hand, are free of spread riskcharges under Solvency II and are therefore likely to be more attractive for insurers.

Horing (2012) uses the external Standard & Poor’s (S&P) rating of a representativeEuropean life insurer to assess whether the market risk requirements under SolvencyII will force insurance companies to alter their asset allocation. The logic is thatan S&P rating of A, which corresponds to a 99.5% confidence of being solvent oneyear ahead coincides with the 99.5% VaR assumption of the Solvency II standardmodel calibration. The author finds that a target S&P rating of A requires 68%more capital than the standard formula. The S&P model is only more restrictiveregarding the asset classes equities, alternatives and EEA bonds with respect to grosscapital charges but the standard formula allows for larger diversification e�ects thuslowering the overall capital charge. Since a target rating of A requires substantiallymore capital than the standard approach, the authors conclude that market riskcharges under Solvency II will be no practical constraint for insurance companiestargeting a higher rating. In fact, insurance companies targeting a rating over BBB,which corresponds to a probability of economic insolvency of 2.8%, should not beconstrained by Solvency II capital charges.

Schwarzbach et al. (2014) evaluates the attractiveness of stock investments fromboth a long-term and a risk-adjusted perspective in connection with the comingchanges to the regulations. Due to the current low-yield environment of governmentbonds, the author proposes that equities could be an important asset for life insurersto meet the payments of the promised returns. The paper shows that German lifeinsurers will have a harder time meeting their payment obligations due to the highercapital requirements on equity investments, especially in a market environment withextremely low interest rates and more volatile risk premiums for interest-bearing as-

11

sets. Relying on investments in fixed income instruments will decrease the potentialfor future investment returns, especially in comparison with mutual funds. Theauthors point out that insurance asset managers have reduced their exposure toequities in the last decade which could be an indication of the adaption to SolvencyII.

3.2 Standard and partial models’ implications on SCR

In addition to using the standard model for SCR calculation in Solvency II, insurershave the option to use full or partial internal models to calculate the SCR. Internalmodels must be approved by the regulator which can be a long process.

Gatzert & Martin (2012) develop an internal model with equity, interest and spreadrisks for a portfolio of stocks and bonds without interest rate sensitive liabilities. Foreach risk, a stochastic model is set up and the aggregation of risks is done througha Gaussian copula with empirical, linear correlation coe�cients. The authors thensimulate 100,000 paths of the internal model to derive a 99.5% VaR consistent withthe aim of the Solvency II standard formula. They conclude that although thestandard approach is easier to use, it both under- and overestimates risk of invest-ments compared to if an insurer uses an internal model. Especially EEA governmentbonds, they point out, have no spread risk charge in the standard model, which maylead to an underestimation of asset risk for lower rated government bonds. In gen-eral, however, an internal model approach can lead to lower capital requirementsdue to higher benefits from diversification. Especially the spread risk and equityrisk components can be lowered with a customized internal model according to theauthors.

Braun et al. (2013) propose yet another internal model for a European life insurer.The general methodology is to create a minimum variance portfolio of BOF. To thatend, empirical risk-return profiles for six asset classes are estimated from historicaldata. Together with the assumption of a flat interest rate for discounting liabilitiesand the assumption of 10 year liability duration, e�cient portfolios for target re-turns can be calculated. The authors use a quadratic optimization program with anumber of constraints to calculate an e�cient frontier from a risk-return perspec-tive and the corresponding SCR for each portfolio. Under this framework, a largepart of the e�cient portfolios are inadmissible under Solvency II which may leadto severe asset management biases according to the authors. They conclude thatSolvency II will heavily restrict the choice of investments, not take the duration gap

12

between assets and liabilities into account and promote ine�cient instead of e�cientportfolios. In addition, the authors point out that in many cases the contractualobligations of insurance companies can be di�cult to fulfil and that a widespreadrestructuring of portfolios of European insurance companies can impact certain assetclasses regarding demand and pricing.

3.3 Asset allocation optimization with regulatory constraints

Fischer & Schlütter (2014) explore the optimal capital and investment strategy ofan insurer that maximizes shareholder value when capital requirements are basedon a standard formula. The authors set up a model that accounts for the limitedliability that shareholders face since the maximum amount they can lose is theendowed equity. The model takes equity and interest rate risks into account andassumes that the insurer can invest assets either in equities or at a risk-free rate.The authors conclude that the calibration of the standard formula’s stock risk hasa strong influence on the capital, investment strategy and default probability ofthe insurer. The weight of assets invested in equities depends largely on capitalrequirements in the standard formula. Low capital charges for equity risk may leadto additional investment risk and vice versa.

13

4 Theoretical framework

This chapter will introduce the reader to all relevant theory used in the remainder ofthe study. First, a theoretical section about quadratic optimization, its properties andhow it is performed is presented. Mean-variance optimization is then explained ina similar fashion and related to the e�cient frontier concept. The following sectiondescribes interest swaps and how they are used to calculate zero rates. The concept ofduration is explained in the next section. This is followed by a thorough descriptionof the market risk module and the calculation of SCR within each sub-module. Alltheory in Section 4.5, including calculations and tables, is specified in the TechnicalSpecifications (EIOPA, 2014a,b).

4.1 Quadratic optimization

One general method used in this thesis is quadratic optimization of convex functions.Consider the quadratic form

f(x) = 12x

T

Qx ≠ x

T

b, (4.1)

where Q is a positive-semidefinite matrix with all eigenvalues Ø 0. The functionf(x) is then also convex. In the case when the matrix Q is positive definite, i.e. itseigenvalues are strictly greater than 0, there exists a unique minimum of the functionf(x). In the case of one ore more eigenvalues that are 0, minimization of f(x)has infinitely many solutions which are linear combinations of the eigenvector(s)associated with eigenvalue(s) 0 (Luenberger & Ye, 2008).

Solving the optimization problem of minimizing f(x) is referred to as quadraticoptimization or quadratic programming.

4.2 Mean-variance optimization

Mean-variance optimization is a quadratic investment principle, based on means andvariances of asset returns, used to find the optimal portfolio (Markowitz, 1952). Aninvestor has a set of financial instruments to invest in, with returns all assumed tobe normally distributed with mean E[R

i

] = µi

and variances V ar(Ri

) = ‡2

i

. Thereturns are defined as the relative change of value of one asset between time t andt + 1. R

p

is the total return of the portfolio and w are the monetary weights of

14

di�erent assets. The expected portfolio return becomes

E[Rp

] =ÿ

n

i=1

E[Ri

]wi

=ÿ

n

i=1

µi

wi

= w

T

µ. (4.2)

The total portfolio variance is expressed as

V ar(Rp

) =ÿ

n

i=1

V ar(Ri

)wi

=ÿ

n

i=1

ÿn

j=1

wi

wj

‡i

‡j

flij

= w

T �w, (4.3)

where flij

= 1 is the correlation coe�cient between asset i and j and � is thecovariance matrix for the asset returns R

i

.

An underlying assumption regarding the optimization problem is that an investor isrational as to invest in a certain number of assets to achieve the lowest possible vari-ance, given a certain expected return threshold (Markowitz, 1952). If the objectiveis to minimize the variance of the investments while receiving a certain expectedreturn, we set up the optimization problem

minw

12w

T �w

subject to w

T

µ Ø µ0

V0

w

T 1 Æ V0

wi

Ø 0,

(4.4)

where µ0

V0

is the lower bound on expected value of the investments.

An investor, such as a life insurer, has certain liabilities, Li

, in the future at timesti

. The value of the sum of the discounted liabilities is calculated with the formulaL = q

n

i=1

Li

di

, where di

represents a discount factor at time i. The optimizationproblem of minimizing variance for a portfolio can also be formulated as maximizingexpected returns under a variance constraint. Since the investor has a liability inthe future, it needs to be incorporated in the expression of maximizing expectedreturns (Hult et al., 2012). The optimization problem (4.5) optimizes the expectedvalue of assets minus liabilities under a variance constraint.

maxw

E[wT

R ≠ L]

subject to w

T �w Æ ‡2V 2

0

w

T 1 Æ V0

wi

Ø 0,

(4.5)

15

where R is the vector of asset returns.

4.2.1 E�cient frontier

The monetary weights of di�erent assets are wi

for i = 1, 2, 3, ..., n and a combinationof weights under the constraint q

n

i=1

wi

Æ V0

make up a portfolio. Under theconstraints, there exists a large and finite number of portfolios, all associated withan expected return µ and a variance ‡2, calculated using formulas presented above.All portfolios can be plotted into a space of µ and ‡ and the e�cient frontier ismade up of all portfolios with the highest return for every variance (or the lowestvariance for every expected return) (Markowitz, 1952). In this thesis, the e�cientfrontier will be an important tool to analyze how the current asset allocation canbecome more e�cient, by moving towards a higher expected return or lower risk.

4.3 Interest rate swaps and zero rates

An interest rate swap is an agreement between two parties to exchange floating andfixed interest payments on some notional amount L. The swap is considered to beat par if the present value of such an agreement is zero. Let the fixed rate paymentsbe cL, i.e. the fixed interest rate be c and the floating rate payments be for exampleStockholm Interbank O�ered Rate (STIBOR). Let d

i

denote the discount factor foryear i. The present value of the floating rate payments is then L(1 ≠ d

n

). Thisequals to the present value of investing the notional amount L at t

0

, collecting thefloating rate payments until t

n

and then receiving the notional L again. The presentvalue of the fixed rate payments cL is cL

qn

i=1

di

. For a swap contract to have aninitial value of zero, the present value of the floating rate payments and fixed ratepayments must be equal. It therefore holds that

c = 1 ≠ dnq

n

i=1

di

. (4.6)

Given the swap rates c for di�erent maturities it is possible to calculate the discountfactors d

i

from equation 4.6. From the discount factors di

one can then calculatethe swap zero rates r

i

asd

i

= 1(1 + r

i

)i

. (4.7)

This procedure can be used to construct a yield curve from swap market data (Hultet al., 2012).

16

4.4 Duration

The duration of an interest rate bearing asset (or liability) is a measure of interestrate sensitivity. Consider a bond with price B, coupon payments c

i

at times ti

andyield y which are related by

B =ÿ

n

i=1

ci

e≠yti .

The duration is then defined as

D =q

n

i=1

ti

ci

e≠yti

B.

For a small change �y in the yield, the following first order linear approximationholds: (Hull, 2006)

�B ¥ ≠BD�y

In this thesis, duration will be used for the calculation of SCR.

