UNIVERSITÉ CLAUDE BERNARD LYON 1

PROFESSIONAL THESIS

Dynamic Duration Steering within ALMModeling

Author:

Sergio ROQUESupervisors:

Anne-Laure JACQUEMARDDr. Ying JIAO

A thesis submitted in fulfillment of the requirements

for the status of

Actuary Associate Member

of the

Institut des Actuaires

in co-operation with

Allianz France

March 31, 2019

iii

Declaration of AuthorshipI, Sergio ROQUE, declare that this thesis titled, “Dynamic Duration Steering withinALM Modeling” and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a research de-gree at this University.

• Where any part of this thesis has previously been submitted for a degree orany other qualification at this University or any other institution, this has beenclearly stated.

• Where I have consulted the published work of others, this is always clearlyattributed.

• Where I have quoted from the work of others, the source is always given. Withthe exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I havemade clear exactly what was done by others and what I have contributed my-self.

Signed:

Date:

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“Try not to become a man of success, but rather try to become a man of value.”

Albert Einstein

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INSTITUT DES ACTUAIRES

AbstractUCBL - Institut de Science Financière et d’Assurances (ISFA)

Allianz France

Dynamic Duration Steering within ALM Modeling

by Sergio ROQUE

The Investment Strategy of an insurance company is a mandate created by the In-vestment Management Department, and given to the Asset Managers, with the aimto steer the investment of premiums earned in the Life & Health and the Property& Casualty segment. Thus, in addition to the asset portfolios management, the In-vestment Management Department is in charge of building the bridge between theinsurance and the investment world.

Life insurers are liability-driven and long-term investors and, as such, their ob-jective is to maximize value. Therefore, shareholder resources are maximized, whileadequate investment returns to policyholders are provided at acceptable levels ofrisk. Moreover, trust and impeccable reputation with partners should be also guar-anteed.

Life Insurance companies use complex Asset-Liability Management (ALM) com-putational models, which reflects essential interactions between asset and liabilitycash-flows as well as the impact of policyholder behaviour and the management ofthe insurer. These models allow to measure accurately risk and performance metricsso that management actions can be executed, in particular a Strategic Asset Alloca-tion (SAA) can be defined and supervised.

In this context, the purpose of this work is to improve the Investment Strategymodule of Allianz ALM model, so called ALIM. Therefore, a framework is providedfor implementing a strategic investment modeling which steers asset duration dy-namically according to market movements and Liability characteristics. More specif-ically, this thesis conduct to:

• Model the Liability duration along the projection within ALIM. Mainly, interest-rates movements and Solvency 2 run-off characteristics of the Liability aretaken into consideration for such purpose;

• Propose new re-balancing mechanics which allow ALIM to steer asset dura-tion, up or down, with the aim to catch liability duration for every projectiontime-step and every simulation.

• Test the impact of the proposed model revision on a variety of risk and man-agement metrics. Specially, a focus on the effects on Own Funds, Risk Capital(RC) and Solvency 2 Ratio will be presented.

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Keywords: Asset-Liability Management (ALM), Investment Strategy, LifeInsurance, Risk, ALM model, Asset Dynamic Duration Steering, Duration,Liability duration modeling, Interest-Rates, Run-Off Portfolios, Solvency2, Risk Capital (RC), Own Funds, Solvency 2 Ratio

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Résumé en Français

La stratégie d’investissement d’une compagnie d’assurance est un mandat donnépar la direction des investissements aux asset managers désigné pour la gestion del’investissement des primes perçues les segments vie et non vie. C’est pourquoi, ladirection des investissements est chargée de la gestion de portefeuilles d’actif ainsique de faire le lien entre le monde de l’assurance et celui de l’investissement.

Les assureurs vie sont des investisseurs institutionnels de long terme quicherchent à optimiser la valeur. Le capital de l’actionnaire est donc maximisé, touten garantissant des rendements adéquats aux assurés et des niveaux de risque pru-dents. De plus, la confiance et réputation avec les partenaires doivent être égalementgaranties.

Les compagnies d’assurance vie utilisent modèles complexes de projection actif-passif qui reflètent les principales interactions entre les flux de l’actif et du passifainsi que le comportement de l’assuré et les decisions du top management. A l’aidede ces modèles, différents indicateurs de risque et de profitabilité peuvent être cal-culés avec précision, ce qui permet au top management de prendre des décisions degestion, plus précisément une allocation stratégique d’actifs est définie et pilotée.

Le présent document a pour objet d’améliorer le module de stratégied’investissement du modèle actif-passif d’Allianz France qui s’appelle ALIM. Uncadre de modélisation est ainsi proposé afin d’incorporer une fonctionnalité dansALIM qui lui permettrait de gérer la duration de l’actif en prenant en comptel’environnement économique et les caractéristiques du passif tout au long de la pro-jection. Notamment, les évolutions suivantes :

• Modélisation de la duration du passif tout au long de la projection dans ALIMen fonction de l’évolution des taux d’intérêt et du vieillissement des porte-feuilles dans le cadre de la valorisation bilantielle de solvabilité 2 ;

• Proposition de nouvelles mécaniques de réinvestissement qui permettraient àALIM de gérer la duration de l’actif par rapport à l’évolution de la durationdu passif à chaque pas de projection et pour toute trajectoire ;

• Test d’impact du changement de modèle sur des indicateurs clés de perfor-mance tels que le fonds propres économiques ainsi que le ratio de solvabilitéet le capital réglementaire de solvabilité.

Mots clés: gestion actif-passif, stratégie d’investissement, assurance vie, risque,modèle actif-passif, gestion de duration dynamique, duration, modélisation dela duration du passif, taux d’intérêt, écoulement de flux du passif, solvabilité 2,capital réglementaire de solvabilité, fonds propres, ratio de solvabilité

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Acknowledgements

This work would not have been possible without the support of my enterprise ad-visor Anne-Laure Jacquemard who has supported me and guided me during myapprenticeship at Allianz France and the drafting of this professional thesis.

I am grateful to all those with whom I have had the pleasure to work in AllianzInvestment Management (AIM) in Paris. Mark Hicks who has heard all of my ques-tions and gave me wise guidance. Coung Nguyen who has always had an answerfor the most challenging problems. I am specially indebted with Roger Prestoz whohas believed in my work and supported my career. I received generous supportfrom Sebastien Lecorrs and Edouard Jozan for my job applications. I want to thankLaurene Lenselle, Marc De La Jonquiere and Mathilde Sabathie, with whom I haveshared good times and worked in many interesting topics.

I owe my gratitude to the ISFA teachers who helped me to build the bridge be-tween the academic and the practical world. Dr. Ying Jiao who has revised andvalidated the work contained in this professional thesis. Yahia Salhi who has showninterest in my work and gave me ideas for further developments.

I would like to thank my friends and colleagues who have given emotional sup-port. I am deeply grateful to Noelle Sadoun, Juan Barragan and Ricardo Jurado whohave been by my side in the good and bad times and who have became very goodfriends of mine. I would like to express the deepest appreciation to Maria AlejandraMartinez who has supported me from the distance.

Nobody has been more important to me to the pursuit of this venture in a foreigncountry than my mother. Your love has been my pillar.

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Allianz Group and Allianz France

The Allianz Group is a worldwide financial services provider, headquartered in Mu-nich, Germany, and founded in 1890, with operations mainly in the insurance andasset management business. On the insurance side, Allianz is the leader in the Ger-man market and has a strong international presence; In fact, Allianz offers a widerange of products, both Property & Casualty and Life & Health, in all business lineswith the aim to protect clients against any risk. On the asset management side, Al-lianz run two distinct investment management businesses, Allianz Global Investors(AllianzGI) and PIMCO.

As of December 31, 2017, the Allianz Group reports:

• 88 million retail and corporate clients in more than 70 countries;

• 1,960 billion euros assets under management;

• Over 140,000 employees worldwide;

• Total revenues of 126.1 billion euros and operating profit of 11.1 billion euros.

FIGURE 1: Allianz global locations

Allianz France is the French holding of the Allianz Group with operations inmost of the Group business lines. As of December 31, 2017, Allianz France reports:

• 5.5 million clients;

• Over 9,000 employees;

• Total revenues of 12.9 billion euros and operating profit of 1.1 billion euros.

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Asset-Liability Management in thecontext of Solvency 2

Modern insurance management and recent regulatory developments, such as Sol-vency 2, evaluate balance-sheet in economic vision with the aim to measure realisticsolvency. As a matter of fact under Solvency 2, the economic value of the com-pany, or Own Funds, is the market value of assets less the market value of liabilities.Given that there is no such a thing as a liabilities market, in the Life segment, itis a common practice to use complex ALM computational models which allows tomap assets and liabilities cash-flows over the course of the business projection in awide range of economic scenarios in order to determine, by Monte-Carlo simulationmethods, the market value of liabilities, or Best Estimate Liability (BEL). As in thecase of financial derivatives pricing, the Monte-Carlo method provides a manner toestimate the Time Value of Options & Guarantees (TVOG) which is the cost of com-plex embedded financial options in Life insurance contracts1.

Beyond the estimation of the available Economic Capital, Solvency 2 assess theamount of Own Funds required to absorb losses from financial markets and insur-ance risk exposure as well as from business operation so that policyholder resourcesare guaranteed. More exactly, the solvency of insurance companies must be guar-anteed in the 99,5% of the cases throughout 1 year; this implies having a sufficientlevel of Own Funds to safeguard policyholder resources, so called Solvency Cap-ital Required (SCR) or, in the context of this thesis, Risk Capital (RC). Nowadays,this amount is a key indicator for the management of an insurer, particularly for theportfolio steering. This way, ALM department steers risk exposure, along with thetrade-off between shareholder profitability and policyholder investment return, bydefining, implementing and supervising a Strategic Asset Allocation (SAA). Apartfrom Risk Capital, the ALM department optimizes other metrics such as EffectiveDuration, Effective Convexity, Return-On-Equity (ROE), among others.

Notably, Effective Duration is a sensitivity used, by ALM analysts and asset man-agers, for the day-to-day portfolio management in the Investment Department withthe aim to protect portfolios from market movements, specially from interest ratesmovements. Consequently, this thesis tackles the modeling of the duration steeringprocess so that Allianz ALM model is improved by replicating more realistic in-vestment practices. Precisely, a framework is provided for identifying the principaldrivers of the duration evolution and for analyzing the impacts of such a modelingon risk and management metrics, particularly on Risk Capital and Present Value ofFuture Profits (PVFP).

1Embedded financial options in Life insurance contracts are mainly generated by the SurrenderOption and the Minimum Guarantee profit participation policies.

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Contents

Declaration of Authorship iii

Abstract vii

Synthesis ix

Synthesis ix

Acknowledgements xi

Allianz Group and Allianz France xiii

Asset Liability Management in the context of Solvency 2 xv

1 Life Insurance, Regulation and Asset Liability Management 11.1 Life Insurance in France . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Solvency 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Market Value Balance-Sheet (MVBS) and Best Estimate Liabil-

ity (BEL) calculations . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Solvency capital Required (SCR) and Solvency 2 Ratio . . . . . 71.2.4 Standard Formula and Internal Model . . . . . . . . . . . . . . . 9

1.3 Asset-Liability Management (ALM) . . . . . . . . . . . . . . . . . . . . 121.3.1 Investment Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Chapter Summary: In-depth focus on Investment Strategy and link toDynamic Duration Steering modeling . . . . . . . . . . . . . . . . . . . 17

2 ALM Model 192.1 ALIM: Allianz ALM model . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Inputs: General, Asset, Liability, Corporate and Economic Sce-narios assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Strategy Model: Investment and Crediting . . . . . . . . . . . . 232.2.3 Outputs and Reporting . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Investment Strategy Model . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Module structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.2 Re-balancing mechanics . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Chapter Summary: Current Investment Strategy modeling and linkto Dynamic Duration Steering sub-model . . . . . . . . . . . . . . . . . 27

3 Dynamic Duration Steering Modeling 293.1 Duration and Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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3.1.2 Effective Duration and Effective Convexity within Solvency 2economic vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.3 Aging of Run-Off Business Portfolios . . . . . . . . . . . . . . . 343.2 Dynamic Duration Steering in ALIM . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Liability Duration Proxys . . . . . . . . . . . . . . . . . . . . . . 373.2.2 Re-balancing mechanics . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Chapter Summary: Liability duration complexities . . . . . . . . . . . . 40

4 Liability Duration Modeling 414.1 Liability Duration as a function of interest rates . . . . . . . . . . . . . . 41

4.1.1 Historical approach . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1.2 Sensitivities Approach without re-calibration of credit spreads . 434.1.3 Sensitivities Approach with re-calibration of credit spreads . . . 474.1.4 Approach retained and modelling formulation . . . . . . . . . . 50

4.2 Portfolio Aging effect on Liability Duration . . . . . . . . . . . . . . . . 514.3 Convexity Analysis for the Sensitivities Approach without re-calibration

of credit spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4 Chapter Summary: Liability Duration Modelling Synthesis . . . . . . . 55

5 Dynamic Duration Steering Implementation and Impact on Key Perfor-mance Indicators (KPIs) 575.1 Dynamic Duration Steering Module Implementation . . . . . . . . . . 57

5.1.1 ALIM Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.1.2 Asset and Liability Duration evolution within ALIM . . . . . . 62

5.2 Results and impacts on Key Performance Indicators (KPI) . . . . . . . . 645.3 Chapter Summary: Investment in practice vs. Investment Strategy

Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Conclusions 69

A Additional Figures and Tables 71

Bibliography 77

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List of Figures

1 Allianz global locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1.1 Estimated Total Revenues in 2017 (Bn†). Source: FFA 2017(e) . . . . . . 21.2 Stock and Revenues evolution in Life Segment in France (Bn†). Source:

FFA 2017(e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Gross Returns in % (before inflation discount). Source: FFA 2017(e) . . 31.4 Three Pillars of Solvency 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Market Value Balance-Sheet (MVBS) approach . . . . . . . . . . . . . . 51.6 One-year Basic Own Funds (BOF1) distribution. Source: EIOPA 2014 . 81.7 One-year Basic Own Funds (BOF1) distribution . . . . . . . . . . . . . . 81.8 SCR estimation approach in Standard Formula and Internal Model.

Source: EIOPA 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.9 Overall structure of the Standard Formula. Source: EIOPA 2014 . . . . 101.10 Risk structure of Allianz Internal Model . . . . . . . . . . . . . . . . . . 111.11 Solvency Capital Required (SCR) risk decomposition before diversi-

fication. Source: Rapport de Solvabilité et la situation fianncière 2016.

Allianz Vie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.12 Organization chart of Allianz Investment Management (AIM) . . . . . 131.13 Cash-Flow Analysis example . . . . . . . . . . . . . . . . . . . . . . . . 141.14 Average asset portfolio allocation of Life insurers in France (without

look-through approach of funds). Source: ACPR S.06.22 . . . . . . . . . 151.15 Average asset portfolio allocation of Life insurers in France (with look-

through approach of funds). Source Enquête AF2i 2018 . . . . . . . . . 161.16 Allianz Investment Management value-creation chain process . . . . . 17

2.1 ALIM Structure Flow-Chart . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 ALIM Code Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 ALIM assumptions tables . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 ALIM asset types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 ALIM strategy process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Investment Strategy Sequence . . . . . . . . . . . . . . . . . . . . . . . . 262.7 Re-balancing mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.8 Strategy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Impact of Interest-rates shock level on sensitivity estimations . . . . . . 343.2 Minimum-Guarantee technical cash-flows projection . . . . . . . . . . 353.3 Macaulay Duration calculated on Minimum-Guarantee technical cash-

flows projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Dynamic Duration Steering within Investment Strategy modeling . . . 36

4.1 Fixed Income Portfolio value profile (Mn†). Data collected from 4Q2014 to Q3 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Asset and Liability Duration profile. Data collected from 4Q 2014 toQ3 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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4.3 Sensitivity Approach without credit spreads re-calibration: Valuationof Asset and Liability Portfolios . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Sensitivity Approach without credit spreads re-calibration: EconomicOwn Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 Sensitivity Approach without credit spreads re-calibration: EffectiveDuration as a function of interest-rates . . . . . . . . . . . . . . . . . . . 45

4.6 Sensitivity Approach with credit spreads re-calibration: Valuation ofAsset and Liability Portfolios . . . . . . . . . . . . . . . . . . . . . . . . 49

4.7 Sensitivity Approach with credit spreads re-calibration: Economic OwnFunds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.8 Sensitivity Approach with credit spreads re-calibration: Effective Du-ration as a function of interest-rates . . . . . . . . . . . . . . . . . . . . . 49

4.9 Minimum-Guarantee technical cash-flows projection . . . . . . . . . . 514.10 Macaulay Duration calculated on Minimum-Guarantee technical cash-

flows projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.11 Effective Convexity calculated by Finite Differences approach . . . . . 534.12 Convexity calculated using theoretical relation . . . . . . . . . . . . . . 54

5.1 Duration evolution within ALIM (only interest-rates modeling) . . . . 625.2 Duration evolution within ALIM (interest-rates and Minimum-Guarantee

modeling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Duration evolution within ALIM (interest-rates and Minimum-Guarantee

modeling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.1 Inflation, OAT TEC 10yr and 10yr US Treasuries evolution . . . . . . . 71A.2 Effective Duration calculated by Finite Differences approach (Pensions

Portfolio) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.3 Effective Convexity calculated by Finite Differences approach (Pen-

sions Portfolio) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A.4 Convexity calculated using theoretical relation (Pensions Portfolio) . . 72A.5 Duration evolution within ALIM . . . . . . . . . . . . . . . . . . . . . . 76

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List of Tables

3.1 Example of maturity baskets of bonds for dynamic duration steeringmodeling purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 ALIM corporate assumptions table (.tbl) dedicated entirely to the set-ting of the Dynamic Duration Module . . . . . . . . . . . . . . . . . . . 58

5.2 Split of the New Bond Profile #4 for the purpose of the Dynamic Du-ration Steering modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 New Bond profiles to be used for purchases . . . . . . . . . . . . . . . . 605.4 Extract form ALIM Assignment table (.tbl) . . . . . . . . . . . . . . . . 615.5 Risk Capital (RC) impacts before and after model change . . . . . . . . 655.6 Solvency 2 Ratio, Own Funds and BEL after an before change . . . . . 675.7 Extract of the New Bond profiles to be used for purchases . . . . . . . . 68

A.1 Determination coefficients of the linear regressions on effective dura-tion figures generated by ALIM using the sensitivities approach with-out re-calibration of spreads . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.2 Determination coefficients of the quadratic regressions on Macaulayduration figures calculated using Minimum-Guarantee technical cash-flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.3 Determination coefficients of the exponential regressions on Macaulayduration figures calculated using Minimum-Guarantee technical cash-flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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List of Abbreviations

ACPR Autorité de contrôle prudentiel et de résolution

AIM Allianz Investment Management

ALIM Asset-Liability Interaction Model: Allianz ALM Model

ALM Asset-Liability Management

BEL Best Estimate Liability

BOF Basic Own Funds

DAA Dynamic Asset allocation

DD Dynamic Duration

ESG Economic Scenario Generator

FI Fixed Income

KPI Key Performance Indicator

MVBS Market Value Balance-Sheet

NAV Net Asset Value

PVFP Present Value of Future Profits

RC Risk Capital

RM Risk Margin

ROE Return-on-Equity

SAA Strategic Asset Allocation

TA Transaction of Assets

TVOG Time Value of Options and Guarantees

UCGL Unrealized Capital Gains and Losses

VaR Value at Risk

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Index

ALIM, 19, 36, 57, 67ALIM inputs, 22ALIM outputs, 24Allianz, xiiiALM, 20ALM Model, 19, 57ALM model, 16, 18, 19, 36asset assumptions, 23asset duration, 62Asset-Liability Interaction Model (ALIM),