4.5 Market risk module and SCR

The market risk module is supposed to capture the e�ect of financial instrumentvolatility on the BOF. The total SCR for market risk, SCR

mkt

, consists of theinterest, equity, property, spread, currency and concentration risk (for an overview,see Figure 2.1). The individual market sub-risks, hereafter called market risk factors,are combined to the total SCR for market risk through the following formula

SCRmkt

=Ûÿ

r,c

CorrMktr,c

· Mktr

· Mktc

, (4.8)

where CorrMktr,c

corresponds to the entries of the correlation matrix CorrMkt

in Table 4.1 and Mktr,c

are the capital requirements of the individual market riskfactors.

The constant A in CorrMkt is equal to 0.5 when considering the risk scenario ofa decrease in interest rate and 0 otherwise. For a typical life insurer, a durationmismatch makes the interest rate down scenario critical since the present value ofliabilities will increase, thus decreasing the BOF. Typically, the duration of liabilitiesis longer than for assets, which makes the insurer’s BOF sensitive to parallel shifts

17

Table 4.1: Market risk correlation matrixThe matrix, CorrMkt, shows the correlation between the sub-modules and is usedto calculate total SCR through aggregating the individual charges. The matrix isdefined in the technical specifications (EIOPA, 2014b, p. 138)

Interest Equity Property Spread Currency ConcentrationInterest 1Equity A 1Property A 0.75 1Spread A 0.75 0.5 1Currency 0.25 0.25 0.25 0.25 1Concentration 0 0 0 0 0 1

in the yield curve. When interest rates decrease, the present value of liabilities willincrease. At the same time, interest rate bearing assets will also increase in value,but usually not enough to cover the increase in value of liabilities. The net e�ect onBOF is then negative.

4.5.1 Interest rate risk

The interest rate risk sub-module includes all assets and liabilities that are sensitiveto changes in the term structure of interest rates or interest rate volatility. Thecapital requirement for interest risk uses two predefined scenarios:

MktUp

int

= �BOFup

MktDown

int

= �BOFdown

,(4.9)

where �BOFup

and �BOFdown

result from the revaluation of interest rate sensitiveassets and liabilities under an interest rate up and an interest rate down scenariosrespectively. Stress scenarios for di�erent maturities are defined according to Table4.2, where sup(t) and sdown(t) are the stress factors of the prevailing yield curve thatare applied in the stress scenario. The interest rate for maturity t, r

t

, is multipliedby the relevant stress factor sup(t) or sdown(t). The stressed rate in the interest ratedown scenario is calculated as rdown

t

= rt

sdown(t). As an example, the prevailing 15-year interest rate would be multiplied by the factor (1+sdown(15)) = 63% accordingto Table 4.2. If the 15-year interest rate r

15

were for example 1%, the stressed raterdown

t

would be 0.63%. For maturities between 20 and 90 years, the stress factorsshall be linearly interpolated.

The interest rate that is used to discount liabilities and calculate the stress scenarios

18

Table 4.2: Interest rate stress factorsThe table shows the relative interest rate change, in percent, of the risk free rate permaturity. The two scenarios shown represent interest rate movement up and down,respectively. The relative change in interest rate of maturities between 20 and 90years are interpolated.

Maturity t (years) Relative change sup(t) Relative change sdown(t)1 70% -75%2 70% -65%3 64% -56%4 59% -50%5 55% -46%6 52% -42%7 49% -39%8 47% -36%9 44% -33%10 42% -31%11 39% -30%12 37% -29%13 35% -28%14 34% -28%15 33% -27%16 31% -28%17 30% -28%18 29% -28%19 27% -29%20 26% -29%

90 or longer 20% -20%

is called the basic risk free rate. The construction of the basic risk free rate fordi�erent maturities is prescribed by the regulator. The general procedure for theSwedish basic risk-free rate uses plain vanilla interest rate swaps up to a maturityof 10 years, which is considered the last liquid point. After that, the Smith Wilsonextrapolation method for convergence to the Ultimate Forward Rate (UFR) 4.2% isused (EIOPA, 2014b, pp. 142). For details on the implementation, refer to Section5.3.

For the revaluation of assets under the interest down scenario, we use a simplified,duration based approach. We calculate the change in value of an interest rate bearingasset A

i

as

19

�Ai

Ai

= (rt

≠ rdown

t

)DAi ,

where rt

and rdown

t

are the interest rates for maturities corresponding to the durationD

Ai .

The present value of liabilities is calculated as

L =ÿ

n

t=1

dt

lt

,

where lt

is the best estimate of future payment and dt

is the discount factor, both attime t. This can instead be formulated using risk-free interest rates r

t

by applyingEquation 4.7

L =ÿ

n

t=1

lt

(1 + rt

)t

,

and we can observe that the change in the present value of liabilities is dependent onthe change in risk-free interest rate (Hult et al., 2012). The stressed present valueof the liabilities is calculated using the stressed yield curve of the interest rate downscenario. The di�erence in value of liabilities becomes

�L =ÿ

n

t=1

lt

(1 + rt

)t

≠ÿ

n

t=1

lt

(1 + rdown

t

)t

.

4.5.2 Equity risk

The equity risk consists of all assets and liabilities that are sensitive to changesin equity prices. The calculation of the capital requirement for equity risk splitsequities into two types: Type 1 and Type 2. Type 1 equities are equities listed inregulated markets in countries which are member of the EEA or the OECD. Type2 equities are equities not covered by Type 1 criteria and comprise all investmentsother than those covered in the interest rate risk sub-module, property risk sub-module or the spread risk sub-module, in the case where a look-through approach isnot possible. This includes for example hedge funds, private equity investments andemerging market equities. The equity shock scenarios for the two types of equityare defined according to Table 4.3.

20

Table 4.3: Equity shocks by equity typeThe table shows the relative movement of the value of assets and liabilities whichare sensitive to changes in equity prices. The two movements correspond to the typeof equity that the assets or liabilities are sensitive to.

Type 1 Type 2Equity shock

i

39% 49%

The capital requirement for each category i is determined as the result of a predefinedstress scenario in the equation

Mkteq,i

= max(�BOF |Equity shocki

; 0).

In a second step, the capital requirement for equity risk is derived by combiningthe capital requirements for the individual categories using a correlation matrix asfollows

Mkteq

=Ûÿ

rxc

CorrIndexrxc · Mktr

· Mktc

,

and with a correlation matrix as defined in Table 4.4.

Table 4.4: Equity shock correlation matrixThe matrix, CorrIndex, is used to multiply the capital requirements of Type 1 andType 2 equity shocks to reach the SCR for equity risk.

Type 1 Type 2Type 1 1Type 2 0.75 1

The equity shocks in Table 4.4 of 39% and 49% can be modified using the ”symmetricadjustment mechanism” on the equity capital charge. The equity capital chargewill be lowered or raised with between 0% and 10% in order to prevent pro-cyclicalbehavior. The size of the adjustment is calibrated as to be high when equity marketshave risen or fallen substantially recently and low otherwise (EIOPA, 2014b, p.147).In this thesis, the standard capital charges of 39% and 49% are used.

21

4.5.3 Property risk

The property risk consists of changes in value of assets, liabilities and financialinvestments and includes the following:

• land, buildings and immovable-property rights

• property investment for the own use of the insurance undertaking

The property shock is the immediate e�ect on the net asset value of asset andliabilities in the event of a 25% decrease in their value. The capital requirement forproperty risk is determined by

Mktprop

= max(�BOF |Property shock;0).

4.5.4 Currency risk

The currency risk consists of the change in the level or volatility of foreign currenciesagainst the local currency. Foreign currencies are defined as currencies not used inthe undertaking of financial statements. The currency position of each currencyshould include any instrument that is not hedged against currency risk. This meansthat investments in foreign currencies of interest rate, equity, spread and propertyrisk has to be taken into consideration. Two predefined scenarios a�ect the capitalrequirement of currency risk: upward shock and downward shock, both with a valueof 25% of the foreign currency C against the local currency. The capital requirementis based on

MktUp

fx,C

= max(�BOF |Fx upward shocki

; 0)MktDown

fx,C

= max(�BOF |Fx downward shocki

; 0).

4.5.5 Spread risk

The spread risk covers di�erent types of bonds, asset-backed securities and tranchesof structured credit products. The capital requirement is dependent on the changesin value of assets, liabilities and other financial instruments due to changes in creditspreads. The capital requirement for spread risk is determined by

Mktsp

= Mktbonds

sp

+ Mktsecuritisation

sp

+ Mktcd

sp

.

22

The aggregated formula above consists of the formulas below, where credit derivativespread risk has two predefined scenarios.

Mktbonds

sp

= max(�BOF |Spread shock on bonds; 0)Mktsecuritisation

sp

= max(�BOF |Direct spread shock on securitisation positions; 0)Mktcd

sp,upward

= max(�BOF |Upward spread shock on credit derivatives; 0)Mktcd

sp,downward

= max(�BOF |Downward spread shock on credit derivatives; 0).

The spread risk e�ect on the net value of asset and liabilities is immediate followinga widening of credit spreads and is calculated as

Spread shock =ÿ

i

MVi

· F up(ratingi

; durationi

),

where F up(ratingi

; durationi

) is a function that depends on the credit quality stepof the credit risk exposure and is calibrated to deliver a shock consistent with 99.5%VaR following a credit spread widening. The factors F up are set out in the SolvencyII technical specifications (see Table B.1, Appendix A). For an AAA rated mortgagebond with duration 5 years, the factor F up is 0.7% resulting in a stress of 3.5% (seeTable A.2, Appendix A). The mapping of S&P ratings to Solvency II credit qualitysteps is presented in Table 4.5.

Table 4.5: Credit quality step by ratingThe table shows the mapping of external ratings to credit quality steps as specifiedin (EIOPA, 2014c, p.19). The credit quality steps are used to calculate the relativemovement of the value of assets, liabilities and other financial instruments due tocredit spreads.

S&P rating Credit quality stepAAA 0AA 1A 2BBB 3BB 4Other 5-6

23

4.5.6 Concentration risk

The concentration risk concerns the accumulation of exposures with the same coun-terparty and not any other types of concentration (e.g. industry). The scope of therisk extends to assets considered in the equity, spread and property risk sub-modulesand excludes all assets covered by the counterparty default risk module to eliminateoverlap in SCR calculations. (EIOPA, 2014c, pp. 168)

Capital charges stemming from concentration risk are not included in the remainderof this thesis. We will assume a well-diversified investment portfolio with no singlename expose over regulatory thresholds.

24

5 Methodology

The method chapter provides the reader with a thorough overview of the di�erentmethods used in the thesis. A discussion regarding the choice of method will bepresented first, followed by a step-by-step method of calculating the SCR of the mar-ket risk module and a section containing the valuation of expected cash flows. Thefollowing sections are concerned with optimization problems with di�erent objectivefunctions. This is followed by a presentation and discussion of the data used. Fi-nally, discussions regarding the limitations, reliability and validity are provided.

5.1 Case study

This thesis performs a case study on a Swedish life insurer for the purpose of de-veloping and evaluating the asset allocation model. To that end, the investmentportfolio for occupational pensions of the insurer is studied. The insurer bears theinvestment risk of the pension plans as opposed to defined contribution schemeswhere the future pension benefits of the policyholder is subject to investment risk.The yearly paid premiums are invested by the insurance company that guarantee aminimum return over a certain time period. The policyholder will also participatein excess returns of the investment portfolio. In this setting, where the insurancecompany bears investment risk, the balance sheet of the insurance company is sub-ject to market risk and Solvency II is applicable. The studied insurance companyhas chosen to calculate the SCR according to the standard formula as specified byEIOPA which makes it suitable for this study.

In order to answer the overall research question regarding the integration of SolvencyII capital requirements into the asset allocation decision, the current asset allocationand solvency of a Swedish life insurer will be used as a starting point. The capitalrequirements can be seen as rules that have di�erent applications for di�erent typesof risk exposures and are designed to be used on e.g. a specific asset allocation. Ifdata on e.g. asset allocation for the life insurance industry as a whole in Swedenwere used, the analysis would yield very high level results that would only apply to ageneral set of asset classes without much granularity. Furthermore, the structure ofpension payment obligations, i.e. liabilities, is proprietary data that is not publiclyavailable. In order to be able to perform an analysis on optimal asset allocation,the interest rate sensitivity of liabilities has to be taken into account and thereforea case study where future liability cash flow data is available is suitable.

25

5.2 Reformulation of SCR formula

In order to be able to perform the calculations of the SCR for market risk and en-able quadratic optimization methods, we reformulate the SCR into vector notation.Equation 4.8 can be formulated in vector notation as

SCRmkt

=Ô

MktT · CorrMkt · Mkt, (5.1)

where Mkt is a column vector containing the di�erent market sub-risks and MktT

is its transpose. The market sub-risks, Mkti

, are the di�erences in BOF for stressscenario i. As a first step Mkt

i

can be formulated as the monetary amount ui

(theexposure to market sub-risk i) that is multiplied by the stress factors for di�erentmarket sub-risks. For example, for Swedish listed equity that is classified as Type1 equity, a stress factor of 39% shall apply to the monetary amount u invested inSwedish equities. Mkt

i

for each sub-risk can then be formulated as the monetaryamount for which the sub-module is applicable multiplied by its stress factor, i.e.Mkt

i

= ui

·SCRrel,i

where SCR

rel

contains the relative stress factor of each marketrisk

SCR

rel

=

Q

ccccccccccccca

InterestType 1 equitiesType 2 equities

PropertySpread

Currency

R

dddddddddddddb

=

Q

ccccccccccccca

1%39%49%25%1%25%

R

dddddddddddddb

. (5.2)

.

With this notation, Equation 4.8 becomes

SCRmkt

=Ûÿ

r,c

CorrMktr,c

· Mktr

· Mktc

=Ûÿ

r,c

CorrMktr,c

· ur

· SCRrel,r

· uc

· SCRrel,c

=Ûÿ

r,c

CorrMktr,c

· SCRrel,r

· SCRrel,c

· ur

· uc

=Ò

u

T �SCR

u,

(5.3)

26

where

�SCRr,c = CorrMkt

r,c

· SCRrel,r

· SCRrel,c

and u =

Q

cccca

u1

...u

n

R

ddddb.

The interest and spread stress factors are set to 1% since they depend on e.g. du-ration and rating of an interest rate bearing asset and will be modified by changingthe weights u

i

.

We now have an expression for SCRmkt

that depends on the di�erent weights ui

and the Solvency II risk factors. Equation 5.3 is exactly the standard deviation of asum of random variables with covariance matrix �

SCR

and weights u. This is not acoincidence; the aggregation of the market sub-risks under Solvency II is based onthe assumption of a multivariate normal distribution. The Solvency II market riskfactors can be treated as normally distributed variables which have the correlationstructure defined by the matrix CorrMkt and the standard deviations of SCR

rel

.In reality though, they are not random variables but fully deterministic.

With the expression in Equation 5.3 we can calculate SCRmkt

in a simpler way. Onequestion that arises, however, is how we can express SCR

mkt

as a function of theallocation to di�erent asset classes directly. It turns out that this is possible, withsome minor modifications. The general procedure is to find the factor sensitivitymatrix A that expresses the SCR of asset classes as sums of market risk factors. Inessence, this is a simplified multifactor model as described in (Campbell et al., 1997,pp. 219).

Take for example SCRmkt

for a portfolio consisting of Type 1 equities denominatedin EUR. The equities contribute to equity risk with 39% of their value and to thecurrency risk with 25% of their value for a Swedish investor. The total SCR

mkt

for a portfolio consisting of only equities of this type, however, must be calculatedaccording to Equation 5.3. Treating the Solvency II market risk factors as normallydistributed random variables Z

i

with mean zero and standard deviation SCRrel,i

, wecan express the Solvency II capital requirement for the Type 1 EUR-denominatedequities as

SCR(EUR-equities) = SCR(1 · Zequity

+ 1 · Zcurrency

).

Then SCRmkt

of this portfolio is equal toÒ

Varimplied

(EUR-equities). Similarly,corporate bonds can be expressed as a sum of the risk factors Z

interest

and Zspread

.The returns of every asset class i can in fact be expressed as a function of the

27

normally distributed risk factors and the expected return µi

of each asset class:

Ri

= ai,1

Z1

+ ai,2

Z2

+ . . . + ai,n

Zn

+ µi

. (5.4)

In vector notation, Equation 5.4 can be expressed as

R = AZ + µ,

where

A =

Q

cccca

a11

· · · a1n

... . . . ...a

k1

· · · akn

R

ddddb, R =

Q

cccca

R1

...R

k

R

ddddband µ =

Q

cccca

µ1

...µ

k

R

ddddb.

It is observed that E[R] = µ. It also holds that Covimplied

(R) = A�SCR

A

T =�

SCR

, which is a general finding for multivariate normal distributions ((Gut, 2009);(Campbell et al., 1997)). �

SCR

can be seen as the Solvency II implied covariancematrix between assets1. SCR

mkt

can now be expressed directly from the asset classesas

SCRmkt

=Ò

w

T �SCR

w, (5.5)

where w is a vector of monetary weights to the di�erent asset classes. The entriesof A are set according to how much each asset class contributes to the Solvency IIrisk factors assuming that the insurer’s assets in each asset class are representativefor the SCR of the asset class (see Table B.1, Appendix B).

5.3 Discounting of expected cash flows in Solvency II

To calculate the technical provisions, the best estimate of future liabilities for alife insurer, the basic risk free rate along with the estimated future cash flows fromliabilities are needed. In Solvency II, the calculation of the basic risk free rateis prescribed by the regulator and is based on swap rates from 1-10 years. Theregulator has determined that 10 years is the last liquid point for Swedish interestrate swaps. After that, the forward rate shall within 10 years converge to the UFRset by the regulator. This reduces the sensitivity of technical provisions to interest

1To simplify calculations and receive a so called base correlation matrix we assume that theSCR for Type 1 equities and Type 2 equities are aggregated in one single correlation matrix andnot in a two-step procedure as proposed in Solvency II (see Section 4.5.2). In general, there isno unique base correlation matrix (see for example FilipoviÊ (2009)). We rely on an approximateprocedure where we introduce Type 1 and Type 2 equities directly in CorrMkt.

28

rate volatility of long term interest rates and the present value of liabilities do notdepend on interest rates longer than 10 years. The procedure for calculating thebasic risk free rate is as follows

1. Extract market data for 1-10 year plain vanilla interest rate swaps. Subtractthe credit risk adjustment in order to correct for the credit risk premium ofswap rates that comes with the 3-month indexation of interest rate swaps. Itis defined as the rate di�erence between the 3-month STIBOR and a 3-monthovernight indexed swap

2. Calculate the zero coupon rates for maturities 1-10 years based on the adjustedswap rates (described in section 4.3)

3. Extrapolate the zero rates from 10 years to e.g. 100 years with the Smith-Wilson extrapolation method. Use a convergence period of 10 years to theUFR. Details on the Smith-Wilson extrapolation can be found in (EIOPA,2014c, pp. 24-26)

4. Discount the expected cash flows with the basic risk free rate

5.4 Calculation of SCR

The first step of the method is to calculate the current SCR of the life insurers’ port-folio using their asset allocation and future liabilities. This will, later in the thesis,be put in relation to the current solvency to see if it is possible to take increasedpositions in assets with higher expected returns or if a major asset allocation shiftto less risky assets is needed. The current SCR is calculated as follows

1. Calculate the present value of the liabilities by discounting future cash flowsusing the basic risk-free rate (see Section 5.3)

2. Map all investments into the asset classes T-bills, Government bonds, AAACredits, IG credits, Swedish Equities, Global equities, Real estate, Hedgefunds, Private equity, EM credits and HY credits

3. Assign market risk factors to each asset class and calculate the SCR of eachmarket risk factor (see Section 4.5)

4. Aggregate the SCR of the risk factors to a total market risk SCR (see Section4.5)

29

With the above calculations, we can also extract the SCR charge per asset classwhich can then be used in the optimization problems in Section 5.6 and for calcu-lation of SCR using asset allocation weights obtained from the mean-variance, SCRand combined optimization.