20Asset-Liability Management (ALM), 12,

13assumption table, 22Autorité de contrôle prudentiel et de ré-

solution (ACPR), 10

Basic Own Funds (BOF), 6Best Estimate Liability (BEL), 6, 19, 20,

44, 48, 64, 67bond maturity, 39

Cash-Flow Matching, 14convexity, 14, 31, 45, 50, 52convexity gap, 14, 50convexity steering, 32corporate assumptions, 23, 57, 60Corporate Bonds, 61Cost Risk, 66Crediting Strategy, 21, 24

determination coefficients R2, 46

deterministic BEL, 7discount factor, 6duration, 14, 30, 45Duration Gap, 46duration gap, 14, 45, 50, 66duration steering, 32duration target, 38, 55Dynamic Asset allocation (DAA), 26Dynamic Duration Steering, 27, 29, 36,

55, 57, 61, 67Dynamic Lapses, 52

Economic Scenario Generator (ESG), 21,23

Economic Scenarios Generator (ESG), 43,47

effective convexity, 33effective duration, 33Euro savings insurance contracts, 2, 33

Fixed Income, 15Fixed Income (FI), 29

general assumptions, 22Governmental Bonds, 61Gross Own Funds, 67

historic probability measure, 6historical approach, 42

interest-rates, 37, 41, 50, 55, 59, 68interest-rates derivatives, 16Interest-Rates Risk, 66Investment Management, 18Investment Strategy, 12, 21, 24, 25, 30, 36

Key Performance Indicator (KPI), 57, 64Key Performance Indicators (KPIs), 17

Lapse Mass Risk, 66Lapse Risk, 66liability assumptions, 23liability duration, 37, 41, 50, 51, 55, 62liability duration proxy, 37Life Insurance, 1Life Non-Market Risk factors, 66linear fitting, 47linear fitting re-levelling, 50linear regression, 46Longevity Risk, 66

Macaulay duration, 30, 52Market Risk factors, 65Market Value Balance Sheet (MVBS), 5Minimum-Guarantee profit sharing, 51,

55Minimum-Guarantee profit shqring, 35Monte-Carlo, 6Mortality Calamity Risk, 66Mortality Risk, 66MoSes, 20, 22

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Net Asset Value (NAV), 9, 67new bond profile, 60New Business, 68

Options and Guarantees (OG), 29Options and Guarantees (O&G), 6, 33Own Funds, 66, 67

Portfolio Aging, 34, 38, 51, 55, 59Portfolio Hedging, 16Present Value of future Profits (PVFP), 66profit sharing, 24

quadratic regression, 52

re-balancing mechanics, 38, 57, 67re-investment, 39replicating portfolio, 11Risk Capital (RC), 7, 20, 57, 64, 67Risk Margin (RM), 6risk-neutral probability measure, 5run-off business, 34, 38, 55, 59, 68

sensitivities approach, 43, 47, 55Solvency 2, 3Solvency 2 Internal Model, 20Solvency 2 Pillars, 4Solvency 2 Ratio, 8, 57, 67, 68Solvency 2 Standard Formula, 9, 20Solvency Capital Required (SCR), 4, 7, 9,

64spreads re-calibration, 48stochastic BEL, 6stochastic process, 6Strategic Asset Allocation (SAA), 15, 17,

20, 29, 36Strategy Model, 23

Time Value of Options & Guarantees (TVOG),7, 19, 20

Time Value of Options and Guarantees(TVOG), 41

Transaction of Assets (TA), 26

Value at Risk (VaR), 7

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Dedicated to my Mother who has always kindly supported me

1

Chapter 1

Life Insurance, Regulation andAsset Liability Management

Since 2014, European interest rates have attained historical low levels (See FigureA.1). Multiple factors have contributed to this economic environment such as the fallin productivity, the excess of savings and the price decline in raw materials. Conse-quently, inflation have remained low and central banks have deployed expansionarymonetary policies just like Quantitative Easing whereby predetermined amounts ofgovernment bonds or other securities are bought in order to stimulate the economyand increase liquidity.

In this context, the operation conditions of Life insurers have been modified.Particularly, the financial assets return of insurers debt portfolios have been sloweddown as well as the policyholder investment return. Despite this unusual environ-ment, insurance companies in France still being a strong engine of the economy dueto the its long-term investor profile and the size of its asset portfolios under man-agement, which amounts to 2,350 Bn†. As a matter of fact, according to the annualreport for 2016 from the FFA, French insurers have invested 1,384 Bn† in 2016 in theprivate sector via corporate debt securities (39%), equity shares (17%) and corpo-rate real estate investments (3%). In addition, the level of Own Funds reported byinsurance companies remains twice the minimum level required by the Solvency 2regulation.

Therefore, this first chapter aims to give the general framework in which anAsset-Liability Management (ALM) computational model has to be developed, alongwith an investment strategy modeling. More exactly, the market specifications ofFrench Life insurance are presented as well as the Solvency 2 regulation context inorder to precise the metrics used in the investment and business management inFrance.

1.1 Life Insurance in France

French insurance has become the leader in Europe after Brexit. Additionally, theFrench population have been historically attracted by the savings and, thus, a widevariety of products have been developed by different national and internationalstakeholders. Apart from the traditional insurance companies, the French markethas also seen a strong presence of insurance contracts proposed by banks and mutualinsurance companies. In fact, due to the solid presence of these actors, the pensionfunds did not prosper in France.

2 Chapter 1. Life Insurance, Regulation and Asset Liability Management

FIGURE 1.1: Estimated Total Revenues in 2017 (Bn†). Source: FFA2017(e)

As of December 31 2016, in France, there are more than 37 million policyholdersin the Life segment which raise a total stock of 1,635 Bn†. Despite the low interestrates environment, the Life market continues to expand with an annual rate of 3%,just like the last year. However, the economic environment have influenced the Lifesegment as it is observed in a funding migration from Euro to Unit Linked supports.

FIGURE 1.2: Stock and Revenues evolution in Life Segment in France(Bn†). Source: FFA 2017(e)

Life savings insurance contracts still are very attractive for French savers becauseof its fiscal advantages and their competitive returns, in comparison with those frombanking saving accounts (Livret A). Moreover, French people have shown more ap-petite for risk in the search of higher returns, increasingly found in Unit Linkedcontracts.

1.2. Solvency 2 3

FIGURE 1.3: Gross Returns in % (before inflation discount). Source:FFA 2017(e)

1.2 Solvency 2

Solvency 2 regulation has been adopted by the European Parliament and Councilsince 2009 due to the need for stronger measures on insurance companies in orderto safeguard the policyholder resources. This was particularly compelling after thefinancial crisis of 2008-2009 in which many financial institutions showed a clear in-sufficiency of own funds to carry their businesses in a prudent fashion. Therefore,Solvency 2 assess the amount of own funds required to absorb losses from financialmarkets and insurance risk exposure as well as from business operation so that pol-icyholder investment is guaranteed.

Solvency 2 was revisited and modified in 2014 and finally applied since January1st 2016. Since its implementation, French insurers have shown solid margins ofsolvency, twice the minimum required by the regulation. However paradoxically,this regulation penalize the long-term investment, specially for small and mid-sizecompanies, as well as infrastructure investment, as it is remarked by the FFA in itsannual report 2016. It means that even when the policy is addressed to long-terminvestors, it holds a short-term nature. Therefore, two revisions are envisaged: Thefirst one, by the end of 2018, aims to revise the calibration of the Standard Formula;the second one, by 2021, aims to examine the long-term investment and equity char-acter of the methodology.

1.2.1 Overview

The solvency of an insurer is its capacity to pay their financial engagements towardthe policyholders during the validity period of the insurance contracts. It dependson the size of the engagements as well as the amount of own funds available to ab-sorb losses from financial markets and business operation in addition to insurancerisk exposure.

Solvency 1 regulation did not include financial markets risk directly in its sol-vency metrics. More exactly, the asset allocation was not considered directly in the

4 Chapter 1. Life Insurance, Regulation and Asset Liability Management

measurements and, thus, the risk profiles of assets were neither included. In con-trast, solvency was determined mainly based on liability characteristics such as rev-enues and technical reserves. As a matter of fact, the financial crisis of 2008-2009 wasa red flag for the manner to asses risk management on financial institutions.

In this way Solvency 2 redefines the acceptable margins of own funds for in-surers. Precisely, it presents the framework in which risk metrics can be calculatedas well as the structure needed in order to define, control and report risk manage-ment of insurance companies. Below it is presented the three pillars structure of theSolvency 2 framework directive.

FIGURE 1.4: Three Pillars of Solvency 2

It is remarked that risk components of Solvency 2 considered for the aims of thisprofessional thesis are related to the Quantitative Pillar, so called Pillar 1. Thus,in section 1.2.2, the economic valuation of the liability and equity of an insurancecompany will be presented. Then, in section 1.2.3, the sufficient level of own fundswill be discussed and the Solvency Capital Required (SCR) will be precisely defined.Finally, in section 1.2.4, two methodologies will be exposed in order to address thesesophisticated calculations.

1.2.2 Market Value Balance-Sheet (MVBS) and Best Estimate Liability(BEL) calculations

Big capital markets are liquid enough to estimate accurately the market value of anysecurity, such as treasury bonds, forwards, public equity shares, among many oth-ers. Therefore, from market data, it is possible to estimate the market value of an

1.2. Solvency 2 5

insurer assets portfolio and, as such, to determine the amount of Unrealized Cap-ital Gains & Loses (UCGL). Several securities can be priced directly from marketdata with deterministic formulas and some others, such as derivatives or structuredand hybrid securities, need a more sophisticated treatment. In any case, the assetpricing process is relatively simple and the difficulty yields mainly in the data man-agement.1

In contrast, there is no such a thing as an insurance liabilities market and, thus,it is a common practice, in the Life segment, to use complex ALM computationalmodels which allows to map assets and liabilities cash-flows over the course of thebusiness projection in a wide range of economic scenarios in order to determine, byMonte-Carlo simulation methods, the market value of liabilities, or Best EstimateLiability (BEL).

Following this path, the Market Value Balance-Sheet (MVBS) approach definesthe economic equity of the company as the difference, at the valuation date, betweenmarket values of assets and liabilities.

FIGURE 1.5: Market Value Balance-Sheet (MVBS) approach

The economic estimation of the liability can be formalized as follows:

BEL = EPaN

Q f

"

Ât�1

d(t)F(t)

#= Â

t�1EPa

NQ f

[d(t)F(t)] (1.1)

where

• Q f is a risk-neutral probability measure for the financial hazard;1There are many data management complexities. On the one hand, Life Insurers own a wide va-

riety of assets which are in continuous evolution and have to be accounted and controlled frequently;Thus, a database system has to be implemented in order to determine the market value and specifi-cations of assets at each valuation time. On the other hand, Life insurers are big long-term investorsand, consequently, they invest in funds that also have large asset portfolios; Moreover, these funds alsoinvest in other funds. Therefore, at the valuation time, it is necessary to apply a look-trough approachin order to determine the asset exposition of the company.

6 Chapter 1. Life Insurance, Regulation and Asset Liability Management

• Pa is a historic probability measure for the insurance hazard. For example, itcontains biometric risks such as mortality, longevity and morbidity as well asbusiness risks such as lapses and costs;

• d(t) is the stochastic discount factor in time t which is equal to exp⇣�R

t

0 rudu

⌘

with ru a short-interest rate Q f -stochastic process;2

• F(t) is the liability cash-flow in time t which is a PaN

Q f -stochastic process.In other words, liability cash-flows are considered dependent of financial andinsurance hazard.

Particularly, the inclusion of financial hazard in the estimation of future liabilitycash-flows is a key component of the MVBS approach. Additionally, a Risk Margin(RM) is integrated in Solvency 2 technical provisions due to the uncertainty of theBEL estimation. According to Article 87 of the Solvency 2 Directive of The EuropeanParliament and of the Council, 2009, Basic Own Funds (BOF) are determined as theexcess of assets over liabilities3, valued in accordance with Article 75, as illustratedin figure 1.5, which is formalized as follows

BOF = Assets � Liabilities = Assets � (Technical Provisions| {z }=BEL+RM

+Other Liabilities).

(1.2)It is a major challenge to estimate the expected value presented in equation 1.1,

in other words to calculate the BEL, for Life Savings savings contracts with Optionsand Guarantees (O&G). Particularly, policyholders surrender and arbitrage behavioras well as the profit sharing are dependent of the capital markets evolution. There-fore, the liability cash-flow F(t) and the discount factor d(t) are dependent underthe measure Q f (financial hazard) and, thus, it is difficult to find theoretical closedformulas to calculate the BEL.

Due to the difficulty of implementing theoretical approaches for economic valueliability estimations, Life insurers develop complex ALM computational modelswith the aim to calculate the BEL, as well as the regulatory capital required andother equity and liability value estimations, by Monte-Carlo simulation methods.More exactly, a set of risk-neutral economic scenarios4 is calibrated, then liabilityand asset cash-flows are projected for each economic trajectory. Finally, liabilityvalue is calculated for each trajectory, as the sum of the present value of future li-ability cash-flows, and the BEL is, then, found as the average of the liability valuesof all trajectories. This Monte-Carlo stochastic estimation can be formulated as

dBELsto

=1M

M

Âi=1

T

Ât=1

di(t)Fi(t)

where M is the number of simulations, T the horizon projection time which is equalto 40 years for the purpose of this professional thesis, and, given a fixed i 2 {1, · · · , M},t 7! F

i(t) is a liability cash-flow simulated trajectory.

2In the modeling, this stochastic process has to be calibrated to meet the reference zero-couponcurve provided by the EIOPA.

3Other subordinated liabilities should be also taken into consideration according to Article 88 ofthe Solvency 2 Directive

4Fixed Income, Equity and Real Estate as well as Spread and Volatility trajectories are providedover a 40 years projection horizon.

1.2. Solvency 2 7

As in the case of financial derivatives pricing, the Monte-Carlo method providesa manner to price the Time Value of Options & Guarantees (TVOG), within thestochastic valuation of the BEL, which is the cost of embedded financial options inLife & Savings insurance contracts5. Additionally, Life insurers also use determinis-tic actuarial models which allows to project liablity cash-flows under a set of techni-cal and financial assumptions in order to estimate the BEL without the asset-liabilityinteraction, and neither the stress derived from the financial hazard. Precisely, it canbe formulated as

\TVOG = dBELsto � dBEL

det

where dBELdet

is the deterministic calculation of the BEL.

All in all, the MVBS approach requires the implementation of an ALM computa-tional model. From this perspective, the purpose of this professional thesis is to im-prove the modeling of the assets re-balancing mechanics within Allianz ALM model.Hence, asset and liability cash-flows, which interact in several stages at all time stepsover the entire projection, could be estimated more accurately. In terms of the formu-lation given in this section, the improvement of the Investment Strategy modelingleads to a better estimation of the liability cash-flows F

i(t).

Finally, it is remarked that one stochastic estimation implies an implementationof heavy computational calculations in powerful machines; It usually takes between5 to 30 minutes, depending on the model complexity and the coding language aswell as the calculation optimization.

1.2.3 Solvency capital Required (SCR) and Solvency 2 Ratio

Once a framework and methodology are provided for calculating the Best EstimateLiability (BEL) and the Basic Own Funds (BOF) of an insurance company under Sol-vency 2 directive, the Solvency Capital Required (SCR) is defined as the amount ofOwn Funds that insurance and reinsurance companies are required to hold in orderto have a 99.5% confidence they could survive the most extreme expected losses overthe course of a year.

From a theoretical point of view, the SCR is the 99.5% one-year Value at Risk(VaR) measure of the BOF distribution, as it is illustrated in the figure below.

5Embedded financial options in Life Savings insurance contracts are mainly generated by theSurrender option and the Minimum-Guaranteed profit sharing in Euro savings contracts

8 Chapter 1. Life Insurance, Regulation and Asset Liability Management

FIGURE 1.6: One-year Basic Own Funds (BOF1) distribution. Source:EIOPA 2014

In practice, the estimation of the BOF distribution in the course of one year isdone by Monte-Carlo approach. More exactly, several levels of BEL and, thus, ofBOF are simulated using a wide range of financial and technical assumptions. Fi-nally, a weighted average of the BOF levels is calculated around the extreme lossesscenarios in order to have a solid estimation of the SCR, as it illustrated in the figurebelow.

FIGURE 1.7: One-year Basic Own Funds (BOF1) distribution

As it was already mention in section 1.2.2, one BEL calculation and, thus, oneBOF estimation usually takes 5 to 30 minutes; For example, if 50,000 BEL simula-tions are needed in order to build the BOF distribution, then it would take almostthree years to make this calculation with current methods and machines. To over-come this issue, Solvency 2 gives two possible SCR estimation frameworks calledStandard Formula and Internal Model, which are covered in the next two sections.

Once the SCR has been somehow calculated, the Solvency 2 Ratio is defined as

Solvency 2 Ratio =NAV

SCR

1.2. Solvency 2 9

where the Net Asset Value (NAV) is the eligible Own Funds accounted for cov-ering regulatory capital requirements under Solvency 2. According to Article 87 ofthe Solvency Directive of The European Parliament and of the Council, 2009, eligi-ble Own Funds shall comprise the sum of basic own funds, referred to in Article 88and ancillary own funds6 referred to in Article 89. Indeed, this ratio is a solvencyindicator that is required to be greater than 100% by the regulation. For example,the solvency ratio of Allianz France is equal to 236%, as reported on the first quarter2018.

1.2.4 Standard Formula and Internal Model

Solvency 2 Standard Formula and Internal Model frameworks estimate the SCR asthe change of Basic Own Funds (BOF) in a shock scenario, formally described as

SCR = BOFcentral � BOF

Shock

= (Assetscentral � Liabilities

central)� (Assetsshock � Liabilities

shock)

which is illustrated in the figure below.

FIGURE 1.8: SCR estimation approach in Standard Formula and In-ternal Model. Source: EIOPA 2014

According to the EIOPA, "The SCR standard formula follows a modular ap-proach where the overall risk which the insurance or reinsurance undertaking (herebyundertaking) is exposed to, is divided into sub-risks and in some risk modules alsointo sub- sub risks. For each sub-risk (or sub-sub risk) a capital requirement is de-termined. The capital requirement on sub-risk or sub-sub risk level is aggregatedwith the use of correlation matrices in order to derive the capital requirement for theoverall risk."

6Ancillary own funds consist of items other than basic own funds which can be called up to absorblosses. These are therefore items that have not yet been paid in or called up. Once an ancillary own-fund item has been paid in or called up, it will be treated as a basic own-fund item and cease toform part of the ancillary own-fund items. Precisely, ancillary own funds are unpaid share capital orinitial fund that has not been called up, letters of credit and guarantees, and any other legally bindingcommitments received by insurance and reinsurance undertakings

10 Chapter 1. Life Insurance, Regulation and Asset Liability Management

FIGURE 1.9: Overall structure of the Standard Formula. Source:EIOPA 2014

Generally, small and middle insurance companies uses the Standard Formula forthe SCR estimation. In contrast, large insurance corporations use the SCR StandardFormula estimation as a reference and, in addition, they define and calibrate theirown risk modules as well as the method of aggregation7 for the modules, so calledInternal Model. Any change proposed and implemented by an insurance company,with respect to the Standard Formula, has to be validated by the Regulatory Institu-tions, for example the Autorité de contrôle prudentiel et de résolution (ACPR) in France.The Internal Model structure of Allianz France is presented below.

7Some insurers adjust the correlation coefficient of the Standard Formula aggregation matriceswhile some others do not use linear correlations at all; Instead, those, who uses an alternative to cor-relation matrices, usually implements a system of copulas in order to capture correlations between therisk modules.

1.2. Solvency 2 11

FIGURE 1.10: Risk structure of Allianz Internal Model

Allianz has developed its own ALM computational model called ALIM (Asset-Liability Interaction Model). ALIM is used for the projection of liability cash-flowsand for the calculation out-of-sample shocks for the calibration of the replicatingportfolio, which is then used to calculate the SCR for the market risk module. Addi-tionally, ALIM is also used to calculate shocks for non-market risk modules and thepolicyholder participation on credit risk.

The replicating portfolio is a set of financial assets8 whose its future cash-flowsare as close as possible to those from the replicated liability, in the largest possiblenumber of economic scenarios. The replicating portfolio is obtained by solving thefollowing optimization problem

minw2D

S

Âs=1

"PV

liab(s)�N

Ân=1

wnPVrep

n (s)

#

where

• PVliab(s) is the present value of liability cash-flows in the scenario s;

• PVrep

n (s) is the present value of the n-th replicating asset in the scenario s;

• wn is the weight of the n-th replicating asset within the replicating portfolio.

As of the first quarter 2018, 5,500 simulations are executed in ALIM for the cal-ibration of the replicating portfolio. Finally, the distribution of the One-year Basic

8Fixed Income (Zero-coupon, Inflation-linked), Interest Rate Derivatives (Swaps, Swaptions,CSM), Equity Derivatives (Forwards, Puts)

12 Chapter 1. Life Insurance, Regulation and Asset Liability Management

Own Funds (BOF1) can be built from 50,000 simulations of the economic value ofliabilities using the replicating portfolio once calibrated, as it is presented in figure1.7. As reported in the SFCR 2016, the SCR risk decomposition of Allianz France Vieis presented below.

FIGURE 1.11: Solvency Capital Required (SCR) risk decompositionbefore diversification. Source: Rapport de Solvabilité et la situation fian-

ncière 2016. Allianz Vie

ALIM is therefore the stochastic projection tool of liability cash-flows used in theBEL and SCR calculation. Thus, the aim of this thesis is to improve the InvestmentStrategy modeling of ALIM which leads to more accurate estimations of Solvency 2metrics as well as other equity valuation and management metrics.

1.3 Asset-Liability Management (ALM)

The Investment Strategy of an insurance company is a mandate designed to steerthe investment of premiums earned in the Life & Health and the Property & Casu-alty segment. Thus, in addition to the asset portfolios management, the InvestmentDepartment is in charge of building the bridge between the insurance and the in-vestment world.

In the case of Allianz France, the Asset-Liability Management (ALM) team lieswithin the Investment Strategy Direction which, in turn, is inside the Investment De-partment so-called Allianz Investment Management (AIM). According to Gilbert,2004, "Asset Liability Management is the ongoing process of formulating, imple-menting, monitoring, and revising strategies related to assets and liabilities to achievefinancial objectives, for a given set of risk tolerances and constraints". In line withthis, Allianz ALM team is in charge of managing the investment strategy for aninsurance-liability-backed portfolio of assets. Below AIM organization chart is pre-sented in order to illustrate the structure of an insurer Investment Department andthe place of an ALM team within.

1.3. Asset-Liability Management (ALM) 13

FIGURE 1.12: Organization chart of Allianz Investment Management(AIM)

On the one hand, Life insurance companies, with long liability durations, areexposed to financial markets and to the economic environment, specially to interestrates movements. On the other hand, Property & Casualty insurers, with short lia-bility durations, are more focus on liquidity steering. Henceforth, for the purpose ofthis professional thesis only the long-term structure of life insurance liabilities willbe treated.

Life insurers are liability-driven and long-term investors and, as such, their ob-jective is to maximize value. Therefore, shareholder resources are maximized, whileadequate investment returns to policyholders are provided at acceptable levels ofrisk. Moreover, trust and impeccable reputation with partners should be also guar-anteed.

Consequently, Asset Liability Management (ALM) is a strategic function withinan insurance company because it aims to steer the assets portfolios so that liabilitycash-flows structure and constraints are respected. More specifically, assets are pur-chased and sold with the aim to attain financial objectives while a prudent risk man-agement and the estimated insurance obligations are ensured, knowing that suchobligations are uncertain due to the options and guarantees owned by policyhold-ers.

The aim of this section is therefore to present the ALM in practice in order to un-derstand the logic behind the modeling of the Investment Strategy within an ALMcomputational model. Common practices of the Investment Strategy Departmentregarding duration and portfolio steering are also presented.

Keywords: Asset-Liability Management (ALM), Investment Strategy, LifeInsurance, Risk, Duration

1.3.1 Investment Strategy

Asset-Liability Management (ALM) function can be considered as the first link in thevalue-creation chain of the investment process. Precisely, ALM takes into accountliability and policyholder constraints as well as risk exposure in order to maximize

14 Chapter 1. Life Insurance, Regulation and Asset Liability Management

the shareholder value via a variety investment solutions. Below some of the mostcommon practices in practical ALM

• Cash-Flow Matching: Future asset and minimum-guaranteed liability cash-flows are estimated with the aim to define the necessary maturity asset reallo-cation for matching liabilities structure.

FIGURE 1.13: Cash-Flow Analysis example

In the example above, in order to re-equilibrate the volume of cash-flows foreach maturity, the ALM team should suggest asset purchases oriented to in-crease the volume of cash-flows on maturities 10, 13, 14 and so on, while theexcess of volume on maturities 2 to 9 should be vacuumed.

Even if the cash-flows analysis above is just an illustration, as a matter of factusually, in asset portfolios managed by Life insurers, there is a deficit of vol-ume in long-term maturities due to the life long-term obligations towards pol-icyholders. For example, annuities in retirement contracts and disability rentsare long-term obligations. This is why in practice Life insurers usually seek toreinvest an important part of their available cash resources in long-term secu-rities.

• Duration Gap Steering: Duration is a metric controlled by the Executive Boardand mainly steered by the ALM team. There are many ways to estimate thissensibility and to use it in practice. As a matter of fact, duration is a centralcomponent of this professional thesis and, thus, it is treated in detail later inChapters 3 and 4 as well as the modeling of dynamic duration steering me-chanics.

Not formally defined yet, duration represents the average life of a portfolioand, at the same time, a sensibility of such portfolio value, whether of assets orliabilities, to interest-rate movements. Therefore, an smaller gap between theasset and liability duration implies a lower risk to interest rates movements.

It is important to remark that duration is a key metric for the investment man-agement of a Life insurer but it still is an unidimensional measurement and,thus, it should not be steered in an isolated fashion. This is why a good matchin duration should be complemented with a good Cash-Flow Matching in or-der to ensure a better risk coverage to interest rates movements as well as abetter liquidity management.

• Convexity Gap Steering: Convexity is a second order sensibility, whether ofassets or liabilities value, to interest rate movements. As such, the ALM team

1.3. Asset-Liability Management (ALM) 15

ensures an appropriate match in duration and convexity in order to cover port-folios value against interest rates curve parallel shifts and slope curve changes.As a matter of fact in order to understand better Duration, Convexity is alsotreated in detail in Chapters 3 and 4.

• Strategic Asset Allocation (SAA): An important part of risk management inthe investment process is the asset class selection. Thus, the ALM team ana-lyzes the risk impact on asset portfolios regarding their liability-backed pro-file. A variety of allocations are tested using projection tools such as an ALMcomputational model as well as qualitative analysis from investment experts.Finally, a set of Key Performance Indicators (KPIs) is presented to the Execu-tive Board with the aim to determine a Strategic Asset Allocation (SAA) whichis the allocation target percentages per asset class. The average asset allocationof Life France insurers is presented below.

FIGURE 1.14: Average asset portfolio allocation of Life insurers inFrance (without look-through approach of funds). Source: ACPR

S.06.22

Within the Fixed Income class, the ALM team studies the Credit and Credit-Spread risks as well as the maturity of interest rates linked securities, as it isillustrated in the figure above. This last point joins the concepts of Duration-Convexity and Cash-Flow Matching presented before. Due to the complexityand variety of variables to take into account in investment for the risk-returnoptimization, it is necessary to implement qualitative analysis in oder to ex-ploit the results obtained from quantitative models.

Life insurers invest also in funds which also have diversified portfolios. Thus,a look-through approach is used in oder to control the risk exposure of theinsurer portfolios in terms of asset class selection. This is mentioned becausethe ALM computational model has to consider the modeling of different typesof assets and funds, in line with real practices in practical ALM. Therefore, themodeling of dynamic duration steering mechanics, presented later in Chapters3 and 4, are developed considering also the portfolio allocation by asset class.

16 Chapter 1. Life Insurance, Regulation and Asset Liability Management

FIGURE 1.15: Average asset portfolio allocation of Life insurers inFrance (with look-through approach of funds). Source Enquête AF2i

2018

• Portfolio Hedging: In addition to Cash-Flow Matching, Duration-ConvexitySteering and Risk-Return Optimization via Strategic Asset Allocation, it is alsonecessary to use derivatives strategies with the aim to cover specific risk sen-sibilities and attain profitability objectives. For example, forwards on bondsare signed in order to reach longer maturities, while interest-rates derivativesare purchased and sold, such as swaptions as well as caps and floors, withthe aim of decreasing the portfolio risk exposure to interest-rates movements.Similarly, derivatives are also implemented on equity shares. the ALM team isusually involved in the definition and steering of the derivatives strategies sothat risk is assessed in line with SAA and the sensibility analyses.

The risk by structured investment solutions, such as hybrid securities, is alsoassessed by the ALM team.

• Liability Constraints Analysis: On the one hand, the actuarial department isthe branch in charge of producing and analyzing the metrics regarding liabil-ity risks, for example the company risk exposure to mortality and longevityas well as surrender. On the other hand, regulatory provisions are steered bythe technical unit. Therefore, the ALM team ensures a prudent investment ap-proach in line with the liability constraints mentioned before. This is why, theALM team extract relevant indicators, such as policyholder profit participationprovision9, with the aim to add them to investment steering analysis.

• ALM modeling: Last but not least, the ALM team is also in charge of translat-ing practical knowledge in mathematical and computational models for risk-return optimization purposes. Usually, these developments are made in co-operation with other risk modeling teams.

9Provision pour Participation aux Excedents (PPE) in French

1.4. Chapter Summary: In-depth focus on Investment Strategy and link toDynamic Duration Steering modeling 17

In fact, the list of ALM common practices presented below is complex and re-quires the participation of a variety of experts from different departments. There-fore, the ALM complete these task in co-operation with other teams and depart-ments. Anyway, it is remarked once again that the global objective is summarizedin the maximization of the shareholder value and, as such, all these tasks can beconsidered as constraints of this target.

1.4 Chapter Summary: In-depth focus on Investment Strat-egy and link to Dynamic Duration Steering modeling

In the case of Allianz Investment Management (AIM), the functions presented in thelast section are developed by the ALM team in co-operation mainly with the Invest-ment Management team with the aim to build a coherent investment strategy, as amatter of fact the two teams form the Investment Strategy Direction of AIM. Giventhat the ALM team is one of the principal actors of the link between the investmentand the liability world, it ensures also a consistent groundwork with the Actuarialand Risk Department objectives as well as with those from the Product & TechnicalUnit.

FIGURE 1.16: Allianz Investment Management value-creation chainprocess

All in all, ALM could be considered as the main manager of the asset portfoliosensitivity to market movements,10 as illustrated in the figure below. As such, inthe case of Allianz France, the ALM team presents a set of Strategic Asset Allocation(SAA) to the Executive Bord. Along with the SAA, the ALM team presents also a setof Key Performance Indicators (KPIs) which are produced using the ALM computa-tional model as well as other quantitative and qualitative analyses. Finally, this setof KPIs allow the Executive Board to chose the SAA for the portfolio steering thatwill be usually fixed during one year.

10It is closely related with the b metric from the CAPM model in classical Portfolio theory

18 Chapter 1. Life Insurance, Regulation and Asset Liability Management

Once the SAA is fixed, the Investment Management team is in charge of choos-ing the best sectors and types of assets in line with the SAA which represent a goodpotential of return at adequate levels of risk.11 This is why the ALM and InvestmentManagement team are in the same operation unit (the Investment Strategy Direc-tion); Precisely, this two teams co-operates for the portfolio macro-management.12

Consequently, the re-balancing mechanics proposed in this professional thesisaims to improve the Allianz ALM computational model, so-called ALIM, by re-producing realistic investment practices. Therefore, a framework is provided forimplementing a strategic investment modeling which steers asset duration dynam-ically according to market movements, precisely to interest rates movements. Morespecifically, this thesis conduct to:

• Model the Liability duration, as a function of interest rates, using the sensitiv-ities of Allianz ALM model;

• Implement re-balancing mechanics in order to attain Strategic Asset Allocationobjectives and, simultaneously, to steer the asset duration;

• Test the impact of the proposed model revision on a variety of risk and man-agement metrics. Specially, a focus on the effects on Basic Own Funds (BOF),and Risk Capital (RC) and Solvency 2 Ratio will be presented.

11So-called Tactical Asset Allocation which derives from the concept of active portfolio managementfrom classical Portfolio Theory. It is closely related with the a metric from the CAPM model, as it isillustrated in figure 1.16

12Portfolio macro-management concerns the asset allocation as well as the investment risk expo-sure management and the asset class selection by sector and credit-risk grades. Once an investmentmandate is defined by Allianz Investment Management (AIM), the portfolio micro-management ac-tions are mainly executed by the European branch of Allianz Global Investors (AGI) and PIMCO, suchactions concerns principally the direct asset-management; For example, asset selection, trading andbrokerage are some of the functions executed on capital markets by asset managers.

19

Chapter 2

ALM Model

Insurance companies employ an Asset-Liability Management (ALM) computationalmodel, which reflects essential interactions between asset and liability cash-flows aswell as the impact of the actions of policyholders and the management of the insurer.This model allows to measure accurately a wide range of risk and performance met-rics so that the business can be supervised and management actions can be executed.

In the Life segment, it is a common practice to use complex ALM computationalmodels which allows to map assets and liabilities cash-flows over the course of thebusiness projection in a wide range of economic scenarios in order to determine,by Monte-Carlo simulation methods, the market value of liabilities, or Best Esti-mate Liability (BEL). As in the case of financial derivatives pricing, the Monte-Carlomethod provides a manner to estimate the Time Value of Options & Guarantees(TVOG) which is the cost of complex embedded financial options in Life insurancecontracts1.

Henceforth for the purpose of this thesis, Allianz ALM model, called ALIM, isgoing to be used in order to illustrate the structure and functionalities of a Life ALMmodel. It should be noted that most models in the market have very similar struc-tures but ALIM has its own particularities the reader should not be able to find else-where.

Consequently, the purpose of this chapter is to show the main components ofan ALM model with the aim to introduce the framework in which the InvestmentStrategy Model operates. Besides, the examples and results of this professional the-sis are implemented in 2017 fourth quarter version of ALIM. Moreover, at the endof the chapter the ALIM Investment Strategy Model itself is presented with the ac-tual modeling before the improvements proposed, later for the aims of this thesis, inChapters 3 and 4.

Keywords: ALIM, ALM model, Best Estimate Liability (BEL), Time Valueof Options & Guarantees (TVOG)

2.1 ALIM: Allianz ALM model

Allianz has developed its own ALM model called ALIM (Asset-Liability InteractionModel). This model has been developed in MoSes, a life modeling software tool ofTowers Watson, which has mainly be selected because of the flexibility to customize

1Embedded financial options in Life insurance contracts are mainly generated by surrender optionand the Minimum-Guaranteed profit sharing

20 Chapter 2. ALM Model

the model to the company specific requirements and needs. It should be noted thatALIM is used by multiple operating entities of the Allianz Group and, thus, theFrench subsidiary has customized parts of the application according to its needs.

ALIM is involved in the following calculation processes:

• Solvency 2 Standard Formula: ALIM is used to calculate the Solvency CapitalRequired (SCR) for the single risk modules. Further processing is performedoutside of ALIM; In particular, the aggregation of risks and reporting are per-formed in an external tool.

• Solvency 2 Internal Model: ALIM is used for the projection of liability cash-flows and for the calculation out-of-sample shocks for the calibration of thereplicating portfolio, which is then used to calculate the SCR for market risk.Additionally, it is also used to calculate shocks for non-market risks and thepolicyholder participation on credit risk. Once again, further processing ofresults is performed outside of ALIM.

• Solvency 2 Technical Provisions: Parts of the BEL are calculated with ALIM,specially the Time Value of Options and Guarantees (TVOG).

• MCEV calculations: ALIM is used to calculate the stochastic Present Value ofFuture Profits (PVFP)

• ALM calculations: ALIM is used to produce a wide range of Key PerformanceIndicators (KPIs) for the Asset-Liability Management and the Strategic AssetAllocation (SAA) as well as the Investment Management of Allianz Franceportfolios.

Most of the metrics mentioned before are calculated by Monte-Carlo Methods.Therefore, ALIM performs stochastic runs with multiple iterations, each iterationrepresenting a realization of a path of the economic environment.

Given that the purpose of this work is to improve a modeling module of ALIM,thus, any further processing outside of ALIM will not be revisited and will be onlyused for the production of certain Key Performance Indicators (KPIs), notably RiskCapital (RC) and TVOG.

2.2 Model structure

Allianz uses ALIM as a tool for the projection of the asset and liability of the com-pany. Therefore, ALIM uses a modern ALM system which captures the essentialinteractions between asset and liability cash-flows, as it is presented in the followingflow-chart.

2.2. Model structure 21

FIGURE 2.1: ALIM Structure Flow-Chart

Assets and liabilities are regrouped in classes and set specifically for ALIM cal-culation. The parameters of these classes are calibrated in order to assure that theyare representative of real assets and liabilities in Allianz portfolios. For this purpose,they are registered in tables, containing mainly cash-flows information, and givenas an input at the beginning of the projection. In the same way, a set of economicscenarios is fixed before to perform any calculation. Nevertheless, assets and liabili-ties tables evolute during the projection time, while the economic scenarios are keptfixed for all the projection.

Additionally, management assumptions, concerning the Investment and Cred-iting strategy, are set at the beginning of the projection and fixed for ulterior cal-culations. More precisely, these hypothesis determine the rules so that assets andliabilities evolute in different economic environments.

It is important to remark that many departments of the company are involved inthe calibration of ALIM assumptions in order to accomplish accurate calculations.Therefore, there are many components of the model that are not treated, not evenmentioned, in this thesis. Just to mention some them, Lapse & Surrender and bio-metric assumptions are also given as an input for ALIM.