5.5 Mean-variance optimization

5.5.1 Optimization problem

To generate capital e�cient portfolios given certain risk-return requirement, we willuse quadratic optimization and minimization of variance outlined in Section 4.2.Let BOF = A ≠ L be our objective function where A denotes the assets and L thetechnical provisions, i.e. the present value of liabilities. Let R

i

, i = 1 . . . k, denotethe returns of each asset class, R

L

the return of liabilities, wi

the relative weight ofthe allocation to each asset class and w

L

the relative weight of liabilities expressedas fraction of the assets. The returns on BOF is then R = q

k

i=1

wi

Ri

+ wL

RL

. Thevariance of the BOF is then

Var(BOF (w)) = w

T �w, (5.6)

where � is the covariance matrix of assets and liabilities, w = (wA

, wL

) = (w1

, . . . , wk

, wL

)T

is the vector of portfolio weights.

We can then set up the optimization problem for k assets as

minw

12w

T �w

subject to w

T

A

1 = 1w

L

= L/A

E[wT

R] = Rrequired

wi

Ø 0 fori = 1, . . . , k.

(5.7)

The optimization problem (Equation 5.7) is sensitive to the assumed expected re-turns, variances and correlation amongst asset classes and di�erent estimates ofthese input parameters will impact the optimal portfolios (Hult et al., 2012).

30

5.5.2 Estimating covariances

To solve the optimization problem in Equation 5.7 for di�erent values of Rrequired

weneed to estimate the covariance matrix � with historical returns. Returns for dif-ferent asset classes can be extracted from relevant index time series. However, theredoes not exist a suitable index for change in value of pension liabilities and thereforethe optimization procedure requires the treatment of liabilities as an asset class forwhich correlations to other asset classes and historical volatilities are needed. Togenerate a series of historical returns for liabilities, we model the liabilities as beingsensitive to interest rates only. As for bonds, the present values of liabilities willincrease with falling interest rates and vice versa. The procedure for generating atime series of liability returns is the following

1. Calculate the basic risk free rate at tj

. To simplify calculations, we use linearinterpolation instead of the Smith-Wilson method. The forward rate betweenyear i and i + 1, i = 11, . . . , 19 is

rforward

i

= rforward

10

+ i ≠ 1010 (rforward

20

≠ rforward

10

), (5.8)

where rforward

10

= d10d9

≠1 is the forward rate between 9 and 10 years and rforward

20

the prevailing 1-year forward rate 20 years from now. EIOPA set this rate tothe UFR 4.2%.

2. Discount the liability cash flows with the prevailing zero coupon rates at tj

.Assume that the expected cash flows are the same over time

3. Do the above for all time points tj

4. Calculate the relative change in value of the liabilities, RL

= Lj≠Lj≠1Lj≠1

betweentj

and tj≠1

To reduce the interest rate sensitivity of liabilities, a life insurer can enter into Swapcontracts as e.g. the fixed rate receiver. In this study an adjustment of the interestrate sensitivity of the liabilities is made to account for swap positions. It is in linewith industry practice assumed that interest rate volatility is caused primarily byparallel shifts in the yield curve. The volatility estimated from historical data istherefore reduced by multiplying it with a factor �V (Swaps+L)

�V (L)

, where �V (·) is thechange in value that results from an interest rate stress of +100bps.

The covariance matrix � can now be estimated from the historical returns of assetsand liabilities.

31

5.6 Minimization of SCR

A prudent allocation of investments under the Solvency II regime could be to mini-mize SCR

mkt

with certain restrictions. From Section 5.2 we know that SCRmkt

=Ôw

T �w. We set up the optimization problem

minw

w

T �w

subject to w

T

A

1 = 1w

L

= L/A

E[wT

R] = Rrequired

wi

Ø 0 for i = 1, . . . , k,

(5.9)

where w are the monetary weights of di�erent asset classes. This optimizationproblem is to be solved using convex optimization techniques.

The function SCR(w) is convex since the matrix �SCR

is positive semidefinite asdiscussed in Section 4.1.2

5.7 Combined optimization

A mean-variance optimized portfolio may prove ine�cient with respect to its con-sumption of regulatory capital, i.e. the SCR of mean-variance optimized portfolios.At the same time, optimal portfolios with respect to minimizing SCR may not beeconomically sound. We therefore set up a combined optimization problem whichincludes both variance and SCR in the objective function. The approach serves asa joint optimization to both risk measured in standard deviation and SCR. Theobjective function can be seen as a simple utility function where a higher value ofthe objective function results in a lower utility, i.e. high risk combined with highSCR is the least desirable for a given return.

2One eigenvalue of �SCR is 0 which would yield infinitely many solutions to the optimizationproblem (Equation 5.9). The associated eigenvector, however, has negative entries. Since we forbidshort-selling, these solutions are not feasible.

32

minw

c · Var(BOF (w)) + (1 ≠ c) · SCR2(w)

subject to w

T

A

1 = 1w

L

= L/A

E[wT

R] = Rrequired

wi

Ø 0 for i = 1, . . . , k.

(5.10)

The objective function of the optimization problem (Equation 5.10) can be refor-mulated as

c · w

T �w + (1 ≠ c)wT �w

=w

T (c� + (1 ≠ c)�)w=w

T �c

w,

(5.11)

where �c

is a linear combination of � and �. Depending on the parameter c, theoptimization will be more or less similar to the mean-variance and SCR optimization.In the cases where c = 1 and c = 0 the combined optimization coincides with theoptimization problems in (5.9) and (5.10). The results of di�erent choices of theparameter c will be discussed thoroughly in Section 6.2.3.

5.8 Data

In order to calculate the SCR for the Swedish life insurer, both internal and externaldata is required. All data is secondary data and is provided by either SEB, the lifeinsurer or external sources e.g. Bloomberg.

5.8.1 Case study data

The life insurer has provided us with its portfolio of assets and liabilities as of 2014-12-31. The current portfolio contains assets, swaps and currency hedges includingadditional information such as e.g. credit rating of corporate bonds, which areimportant for the optimization calculations. The current portfolio will be a point ofreference in relation to the optimal portfolios, both from a mean-variance and SCRperspective. The life insurer has also provided us with estimated future liabilitycash flows. The data is given in five year intervals as presented in Figure 5.1. It isassumed that the cash flows are equally distributed within the time intervals.

33

Figure 5.1: Liability cash flowsThe figure shows the life insurer’s future liabilities, distributed in five year intervals,as of 2014-12-31.

5.8.2 Index data

In order to solve the portfolio optimization problems (5.7) and (5.10), we need accessto historical time series data for some generic asset classes that life insurers typicallyinvest in. The indices have been chosen in collaboration with SEB and are presentedin Table 5.1.

34

Table 5.1: Index mapping of asset classesThe table shows the assigned benchmark indices for each asset class as well as indexname, Bloomberg ticker and currency.

Asset class Index name Bloombergticker

Cur-rency

T-bills OMRX Treasury Bill Index rxvx index SEK

Governmentbonds

OMRX Treasury Bond Index omrx index SEK

AAA Credits OMRX Mortgage Bond Index rxmb index SEK

IG credits iBoxx e Corporate Bond TR 3-5years

QW5E Index Local

Swedishequities

OMX Stockholm Benchmark Index sbx index SEK

Globalequities

MSCI World Index Daily Net TR nddlwi index Local

Real estate FTSE EPRA/NAREIT DevelopedEurope

epra index Local

Hedge funds HFRX EH: Equity Market NeutralIndex

hfrxemnindex

Local

Privateequity

LPX Europe TR lpxeurtrindex

Local

EM credits Morningstar Emerging MarketsCorporate Bond Index

msbiertrindex

SEK

HY credits Markit iBoxx EUR Liquid High Y iboxxmjaindex

Local

The OMRX Treasury Bill Index will be used as a proxy for interest bearing assetswith short duration, e.g. cash, repos and treasury bills. Life insurers also investin government bonds, for which the OMRX Treasury Bond Index is chosen. Theindex is a total return index of Swedish government bonds. For AAA Credits,we use the OMRX Mortgage Bond index which consists of a portfolio of Swedish

35

Mortgage bonds that typically have AAA or AA S&P rating. For IG credits weuse the iBoxx Corporate Bond Index for EUR-denominated corporate bonds. Themarket for corporate bonds in Sweden is small which forces institutional investorsto invest in corporate bonds denominated in EUR or USD. Furthermore, there is nosuitable index for Swedish corporate bonds. For the exposure to Swedish equities,we rely on the OMX Stockholm Benchmark Index, which is a total return index forNasdaq OMX Stockholm. MSCI World Index contains captures equities in manydeveloped countries and a large part of the floated market cap, making it a suitableproxy for Global equities. The FTSE EPRA/NAREIT index of listed real estatefrom developed Europe is chosen as benchmark index for Real estate. The assetclass Hedge funds is mapped to the HFRX EH: Equity Market Neutral Index. LPXEurope TR index consists of listed European private equity companies and is thebenchmark index for Private equity. For EM credits, the Morningstar EmergingMarkets Corporate Bond Index is suitable. High yield corporate bonds that haveratings lower than BBB get assigned the iBoxx EUR Liquid High Yield Index.

The index data is monthly data from 2006-01-30 to 2014-12-31 since all indices havedata during that period. It has not been possible to use longer time series datasince there is a lack of suitable corporate bond indices. From the index values,monthly returns defined as Indext≠Indext≠1

Indextare calculated in order to extract variances

and covariances.

A Swedish investor investing in assets denominated in foreign currencies will beexposed to currency risk. If SEK depreciates against foreign currencies, assets de-nominated in foreign currencies will increase in value (measured in SEK) and viceversa. Institutional investors, however, have the ability to hedge most of their cur-rency exposure through entering forward rate agreements on foreign exchange rates.It is common practice among life insurers to hedge a large part of the currencyrisk. In our case we assume that all currency risks are hedged except investments inEM credits. For our life insurer, EM credits are denominated in emerging marketcurrencies which are sometimes di�cult and most often expensive to hedge. Theseassumptions are reflected in the choice of index data by choosing the local currencyof the index, which is equal to assume no FX risk.

5.8.3 Expected index returns

The index data is used to calculate variances and covariances between asset classes.However, in order to estimate accurate future expected returns of asset classes, the

36

historical data is not enough. As an example, taking the mean of the monthlyreturns to be the expected return would be a crude estimate and also dependent onthe start and end date of the time series. Instead of using historical data, expectedreturn estimates are provided by research at SEB. SEB continuously estimate futureexpected returns of asset classes. The estimates used in this thesis are dated as of2014-12-31 and have a 12-month horizon. The estimates shown in this thesis arenot publicly available.

Table 5.2: Asset classes: expected return and standard deviationThe table shows the asset classes used in the optimization with their respectiveestimated expected return and historical standard deviation.