22 Chapter 2. ALM Model

FIGURE 2.2: ALIM Code Architecture

MoSes is based on a object-oriented programming language. Following thislogic, the modeling architecture of ALIM is presented in the diagram above, Figure2.2, and details on particular functionalities are provided in the next sections.

2.2.1 Inputs: General, Asset, Liability, Corporate and Economic Scenariosassumptions

In ALIM, assumptions are assigned to MoSes variables on basis of assumption ta-bles. It is a core aspect of ALIM to be able to vary assumptions in an easy and effi-cient way with the aim to define and run multiple calculations while only the set ofassumptions is changing from step to step. That is why assumptions are organizedin meaningful assumption tables that bundle hypothesis related to specific areas orto a specific sub-model, as it is illustrated in the following diagram.

FIGURE 2.3: ALIM assumptions tables

Inputs are closely linked to the modeling architecture presented before in Figure2.2. To sum up, assumptions in ALIM are the following:

• General assumptions: mainly linked to the Top model which is in charge ofsetting up the model structure, initializing assumptions of underlying mod-els and controlling the calculation order. They are also linked to the Company

2.2. Model structure 23

which performs top level company and tax calculations. Finally, these assump-tions contain also Fund model information for accounting related calculationsfor the underlying funds of the company.

• Asset assumptions: included in tables of each asset sub-model, the Asset andthe Fund model. A wide variety of assets are modeled in ALIM, furthermore,the following diagram show the different kind of asset types present in themodel

FIGURE 2.4: ALIM asset types

• Liability assumptions: included in tables of the Top model, the Liab model,the Rollup model and the Credstr model. In particular, the Liab model pro-vides an interface for the liability results and the Rollup model uses pre-generatedresults for fast scenario projections.

• Corporate assumptions: contain information related with the Strategy modelsand the general management of the company2.

• Scenario assumptions: mostly linked to the Top model, these tables containsthe economic environment in which projections are performed. More preciselyfor the Economic Scenario Generator (ESG), each row corresponds to one timestep of a scenario and it is ordered first by scenario and second by time step;In addition, each column corresponds to one output, for example, Euro zonecoupon bond prices with time to maturity 10 years. Furthermore, differentESG files can be used for the initial repricing of assets and projections inde-pendently.

2.2.2 Strategy Model: Investment and Crediting

ALIM performs multiple iterations in the stochastic mode, each iteration represent-ing a realization of a path in the economic environment. Additionally, ALIM followsa calculation order each time an iteration is executed, this process is illustrated be-low.

2In particular, the Dynamic Duration module within the Investment Strategy Model of ALIM isparametrized mainly within the Corporate Assumptions

24 Chapter 2. ALM Model

FIGURE 2.5: ALIM strategy process

On the one hand, the Crediting Strategy model reflects management actions con-cerning the profit sharing between the policyholder and the shareholder. Mainly,this model is in charge of the realization of capital gains and the allocation of bonusreserves. Likewise, the amount of statutory solvency coverage and the net incomebefore strategy decisions represent also a scope of management treated by the Cred-iting Strategy model.

On the other hand, the Investment Strategy model controls the development ofthe asset portfolio by executing purchases and sales according to defined priorities.This model will be explained in detail later, in section 2.3, because the purpose ofthis professional thesis is to modify and improve such a modeling.

2.2.3 Outputs and Reporting

It is important to remark that in the stochastic mode ALIM processes a huge amountof data and only certain part of this information is registered for further calcula-tions. As it was already mentioned in section 2.1, ALIM is involved in calculationsprocesses of Solvency 2, MCEV and ALM studies. Therefore, there are specific mod-ules of ALIM specifically created for reporting purposes. More exactly tis modulesallows to extract the information used for the calculations mentioned before as wellas to minimize the run time of the entire model, by only writing information that isneeded to a file.

For instance, the key values extracted from ALIM for Solvency 2 calculations arethe present value of asset and liability cash-flows; A theoretical approach on Best Es-timate Liability (BEL) calculations is provided in section 1.2.2. On the other hand, thePresent Value of Future Profits (PVFP) defined for MCEV (without interest earnedon shareholder equity) are also extracted from ALIM as well as the Present Value ofShareholder Profit (including interest on shareholder equity).

Additionally, a wide variety of values can be selected by the user of ALIM andextracted in different reports in order to perform further calculations processes suchas Net Asset Value (NAV) and Shock sensitivity Analysis.

2.3. Investment Strategy Model 25

2.3 Investment Strategy Model

The Investment Strategy model aims to reflect management actions taken in thepractical world for the asset portfolio steering. In a general fashion, it assess the as-sets re-balancing mechanics (i.e. buying and selling) required to reach a predefinedtarget asset allocation. The target asset allocation and the re-balancing mechanicsin ALIM correspond in the practical world to the process of Strategic Asset Alloca-tion (SAA) presented in 1.3.1 which, in the case of Allianz France, is defined andexecuted by the ALM team in co-operation with the Investment Management teamwithin the Investment Strategy Unit. The SAA is the main driver of the long-terminvestment and, as such, the target allocation in ALIM is the principal objective ofthe investment strategy model.

Currently, the Investment strategy model comprises three main purposes:

• To control the development of the asset portfolio in order to guarantee restric-tions and limits and, at the same time, to follow a target asset allocation;

• To perform sales and/or strategic realizations according to defined priorities;

• To control the duration of the asset portfolio.

ALIM inputs allows to parameter precisely the type of assets involved in there-balancing and the target allocations. Among others, the method to perform thepurchases and sales can be also parametrized as well as the profit realization con-straints. However, all the parametrization is fixed during all the projection horizonand the current model does not have the possibility to readjust the parameters in astrategic way to market movements. In particular, the duration of assets involved inthe re-balancing is calibrated and fixed to meet current and expected durations overthe projection years but they do not react strategically to market movements.

In this context, the purpose of this work is to improve the Investment Strategymodel of ALIM. Therefore, a framework is provided, in chapter 3 and 4, for imple-menting a strategic investment modeling which reacts dynamically to market move-ments, precisely to interest rates movements.

2.3.1 Module structure

The Investment Strategy occurs at the end of each projection year, after cash-flowscalculation and market value revaluation, and before the Crediting Strategy. How-ever due to the Crediting Strategy, further transactions are necessary to realize Un-realized Capital Gains and Losses (UCGL).

26 Chapter 2. ALM Model

FIGURE 2.6: Investment Strategy Sequence

In the diagram above, the order of calculations are presented in addition to thetwo sub-models operating into the Investment Strategy model. Firstly, the DynamicAsset allocation (DAA) sub-model computes the amounts to buy or sell, per assetclass, in order to reach the target allocation. Lastly, the Transaction of Assets (TA)sub-model determine which assets have to be sold and buy, for each asset class,according to the re-investment parameters.

2.3.2 Re-balancing mechanics

The re-balancing process covers the purchase and sale of assets so that target al-location rates are reached. Furthermore, this professional thesis aims to give theframework in which a duration strategy can be executed also within the Transactionof Assets (TA) sub-model.

FIGURE 2.7: Re-balancing mechanics

As it is illustrated in the diagram above, the core algorithm works as follows:

2.4. Chapter Summary: Current Investment Strategy modeling and link toDynamic Duration Steering sub-model 27

1. At a user-defined frequency, it is checked if the actual market value allocationrates of the involved asset classes are within defined tolerance limits of theirtarget allocation rates;

2. If at least one asset class is out of it tolerance limits, a re-balancing is triggered,i.e. the volume of asset classes exceeding their maximum limits are decreasedby sales,a and asset classes lying below of minimum limits will be increasedby purchases;

3. The extent of re-balancing is closely linked to the re-balancing method chosenby the user. Henceforth, it is possible to focus on minimizing the realization ofcapital gains or the focus can be placed on meeting the target allocation rates.

2.4 Chapter Summary: Current Investment Strategy model-ing and link to Dynamic Duration Steering sub-model

Within the entire calculation process, at the end of each projection year, the re-balancing is performed before the Crediting Strategy. The reason is mainly that there-balancing usually affects the book return of a year and, thus, directly all the tar-gets related to that. Therefore, the strategies related to profit sharing or solvencyfigures should be executed thereafter.

The realization process draws upon decisions derived from the Crediting Strat-egy. In particular, unrealized capital gains are management buffers to steer profits.Then, the Investment Strategy translates the decision on the amounts to realize intodecisions on single asset level.

FIGURE 2.8: Strategy model

The re-balancing process covers the purchase and sale of assets so that targetallocation rates are reached. Including the Dynamic Duration Steering proposed inthis professional thesis, the entire re-balancing process is illustrated in the diagramabove and described below:

28 Chapter 2. ALM Model

1. Information aggregation: In order to perform the strategies, the InvestmentStrategy sub-model requires a lot of information on single asset level as wellas on asset class level. At the start of each re-balancing period, this informationis requested from the asset models and further aggregated on asset class level;

2. Target rates per class: When an emergency arises, the target rates for re-balancingare defined and applied;

3. Sales/Purchases per class: Once targets per class have been set, the amountsto be sold or purchased per class are calculated.

4. Prioritized sales: The Investment Strategy model has access to all relevantcharacteristics and values on single asset level to determine sales based onsorted priorities;

5. Purchases: The amounts to be bought per asset class are translated into thepurchase of single assets or in scaling factors;

6. Duration Strategy: If Duration Strategy is applied, the priorities for sales andthe profiles for purchases might depend on the current duration situation.Thus, the standard priorities for sales and the standard profiles for purchasesare revised and adjusted if necessary.

7. Perform transactions: Lastly, transactions are executed and the asset and com-pany profiles are updated for further calculations.

Finally, it is remarked that the Dynamic Duration Steering interacts inside eachasset class separately. Thus, re-balancing between asset classes remains untouched.In other words, the Duration Strategy respects the Strategic Asset Allocation (SAA)Steering, which is usually the case in the practical world. More exactly, the DynamicDuration (DD) module operates inside the Transaction of Assets (TA) sub-model,once the the amounts to be bought or sold are already defined in the Dynamic AssetAllocation (DAA) sub-model.

29

Chapter 3

Dynamic Duration SteeringModeling

One common practice of the ALM Department concerns the steering of the matu-rities of the Fixed Income and interest-rates securities within the asset portfolio inorder to close the gap between asset and liability duration somehow estimated, socalled Duration Steering. Precisely, securities with long maturities could be pur-chased or securities with short maturities could be sold with the aim to push du-ration up. Conversely, duration can be steered down by selling longer securities orbuying shorter ones.

Asset and Liability duration reacts to interest-rates movements but the cash-flows structure of Life insurance liabilities depends also of the fund-specific mini-mum guarantee level and the Surrender behavior. In fact, embedded financial op-tions in Life and Savings policies, so called Options and Guarantees (OG), are esti-mated using ALM calculation models and, thus, the estimation of liability durationalso requires such modelling. Consequently, one of the objectives of the StrategicAsset Allocation (SAA) is to define the Fixed Income and interest-rates related secu-rities weight within the asset portfolio in order to steer duration, among other pur-poses. All these common practices of the ALM & Investment Management team1

are presented in detail in section 1.3.

The re-balancing mechanics of the current version of Allianz ALM computationalmodel, so-called ALIM, seeks to replicate common investment practices:

• Asset Allocation: An asset class target allocation, within pre-defined leeways,is set per time-step in ALIM inputs, equal for all scenarios2, in line with thereal SAA implemented in practice;

• Asset Management: During the projection, assets are sold or bought to get thetarget asset allocation in line with the permitted asset allocation leeways.

• ALM: asset transactions respect liability constraints such as policyholder profitsharing as well as specific-fund clauses.

The current version of ALIM performs purchases according to static asset profilesand allows only a pre-defined asset allocation with pre-defined weights. Indeed, thematurity allocation of Fixed Income assets is fixed and still being the same during

1In the case of Allianz Investment Management (AIM), ALM & Investment Management teamform the Investment Strategy Unit.

2Allianz France has chosen to use static SAA modelling for all funds except for one portfolio whichis steered with a dynamic SAA

30 Chapter 3. Dynamic Duration Steering Modeling

the entire projection. Therefore, duration is only a result of the modeling and it isnot actively used to control the investment process.

Consequently, the objective of this chapter, and in general of this professionalthesis, is to provide a framework in which duration steering is implemented withinthe Investment Strategy modeling. This feature allows the model to react dynam-ically to market movements, more exactly to interest-rates movements. It furtherprovides a more accurate tool to measure the impact of different long-term ALMand investment strategies on a variety of Key Performance Indicators (KPIs); Laterin Chapter 5, some impact analyses on Own Funds are presented as well as on Sol-vency Capital Required (SCR)and Solvency 2 Ratio.

3.1 Duration and Convexity

According to Hull, 2009, "the duration of a bond, as it name implies, is a measureof how long on average the holder of the bond has to wait before receiving cashpayments". In general for a financial asset portfolio, duration measures how longon average the company has to wait to wind up the business in order to recover theinvestment together with all profits (or losses) pulled from the financial activity.

Duration has become a popular metric, since it was first suggested by Macaulayin 1938, because it also allows to establish a relationship between the percentagechange in bond prices and change in interest rates. This is still a valid statementfor a Fixed Income portfolio once cash-flows have been aggregated per time step.Given that Duration is a first order sensitivity, there exist a second order sensitivityso-called Convexity. It is a common mistake to state that Duration and Convexityare the first and second order derivatives of the Bond Price respectively. However,there exist a close relation between derivatives of the Bond Price and Duration aswell as Convexity.

Therefore, the fundamentals of Duration and Convexity are presented in this sec-tion and, in the context of Solvency 2 economic vision, the extension of this conceptsto modern portfolio valuation practices are also developed.

3.1.1 Fundamentals

Firstly, the price of a bond is denoted as P and defined as

P :=T

Ât=1

Cte�yt

where T is the time to maturity, Ct is the bond cash-flow at time t, and y is the con-tinuously compounded bond yield.

Macaulay duration, denoted as Dmac, is often presented as a weighted averagetime to maturity of a bond as it follows

Dmac :=T

Ât=1

tCte

�yt

P=

T

Ât=1

tCte

�yt

ÂT

t=1 Cte�yt

=T

Ât=1

twt (3.1)

3.1. Duration and Convexity 31

where wt =Cte

�yt

ÂT

t=1 Cte�yt

is a weight with ÂT

t=1 wt = 1.

On the other hand, using the same notation, convexity, denoted C, is defined as

C :=T

Ât=1

t2 Cte

�yt

P=

T

Ât=1

t2wt. (3.2)

Furthermore, the first order derivative of the bond price with respect to the bondyield are respectively

∂P

∂y= �

T

Ât=1

tCte�yt (3.3)

and∂2

P

∂y2 =T

Ât=1

t2Cte

�yt. (3.4)

A series of remarks and relations regarding duration and convexity as well as thefirst and second order derivatives are presented below:

• The first order derivative of the bond price with respect to the bond yield is notequal to the negative of Macaulay duration. Indeed from equations 3.1 and 3.3,the following relation is provided

Dmac = � 1P

∂P

∂y(3.5)

• The second order derivative of the bond price with respect to the bond yieldis not equal to convexity. Indeed from equations 3.2 and 3.4, the followingrelation is provided

C =1P

∂2P

∂y2 (3.6)

• The first order derivative of Macaulay duration with respect to the bond yieldis not equal to convexity. Indeed from equation 3.5, it follows that

∂Dmac

∂y=

∂

∂y

�

∂P

∂y

P

!=

✓1P

∂P

∂y

◆2� 1

P

∂2P

∂y=

✓� 1

P

∂P

∂y

◆2� 1

P

∂2P

∂y.

Therefore∂Dmac

∂y= D

2mac � C. (3.7)

Indeed, the relationship presented in equation 3.7 will be used later in Chapter4 for further developments of duration steering modeling in ALIM.

• Hereafter DP and Dy denote a small change in bond price and yield respec-tively. Thus, a Taylor series approximation truncated to the first term gives

DP ⇡ ∂P

∂yDy

and, using equation 3.5, this can be written as

DP

P⇡ �DmacDy (3.8)

32 Chapter 3. Dynamic Duration Steering Modeling

Equation 3.8 allows to see duration as a sensitivity, precisely the percentagechange in the bond price is approximately equal to minus its Macaulay dura-tion multiplied by the size of the parallel shift in the yield curve. However,yield curves movements over time are not uniquely parallel shifts and, thus,higher order approximations are required to capture the sensitivity of bondprices to more complex deformations of yield curves.

By matching duration, Life insurance companies ensure that small parallelshifts in the yield curve will impact evenly asset and liability portfolios and,thus, risk profile to small parallel shifts in interest-rates curves remains un-changed.

• A Taylor series approximation truncated at the second term gives

DP ⇡ ∂P

∂yDy +

12

∂2P

∂y2 (Dy)2

and, using equations 3.5 and 3.6, this can be written as

DP

P⇡ �DmacDy +

12

C(Dy)2 (3.9)

Therefore, percentage changes in bond price can be more accurately approxi-mated using duration and convexity together with the help of equation 3.9.

By matching duration as well as convexity, Life insurance companies can makeitself immune to relatively large parallel shifts in interest-rates curves. How-ever, they are still exposed to non-parallel shifts. This is why, duration and con-vexity gap steering should be complemented by other management actions, asit was presented in section 1.3.

It is remarked that Macaulay duration and convexity, as they were presented sofar, only work for certain cash-flows at the valuation time. However, several cash-flows of asset and liability portfolios of Life insurers are uncertain due to financialand insurance embedded options. On the one hand, asset cash-flows can be esti-mated uniquely from asset and market data and, thus, its duration and convexitycan be found subsequently. On the other hand, liability cash-flows depend of sev-eral variables such as mortality and longevity as well as surrender; In particular,they also depend of the level of return on the investment to policyholders and, thus,this cash-flows depend also of the economic environment and the asset cash-flows.

To asses complex liability cash-flows, effective duration and effective convexityare then defined using the overall portfolio valuation. For the purpose of this thesis,the extension of duration and convexity concepts is done within the context of theMarket-Value Balance-Sheet (MVBS) valuation and the solvency 2 regulation, as it ispresented in Section 1.2.2.

3.1. Duration and Convexity 33

3.1.2 Effective Duration and Effective Convexity within Solvency 2 eco-nomic vision

In Life insurance business, part of company’s capital and profits (or losses) belongsto the policyholder3. Due to the intrinsic uncertainty of Life insurance contracts,part of future expected liability cash-flows is estimated using actuarial deterministicmodels while the part concerning financial embedded options has to be calculatedby Monte-Carlo simulation methods using an ALM computational model.

In France, Life insurance savings contracts labeled "Euro" give the right to thepolicyholder to a minimum investment return as well as to surrender the contractat any time. On the one hand, French Life insurance business still is profitable evenif the surrender guarantee is offered because policyholders have tax incentives tohold the policy4; On the other hand, the policyholder and the shareholder have anasymmetric risk profile because the policyholder have a guarantee on a minimumlevel of return while the shareholder would absorb all losses if they exist.

These Options and Guarantees (O&G) in Euro savings insurance contracts havean impact on the structure of expected future cash-flows and, thus, on duration aswell as convexity. Using a finite-differences approach on equations (3.5) and (3.6),effective duration and effective convexity are respectively estimated as follows

De f f ⇡V�Dy � V+Dy

2V0Dy(3.10)

andCe f f ⇡

V�Dy + V+Dy � 2V0

V0(Dy)2

where, in the context of Solvency 2,

• Dy is a parallel shift on the EIOPA reference interest-rate curve. In this con-text, the parallel shift is applied in the same way as interest-rates shocks in theStandard Formula (or Internal Model);

• V�Dy and V+Dy are the Standard Formula (or Internal Model) valuations in theparallel shift interest-rates shocked scenarios; The parallel shift size is �Dy

and �Dy respectively. As it was presented in section 1.2.2 and 1.2.3, liabilityvaluation is equal to the sum of the Best Estimate Liability (BEL) and the RiskMargin (RM); While asset valuation is equal to the market value of the assetportfolio.

• V0 is the market value estimation of assets in both cases. Liability and assetsensitivity estimations are normalized to the same value V0 with the aim tohave comparable metrics and, subsequently, determine the duration and con-vexity gap.

In order to illustrate the finite differences approach, equation 3.10 can be seen asan average of percentage changes of the value with respect to small parallel shift on

3the French regulation "Code des Assurances L. 331-3, A. 331-3 et A. 331-4" globally imposes theminimum financial and technical profits/losses sharing to be provided by the Life Insurers to thepolicyholder, 85% and 90% respectively,

4The first tax incentive is given after 4 years and the most significant one after 8 years

34 Chapter 3. Dynamic Duration Steering Modeling

interest rates, as presented below

De f f ⇡

V�Dy�V0

V0Dy

+V0�V+Dy

V0Dy

2=

V�Dy � V+Dy

2V0Dy.

It is important to remark that these metrics are local numerical estimations and,as such, their accuracy depends on the accuracy of the economic valuations as wellas the size of the interest-rates curve shock Dy. Moreover, as Dy becomes smaller,theoretically speaking, effective duration and convexity estimation becomes moreprecise. However, in practical implementation, Dy should not be chosen too smallin order to avoid model bias.5.

(A) Effective Duration as a function of Dy (B) Effective Convexity as a function of Dy

FIGURE 3.1: Impact of Interest-rates shock level on sensitivity estima-tions

Estimations on the figure below were produced in ALIM. It is remarked that theduration and convexity estimations as well as the gaps change with Dy. Hereafter,Dy is fixed to 1% as the level of shock proposed for the calculation of the interest-rates risk module of the Standard Formula. Even if it seems as a too big level fora local numerical estimation, this choice is appropriate because estimations do notvary significantly for levels below 1%.

Finally it is remarked that sensitivity estimations such as duration and convexityare not unique. As a matter of fact, there are several approaches to calculate thesemetrics and many variables that impact directly these estimations. It is important tokeep this point on mind for safeguarding the coherence of analyses regarding suchmetrics.

3.1.3 Aging of Run-Off Business Portfolios

A key point of the economic vision in Solvency 2 is that all portfolios valuationsare estimated on a run-off business approach. This is coherent with the risk-neutralframework in which the premium of financial embedded options is priced using amartingale approach. Thus, new-business assumptions are avoided with the aim torespect the risk-neutral return and, thus, the martingale nature6.

5Due to the complexity of ALIM, if the interest-rate shock is too small, the model could be notcapable of catching the adequate change in the valuation

6As of today, according to EIOPA guidelines and contract boundaries, only regular premiums onindividual savings and pensions are projected.

3.1. Duration and Convexity 35

More exactly, business hazard is not incorporated in the valuation which meansthat expected premiums of future contracts are avoided. In addition, regulatory in-stitutions, such as the EIOPA, provide the guidelines for the calibration of EconomicScenarios Generator (ESG). As a result, valuations are comparable from one insurerto another. Nevertheless, it is important to remark that in-force business future ex-pected premiums of valid contracts at the valuation time are taken into account inthe pricing. The projection of the technical Minum-Guarantee cash-flows is pre-sented below with the aim to exhibit the portfolio aging.

FIGURE 3.2: Minimum-Guarantee technical cash-flows projection

Additionally, Macaulay duration is calculated, and presented in the figure below,on in-force business Minimum-Guarantee technical cash-flows in order to illustratethe duration evolution in a run-off portfolio.

FIGURE 3.3: Macaulay Duration calculated on Minimum-Guaranteetechnical cash-flows projection

In fact, interest-rates movements and minimum-guarantee projection are the prin-cipal drivers of duration. Given that this is a central fact of this professional thesis,it will be further developed and illustrated and will be also included in the steeringduration modeling which is presented henceforth.

36 Chapter 3. Dynamic Duration Steering Modeling

3.2 Dynamic Duration Steering in ALIM

The investment strategy modeling aims to replicate the essential practical functionsof the Investment Strategy Department. Therefore, the re-balancing mechanics ofAllianz ALM computational model, so-called ALIM, are in line with the practicalmechanism of Strategic Asset Allocation (SAA), presented in section 1.3.1, which isthe main driver of the long-term investment results, for every insurance portfolio.As such, the subsequent changes proposed in this professional thesis keep the ex-isting re-balancing mechanics of ALIM in place as far as possible. In fact, dynamicduration steering does not interfere on the asset class weights defined in the existingre-balancing algorithm, but interact inside each asset class separately.

FIGURE 3.4: Dynamic Duration Steering within Investment Strategymodeling

The figure above illustrates the investment strategy algorithm presented in sec-tion 2.3 as well as the placement of the dynamic duration steering model.

In order to re-balance the assets inside the model for duration steering purposes,it is essential to calculate asset and liability duration at each time step along the busi-ness projection for every trajectory. As a matter of fact, asset duration cash-flows canbe accurately estimated by closed formulas within ALIM as well as duration7. Onthe contrary, projection of liability cash-flows is more sophisticated and it requiresthe use of an ALM model; Consequently, liability duration can only be accuratelydetermined by a sensitivity approach as the one presented in section 3.1.2, preciselyeffective duration should be calculated. Nevertheless, effective duration implies atleast three stochastic runs for just one estimation; Thus, with current machines andalgorithms, it would take several years to perform this calculation at each time stepalong the business projection for every trajectory, which bring us back to the "sim-ulation within a simulation problem" already mentioned in section 1.2.3 for the Sol-vency Capital Required (SCR) calculation.

7Macaulay duration formula presented in section 3.1.1 can be refined in order to have more accu-rate estimations

3.2. Dynamic Duration Steering in ALIM 37

As first step of duration steering, a target duration has to be defined each timestep, which corresponds to the liability duration. This target duration can be thencompared with the Fixed Income portfolio duration at each time step with the aimto determine the duration gap. Finally, the duration gap is split up into all assetclasses which participate in the dynamic duration steering. For the purpose of thisprofessional thesis, only bonds, including inflation-linked and excluding forwards,are used for this modeling. Payer and receiver swaps could be also included in fu-ture improvements.

This section aims to introduce the framework in which Dynamic Duration Steer-ing is integrated in the investment strategy modeling of ALIM. Firstly, the funda-mentals for the modeling of the liability duration are developed. Lastly, the func-tional specifications of such modelings within the investment strategy and assetmodels in ALIM are presented.

3.2.1 Liability Duration Proxys

Calculation of liability effective duration implies a determination via sensitivity runsin every point in time for each run and for each economic scenario within the model(simulation within a simulation problem). Since this is not implementable for runtime reasons, liability duration has to be approximated for every point in time andevery iteration.

As presented in section 3.1, duration is directly impacted by interest rates as wellas the structure of the cash-flows. It is then proposed to approximate liability dura-tion as an empirical function of interest rates level combined with an empirical ratein order to incorporate also the aging/structure of the cash-flows. Such empiricalformula is built using the following algorithm:

1. Interest-rates: Liability duration is firstly estimated for each simulation i at theprojection time step t as a function of the interest rates (or forward rates) levelx(i, t), formally written as it follows

durtgt(i, t) = f (x(i, t)).8

The choice of such function f will be further discussed in sections 4.1.2, 4.1.2and 4.1.3 as well as the choice of the interest rate level t 7! x(i, t)9 at time stept.

2. Limits Control: The duration target is adjusted according to duration goallimits, i.e.

durtgt(i, t) = max�min

�durtgt(i, t), durmax

�, durmin

�

where durmax and durmax are respectively the maximum and minimum dura-tion allowed targets which are parameters defined in ALIM inputs;

8Several interest rates could be considered for this first estimation. For example, a model with twoarguments could be implemented, such as durtgt(i, t) = f (x(i, t), y(i, t)), where x(i, t) and y(i, t) couldbe interest rates in different currencies.

9Given a fixed trajectory i, the most natural choice for t 7! x(i, t) are zero-coupon rates given thatthey are the drivers of the interest rates in the risk-neutral framework of Solvency 2. A deeper analysisregarding this choice will be presented in Chapter 4

38 Chapter 3. Dynamic Duration Steering Modeling

3. Portfolio Aging: duration target is modified in order to capture the portfolioaging caused by a run-off business:

durtgt(i, t) = durtgt(i, t) ⇤ g(t)

where g(t) is a dynamic rate which aims to capture the evolution of the tech-nical cash-flows regarding the run-off business characteristic, presented in sec-tion 3.1.3 and figure 3.3, isolated from the interest rates movements. It is no-ticed that aging of the portfolio is assumed independent of the of the trajectoryand, thus, the function g does not depend on the iteration i but only on thetime step t. The calibration of such a function g(t) will be presented in section4.2.

4. Limits Control: Limits control is once again executed,i.e.

durtgt(i, t) = max�min

�durtgt(i, t), durmax

�, durmin

�;

Certainly, it is not a simple task to calibrate the functions f and g described inthe algorithm above, neither to choice the interest rate level t 7! x(i, t). Indeed, thisprofessional thesis aims to provides different approaches for these calibrations anddiscuss the accuracy of such modelings. Therefore, deepen analysis regarding the li-ability duration proxys and portfolio aging of run-off businesses is further discussedin detail in chapter 4.

3.2.2 Re-balancing mechanics

A set of parameters are proposed in order to steer duration within ALIM. Moreprecisely, Dynamic Duration Steering module comprises:

• Target duration: Target duration input via an estimation of liability durationby empirical formulas, presented in section 3.2.1, mainly depending on interestrates and minimum guarantee levels. The calibration of the target duration isfurther presented in detail in chapter 4;

• Re-allocation: Re-balancing between asset classes (asset class weight steering)remains untouched, as happens in highest level;

• Steering mode: Steering mode can be set to Active or Passive, where Pas-sive excludes extra sales for duration steering purposes, and just uses availablecash: Only Passive mode is tested for the aim of this professional thesis;

• Asset classes: purchased or sold assets have to be chosen adequately to man-age duration. Only governmental bonds are tested for the aim of this profes-sional thesis;

• Asset profiles: Purchases happen via pre-defined buy profiles for durationincrease/decrease purpose.

• P&L constraints: If bond sales happen via prioritisation of realization of Un-realized Capital Gains & Losses (UCGL), it is possible to set UCGL limits inorder to respect P&L targets. This can cause the duration not to be met incertain time steps.

3.2. Dynamic Duration Steering in ALIM 39

Current ALIM modeling uses a fixed re-investment bond profile which mainlyhas information on the maturities of the assets to be purchased, denoted NormalBasket and illustrated in the figure below, as well as information of the type of bond(e.g. governmental or corporate bond). Once the amount to be re-invested is de-termined by the Dynamic Asset Allocation (DAA) module in line with the StrategicAsset Allocation, the Transaction of Assets (TA) module performs the purchase us-ing the percentages set in the Normal Basket of maturities10.

Bonds Maturity Normal Basket Shorter Basket Longer Basket

2yr 5% 10% 0%5yr 10% 40% 10%10yr 60% 40% 40%30yr 25% 10% 50%

Total 100% 100% 100%

TABLE 3.1: Example of maturity baskets of bonds for dynamic dura-tion steering modeling purposes

In order to steer duration, the single bond profile of re-investment is replaced bytwo profiles, one with a Shorter Basket of maturities and the other one with a LongerBasket as illustrated in the table above. When the Passive Steering Mode is selected,ALIM uses the amount to be re-invested to purchase longer or shorter maturity bas-kets depending on the gap between asset and liability duration.

The Passive Steering Mode re-investment algorithm is formally presented, usingthe notation introduced in section 3.2.1, as follows

• If durasset(i, t) > durtgt(i, t), then ALIM purchases bonds with the Shorter Bas-ket of maturities;

• If durasset(i, t) durtgt(i, t), then ALIM purchases bonds with the Longer Bas-ket of maturities.

As a matter of fact, there are several parameters to be calibrated and coherencehave to be also guaranteed between the different taken decisions. On the one hand,only the Passive Steering Mode is tested in this professional thesis and the only assetclass used for duration steering purposes is governmental bonds. Moreover, only aset of two new bonds profile is introduced. On the other hand, the calibration ofthe empiric functions f and g, presented in section 3.2.1, is very time demanding.Therefore, there are several possible parametrisations that could be tested in orderto enlarge the scope of this study. More precisely, the following could be tested

• Target duration: Several approaches are presented in chapter 4 for the calibra-tion of the target duration. Consequently, other approaches could be consid-ered or, even, the modelling presented in the previous section 3.2.1 could bereformulated;

• Re-allocation: Re-balancing between asset classes (asset class weight steering)could be modified in order to steer duration. This would be a major modelchange because it would touch directly the Strategic Asset Allocation which

10More details on the Investment Strategy Model are presented in section 2.3

40 Chapter 3. Dynamic Duration Steering Modeling

could potentially have significantly big impacts on Reserves and Risk Capitalestimations;

• Steering mode: An Active mode could be implemented in order to allowALIM to perform extra sales for steering duration purposes;

• Asset classes: Apart from governmental bonds, other assets could be used forsteering duration such as corporate bonds11 swaptions;

• Asset profiles: Apart from the increase/decrease profiles, several profiles ofassets could be set in order to control the maturity of the re-investment.

• P&L constraints: If the Active mode is activated, it would be, then, immedi-ately necessary to take into consideration P&L constraints for the realization ofUCGL buffers.

3.3 Chapter Summary: Liability duration complexities

Duration and Convexity are very popular metrics on Portfolio Management. In par-ticular, Life insurance companies, which are long-term and liability-driven investors,use these two sensitivities as key indicators of their portfolios for ensuring a prudentmanagement in terms of liquidity and risk to interest-rates movements. Even if theyare synthetic and unidimensional metrics, they provide an immediate insight in therisk and engagements profile of asset and liability portfolios of Life insurance com-panies.

This is why it is a common practice for Life insurers to keep the duration and con-vexity gap closed as much as possible. Thus, this professional thesis aims to extendAllianz ALM computational model, so-called ALIM, in order to replicate practicalmanagement actions. More exactly, a framework for duration steering modeling isprovided.

Effective duration and effective convexity can be calculated accurately at the val-uation date (i.e. t = 0) using ALIM and, thus, management actions can be executedbased on this metrics. However, the calculation of liability effective duration, via thesensibilities approach presented in section 3.1.2, can not be reproduced along thevaluation projection within ALIM (simulation within a simulation problem). Sincethis is not implementable for run time reasons, liability duration has to be approxi-mated for every time-step and every simulation. As a matter of fact, it is not certainlya simple task to model liability duration track within ALIM and, thus, Chapter 4 ofthis professional thesis tackles this problem.

There are several parameters to be considered and calibrated for the implemen-tation of the Dynamic Duration Steering module as may be gathered from section3.2. Indeed, the re-balancing mechanics are set to the most simple possible config-uration in order to understand the impacts of the execution of this new module onsome Key Performance Indicators (KPIs). More details about the calibration andparameters are provided in sections 4.4 and 5.1.

11Indeed, corporate bonds have been tested together with governmental bonds but such tests indeterministic projection vision have produced the same results as using only governmental bonds

41

Chapter 4

Liability Duration Modeling

It is a common practice in Life Insurance to determine the Time Value of Options andGuarantees (TVOG) by Monte-Carlo methods. Thus, it is necessary to implement astochastic valuation approach in order to determine an accurate estimation of theliability, so called BEL. Liability effective duration, as presented in section 3.1.2, canbe seen as a sensitivity of the BEL, and, as such, it is necessary to estimate this metricusing stochastic valuations in order to accurately capture BEL movements.

In practice with the use of the ALM model and stochastic valuations, effectiveduration is calculated by the ALM team, quarterly as well as in the context of avariety of studies, with the aim of steering the assets portfolio. However, it is impos-sible to replicate this practice into the ALM model because it would imply stochasticvaluations within a stochastic valuation, so called simulation within a simulationproblem.

As explained in section 3.1, interest rates and the structure of the cash-flows arethe main drivers of duration. Consequently, this chapter aims to propose differ-ent approaches for the calibration of the empirical functions presented in section3.2.1, which would subsequently allow ALIM to approximate the liability duration.More precisely, ALIM would be capable to calculate liability duration as a functionof interest rates for every time step and for every iteration of a stochastic valuation,taking also into consideration the Run-off business nature of Solvency 2 economicvaluation.

Moreover, the appropriateness and accuracy of such liability duration model-ings is also discussed and back-tested on historical data as well as on data producedby the ALM model. Finally, derived from the liability duration modeling, new ap-proaches for the calculation of convexity are also presented.

4.1 Liability Duration as a function of interest rates

As presented in section 3.2.1, liability duration is approximated for each simulation i

at the projection time step t as a function of the interest rates (or forward rates) levelx(i, t), formally written as follows

durliab(i, t) = f (x(i, t)). (4.1)

Therefore, the aim of this section is to calibrate the function f empirically andto choose the interest rate t 7! x(i, t) based on historical data as well as on dataproduced by ALIM model.

42 Chapter 4. Liability Duration Modeling

4.1.1 Historical approach

In order to calibrate the empirical function f , a simple historical approach can beeasily implemented based on data collected from quarterly closing accounts. Pre-cisely, duration figures and interest-rates are collected quarterly and, subsequently,a statistical fitting is applied on them.

FIGURE 4.1: Fixed Income Portfolio value profile (Mn†). Data col-lected from 4Q 2014 to Q3 2017

FIGURE 4.2: Asset and Liability Duration profile. Data collected from4Q 2014 to Q3 2017

The graphics above show respectively the Market Value of the Asset Portfolioand the Effective Duration figures as a function of zero-coupon interest rates. Re-marks on the historical approach are then presented below:

• Asset Market Value and Duration data is collected quarterly from 4Q 2014 toQ3 2017

• Figure 4.1 shows the typical convex profile of a Fixed Income portfolio. Giventhat the asset portfolio of a Life insurer is manly made up of Fixed Incomeinstruments, then it is coherent to see this kind of shape in the graphic;

• There exist a bias in short maturities due to the lack of negative interest-ratesmodeling in the Economic Scenarios Generator (ESG) before 2017Q3;

4.1. Liability Duration as a function of interest rates 43

• As shown in Figure 4.2, the determination coefficient R2 of the linear fitting of

effective duration levels on 10 years EIOPA zero-coupon rates is too weak tovalidate this approach;

• Graphically it can be seen that any other fitting would not work either dueto a high volatility. Indeed, duration figures are calculated quarterly usingthe current version of ALIM at the estimation time. Therefore, model changeslead to duration estimation changes and, thus, historical duration figures arenot comparable with each other.

It is important to remark that the bias on historical data due to model changes issignificantly strong and, thus, the historical approach can not be validated. There-fore, the approaches presented in the next two sections use a fixed model and alsodata on a fixed closing period.