Asset class Expected return Standard deviation(estimate) (historical)

T-bills 0.10% 0.40%Government bonds 0.40% 4.53%AAA credits 0.90% 1.91%IG credits 1.40% 3.66%Swedish equities 6.20% 18.16%Global equities 3.40% 14.89%Real estate 3.20% 19.61%Hedge funds 2.10% 3.96%Private equity 6.30% 25.22%EM credits 4.00% 12.25%HY credits 3.00% 11.77%Liabilities 0.00% 4.96%

5.8.4 Interest rate swap data

Interest rate swap data is used to create the ”basic risk free rate” for discountingthe liabilities. To that end we collect monthly swap rate data for SEK interest rateswaps from 2006-01-30 to 2014-12-31.

5.9 Limitations

The chosen methods and the approach to which this study aims to solve the lifeinsurers’ problem is associated with certain limitations. The asset classes chosen is

37

a limitation for generalizability since other life insurers might hold assets in theirportfolio which do not fit into the chosen asset classes. However, the asset classestogether cover most of the important assets that insurers invest in e.g. equities andfixed income. The asset classes were given certain expected returns and standarddeviations to make the optimization viable. If the parameters change, the resultwould be di�erent. The chosen risk measure in this study is the standard deviationwhich is another limitation. Other risk measures exist and the use of them in theoptimization could yield other results. Although the aforementioned limitations ex-ist in this study, the chosen methods are considered su�cient to answer the researchquestions.

5.10 Reliability and validity

The aim with the thesis is to develop an investment decision framework for a lifeinsurer with respect to return, risk and capital requirements from Solvency II. Inorder for the result of the method to be relevant, the level of reliability and validityhas to be deemed su�cient.

The reliability is assured by the use of public regulations to calculate SCR. Variancesand covariances of the chosen asset classes are estimated from publicly availableindices. All methods used in the thesis have been presented in this chapter, ensuringthat the study can be repeated by following them as stated. Data for future paymentobligations are proprietary and reduce the level of reliability, while being critical forthe study. Other insurance companies may have liabilities that di�er from the onesstudied in this thesis.

The validity of the thesis in answering the research questions has been assured byidentifying relevant risks factors assigned to asset classes. The risks used in thethesis correspond to the risks described in the Solvency II framework, ensuring thatthe thesis measures what it intends to do. The asset classes were chosen to cover allassets that the life insurer holds today but still be general enough for other insurers.The method includes quadratic optimization using the estimated correlations be-tween the asset classes. Quadratic optimization is used since all objective functionsin the considered optimization problems are quadratic. With regards to the purpose,the investment decision framework must be tested on real allocations to ensure itsvalidity. Therefore, a case study is deemed appropriate for answering the researchquestion. The chosen risk measure is the standard deviation, a measure frequentlyused, both in theory and practice, to measure risk of investments.

38

6 Results and analysis

This chapter contains the results produced using the developed models and the dataspecified in previous chapters. The first subsection gives a presentation of the currentallocation for the life insurer, including the asset allocation and SCR contributionby asset class. Subsequently, the results from the mean-variance, the SCR and thecombined optimizations are shown through e�cient frontiers and respective assetallocations. The same optimizations are performed using investment constraints inthe following section. The results in this chapter are continuously analyzed with athorough discussion at the end of the chapter.

6.1 Current allocation

6.1.1 Allocation per asset class

The insurer’s portfolio is mapped to the predefined asset classes resulting in therelative allocation displayed in Table 6.1. The allocation to Hedge funds and Privateequity can be considered to be relatively high amounting to 16.55% and 5.95%respectively. Furthermore, the portfolio contains substantially more Credits thanGovernment bonds. From a Solvency II perspective, this means that the insurancecompany has allocated its assets into asset classes that are expensive in terms ofSCR. The liabilities amount to 85.16% of the assets.

39

Table 6.1: Current asset allocationThe table shows the life insurer’s current allocation to asset classes as percent oftotal assets.

Asset class AllocationT-bills 1.21%Government bonds 10.59%AAA credits 21.72%IG credits 19.67%Swedish equities 5.20%Global equities 13.18%Real estate 1.59%Hedge funds 16.55%Private equity 5.95%EM credits 1.88%HY credits 2.46%Liabilities 85.16%

6.1.2 SCR contribution

The calculated SCR for the insurer is displayed in Figure 6.1 and amounts to 23.73%of the assets. This means that the life insurer needs to have 23.73% more assets thanliabilities with the current asset allocation. A substantial amount of the SCR stemsfrom equity risk, whereas the contribution from interest rate risk is small. One wouldexpect the contribution from interest rate risk to be larger, but the low interest rateenvironment at the asset and liability valuation date, 2014-12-31, results in smallabsolute interest rate stresses in the interest rate down scenario. The reason for thatis that the stressed interest rate is expressed as a percentage of the basic risk freerate as described in Section 4.5.1. In contrast to the interest rate up scenario, thereis no floor on the downward stress scenarios, that is, the absolute stressed interestrate can deviate very little from the basic risk-free rate when it is close to 0%. Inthe interest rate up scenario, however, it must at least be 1%. As described inEIOPA (2014c), the absolute shifts in the term structure are assumed to be smallerin low interest rate environments. Another reason for the relatively small interestrisk, given its allocation, are receiver swap positions held by the insurer. The SCRgenerated by Equity Type 1, Equity Type 2, Property, Spread and Currency riskamounts to 7.17%, 11.94%, 0.4%, 2.5% and 1.13% respectively. The diversification

40

results in a negative SCR of 4.36%.

Figure 6.1: SCR contribution by risk factorThe figure shows the calculated SCR per risk factor, as percent of assets, aggregatedto total SCR.

In order to generate Solvency II implied covariance matrix � the contribution ofeach asset class to the di�erent market risk factors is estimated according to themethod set out in Section 5.2. The mapping of the insurer’s assets into asset classesleads to the average contribution to the market risk factors presented in Table B.1(see Appendix B), which are also the entries of matrix A.

Table 6.2 shows how changes in allocation to di�erent asset classes changes the SCR.Government bonds, for example, lower the SCR since they have no spread risk andthe interest rate risk they carry is opposite to that of the liabilities. One generalresult is that the SCR contribution is lower than the stand-alone SCR for di�erentasset classes. For example, the SCR for Swedish equities is 34.62% instead of thecurrent 39.00% without diversification e�ects. Expected return / SCR Contributionindicates how SCR-e�cient di�erent asset classes are in the current allocation. T-bills and Government bonds are highly capital e�cient since the former has no SCRcontribution and the latter a negative one. AAA credits is the third most capitale�cient asset class with a low expected return and low SCR contribution. The assetclass contributes to interest rate risk diversification at the same time as the spreadrisk charge remains low. In comparison, IG Credits carry a significantly higherspread risk charge. Swedish equities perform better than Global equities which ismainly attributable to a higher expected return with a similar SCR contribution.

41

Hedge funds do not look attractive from a capital e�ciency perspective, nor do HYcredits and EM credits.

Table 6.2: Asset classes: expected return and SCR contributionThe table shows the SCR contribution, expected return and ratio between them foreach asset class as seen in the current allocation. The SCR contribution is calculatedas �SCR

�wi.

Asset class SCRcontribution

Expected return Expected return/ SCR

contributionT-bills 0.00% 0.1% -Government bonds -0.87% 0.4% -0.46AAA credits 1.44% 0.9% 0.63IG credits 6.43% 1.4% 0.22Swedish equities 34.62% 6.2% 0.18Global equities 36.28% 3.4% 0.09Real estate 19.75% 3.2% 0.16Hedge funds 45.96% 2.1% 0.05Private equity 45.96% 6.3% 0.14EM credits 54.25% 4.0% 0.07HY credits 14.29% 3.0% 0.21Liabilities 4.25% 0.0% -

6.2 Optimization without investment constraints

6.2.1 Mean-variance optimization

This section provides the results of minimizing risk measured in standard deviationfor di�erent portfolio returns (see Equation 5.7 in Section 5.5.1). The quadraticprogram is run in steps of 0.05% resulting in 125 feasible portfolios with expectedreturn values ranging from 0.10% to 6.30%. No constraints regarding allocation toasset classes are used in the following optimizations.

Figure 6.2 shows the e�cient frontier from the mean-variance optimization with thevariance of BOF as objective function. The boundary values of standard deviationare 2.19% and 27.50% for an expected return of 0.10% and 6.30% respectively.

Figure 6.3 shows the asset allocation per portfolio. Since there are no constraintsregarding asset allocation in the optimization, the portfolios need not include allassets. For instance, the lower boundary portfolio consists of 100.00% T-bills andthe upper boundary portfolio consists of 100.00% Private equity. Global equities,

42

Real estate and HY credits are not included in any portfolio, measured as relativeallocation of the total portfolio equal to or larger than 0.005%, due to their unfa-vorable expected return and risk ratio. The relationship between expected returnand standard deviation for the asset classes is presented in Section 5.8.3. Govern-ment bonds, Hedge funds, EM Credits and Swedish equities are, on the other hand,frequently included.

The initial shift from T-bills to Government bonds and AAA credits increases theinterest rate sensitivity of the assets which in combination with a negative correlationwith the Liabilities actually lowers risk and increases the expected return, which canbe seen at the bottom end of the graph in Figure 6.2.

Figure 6.2: Mean-variance optimization: expected return and standard deviationThe figure shows the e�cient frontier, in expected return and standard deviation,for mean-variance portfolios.

43

Figure 6.3: Mean-variance optimization: asset allocationThe figure shows the relative portfolio allocation per expected return for mean-variance optimized portfolios.

Table 6.3: Mean-variance optimization: asset class frequencyThe table shows the frequency of asset classes in mean-variance optimized portfolios.Measured as relative allocation of the total portfolio larger or equal to 0.005%.

Asset class FrequencyT-bills 11Government bonds 63AAA credits 52IG credits 25Swedish equities 111Global equities 0Real estate 0Hedge funds 97Private equity 14EM credits 109HY credits 0

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6.2.2 Optimization with respect to SCR

This section provides the results of minimizing SCR for di�erent portfolio returns(see Equation 5.9 in Section 5.6).

Figure 6.4 shows the e�cient frontier from the quadratic optimization with respectto SCR. The boundary values of the SCR are 5.30% and 51.85% for an expectedreturn of 0.10% and 6.30% respectively.