4.1.2 Sensitivities Approach without re-calibration of credit spreads

As shown in the last section , a historical approach is not recommended due mainlyto the bias generated by to the model changes between closing periods. As a matterof fact, model and assumption changes lead to significant changes in estimations ofduration figures and, thus, figures collected from quarterly closings are not compa-rable with each other.

In order to collect data with comparable liability duration figures, the followingmethodology is implemented:

1. One model parametrisation is fixed as well as the liability, asset and corporateassumption tables presented in section 2.2.1. In practice, the model version aswell as the assumption tables used for producing official closing statementson 4Q 2017 are fixed. Therefore, every liability estimation is calculated usingthese fixed assumptions.

2. ALIM uses two Economic Scenarios Generators (ESG) for every stochastic val-uation :

• The first ESG is used for calibrating the credit spreads according to thegap between the accounted market value of the assets and the asset port-folio valuation made by ALIM using the average interest rates curves ofthis set of economic scenarios:

In the approach presented in this section, the first set of economic scenar-ios is fixed. More exactly, the ESG used for the calculation of the BEL,for official statements purposes, is fixed for the calibration of the creditspreads for every liability valuation via interest-rates shocks.

• The second ESG is used for the valuation of the assets for every time-stepand for every stochastic trajectory.

In this approach, 17 ESG are used for producing 17 liability valuations.Precisely, the ESG used for the BEL calculation is included as well as16 interest-rates parallel shifts of the BEL Economic Scenarios Generator.

44 Chapter 4. Liability Duration Modeling

These parallel shifts are created with the same methodology used to cre-ate the +/- 100bps1 interest-rates shocked ESG for the Standard Formularates stressed scenarios2.

The parallel shifts retained are: +/- 200bps, +/- 175bps, +/- 150bps, +/-125bps, +/- 100bps, +/- 75bps, +/- 50bps, +/- 25bps.

3. Several figures of liability effective duration are calculated using the sensitivityapproach presented in section 3.1.2 as follows

De f f

r ⇡Vr�Dy � Vr+Dy

2VrDy(4.2)

where Dy is the parallel shift on the interest-rates of the ESG for the BEL valu-ation, V0 is the BEL and, in a general fashion, Vr is the liability valuation calcu-lated with the stresses ESG with a parallel shift of the interest-rates of magni-tude r; all valuations Vr are calculated with the fixed assumptions presented initem 2. In line with the parallel shifts chosen in item 2, Dy is set equal to 25bpsand the set of 17 liability valuations {V�200bps, V�175bps, · · · , V+175bps, V+200bps}is then calculated. Finally, the set of 15 liability duration estimations

nD

e f f

�175bps,

De f f

�150bps, · · · , D

e f f

+150bps, D

e f f

+175bps

ois calculated using the approach presented in

equation (4.2), for example

De f f

+125bps⇡

V+150bps � V+100bps

2 ⇥ V+125bps ⇥ 25bps

As remarked in section 3.1.2, the interest-rates parallel shift Dy could be chosenbetween 25bps and 100bps with a minor impact on duration estimations but ithas to be fixed for all valuations in order to keep the coherence of the approach.

The set of asset and liability valuations as well as the duration figures describedin the items above are represented in the graphics below.

FIGURE 4.3: Sensitivity Approach without credit spreads re-calibration: Valuation of Asset and Liability Portfolios

1Basis point (BPS) refers to a common unit of measure for interest rates and other percentages infinance, e.g. 100bps = 1.00%

2Interest rates shocked scenarios for Solvency 2 purposes in average are parallel shifts of the EIOPArisk-free interest rates curve

4.1. Liability Duration as a function of interest rates 45

FIGURE 4.4: Sensitivity Approach without credit spreads re-calibration: Economic Own Funds

FIGURE 4.5: Sensitivity Approach without credit spreads re-calibration: Effective Duration as a function of interest-rates

Several financial and statistical remarks can be done on the data presented in thegraphics above:

• Asset and Liability valuations show a convex profile in the figure 4.3 duemainly to the fact that an important part of the Asset portfolio is composedof Fixed Income securities and the long term structure of the cash-flows on theLiability side.

• The Liability curve is more convex than the Asset one.

On the one hand, as it can be seen in figure 4.4, the gap between Asset and Lia-bility valuations is concave and, thus, Life Insurers are usually exposed to OwnFunds losses with interest-rates movements in both directions, increase or de-crease of interest-rates. Own Funds losses are due to the Minimum-Guaranteein case of interest-rates drop, and to the Surrender Option in case of interest-rates raise.

On the other hand, as shown in figure 4.5, the slope of the regression on liabil-ity duration figures is greater than the one from the asset ones. This is directly

46 Chapter 4. Liability Duration Modeling

related with a greater convexity which is coherent with the point presentedbelow.

• The 10y EIOPA zero-coupon rate was 0,86% at the valuation date Q4 2017.Thus, the duration gap between liability and asset duration is approximately-0,2 years as shown in figure 4.5.

Asset duration is shorter than liability duration which is usually true in Life In-surance because liability cash-flows are regularly updated with new businessand regular deposits, which makes liability duration stable over time, whileFixed Income securities reach repayment dates, pushing asset duration down.This is why Life insurers portfolio managers usually look to steer asset dura-tion up in order to catch liability duration.

Some business have very long liabilities, such as pensions which demand aconstant steering of the asset portfolio duration to push it up as much as pos-sible.

• Duration steering aims to close the duration Gap by re-investing in securitieswith longer or shorter duration. In practice, Life insurers usually buy bondswith long maturities or Forward Bonds in order to steer the asset duration up.

• Duration steering have to be complemented by cash-flow matching and con-vexity steering as explained in section 1.3.1. In figure 4.5, it can be seen that,with an increase in interest-rates, the duration gap could pass from negativeto positive. In contrast, a decrease in interest-rates could lead to a sharpestmismatch in duration. Therefore, a sensitivities analysis is usually applied inorder to anticipate adverse scenarios and implementing hedging strategies.

• Given that the results are coherent, in a financial fashion, a linear regression isapplied on liability duration figures as presented in figure 4.5. It is remarkedthat the determination coefficient R

2 is significantly high for this example and,thus, the linear regression is suitable for this set of data.

• The results shown above correspond to one3 out of twenty-one asset portfo-lios of the Allianz Life segment4. The analyses presented in the items abovewere reproduced for all portfolios and lead to the same conclusions. More ex-actly, the determination coefficient R

2 is significantly high for all portfolios. Asshown in table A.1 in appendix A, only four portfolios have lower determina-tion coefficients R

2, respectively 0.93, 0.94, 0.96 and 0.93, which still are goodenough to validate the linear regression approach.

• Other polynomial and exponential fittings were tested but the linear one isretained due to the significant high determination coefficients R

2 and the prac-ticality of the calibration.

• The slope of the linear regression is strongly related with Convexity as pre-sented in equation 3.7 in section 3.1.1. Therefore, the linear coefficient of thelinear regression has a financial interpretation which is another reason to retain

3Figures are modified in order to avoid Allianz France exposition4Unit-Linked portfolios are out of scope because in practice duration steering is only performed

on Pensions and Euro-Savings Portfolios

4.1. Liability Duration as a function of interest rates 47

the linear regression approach. More details of this relationship are further dis-cussed in section 4.3.

Consequently, a linear fitting could be proposed to calibrate the empirical func-tion f , from equation (4.1), in order to approximate the liability duration accordingto this sensitivity approach. Moreover, the approach has been validated in all portfo-lios and is coherent in a financial fashion and statistically suitable. However, anothersensitivity approach is studied in the next section with the the aim of having a solidbasis of analysis before to definitely chose an approximation approach for the func-tion f .

4.1.3 Sensitivities Approach with re-calibration of credit spreads

This approach is very similar to the one presented in the last section in which thedata set of duration figures is produced using ALIM. However, this time a re-calibrationof the credit-spreads is implemented for the calculation of each Liability effective du-ration.

Effective Duration figures are calculated, as in the last section, using equation5

De f f

r ⇡Vr�Dy � Vr+Dy

2VrDy(4.3)

but with few changes in the methodology which lead to a different approach:

1. Exactly in the same way as in item 1 in the last section, one model parametri-sation is fixed as well as the liability, asset and corporate assumption tablespresented in section 2.2.1. In practice, the model version as well as the as-sumption tables used for producing official closing statements on 4Q 2017 arefixed. Therefore, every liability estimation is calculated using these fixed as-sumptions.

2. The difference between the two sensitivities approaches is generated by theway the Economic Scenarios are set-up in order to produce the Liability valu-ations. In fact, ALIM uses two Economic Scenarios Generators (ESG) for everystochastic valuation :

• The first ESG is used for calibrating the credit spreads according to thegap between the accounted market value of the assets and the asset port-folio valuation made by ALIM using the present value of future cash-flows, with the average interest-rates curve of this set of economic scenar-ios.

In the approach presented in this section, the first set of economic sce-narios is variable6. More exactly, the ESG used for the valuation of thecentral Liability Vr, in equation (4.3), is also used for the re-calibration ofthe credit spreads for every duration estimation.

5where Dy is the parallel shift on the interest-rates of the ESG for the BEL valuation, V0 is the BELand, in a general fashion, Vr is the liability valuation calculated with the stresses ESG with a parallelshift of the interest-rates of magnitude r;

6While it is fixed for the approach presented in the last section

48 Chapter 4. Liability Duration Modeling

• The second ESG is used for the valuation of the assets for every time-stepand for every stochastic trajectory.

In this approach, 17 ESG are used for producing 45 Liability valuations7.Precisely, the ESG used for the BEL calculation is included as well as16 interest-rates parallel shifts of the BEL Economic Scenarios Generator.These parallel shifts are created with the same methodology used to cre-ate the +/- 100bps8 interest-rates shocked ESG for the Standard Formularates stressed scenarios9.

The parallel shifts retained are: +/- 200bps, +/- 175bps, +/- 150bps, +/-125bps, +/- 100bps, +/- 75bps, +/- 50bps, +/- 25bps.

3. Equation (4.3) is re-formulated in order to include an index for the ESG usedfor the re-calibration of the spreads

De f f

r1 ⇡V

r2r1�Dy

� Vr2r1+Dy

2Vr2r1 Dy

(4.4)

where r1 is the size of the parallel shift on the central ESG used for the Liabilityvaluation Vr1 while r2 is the size of the parallel shift on the central ESG used forthe re-calibration of the spreads, all valuations Vr1 are calculated with the fixedassumptions presented in item 2. In line with the parallel shifts chosen in item2, Dy is set equal to 25bps and a set of 45 liability valuations is then calculated

nV

�175bps

�200bps, V

�175bps

�175bps, V

�175bps

�150bps,

V�150bps

�175bps, V

�150bps

�150bps, V

�150bps

�125bps,

· · ·

V+175bps

+150bps, V

+175bps

+175bps, V

+175bps

+200bps

o

Finally, the set of 15 liability duration estimationsn

De f f

�175bps, D

e f f

�150bps, · · · ,

De f f

+150bps, D

e f f

+175bps

ois calculated using the approach presented in equation (4.4),

for example

De f f

+125bps⇡

V+125bps

+150bps� V

+125bps

+100bps

2 ⇥ V+125bps

+125bps⇥ 25bps

.

As remarked in section 3.1.2, the interest-rates parallel shift Dy could be chosenbetween 25bps and 100bps with a minor impact on duration estimations but ithas to be fixed for all valuations in order to keep the coherence of the approach.

The set of asset and liability valuations as well as the duration figures describedin the items above are represented in the graphics below.

7While only 17 Liability valuations are produced in the last section8Basis point (BPS) refers to a common unit of measure for interest rates and other percentages in

finance, e.g. 100bps = 1.00%9Interest rates shocked scenarios for Solvency 2 purposes in average are parallel shifts of the EIOPA

risk-free interest rates curve

4.1. Liability Duration as a function of interest rates 49

FIGURE 4.6: Sensitivity Approach with credit spreads re-calibration:Valuation of Asset and Liability Portfolios

FIGURE 4.7: Sensitivity Approach with credit spreads re-calibration:Economic Own Funds

FIGURE 4.8: Sensitivity Approach with credit spreads re-calibration:Effective Duration as a function of interest-rates

The following financial and statistical facts are remarked:

50 Chapter 4. Liability Duration Modeling

• The asset valuation remains stable to interest-rates movements because the re-calibration of the spreads in each valuation implies a re-adjustment to the mar-ket value of assets at the valuation time. This is not consistent with interest-rates sensitivities of a portfolio that is mainly composed by Fixed Income se-curities which should exhibit a convex profile.r

• Contrary to the asset valuation, the liability valuation has a convex profile.However the gap between the asset and liability valuations which representsthe Economic Own Funds should be more concave. This is not coherent withthe Own Funds profile in practical world which should shows a risk expositionto increases and decreases of interest-rates as shown in figure 4.4.

• There is an important difference between the slopes of the regressions on liabil-ity and asset duration as shown in figure 4.8. Thus, there is a marked convexitygap which is not coherent with the practical world, mainly, because the assetduration slope is too weak for a Life Insurance asset portfolio.

• The determination coefficients R2 of the linear regressions shown in figure 4.8

are suitable to validate this approach from a statistical point of view; However,this approach should not be considered due to financial incoherence with thepractical world.

In conclusion, this approach should not be retained. In fact, there is a modelingbias in the asset valuation due to the re-calibration of the spreads which lead to thesame valuation for every interest-rates level.

4.1.4 Approach retained and modelling formulation

It is remarked that the sensitivity approach presented in section 4.1.2 is accurate, asshown in table A.1 in appendix A, and the most consistent with the practical world.Therefore, the sensitivity approach without re-calibration of the spreads is retainedand the formulation for the calibration of the function f from equation (4.1) is thefollowing

f (i, t) = b1x(i, t) + eb0 (4.5)

where b1 and b0 are the coefficients of the linear regression while eb0 is a levelling ofb0. It is important to remark that ALIM calculates asset duration for every trajectoryi at each time-step t using the analytic function of Modified Duration and, thus, thecoefficient b0 have to be re-leveled in order to match the effective duration gap atthe beginning of the projection t = 0. Precisely, the following equation have to besatisfied

De f f

asset� D

e f f

liab| {z }=D

e f f

gap

= DMod

asset � f (i, 0)

De f f

gap = DMod

asset �⇣

b1x(i, 0) + eb0

⌘

) eb0 = DMod

asset � b1x(i, 0)� De f f

gap

where DMod

assetand D

e f f

gap are known values. Indeed, De f f

gap is calculated quarterly by theALM department and D

Mod

assetis directly calculated by a closed formula.

4.2. Portfolio Aging effect on Liability Duration 51

4.2 Portfolio Aging effect on Liability Duration

The main market driver of the asset and liability duration is the interest-rates level.Additionally, it is important to incorporate also a technical driver. As explained insection 3.1.3, a projection of the technical minimum-guarantee cash-flows on the in-force business allows to capture the structure of the Liability in the Run-off approachof Solvency 2 economic valuation.T As presented in section 3.2.1, liability duration is scaled for each simulation i at theprojection time step t as follows

durtgt(i, t) = durtgt(i, t) ⇤ g(t) (4.6)

where g(t) is a function to be calibrated from technical cash-flows analysis.

Minimum-Guarantee cash-flows calculated by the Reserving teams are used inorder to estimate the adjustment of the liability duration, to be implemented withinALIM, in line with the velocity of the draining of the in-force business. Precisely, apolynomial regression is implemented on these technical cash-flows as illustrated inthe figure below.

FIGURE 4.9: Minimum-Guarantee technical cash-flows projection

FIGURE 4.10: Macaulay Duration calculated on Minimum-Guaranteetechnical cash-flows projection

52 Chapter 4. Liability Duration Modeling

Some technical and statistical remarks are presented below:

• As expected, there is a draining of the in-force business as shown in figure 4.9due to the run-off valuation assumption. This is translated into a decreasingliability duration over the projection time as shown in figure 4.10.

• Some regressions were tested and the best fitted one in the majority of portfo-lios is the quadratic regression, as shown in tables A.2 and A.3 in appendix A.Moreover, the determination coefficient R

2 is significantly high, over 0,97 withonly three portfolios under this threshold (i.e. respectively 0.92, 0.92 and 0.95).Therefore, the quadratic regression is retained

• The Macaulay duration does not capture the Options and Guarantees (O&G)of Life and Savings insurance contracts. In fact, the Minimum-Guarantee cash-flows projections are determined using deterministic models and the Macaulayduration is calculated by the analytic formula

Dmac :=T

Ât=1

tCte

�yt

P

presented in section 3.1.1.

• Macaulay duration is longer than effective duration due to the lack of the in-corporation of the O&G which shorten the Liability duration. Therefore, thesetwo approaches for the calculation of the duration are not directly comparable.However, the aim is to fit the function g in equation (4.6) and, thus, only thedecreasing velocity is needed from the quadratic fitting presented in the itemsabove.

Once a quadratic function q(t) = a2t2 + a1t + a0 is determined by the fitting

presented in the items above, the following approach is proposed for the calibrationof the function g

g(t) =q(t)

q(t � 1)=

a2t2 + a1t + a0

a2(t � 1)2 + a1(t � 1) + a0. (4.7)

It is remarked again that there is an implicit assumption in the formulation presentedin equation (4.6). Precisely, the technical driver (Minimum-Guarantee projection)does not depend on the stochastic trajectory i and only depends on the time-step t.It means that the velocity of the in-force business draining is considered equal for allthe stochastic trajectories.

Indeed, the assumption remarked above is a simplification because certain eco-nomic scenarios could lead to a faster shortening of the Liability duration due tothe surrender option. Interactions between the Dynamic-Lapses and the Dynamic-Duration module of ALIM could be studied in more detail but this topic is out of thescope of this professional thesis.

4.3 Convexity Analysis for the Sensitivities Approach with-out re-calibration of credit spreads

As presented in section 4.1.2, the sensitivities approach without re-calibration ofspreads is suitable for the calibration of an empiric function for the modelling of

4.3. Convexity Analysis for the Sensitivities Approach without re-calibration ofcredit spreads 53

the liability duration as a function of the interest-rates level. Consequently, this sec-tion aims to analyze convexity for this sensitivities approach.

As a matter of fact, there is a relation between the linear coefficient of the linearregression presented in equation (4.5) and Convexity. Taking up the equation (3.7),it is assumed that the same relation10 holds for every type of duration calculation3.1.1

∂D

∂y= D

2 � C. (4.8)

Once a linear approach is retained for the modelling of duration, then it followsthat

∂D

∂y⇡ b1

where b1 is the linear coefficient presented in equation (4.5). Given that the deriva-tive in the left side of the equation above can be determined by the methodologypresented in section 4.1.2 , a proxy for Convexity is stated as follows

C = D2 � ∂D

∂y⇡ D

2 � b1. (4.9)

On the one hand, a set of 15 effective convexity estimationsn

Ce f f

�175bps, C

e f f

�150bps,

· · · , Ce f f

+150bps, C

e f f

+175bps

ois calculated using the same set of valuations {V�200bps, V�175bps,

· · · , V+175bps, V+200bps} shown in item 3 of the methodology presented in section 4.1.2and the finite differences approach presented in section 3.1.2. Precisely, the followingformulation is used

Ce f f

r ⇡Vr�Dy + Vr+Dy � 2Vr

Vr(Dy)2 ;

For example,

Ce f f

+125bps⇡

V+100bps + V+150bps � 2V+125bps

V+125bps(25bps)2 .

This first set of convexity figures is presented in the graphic below.

FIGURE 4.11: Effective Convexity calculated by Finite Differences ap-proach

10The relation was proved for Macaulay duration in section 3.1.1 and it is assumed that it holds forthe effective and modified approaches

54 Chapter 4. Liability Duration Modeling

On the other hand, another set of 15 convexity figures {C�175bps, C�150bps, · · · ,

C+150bps, C+175bps} is calculated using the 15 effective duration figuresn

De f f

�175bps,

De f f

�150bps, · · · , D

e f f

+150bps, C

e f f

+175bps

ofound in item 3 of the methodology presented in

section 4.1.2 and the relation on equation (4.3). More exactly, the formulation usedis the following

Cr =⇣

De f f

r

⌘2� b1;

For example,

C+125bps =⇣

De f f

+125bps

⌘2� b1;

This second set of convexity figures is presented in the graphic below.

FIGURE 4.12: Convexity calculated using theoretical relation

There are some important remarks on the two graphics below:

• Once more, liability convexity is greater than the asset one. Thus, liability ismore sensible to interest-rates movements than the asset portfolio.

• Regressions on both sets of convexity figures lead to almost the same regres-sion coefficients and, thus, the two approaches are coherent. Moreover, this isa back-testing of the regression linear coefficient b1 on effective convexity fig-ures. Therefore, this is another argument to validate the chosen sensitivitiesapproach without re-calibration of spreads.

• Effective convexity is harder to estimate accurately than effective duration. In-deed, effective convexity figures shown in figure 4.12 are more volatile thaneffective duration figures shown in figure 4.5. The reason is that the finite dif-ferences approach is less accurate for second order derivatives than for firstorder derivatives.

• Consequently, the formulation presented in equation (4.3) is an improvementof the calculation of effective convexity. In the appendix A, duration and con-vexity figures of a pension portfolio are presented (graphics A.2, A.3 and A.4)in which it is evidenced a significant improvement of the convexity calculation.

All in all, analysis on convexity provides a better understanding of the durationresults presented in section 4.1. Remarkably, the analysis provided in this sectionallows to back-test the sensitivities approach retained for duration modelling, inaddition to improvement on accuracy of convexity calculation.

4.4. Chapter Summary: Liability Duration Modelling Synthesis 55

4.4 Chapter Summary: Liability Duration Modelling Syn-thesis

In conclusion, the sensitivities approach presented in section 4.1.2 is retained forthe approximation of the liability duration as a function of the interest-rates levelevolution along the projection horizon, within ALIM for every iteration i at eachtime-step t. Moreover, a technical driver based on Minimum-Guarantee cash-flowsprojection is also chosen in order to capture the draining of the in-force business inthe run-off approach of Solvency 2 valuation. The algorithm presented in section3.2.1 for the proxy of the liability duration is then re-written as follows

1. Interest-rates: Liability duration is firstly estimated for each simulation i at theprojection time step t as a function of the interest rates (or forward rates) levelx(i, t), formally written as it follows

durtgt(i, t) = b1x(i, t) + eb0| {z }= f (x(i,t))

with b1 and eb0 as presented in section 4.1.4.

2. Limits Control: The duration target is adjusted according to duration goallimits, i.e.

durtgt(i, t) = max�min

�durtgt(i, t), durmax

�, durmin

�

where durmax and durmin are respectively the maximum and minimum dura-tion allowed targets which are parameters defined in ALIM inputs;

3. Portfolio Aging: duration target is modified in order to capture the portfolioaging caused by a run-off business:

durtgt(i, t) = durtgt(i, t) ⇤ a2t2 + a1t + a0

a2(t � 1)2 + a1(t � 1) + a0| {z }=g(t)

where a2, a1 and a0 are the coefficients of the quadratic regression presented insection 4.2. It is noticed that aging of the portfolio is assumed independent ofthe of the trajectory and, thus, the function g does not depend on the iterationi but only on the time step t.

4. Limits Control: Limits control is once again executed,i.e.

durtgt(i, t) = max�min

�durtgt(i, t), durmax

�, durmin

�.

Consequently, a2, a1, a0, b1, eb0, durmax and durmin are parameters to be calibratedfor the implementation of the Dynamic Duration Steering module of ALIM. Indeed,this chapter provides the methodology for the calibration of these parameters.11

11durmax and durmin are set by expert judgment

57

Chapter 5

Dynamic Duration SteeringImplementation and Impact on KeyPerformance Indicators (KPIs)

Once Liability duration is modelled as presented in the last chapter and summarizedin section 4.4, it is possible to setting Allianz ALM model, so called ALIM, in orderto implement the Dynamic-Duration steering algorithm presented in section 3.2.2.As such, ALIM would be capable to steer Asset duration, up or down, in order tocatch Liability duration for every projection time-step and every simulation.

On the one hand, this chapter aims to present the operational requirements andthe model setting for the implementation of the Dynamic-Duration module. On theother hand, the impacts of this modelling on some of the Key Performance Indica-tors (KPIs) are also presented such as Own Funds and Risk Capital (RC) as well asSolvency 2 Ratio.

Indeed, a more accurate ALM modelling which reflects better the essential prac-tices of Life insurers should conduct to a more accurate calculation of risk and per-formance metrics, so called KPIs. Therefore, this chapter intends also to analyze theeffect of duration steering on the KPIs mentioned above.

5.1 Dynamic Duration Steering Module Implementation

Apart from the liability duration modelling, it is necessary to set-up several param-eters for the Dynamic Duration Steering algorithm. Some of them are covered insection 5.1.1 with the aim to give an overview of the complexity of the module im-plementation as well as to provide further details on the steering modeling. Addi-tionally, the results of the asset and liability duration evolution along the valuationprojection within ALIM are presented in section 5.1.2.

5.1.1 ALIM Setting

The parameters required by the Dynamic Duration Steering module are read byALIM from the corporate assumptions presented in section 2.2.1. More exactly, somespecific tables are filled with the parameters required by the duration steering algo-rithm and placed in the corporate assumptions folder.

58 Chapter 5. Dynamic Duration Steering Implementation and Impact on KeyPerformance Indicators (KPIs)

fund_name Portfolio1 Portfolio2 Portfolio3 Portfolio4apply_dyn_dur Y Y N Ydur_max_step_up 1 1 1 1dur_max_step_down 1 1 1 1max_swap_participation 0 0 0 0max_time_duration_used 60 60 60 60write_text_files N N N Nmin_text_period 1 1 1 1max_text_period 60 60 60 60dur_min_goal 1 1 1 1dur_max_goal 8 8 20 7dynamic_mode_for_bonds PASSIVE ACTIVE PASSIVE PASSIVErcgl_upper_limit 0,05 0,05 0,05 0,05rcgl_lower_limit -0,05 -0,05 -0,05 -0,05dyn_dur_steering_frequency 1 1 1 1gap_closure_rate 1 1 1 1a0 12,9665 22,1070 11,2912 11,9425a1 -0,2114 -0,6250 -0,0478 0,1614a2 -0,0016 0,0032 -0,0048 -0,0109b0 -0,3 -0,5 -1,2 -0,2b1 7,1 8,25 17 6,25c0 0 0 0 0c1 0 0 0 0f1 1 1 1 1f2 0 0 0 0curr1 EUR EUR EUR EURcurr2 EUR EUR EUR EURterm1 10 10 20 10term2 30 30 30 30fwd_start1 0 0 0 0fwd_start2 0 0 0 0alpha 0 0 0 0f_t 1 1 1 1min_delta_dur_active_mode 5 5 5 5

TABLE 5.1: ALIM corporate assumptions table (.tbl) dedicated en-tirely to the setting of the Dynamic Duration Module

The parameters in the table above cover primarily:

• fund_name: The name of the portfolio in which the Dynamic Duration Steer-ing is going to be applied on. The entire setting of the module is thereforeindependent for each portfolio;

• apply_dyn_dur: The duration steering algorithm can be activated indepen-dently for each portfolio. The steering algorithm has been tested on all port-folios in which the calibration of the parameters of the liability duration proxyare significantly accurate

• The purchase decisions include limitations in order to avoid too siginificantchanges. The assumptions dur_max_step_up and dur_max_step_up definethe maximum changes that can be caused by purchases of bonds or swaps;

5.1. Dynamic Duration Steering Module Implementation 59

• max_time_duration_used: The maximum time-step when duration steering isswitched on. For the purpose of the professional thesis, the algorithm is alwaystested along the entire projection;

• dur_min_goal/dur_max_goal: Floor/Cap for the calculated target duration aspresented in sections 3.2.1 and 4.4. On the one hand, dur_min_goal is set equalto 1 (or 0) because the run-off business assumption conduct the liability dura-tion towards 0. On the other hand, dur_max_goal is set by expert judgmentaccording the duration characteristics of the liability backing the asset portfo-lio.

• dynamic_mode_for_bonds: In the "ACTIVE" mode, if the standard sales andpurchases are not sufficient to reach the target duration, then additional bondsare sold and bought to further close the duration gap. While in the "PASSIVE"mode, only the cash available from the traditional re-balancing module is usedfor duration steering purposes;

• dynamic_dur_steering_frequency: It is set equal to 1 in order to force thesteering algorithm to be used in every time-step;

• a0, a1, a2: Coefficients of the quadratic regression presented sections 4.2 and4.4 which aim to model, via the function g, the rate of decrease of the liabilityduration due to the run-off business specificity of the Solvency 2 valuation;

• b0, b1: Coefficients of the linear regression presented in sections 4.1.4 and 4.4which aim to model, via the function f , the liability duration as a function ofthe interest-rates level for every time-step and every simulation1;

• curr1, term1, fwd_start1: Respectively, the currency, maturity and the forwardstart2 of the interest-rates used in the function f presented in sections 4.1.4 and4.4.

Additionally, some other tables within the corporate assumptions have to bemodified. Specially, the specific table used to provide the information of the assetsto be purchased by ALIM have to be splited into two tables; One with longer matu-rities and the other one with shorter ones with the aim to push duration up or downrespectively. Indeed, the table 3.1 in section 3.2.2 presents the split considered forone profile3 of new bonds (profile #4) to be purchased by ALIM when needed. Be-low, the same table is presented with the weighted average maturity for each basketand the percentage change with respect to the normal one.

1As mentioned in section 3.2.1, two interest-rates could be considered for the estimation of theliability duration. For example, interest-rates in different currencies could be implemented. Therefore,parameters f1 and f2 are the weights of those two factors and c0 and c1 the coefficients of the linearfunction that model liability duration in another currency. Nevertheless, only one factor is tested forthe purpose of this professional thesis

2Instead of zero-coupon rates, forward rates could be also used for the modeling of the liabilityduration. The setting equal to 0 implies that zero-coupon rates are used in the modeling.

3There is a total of 18 profiles/baskets of new bonds

60 Chapter 5. Dynamic Duration Steering Implementation and Impact on KeyPerformance Indicators (KPIs)

Bonds Maturity Normal Basket Shorter Basket Longer Basket2yr 5% 10% 0%5yr 10% 40% 10%10yr 60% 40% 40%30yr 25% 10% 50%

Weighted AverageMaturity 14,1 9,2

(D = -34,75%)19,5

(D = +38,30%)

TABLE 5.2: Split of the New Bond Profile #4 for the purpose of theDynamic Duration Steering modeling

In accordance with the last paragraph, the split has to be done for each of thenew bond profiles which are going to be taken into consideration for the DynamicDuration Steering algorithm. The table below summarizes the profiles of new bondsconsidered for the Dynamic Duration module as well as the percentage change withrespect to the normal basket.

NewBondProfile

Issuer

NormalBasket(Weightedaveragematurity)

ShorterBasket(Weightedaveragematurity)

LongerBasket(Weightedaveragematurity)

DownProfile(% change)

UpProfile(% change)

profile1 CO 6,4 4 8,5 -37,50% 32,81%profile2 CO 7,5 7,5 7,5 0,00% 0,00%profile3 CO 10 10 10 0,00% 0,00%profile4 GO 14,1 9,2 19,5 -34,75% 38,30%profile5 GO 22 14 28 -36,36% 27,27%profile6 CO 2 2 2 0,00% 0,00%profile7 GO 10 10 10 0,00% 0,00%profile8 CO 6,4 6,4 6,4 0,00% 0,00%profile9 CO 7,5 7,5 7,5 0,00% 0,00%profile10 CO 10 10 10 0,00% 0,00%profile11 GO 14,1 9,2 19,5 -34,75% 38,30%profile12 GO 22 14 28 -36,36% 27,27%profile13 CO 2 2 2 0,00% 0,00%profile14 GO 10 10 10 0,00% 0,00%profile15 CO 17,75 12,75 23 -28,17% 29,58%profile16 GO 17 12 22 -29,41% 29,41%profile17 CO 12 8,5 16 -29,17% 33,33%profile18 GO 14 10 19 -28,57% 35,71%

*Highlighted profiles have been modified for duration steering purposes

TABLE 5.3: New Bond profiles to be used for purchases

The maturities in the "up" and "down" profiles are defined by expert judgmentin order to obtain a percentage change in the weighted average maturity of eachprofile close to 30%4. As a matter of fact, the weights of the maturities of each "up"and "down" new bond profile are parameters to be tested and calibrated.

430% is a good trade-off between the convergence to the original setting and the effectiveness ofthe dynamic duration module. Indeed, a higher percentage change would imply a stronger durationsteering but the duration track would most likely be farther from the original setting

5.1. Dynamic Duration Steering Module Implementation 61

Finally, the duration steering can be used partially for each type of asset andfund. In the assignment setting of the Fund/Portfolio presented in the table below,Governmental and Corporate Bonds are used fully (100%) by the Dynamic Durationmodule.

AssetClass

AssetLabel

BuyProfile

Dur UpProfile

Dur UpPercentage

Dur DownProfile

Dur DownPercentage

1 B_FI_CO 1 1 50 1 502 B_FI_GO 5 5 50 5 503 B_FR 2 0 0 0 04 B_LO 3 0 0 0 05 E_PA 1 0 0 0 06 E_PE 2 0 0 0 07 E_S 3 0 0 0 08 E_D_EM 7 0 0 0 09 E_D_HZE 6 0 0 0 010 E_D_PA 10 0 0 0 011 E_D_US 5 0 0 0 012 E_D_ZE 4 0 0 0 013 E_F 8 0 0 0 014 RE_D 9 0 0 0 015 IF 1 0 0 0 016 B_FWD 6 0 0 0 017 B_INFL 7 0 0 0 018 E_FWD 21 0 0 0 019 RE_FWD 22 0 0 0 020 CAP 0 0 0 0 0

TABLE 5.4: Extract form ALIM Assignment table (.tbl)

The "buy profiles" 1 and 5 in the table above5 correspond to the "New bod pro-file" in the figure 5.3. For the purpose of this professional thesis, the percentages areset equal in total to 100% (50% Corporate Bonds + 50% Governmental Bonds)6 withthe aim to conduct the duration steering algorithm to use all the amount availablefor purchases, calculated by ALIM at each time-step.

All in all, the calibration of the parameters of the duration target (or liabilityduration proxy) are derived from the quantitative analysis provided in chapter 4and summarized in section 4.4. Nevertheless, there are several parameters that haveto be set by expert judgment, as presented in this section, which directly impact there-balancing mechanics and, then, the liability valuation. Given that there is a widevariety of possibilities to test, the parameters have been mainly set in order to havea significant impact due to the implementation of the Dynamic Duration Steeringmodule but, at the same time, trying to not to go too much away from the originalmodeling.

5Some other asset classes could be tested for the purpose of duration steering, such as the forwardbond in the row 16 in figure 5.4

6Indeed, the setting 50% Corporate Bonds + 50% Governmental Bonds has been tested as well as100% Governmental Bonds

62 Chapter 5. Dynamic Duration Steering Implementation and Impact on KeyPerformance Indicators (KPIs)

5.1.2 Asset and Liability Duration evolution within ALIM

Given that all the parameters required for the implementation of the Dynamic Du-ration modelling have been calibrated or set by expert judgment; Then, it is possibleto activate this module and measure its impact on the cash-flows projection withinALIM and, subsequently, on the valuation. Before to measure the impacts on someKey Performance Indicators (KPIs), it is firstly illustrated duration evolution alongthe valuation projection within ALIM.

FIGURE 5.1: Duration evolution within ALIM (only interest-ratesmodeling)

FIGURE 5.2: Duration evolution within ALIM (interest-rates andMinimum-Guarantee modeling)

5.1. Dynamic Duration Steering Module Implementation 63

FIGURE 5.3: Duration evolution within ALIM (interest-rates andMinimum-Guarantee modeling)

Indeed, the re-balancing mechanics presented in section 3.2.2 change the courseof the asset portfolios duration as shown in the figures above. More exactly, thefollowing is noticed:

• In each of the three graphics above, there is a representation of the asset du-ration trajectory without duration steering (red line), the duration target orliability duration (blue continuous line), and the asset duration trajectory withduration steering (blue dotted line).

• Three different modeling approaches are presented in the figures above.

In figure 5.1, only interest-rates7 are taken into consideration for the durationtarget calculation within ALIM (liability duration proxy). It means that onlythe function f presented in sections 3.2.1 and 4.4 is used without taking intoaccount the function g.

In figure 5.2, interest-rates and portfolio aging, as presented in sections 3.2.1and 4.4, are taken into consideration for the duration target calculation withinALIM (liability duration proxy).

In figure 5.3, interest-rates and portfolio aging are taken into consideration forthe duration target calculation within ALIM (liability duration proxy). How-ever, a function g different from the one presented in sections 4.2 and 4.4 issuggested, i.e.

g(t) = k

where k is a constant decreasing rate of the portfolio duration. This constantrate has been fixed based on one Allianz ALM study about portfolio aging;However, such approach was not retained for operational reasons8.

• Given that the duration target in figure 5.1 is a linear transformation of the 10years zero-coupon rate, it is then logical to identify the EIOPA zero-coupon

710yr zero-coupon rates within ALIM8Regarding the approach used in this ALM study, it is possible to implement it only on the first

years of projection, on the contrary of the approach retained for this professional thesis in which theAging is determined along the entire projection

64 Chapter 5. Dynamic Duration Steering Implementation and Impact on KeyPerformance Indicators (KPIs)

rates curve shape on it. Similarly, the duration target in figures 5.2 and 5.3 areshaped by the EIOPA zero-coupon curve but with a deformation due to theeffect of Aging.

• The duration gap is respected at the beginning of the projection thanks to there-leveling of the function f presented in section 4.1.4.

• In all three figures, it can be noticed that the duration algorithm succeed todecrease asset duration following the duration target. However, there has notbe seen a significant effect when it comes to raising the asset duration up. In-deed, in figure A.5 in appendix A, another two portfolios present the samephenomenon.

In practice, it is harder to steer duration up given that asset cash-flows will tendto shorten when debt securities mature. Therefore, the phenomenon observed,and described in the paragraph above, is consistent with the practical world.

• Duration target in figures 5.2 and 5.3 decrease faster than in figure 5.1. Consis-tently, the asset duration at a different pace in line with the duration target butthere is not a substantial change. Once more, the same phenomena is observedin the other two portfolios presented in figure A.5 in appendix A.

• Duration steering has been tested with re-investment on a combination of Gov-ernmental and Corporate bonds as presented in the table 5.4 as well as on onlyon Governmental bonds. However, the impact observed on the asset durationtrack is the same.