Figure 6.5 shows the asset allocation per portfolio. The lower boundary portfolioconsists of 100.00% T-bills and the upper boundary portfolio consists of 100.00%private equity. The asset classes Global equities, Hedge funds, IG credits and EMcredits are not included in any portfolio while Real estate is only included in 2portfolios (see Table 6.4), which is a result of the low capital e�ciency measured asSCR contribution / Expected return, illustrated in Table 6.2.

As observed from the mean-variance optimization, the allocation starts with T-billsand Government bonds for low expected returns and then shifts over to AAA credits,Swedish equities and HY credits. For the highest returns, the allocation shifts toSwedish equities and Private equity. The large di�erences in SCR contribution forthe assets creates a large range of SCR values for corresponding expected returns,highlighting the importance of choosing assets correctly.

Figure 6.4: SCR optimization: expected return and SCRThe figure shows the e�cient frontier, in expected return and SCR, for SCR-optimized portfolios.

45

Figure 6.5: SCR optimization: asset allocationThe figure shows the relative portfolio allocation per expected return for portfoliosoptimized with respect to SCR.

Table 6.4: SCR optimization: asset class frequencyThe table shows the frequency of asset classes in portfolios optimized with respect toSCR. Measured as allocation in relation to total portfolio larger or equal to 0.005%.

Asset class FrequencyT-bills 6Government bonds 17AAA credits 77IG credits 0Swedish equities 117Global equities 0Real estate 2Hedge funds 0Private equity 32EM credits 0HY credits 107

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6.2.3 Combined optimization

This section provides the results of the optimization problem (5.10) in Section 5.7 forcertain values of the parameter c. Figure 6.6 illustrates how the e�cient portfoliosof the combined optimization problem perform with respect to standard deviationand expected return. ”SD portfolios” corresponds to c = 1 and ”SCR portfolios”to c = 0. The parameter c can be chosen to be between 0 and 1 and determineshow much weight is given to the portfolio risk, measured in standard deviation, andportfolio SCR in the objective function. Portfolios with c < 1 that take into accountthe capital e�ciency have e�cient frontiers that lie in the ine�cient area of the SDportfolios. The frontiers coincide close to the boundary values where they have thesame proportions T-bills and Government bonds and Swedish equities and Privateequity respectively.

Figure 6.7 illustrates the e�cient frontiers in a space of SCR / Expected return.The mean-variance optimized portfolios generally have substantially higher SCRcompared with SCR portfolios that have the same expected return. Reducing theparameter c from 1 to e.g. 0.98 reduces the SCR and the e�cient frontiers movetowards portfolios optimized with respect to SCR. For c = 0.5, the e�cient frontieris close to the frontier of SCR portfolios. This e�ect follows from the fact thatthe portfolio standard deviation is much smaller than the portfolio SCR measuredin percent for the same allocation. In addition, the portfolio standard deviationand SCR enter the objective function of the combined optimization problem (5.10)quadratically. The parameter c does therefore not have a linear behavior. This is aconsequence of how the optimization problem is set up and that the parameter c isnot scaled.

The parameter c could be calibrated in a way that allows for a significant reduction ofthe SCR without compromising portfolio performance measured in expected returnand standard deviation. In this case, it is possible to improve both the risk-returnperformance of the portfolio and at the same time reduce SCR. In Figure 6.6 and6.7, the portfolios generated with c = 0.98 outperform the current allocation bothwith respect to standard deviation and SCR for the same expected return.

47

Figure 6.6: Combined optimization: expected return and standard deviationThe figure shows the e�cient frontiers, in expected return and standard deviation,for portfolios optimized with respect to both standard deviation and SCR. Thefrontiers represent di�erent values of the parameter c, ranging from 0 to 1.

Figure 6.7: Combined optimization: expected return and SCRThe figure shows the e�cient frontiers, in expected return and SCR, for portfoliosoptimized with respect to both standard deviation and SCR. The frontiers representdi�erent values of the parameter c, ranging from 0 to 1.

48

Figure 6.8: Combined optimization: asset allocationThe figure shows the relative portfolio allocation per expected return for portfoliosoptimized with respect to both standard deviation and SCR, for parameter c=0.98.

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6.2.4 Results overview

To illustrate how di�erent allocations perform with respect to expected return, SCRand standard deviation, six selected allocations are compared to the current allo-cation, presented in Table 6.5. Changes in the current asset allocation, by eitherincreasing the expected return, decreasing standard deviation or decreasing SCR,lead to a shift to the three di�erent e�cient frontiers. The selected portfolios e�-ciency with regards to standard deviation and SCR are plotted in Figures 6.9 and6.10 respectively. A summary of the current allocation and selected portfolios ispresented in Table 6.6. Portfolio expected returns, standard deviation and SCRwill not be exactly equal to their target values since the optimization problems aresolved numerically for di�erent expected returns in steps of 0.05%. Instead, theclosest possible value is chosen. For c = 0.98, Portfolio 6: Min risk for currentreturn reduces the SCR from 23.71% to 18.89% and the standard deviation from7.57% to 5.91%. This is achieved through increased allocation to Government bonds,AAA credits, Swedish equities and EM credits and decreased allocation to mainlyIG credits, Global equities, Hedge funds and Private equity.

Table 6.5: Selected portfolios: definitionThe table shows the description of the selected portfolios that are compared in thefollowing section.

Portfolio number Description1 Current allocation2 Minimize risk for current return3 Maximize return for current risk4 Minimize SCR for current return5 Maximize return for current SCR6 Minimize risk for current return (c=0.98)7 Maximize return for current risk (c=0.98)

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Figure 6.9: Selected portfolios: expected return and standard deviationThe figure shows the expected return and standard deviation for selected portfolios.

Figure 6.10: Selected portfolios: expected return and SCRThe figure shows the expected return and SCR for selected portfolios.

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Table 6.6: Selected portfolios: resultsAsset allocation, expected return, standard deviation and SCR for selected portfolios without investment constraints.Asset Class 1: Current

allocation2: Min riskfor current

return

3: Maxreturn for

current risk

4: MinSCR forcurrentreturn

5: Maxreturn for

currentSCR

6: Min riskfor current

return(c=0.98)

7: Maxreturn for

current risk(c=0.98)

T-bills 1.21% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%Government bonds 10.59% 31.04% 3.02% 0.00% 0.00% 17.77% 3.26%AAA credits 21.72% 0.02% 0.08% 63.75% 16.99% 48.48% 51.95%IG credits 19.67% 0.00% 0.01% 0.00% 0.00% 0.00% 0.00%Swedish equities 5.20% 8.14% 12.61% 16.84% 33.02% 18.07% 23.95%Global equities 13.18% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%Real estate 1.59% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%Hedge funds 16.55% 45.31% 58.75% 0.00% 0.00% 2.89% 2.59%Private equity 5.95% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%EM credits 1.88% 15.49% 25.52% 0.00% 0.00% 12.79% 18.25%HY credits 2.46% 0.00% 0.00% 19.41% 49.99% 0.00% 0.00%Expected return 2.21% 2.20% 3.05% 2.20% 3.70% 2.20% 2.75%Standard deviation 7.57% 5.31% 7.53% 7.31% 13.54% 5.91% 7.49%SCR 23.71% 36.24% 50.10% 13.86% 23.66% 18.89% 24.00%

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6.3 Optimization with investment constraints

6.3.1 Investment constraints overview

The optimization procedures are also performed considering the investment restric-tions of the Swedish insurer. The minimum and maximum constraints of the assetclasses are displayed in Table 6.7.

Table 6.7: Investment constraintsThe table shows the investment constraints by asset class or a combination of assetclasses, currently used by the life insurer.

Asset Class Min MaxReal estate 0% 5%Hedge funds 5% 20%Private equity 0% 10%EM credits 0% 5%HY credits 0% 10%Sum of T-bills, Government bonds, AAA Credits and IG Credits 45% 75%Sum of Swedish equities and Global equities 0% 25%

The constraints including a sum of two or more asset classes can have an allocationin which only one asset has a relative weight larger than zero, e.g. an allocationof 75% T-bills, 0% Government bonds, 0% AAA Credits and 0% IG Credits wouldsatisfy the sixth constraint in Table 6.7.

6.3.2 Mean-variance optimization

Figure 6.11 shows the e�cient frontier for the mean-variance optimization withpreviously specified investment constraints. The investment constraints reduce thenumber of feasible portfolios from 125 to 56. The boundary values for expectedreturn are 0.65% and 3.40%, corresponding to standard deviations of 4.67% and11.22%.

The asset allocation along the e�cient frontier is shown in Figure 6.12. The lowerboundary portfolio consists of 73.33% T-Bills, 1.67% Government bonds, 20.00%Hedge funds and 5.00% HY credits. The upper boundary portfolio consists of 45.00%IG credits, 25.00% Swedish equities, 6.67% Hedge funds, 10.00% Private equity,5.00% EM credits and 8.33% HY credits. As seen in Table 6.8, Real estate and

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Global equities are only included in a few portfolios while Hedge funds are includedin all, with EM credits, Swedish equities, AAA credits and Government bonds haveallocation in almost all portfolios.

The decrease in standard deviation for increasing expected return at low values iscaused by the asset allocation shift from T-bills and HY credits to Governmentbonds, which have a larger negative correlation with the liabilities, lowering thetotal standard deviation.

Figure 6.11: Mean-variance optimization with investment constraints: expected re-turn and standard deviationThe figure shows the e�cient frontier, in expected return and standard deviation,for mean-variance optimized portfolios.

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Figure 6.12: Mean-variance optimization with investment constraints: asset alloca-tionThe figure shows the relative portfolio allocation per expected return for mean-variance optimized portfolios.

Table 6.8: Mean-variance optimization with investment constraints: asset class fre-quencyThe table shows the frequency of asset classes in mean-variance optimized portfolioswith investment constraints. Measured as relative allocation of the total portfoliolarger or equal to 0.005%.

Asset class FrequencyT-bills 7Government bonds 47AAA credits 49IG credits 13Swedish equities 50Global equities 5Real estate 2Hedge funds 56Private equity 29EM credits 51HY credits 9

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6.3.3 Optimization with respect to SCR

Figure 6.13 shows the e�cient frontier from the quadratic optimization with respectto SCR. The boundary values of SCR are 13.75% and 25.62% for expected returnsof 0.65% and 3.40%.