All in all, the duration steering algorithm consistently influence the asset du-ration track. Even if this modeling is not completely accurate, there still is an im-provement of the current model. Consequently, an impact analysis on some KeyPerformance Indicators (KPIs) is presented in the next section.

5.2 Results and impacts on Key Performance Indicators (KPI)

At this point, there has been developed a significantly accurate modeling of the lia-bility duration, as presented in section , as well as re-balancing mechanics within theDynamic Duration Steering module which are consistent with the practical world.On this basis, the impact of such model change on some Key Performance Indicators(KPIs) is measured.

There is a heavy workload within the Risk Department every reporting quar-ter in order to calculate accurately the Solvency Capital Required (SCR), so calledRisk Capital (RC) for the purpose of this professional thesis. For this reason, AllianzFrance uses a simplified process when the impact on RC due to model changes haveto be tested. As of Q4 2017, this simplified process is based on several stochasticvaluations of Best Estimate Liability (BEL) in a variety of stressed scenarios. Moreexactly, 25 stressed BEL are calculated using only ALIM9 with 12 financial10 stressedeconomic scenarios and 13 technical11 stressed liability assumptions. Later, Market

9The Risk Margin is not considered in this calculation10Interest-Rates, Equity, Real Estate, volatility sensitivities, inflation, credit sensitivities, among oth-

ers, are considered11Biometric and business stressed sensitivities are considered

5.2. Results and impacts on Key Performance Indicators (KPI) 65

and Life Non-Market Risk factors are calculated. Finally, correlation and diversifica-tion are applied on these Risk factors using a shorten procedure in order to approxi-mate the Risk Capital impact. These results are summarized in the table below.

Estimated RCAllianz VieBeforeChange

Allianz VieAfterChange

VariationAmount

Variation%

Equity Risk 52,50 52,42 -0,08 -0,16%Interest Rate Risk 31,89 32,56 0,66 2,08%Inflation Risk 30,16 30,20 0,04 0,14%Real Estate Risk 18,30 18,14 -0,16 -0,85%Foreign ExchangeRate Risk (1) 13,29 13,29 0,00 0,00%

Equity Volatility Risk 17,46 18,71 1,24 7,13%Interest Rate Volatility Risk 6,95 6,96 0,01 0,14%Credit Risk 32,87 33,02 0,16 0,47%Credit Spread Risk 53,72 53,26 -0,46 -0,85%Premium Non Cat Risk (1) 0,00 0,00 0,00 0,00%Premium Nat Cat Risk (1) 0,00 0,00 0,00 0,00%Premium Terror Risk (1) 0,00 0,00 0,00 0,00%Reserve Risk (1) 0,00 0,00 0,00 0,00%Mortality Risk 1,32 2,75 1,43 108,28%Mortality Calamity Risk 0,93 2,05 1,11 119,47%Morbidity Risk (1) 2,66 2,66 0,00 0,00%Morbidity Calamity Risk (1) 5,15 5,15 0,00 0,00%Longevity Risk 39,55 39,09 -0,46 -1,17%Lapse Risk 14,34 14,95 0,61 4,27%Lapse Mass Risk 16,52 19,20 2,68 16,20%Cost Risk 23,69 23,73 0,04 0,16%Operational Risk (1) 14,37 14,37 0,00 0,00%

Risk Capital BeforeDiversification(Sum of the StandaloneRisk Factors)

375,67 382,50 6,83 1,82%

Diversification Impact -215,08 -220,78 -5,69 2,65%Capital Charge 22,53 22,53 0,00 0,00%Tax Relief Impact -49,00 -49,00 0,00 0,00%

Risk Capital (RC) 134,11 135,25 1,14 0,85%(1) Risk Capital charge non-restatedEach Risk factor is the VaR(99,5%) of the standalone loss probability distribution

TABLE 5.5: Risk Capital (RC) impacts before and after model change

Indeed, it is not a simple task to accurately explain the movements in Risk factorsdue to model changes but some elements of analysis, on the results shown in thetable above, are presented below:

• Market Risk factors are displayed from row 2 to row 11 and Life Non-MarketRisk factors from row 12 to row 23. Amounts before and after change are theVaR(99,5%) of the standalone loss probability distributions. The calculation

66 Chapter 5. Dynamic Duration Steering Implementation and Impact on KeyPerformance Indicators (KPIs)

of the risk factors labeled (1) in the table require treatments outside of ALIMand, thus, their impacts are not considered for the purpose of this professionalthesis.

• Lapse and Lapse Mass Risk: Lapse Mass is the risk factor with the highest ma-teriality (2,68). As a matter of fact, duration steering tends to re-invest in bondbaskets with shorter maturities, as presented in section 5.1.2, which leads tolower return and, consequently, lower policyholder profit sharing. This lowerprofit sharing trend should conduct to more lapses.

• Mortality and Mortality Calamity Risk: Mortality risks are the factors withthe highest percentage variation, 108,28% and 119,47% respectively. In linewith the item above, lower returns on re-investments conduct to a lower finan-cial profits along the projection. Thus, early claims due to mortality stressedassumptions should derive in a negative impact on the Present Value of futureProfits (PVFP), in other words a negative impact on the Own Funds.

• Cost Risk: Drivers for maintenance expenses, along the projection, are mainlyreserves and number of policies. In line the items above, the lower return trendconduct to a lower revaluation of the liability cash-flows along the projection.This lower revaluation is realized in both scenarios, central and cost stressedscenario, and, thus, there should not be an important impact regarding ex-penses.

• Longevity Risk: In contrast to Mortality risks, a lower revaluation of the lia-bility cash-flows along the projection conduct to lower risk in long liabilitiessuch as annuities.

• All the standalone impact amounts of the Life Non-Market12 Risk factors sumup to 5,41 which represents 79,17% of the total standalone impact (6,83).

• Market Risk Factors: All the standalone impact amounts of the Market Riskfactors sum up to 1,42 which represents 20,83% of the total standalone impact(6,83). In general, a lower revaluation of the liability cash-flows could conductto more supplementary realization of capital gains on a variety of securities.In particular, the Equity volatility is the Market Risk factor with the highestpercentage variation which should be explained mainly by the additional re-alization of capital gains. The rest of the Market Risk factors have a weakmateriality.

• Duration Gap: The duration gap was re-calculated using ALIM with the Dy-namic Duration Steering model change implemented. There is not a signifi-cant change in the duration gap due to model change, mainly explained by theweak impact in the Interest-Rates Risk factor. Precisely, there are not signifi-cant changes in the Interest-Rates sensitivities and, thus, the finite differencesapproach used for the effective duration calculation, presented in section 3.1.2,gives the same results.

• In the simplified approach for the estimation of the impact on the Risk Capital,it is assumed that the Capital Charge is the same before and after change. Onthe other hand, the market value of assets is also calculated in the 25 stressedscenarios in order to determine if it is necessary to do tax relief adjustments. In

12Biometric and business risk factors

5.3. Chapter Summary: Investment in practice vs. Investment Strategy Modelling67

this case, there are not variations observed in the asset valuations and, conse-quently, the tax relief remains the same.

All in all, the Market Risk factors have a weak materiality which should bemainly explained by the fact that amounts to be reinvested defined by the DynamicAsset Allocation (DAA) module of ALIM, shown in section 2.3, are preserved andonly maturities of the new bond reinvestment profiles are changed, as presentedin section 3.2.2. In contrast on the Liability side, lower return from shorter re-investments are reverted on a lower revaluation of the liability cash-flows and, con-sequently, most of the impact is reflected on the Life Non-Market Risk factors.

Finally, the influence of the Dynamic Duration Steering module on the OwnFunds and Solvency 2 Ratio is displayed in the table below.

Allianz VieBefore Change

Allianz VieAfter Change

VariationAmount

Variation%

BEL (ALIM) 7 235,48 7 237,31 1,83 0,025%Gross Own Funds (ALIM) 706,94 705,11 -1,83 -0,259%Net asset Value (NAV)* 489,73 488,46 -1,83 -0,259%

Ratio SII (NAV/RC) 236,3% 233,7% -2,6% -1,1%*Accounted Own Funds eligible for covering the regulatory capital requirements

TABLE 5.6: Solvency 2 Ratio, Own Funds and BEL after an beforechange

In this methodology, the impact on the BEL13 is directly allocated on the GrossOwn Funds14. Subsequently, it is assumed that the percentage variation on the NetAsset Value (NAV)15 is the same that has been calculated on the Gross Own Funds.

In conclusion, the Dynamic Duration model change conduct to a loss in OwnFunds, while the Risk Capital increases. Therefore, the resulting cumulative riskderives in a loss on the Solvency 2 Ratio of 2,6%.

5.3 Chapter Summary: Investment in practice vs. InvestmentStrategy Modelling

As presented in section 5.1.2, the implementation of the Dynamic Duration Steeringmodule conducts a more realistic modelling of the re-balancing mechanics of assetswithin ALIM, precisely the maturity steering of bonds re-investments. However, itshould not be directly inferred that the results obtained, in section 5.2, on Risk Cap-ital and Own Funds are more accurate.

On the one hand, most strategic investment practices of Life insurers, includ-ing duration steering, are driven by long term liability obligations as explained insection 1.3.1. On the other hand, the Market Consistent framework of Solvency 2

13All BEL have been calculated using ALIM without reconciliations with accounted reserves14Gross Own Funds are the excess of assets over the BEL. Taxation is not considered in this calcula-

tion, neither Risk Margin or other liabilities15Accounted Own Funds eligible for covering the regulatory capital requirements as presented in

1.2.3

68 Chapter 5. Dynamic Duration Steering Implementation and Impact on KeyPerformance Indicators (KPIs)

is based on a run-off business valuation. In practice, New Business continuouslypushes liability obligations further and, thus, the Investment Department usuallyseeks to steer duration up. This is why, the implementation of realistic investmentmanagement actions, such as steering duration, within ALIM could be in contradic-tion with the liability cash-flows dynamic due to the run-off business framework inwhich only the draining of the In-Force Business16 is observed along the projection.

As a matter of fact, the loss on the Solvency 2 Ratio due to implementation ofthe Dynamic Duration Steering module should not be considered as a prudentialmeasure due to a more accurate modeling. Indeed, the disconnection between thepractical world and the modeling framework introduces a bias that has to be as-sessed before any implementation of this new modelling. Nevertheless, the durationsteering algorithm still is an improvement of the re-balancing mechanics within theInvestment Strategy module and, thus, some other settings of the Dynamic Durationmodule could conduct to more cohesive results.

In order to reconciliate the practical and modelling frameworks, an alternativesettings of Dynamic Duration Steering module are suggested. In the current mod-elling, new bond profiles are already calibrated seeking to maintain duration stablealong the projection. Therefore, the Dynamic Duration module could be set in orderto only to steer duration up if needed in trajectories with low interest-rates. Thiswould be traduced in the weights set on maturities within the new bond profiles ta-bles which lie on the corporate assumptions. Precisely, the summary table 5.3 wouldchange as follows

NewBondProfile

Issuer

NormalBasket(Weightedaveragematurity)

ShorterBasket(Weightedaveragematurity)

LongerBasket(Weightedaveragematurity)

DownProfile(% change)

UpProfile(% change)

profile1 CO 6,4 6,4 8,5 0,00% 32,81%profile2 CO 7,5 7,5 7,5 0,00% 0,00%profile3 CO 10 10 10 0,00% 0,00%profile4 GO 14,1 14,1 19,5 0,00% 38,30%profile5 GO 22 22 28 0,00% 27,27%profile6 CO 2 2 2 0,00% 0,00%

*Highlighted profiles are modified in order to steer duration up

TABLE 5.7: Extract of the New Bond profiles to be used for purchases

Finally, it is noticed that duration steering could be also implemented in a realworld projection framework. Indeed, real-world is oriented to measure profitabilitymetrics rather than risk metrics such as Risk Capital and Solvency 2 Ratio. Thisis why, investment management actions could lead to more accurate results of thefinancial investment return and profit sharing along the real-world projections.

16Projections of the New Business are not allowed in the Solvency 2 valuation framework for BELand SCR calculation purposes; Only the New Business of the first projection year is taken into ac-count which should correspond to the Business realized when passing from the Opening-Balance tothe Closing-Balance

69

Conclusions

This professional thesis is the result of a back and forth between the academic worldand real investment and risk management practices. On the one hand, I have usedand studied Allianz ALM model (ALIM) during my apprenticeship which have al-lowed me to have a better understanding of the risk management metrics that I haveapproached theoretically at ISFA. On the other hand, the professional focus of ISFAhave gave the tools to analyze the models used in practice and to suggest improve-ments.

This professional thesis succeed to present the framework in which the invest-ment strategy modeling of Allianz ALM model, so called ALIM, is improved byreplicating more realistic investment practices. Precisely, more sophisticated re-balancing mechanics are presented in order to dynamically steer duration withinALIM.

Duration and Convexity are very popular metrics on Portfolio Management. Inparticular, Life insurance companies, which are long-term and liability-driven in-vestors, use these two sensitivities as key indicators of their portfolios for ensuringa prudent management in terms of liquidity and risk to interest-rates movements.Even if they are synthetic and unidimensional metrics, they provide an immediateinsight in the risk and engagements profile of asset and liability portfolios of Lifeinsurance companies. This is why it is a common practice for Life insurers to keepthe duration and convexity gap closed as much as possible. Consequently, this pro-fessional thesis extend the Investment Strategy modeling of ALIM with a durationsteering functionality.

The algorithms presented for duration steering are just the beginning of the mod-eling. As a mater of fact, Life insurance liabilities are complex financial portfoliosand, thus, liability duration modeling is a major challenge to be overcome beforeto implement the Dynamic Duration Steering module. A sensitivities approach isretained for the approximation of the liability duration as a function of the interest-rates level evolution along the projection horizon, within ALIM for every iterationi at each time-step t. Moreover, a technical driver based on Minimum-Guaranteecash-flows projection is also chosen in order to capture the draining of the in-forcebusiness in the run-off approach of Solvency 2 valuation.

This professional thesis presents real ALM and investment management prac-tices. This allows to build the bridge between Life insurance investment and ALMmodeling. Moreover, Solvency 2 valuation framework is also presented so that spe-cific metrics are determined with the aim to measure model changes influence. Moreexactly, the Dynamic Duration Steering module impact have been measured on RiskCapital (RC) and Solvency 2 Ratio as well as Net Asset Value (NAV).

70 Chapter 5. Dynamic Duration Steering Implementation and Impact on KeyPerformance Indicators (KPIs)

As a matter of fact, the loss on the Solvency 2 Ratio due to implementation ofthe Dynamic Duration Steering module should not be considered as a prudentialmeasure due to a more accurate modeling. Indeed, the disconnection between thepractical world and the modeling framework introduces a bias that has to be as-sessed before any implementation of this new modelling. Nevertheless, the durationsteering algorithm still is an improvement of the re-balancing mechanics within theInvestment Strategy module and, thus, some other settings of the Dynamic Durationmodule could conduct to more cohesive results.

Dynamic Duration Steering module has been originally proposed in 2010 by theAllianz Group. Indeed, a historical approach was suggested, in Allianz internaldocumentation, for its calibration. As presented in this professional thesis, suchhistorical approach is not suitable because duration figures are not comparable witheach other across different reporting quarters due to model and assumption changes.Later in 2014, Allianz France ALM team has tested Dynamic Duration using a settingbased mainly on expert judgment but the model change was never implemented.Recently in 2018, I was given the role to provide a quantitative support to the studyin the framework of the Mémoire d’Actuaire de l’Institut des Actuaires in order to in-vestigate the possibility of implementing the Dynamic Duration module within theAllianz France internal model.

Consequently, this professional thesis provides a methodology for the calibrationof the liability duration proxy and the re-balancing mechanics. Moreover, analysisof the results within the Solvency 2 framework are also presented. Therefore, mostaspects of the module implementation have been covered and tested with the aim toprovide essential items of analysis for future studies regarding this topic.

71

Appendix A

Additional Figures and Tables

FIGURE A.1: Inflation, OAT TEC 10yr and 10yr US Treasuries evolu-tion

FIGURE A.2: Effective Duration calculated by Finite Differences ap-proach (Pensions Portfolio)

72 Appendix A. Additional Figures and Tables

FIGURE A.3: Effective Convexity calculated by Finite Differences ap-proach (Pensions Portfolio)

FIGURE A.4: Convexity calculated using theoretical relation (Pen-sions Portfolio)

Appendix A. Additional Figures and Tables 73

Portfolio R2

Portfolio 1 0,98Portfolio 2 1,00Portfolio 3 0,97Portfolio 4 0,93Portfolio 5 0,99Portfolio 6 0,99Portfolio 7 0,94Portfolio 8 0,99Portfolio 9 0,99Portfolio 10 1,00Portfolio 11 0,99Portfolio 12 0,99Portfolio 13 1,00Portfolio 14 0,99Portfolio 15 0,96Portfolio 16 0,99Portfolio 17 0,93Portfolio 18 1,00Portfolio 19 0,98Portfolio 20 0,98Portfolio 21 0,99

AZ L&H (without UL portfolios) 0,99*Unit Linked portfolios are out of scope

TABLE A.1: Determination coefficients of the linear regressions oneffective duration figures generated by ALIM using the sensitivities

approach without re-calibration of spreads

74 Appendix A. Additional Figures and Tables

Portfolio R2

Portfolio 1 0,97Portfolio 2 0,98Portfolio 3 0,99Portfolio 4 0,98Portfolio 5 0,98Portfolio 6 0,99Portfolio 7 1,00Portfolio 8 0,98Portfolio 9 NAPortfolio 10 0,97Portfolio 11 NAPortfolio 12 NAPortfolio 13 0,99Portfolio 14 0,98Portfolio 15 0,99Portfolio 16 0,92Portfolio 17 0,92Portfolio 18 1,00Portfolio 19 0,98Portfolio 20 0,97Portfolio 21 0,95

*Unit Linked portfolios are out of scope**There is not reliable data for portfolios 9, 11 and 12

TABLE A.2: Determination coefficients of the quadratic regressionson Macaulay duration figures calculated using Minimum-Guarantee

technical cash-flows

Appendix A. Additional Figures and Tables 75

Portfolio R2

Portfolio 1 0,70Portfolio 2 0,59Portfolio 3 NAPortfolio 4 NAPortfolio 5 0,66Portfolio 6 0,80Portfolio 7 0,85Portfolio 8 0,83Portfolio 9 NAPortfolio 10 NAPortfolio 11 NAPortfolio 12 NAPortfolio 13 0,80Portfolio 14 0,85Portfolio 15 0,73Portfolio 16 0,63Portfolio 17 0,63Portfolio 18 0,89Portfolio 19 0,77Portfolio 20 0,78Portfolio 21 0,89

*Unit Linked portfolios are out of scope**There is not reliable data for portfolios 9, 11 and 12***Exponential regression do not work on portfolios 3, 4 and 10

TABLE A.3: Determination coefficients of the exponential regressionson Macaulay duration figures calculated using Minimum-Guarantee

technical cash-flows

76 Appendix A. Additional Figures and Tables

(A) Savings Portfolio (B) Pensions Portfolio

(C) Savings Portfolio (D) Pensions Portfolio

(E) Savings Portfolio (F) Pensions Portfolio

FIGURE A.5: Duration evolution within ALIM

77

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