Figure 6.14 shows the asset allocation per portfolio on the e�cient frontier. Thelower boundary portfolio consists of 75.00% T-bills, 19.44% Hedge funds and 5.56%HY credits. The upper boundary portfolio consists of 45.00% IG credits, 25.00%Swedish equities, 5.00% Real estate, 5.00% Hedge funds, 10.00% Private equity,2.50% EM credits and 7.50% HY credits. All asset classes are included in at leastone portfolio, with EM credits being represented the fewest times (1) and Hedgefunds being allocated to all portfolios (see Table 6.9).

Figure 6.13: SCR optimization with investment constraints: expected return andSCRThe figure shows the e�cient frontier, in expected return and SCR, for portfoliosoptimized with respect to SCR.

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Figure 6.14: SCR optimization with investment constraints: asset allocationThe figure shows the relative portfolio allocation against expected return for port-folios optimized with respect to SCR.

Table 6.9: SCR optimization with investment constraints: asset class frequencyThe table shows the frequency of asset classes in portfolios optimized with respectto SCR using investment constraints. Measured as relative allocation of the totalportfolio larger or equal to 0.005%.

Asset class FrequencyT-bills 8Government bonds 17AAA credits 44IG credits 5Swedish equities 48Global equities 9Real estate 36Hedge funds 56Private equity 19EM credits 1HY credits 56

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6.3.4 Combined optimization

Figure 6.15 illustrates how the e�cient portfolios of the combined optimizationproblem perform with respect to standard deviation and expected return. ”SDportfolios” corresponds to c = 1 and ”SCR portfolios” to c = 0. The parameter c

can be chosen to be between 0 and 1 and determines how much weight is given to theportfolio risk, measured in standard deviation, and portfolio SCR in the objectivefunction. Portfolios with c < 1 that take into account the capital e�ciency havee�cient frontiers that lie in the ine�cient area of the SD portfolios. The frontiersdo not coincide in the boundary values, as they did for the combined optimizationwithout investment constraints, but have similar asset allocations.

Figure 6.16 illustrates the e�cient frontiers in a space of SCR / Expected return.The mean-variance optimized portfolios generally have substantially higher SCRcompared with SCR portfolios that have the same expected return. Reducing theparameter c from 1 to e.g. 0.98 reduces the SCR and the e�cient frontiers movetowards portfolios optimized with respect to SCR. Due to the nature of the invest-ment constraints, c = 0.5 has almost identical asset allocations as the SCR portfolios.Similarly, c = 0.9 has the same asset allocation as c = 0 for some values of expectedreturn while c = 0.98 has the same asset allocation as c = 1 (SD portfolios) for veryhigh and very low values of expected return.

The parameter c could be calibrated in a way that allows for a significant reduc-tion of the SCR without compromising portfolio performance measured in expectedreturn and standard deviation. In this case, it is possible to improve both the risk-return performance of the portfolio and at the same time reduce portfolio SCR. InFigure 6.15 and 6.16, the portfolios generated with c = 0.98 outperform the cur-rent allocation both with respect to standard deviation and SCR for every expectedreturn.

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Figure 6.15: Combined optimization with investment constraints: expected returnand standard deviationThe figure shows the e�cient frontiers, in expected return and standard deviation,for portfolios optimized with respect to both standard deviation and SCR. Thefrontiers represent di�erent values of the parameter c, ranging from 0 to 1.

Figure 6.16: Combined optimization with investment constraints: expected returnand SCRThe figure shows the e�cient frontiers, in expected return and SCR, for portfoliosoptimized with respect to both standard deviation and SCR. The frontiers representdi�erent values of the parameter c, ranging from 0 to 1.

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Figure 6.17: Combined optimization with investment constraints: asset allocationThe figure shows the relative portfolio allocation per expected return for portfoliosoptimized with respect to both standard deviation and SCR, for parameter c=0.98

6.3.5 Results overview

As in Section 6.2.4, this section provides illustrations for how selected allocationsperform with respect to expected return, SCR and standard deviation comparedto the current allocation of the life insurer, shown in Table 6.10. Changes in thecurrent asset allocation by either increasing the expected return, decreasing stan-dard deviation or decreasing SCR, lead to a shift to three di�erent e�cient frontiers.The selected portfolios e�ciency with regards to standard deviation and SCR areplotted in Figures 6.18 and 6.19 respectively. A summary of the current allocationand selected portfolios is presented in Table 6.10. Portfolio expected returns, stan-dard deviation and SCR will not be exactly equal to their target values since theoptimization problems are solved numerically for di�erent expected returns in stepsof 0.05%. Instead, the closest possible value is chosen.

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Figure 6.18: Selected portfolios with investment constraints: expected return andstandard deviationThe figure shows the expected return and standard deviation for selected portfolios,optimized with investment constraints.

Figure 6.19: Selected portfolios with investment constraints: expected return andSCRThe figure shows the expected return and SCR for selected portfolios, optimizedwith investment constraints.

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Table 6.10: Selected portfolios with investment constraints: resultsAsset allocation, expected return, standard deviation and SCR for selected portfolios with investment constraints.

1: Currentallocation

2: Min riskfor current

return

3: Maxreturn for

current risk

4: MinSCR forcurrentreturn

5: Maxreturn for

currentSCR

6: Min riskfor current

return(c=0.98)

7: Maxreturn for

current risk(c=0.98)

T-bills 1.21% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%Government bonds 10.59% 31.56% 3.36% 0.00% 0.00% 21.97% 0.06%AAA credits 21.72% 23.38% 44.85% 65.58% 15.00% 42.23% 48.93%IG credits 19.67% 0.00% 0.00% 0.00% 30.00% 0.00% 0.00%Swedish equities 5.20% 20.05% 25.00% 19.44% 25.00% 21.59% 25.00%Global equities 13.18% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%Real estate 1.59% 0.00% 0.00% 0.00% 5.00% 0.00% 0.00%Hedge funds 16.55% 20.00% 20.00% 5.00% 5.00% 9.21% 19.38%Private equity 5.95% 0.00% 1.79% 0.00% 10.00% 0.00% 1.62%EM credits 1.88% 5.00% 5.00% 0.00% 0.00% 5.00% 5.00%HY credits 2.46% 0.00% 0.00% 9.98% 10.00% 0.00% 0.00%Expected return 2.21% 2.20% 2.70% 2.20% 3.30% 2.20% 2.70%Standard deviation 7.57% 5.81% 7.50% 6.78% 11.49% 6.03% 7.51%SCR 23.71% 22.56% 25.65% 15.43% 23.92% 18.58% 25.38%

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6.4 Discussion

6.4.1 Mean-variance optimization

This section will discuss the results that are used to answer the first of the threesub-research questions:

i) How can a portfolio be optimized from a risk perspective?

In the optimization problem, the weights of assets are chosen as to minimize thestandard deviation of a portfolio with a certain expected return. It can be ob-served that the standard deviation of the portfolio could be reduced further withthe same expected return. The main way to achieve this would be to allocate a largeportion of the assets to Hedge funds and Government bonds, which both carry alow standard deviation and the rest in EM credits and Swedish equity. The otherasset classes would all have zero percent of the allocation. However, by only tak-ing standard deviation into account, the SCR of the total portfolio would increasedramatically. The portfolios along the mean-variance e�cient frontier are generallycapital ine�cient from an SCR standpoint. The allocations of selected portfolio 2and 3 in Table 6.6 have a SCR of 36.24% and 50.10% respectively, compared to23.71% of the current allocation. The di�erence in judgment of asset class risk inSolvency II and estimated risks from historical returns create this large disparityof capital ine�ciency. For example, Hedge funds are considered a low-risk asset byits benchmark index with a standard deviation of 3.96%, which is one of the lowestamong the asset classes, while the regulatory framework charges the asset class witha relative, marginal SCR of 45.96% in the life insurer’s current portfolio, which isone of the highest values among the asset classes.

The optimization with regards to standard deviation is highly sensitive to the as-signed expected return values to the di�erent asset classes. Small changes in theexpected returns have a large impact on the asset allocation. This also leads to largere-allocations along the e�cient frontier. If the life insurer was to invest solely onthe basis of standard deviation and expected return, it could lead to large changesin asset allocation if the expected return estimate is altered. At boundary values ofthe optimization, the portfolio would only consist of a few assets, which is highlyunlikely in reality since the insurer has invested in all asset classes today and wouldprobably not seek to concentrate the portfolio to a few assets.

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6.4.2 Optimization with respect to SCR

This section will discuss the results that are used to answer the second of the threesub-research questions:

ii) How can a portfolio be optimized from a capital requirement perspective?

In the optimization problem, the weights of assets was chosen as to minimize the SCRof a portfolio with a certain expected return. It can be observed that the SCR of theportfolio could be reduced further, with the same expected return. The main way tooptimize the current portfolio would be to shift allocation to AAA credits, Swedishequities and HY credits. The SCR for portfolio 4 in Table 6.6 is roughly 10%-pointslower than the current allocation, while also reducing the standard deviation. Onecould also move upwards on the e�cient frontier of expected return and SCR toreach an expected return of 3.70% compared to 2.20% for the current allocation.However, to increase the expected return by such a high amount, the portfolio alsogets a much higher standard deviation, 13.54% compared to 7.31%. In addition, theportfolio would consist of only three asset classes exposing the portfolio to a highdegree of individual asset class risk. For example, the selected portfolio 4 in Table6.6 would allocate 63.75% of its capital to AAA credits, which is an unreasonablyhigh amount.

The optimization with respect to SCR is also, in similarity to the mean-varianceoptimization, highly dependent on the expected return values for the di�erent assetclasses. The relation between expected return and SCR contribution can be seen inTable 6.2. The values of SCR contribution and expected return creates a skewnesstowards AAA credits, which have a very low SCR contribution compared to the otherasset classes. In addition, by optimizing with respect to SCR, some assets are notinvested in at all, which is not practical for a life insurer that wants achieve long-termdiversification for the portfolio. The high correlation between assets under SolvencyII reduces the benefits of diversification for other than interest bearing assets.

One important insight from the optimization is that government bonds are a ”perfecthedge” of the liabilities since they have a negative SCR contribution at portfoliolevel. Therefore, at low values for expected returns, the portfolio might consist ofonly government bonds and have no SCR contribution.

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6.4.3 Combined optimization

This section will discuss the results that are used to answer the third of the threesub-research questions:

iii) How can a portfolio be optimized to satisfy both risk and capital requirements?

In the optimization problem, the asset weights are chosen to minimizec · Var(BOF (w)) + (1 ≠ c) · SCR2(w) of a portfolio for di�erent required returns.In the same way as the portfolio optimizations with respect to standard deviationand SCR, e�cient frontiers are modeled but now for di�erent values of the trade-o�parameter c. Choosing c to be slightly below 1, i.e. introducing a small punishmentfor capital requirements to mean-variance portfolio optimization, yields a substantialimprovement of the total SCR for market risk. In the setting of this thesis, c = 0.98su�ces to exclude asset classes with high SCR contribution. Portfolio 6 in Table 6.6illustrates that minimizing the objective function and keeping the expected returnat the initial level of 2.20% results in a portfolio standard deviation of 5.91% andSCR of 18.89%. Portfolio 2 in Table 6.6 which only includes standard deviation inthe objective function, on the other hand, reduces standard deviation to 5.31% butat the cost of a high SCR, of 36.24%. The rather large discrepancy in SCR is mainlya result of a reallocation from Hedge funds to AAA Credits. The introduction ofthe parameter c e�ectively excludes capital expensive asset classes at the same timeas a good risk-return trade-o� is maintained.

6.4.4 Choice of c

The choice of the parameter c depends on di�erent factors, often related to thebusiness situation of the insurer. Some situations that should influence the choiceof c include

1. The relative attractiveness of asset classes in terms of volatility and SCR. Therisk of one asset estimated with historical data can di�er from its riskiness asviewed by the regulator, i.e. the SCR. As already mentioned, Hedge fundsare charged with 49% capital charge in Solvency II whereas their historicalvolatility measured in standard deviation is estimated to be 3.96%. Whenthere are large discrepancies between the risk assessment in Solvency II andhistorical volatility, choosing c slightly below 1 will yield substantially lowerallocations to these assets. When the risk assessment and correlations aresimilar in Solvency II and through estimations from historical data, c will not

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a�ect the results as much since the SCR e�cient portfolios and mean-varianceportfolios are more similar

2. The solvency of the insurer. Insurers with a high solvency ratio, A

L

, could beable to cover any potential level of SCR from their asset allocation and mightnot be needed to take the SCR into account when deciding on investments

3. The target expected return for the asset allocation. A life insurer can, insteadof varying the parameter c, lower the SCR by lowering the target expectedreturn since a higher expected return is generally associated with a higherSCR

6.4.5 Investment constraints

The quadratic optimizations performed in Sections 6.4.1, 6.4.2 and 6.4.3 were con-ducted using investment constraints that the life insurer has today. The resultsshow more balanced portfolios from a diversification perspective since most port-folios contain a larger variety of assets than the ones created without investmentconstraints. Also, the selected portfolios in general have somewhat higher values ofstandard deviation but a reduced SCR charge, especially seen for portfolios 2 and3 (Tables 6.6 and 6.10). The selected portfolios with c = 0.98 have a similar assetallocation both with and without investment constraints.

As seen in Tables 6.6 and 6.10, many assets reach the upper and lower bounds ofthe investment constraints, reducing the e�ectiveness of the portfolio. This indicatesthat the investment constraints might need revision, in either direction dependingon the allocation.

6.4.6 Sustainability

The two main purposes of Solvency II, to limit systemic risk and protect policyhold-ers, contribute to the long-term stability of the insurance industry and therefore itscontribution to a sustainable society. Introducing SCR that vary with the riskinessof insurance companies’ investment and activities is a sound way of ensuring thatinsurance protection will be available when most needed. At the same time, poli-cyholders of occupational pension plans rely on good returns on paid premiums fortheir future pensions. Strict capital requirements restrict more risky investments ine.g. stocks that have shown to generate higher returns over long time periods thaninterest-rate bearing assets e.g. government and corporate bonds. In addition, the

66

stress tests for di�erent assets in Solvency II may not lead to economically sensibleportfolios. In fact, the results of the optimization problem to minimize SCR leadsto portfolios that include only a few asset classes and are skewed to fixed incomeinstruments. The development of a combined optimization model in this thesis in-troduces the possibility to construct portfolios that satisfy the objectives of multiplestakeholders, primarily the regulator and the policyholders that want their invest-ment to grow but also the shareholders of the insurance company that are limitedin how much capital they can and want to hold.

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7 Conclusion

This final chapter will present the conclusions that can be deducted from the analysiswith regards to the research question posed in the beginning of the thesis. Finally,recommendations for further studies are presented.

The aim with this study was to examine the optimal asset allocation for a life insurerwith respect to capital requirements imposed by Solvency II. This thesis finds thatit is possible to lower the standard deviation and SCR for the same expected return,compared to the current allocation of the life insurer. The three optimizationsproduce largely di�erent results, since the relationship between expected return,standard deviation and SCR di�er across the asset classes. The results also showthat small changes to the trade-o� parameter c yields large di�erences in the type ofassets that are invested in and the associated expected return, standard deviationand SCR.

7.1 Summary of findings

In order to fulfill the purpose of this thesis, to provide an investment decision frame-work for a Swedish life insurer that includes solvency capital charges for asset allo-cation as well as internal risk-return requirements, the following research questionwas posed:

• How can a life insurance company integrate Solvency II capital requirementsinto the asset allocation decision?

To answer the main research question, three models for optimal portfolios wereconstructed with di�erent objectives. The first uses mean-variance optimizationwith the parameters expected return and standard deviation to minimize the riskin the portfolio for certain expected returns. The second optimization has SCR asobjective function and minimizes the value of it for every expected return. The lastoptimization has an objective function that is a linear combination of both standarddeviation and SCR for a given return, using a parameter c. The results show thatminimizing SCR generates portfolios that di�er substantially from portfolios whichare optimal when risk is measured by historical volatilities of asset classes. Especiallyassets classified as Equity Type 2 in Solvency II, e.g. Hedge funds, are more favoredin the optimization of portfolio standard deviation. Despite this, it is possible toconstruct portfolios that take both risk and SCR into consideration and balance theoptimality between the two. The combined optimization produces more practical

68

and useful asset allocations than the optimizations which looks at the two parametersseparately.

The optimization methods were also performed using investment constraints whichthe insurer has today. The e�cient frontiers for all optimizations are significantlyshorter but produce more diversified results. For many values of expected return,the upper and lower bound of the investment constraints are met, showing that cal-ibration of the constraints is probably needed to achieve more mixed and diversifiedresults.

In comparison to the current allocation of the insurer, the results provide portfolioswhich have a more favourable relation between expected return, standard deviationand SCR. One reason that the SCR-e�ciency of the optimized portfolios is signifi-cantly higher than the current portfolio is because the life insurer does not measureSCR of their asset allocation to the same extent as this thesis. Also, the insurer hashad an aim to diversify its portfolio by investing in all asset classes simultaneouslyin contrast to the optimized portfolios which seek to find the best relation betweenthe input parameters.

One limitation to the results is that the portfolios that were constructed in this thesisare sensitive to input parameters, especially the assumed future expected returns ofasset classes on a one-year horizon. Nevertheless, we see the introduced optimizationtechniques as valuable instruments for investment managers to construct portfolioswith their own views on risk and expected return of di�erent asset classes.

7.2 Contribution and further research

The results of this study are of importance to life insurers and other financial in-stitutions that are a�ected by Solvency II. The thesis shows the relation betweenexpected return, standard deviation and SCR of certain asset classes and how thetotal investment portfolio is a�ected by changes in allocation. The theoretical contri-bution is the investment decision framework for asset allocation which incorporates aparameter from the regulatory perspective, namely SCR. The thesis presents an op-timization method that combines internal business criteria and external regulatorymeasures.

We propose that further research set out to minimize the optimization results’ de-pendency on individual parameter estimated. Resampling techniques as proposedby Michaud & Michaud (2008) to generate optimization results that are more stable

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over time and less sensitive to estimation errors can be used. Furthermore, otherrisk measures than standard deviation could be of interest since mean-variance op-timization could underestimate tail-risk, potential catastrophe-like events which theinsurer needs to avoid. To incorporate a capital requirement perspective into opti-mization techniques other than mean-variance optimization would be of great valuefor insurance companies a�ected by Solvency II.

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A Solvency II regulation

Figure A.1: F up credit spread factors for corporate bonds

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Figure A.2: F up credit spread factors for covered bonds

B Additional results

The entries of Table B.1 correspond to the matrix A. For example, Governmentbonds have an interest rate stress of ≠1.3% of their value, i.e. an increased allocationto government bonds actually reduces the SCR. Global equities contribute to theType 1 equities risk factor as well as the currency risk factor. The currency exposureis set to 20% of its value since the currencies USD, EUR, GBP and JPY are assumedto be hedged.

Table B.1: Risk factor sensitivity per asset class

Interest Type 1equities

Type 2equities

Property Spread Currency

T-bills 0.0 0.0 0.0 0.0 0.0 0.0Government bonds -1.3 0.0 0.0 0.0 0.0 0.0AAA credits -0.3 0.0 0.0 0.0 1.9 0.0IG credits -0.8 0.0 0.0 0.0 8.4 0.0Swedish equities 0.0 1.0 0.0 0.0 0.0 0.0Global equities 0.0 1.0 0.0 0.0 0.0 0.20Real estate 0.0 0.0 0.0 1.0 0.0 0.0Hedge funds 0.0 0.0 1.0 0.0 0.0 0.0Private equity 0.0 0.0 1.0 0.0 0.0 0.0EM credits 0.0 0.0 1.0 0.0 0.0 1.0HY credits -0.7 0.0 0.0 0.0 17.7 0.0Liabilties 6.2 0.0 0.0 0.0 0.0 0.0

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Figure B.1: SCR implied correlation matrix between asset classes

Figure B.2: Estimated correlation matrix between asset classes

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C Asset class mapping

Table C.1: Asset classesThe table shows what kind of assets that are mapped into the asset classes used inthe optimization.

Asset Mapping

T-bills Cash, Repos and other short term interest bearing assets

Governmentbonds

Swedish governemnt bonds

AAA Credits Credits with S&P rating AAA, typically Swedish mortgagebonds

IG credits Investment grade credits with S&P rating AAA, AA, A andBBB

Swedish equities Swedish, listed equities

Global equities Listed equities in developed markets

Real estate Real estate investments

Hedge funds Hedge funds

Private equity Private equity funds which invest in unlisted equity

EM credits Emerging market credits

HY credits High yield credits with S&P rating below BBB

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